The philosophical and mathematical commentaries of Proclus on the first book of Euclid_s elements (Vol. 1 of 2) _ To which are added_ A history of the restoration of Platonic theol

The Project Gutenberg eBook of The philosophical and
mathematical commentaries of Proclus on the first book of
Euclid's elements (Vol. 1 of 2)
This eBook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this eBook or online at
www.gutenberg.org. If you are not located in the United States, you
will have to check the laws of the country where you are located
before using this eBook.

Title: The philosophical and mathematical commentaries of Proclus on
the first book of Euclid's elements (Vol. 1 of 2)
To which are added, A history of the restoration of Platonic
theology, by the latter Platonists: And a translation from the
Greek of Proclus's Theological elements

Author: Proclus

Translator: Thomas Taylor

Release date: August 14, 2024 [eBook #74253]

Language: English

Original publication: London: Printed for the author, 1791

Other information and formats: www.gutenberg.org/ebooks/74253

Credits: Wouter Franssen, Karin Spence and the Online Distributed
Proofreading Team at https://www.pgdp.net (This file was
produced from images generously made available by The
Internet Archive)

*** START OF THE PROJECT GUTENBERG EBOOK THE
PHILOSOPHICAL AND MATHEMATICAL COMMENTARIES OF
PROCLUS ON THE FIRST BOOK OF EUCLID'S ELEMENTS (VOL. 1
OF 2) ***

THE

PHILOSOPHICAL and
MATHEMATICAL
COMMENTARIES OF PROCLUS,
ON

THE FIRST BOOK OF EUCLID’S
ELEMENTS.
TO WHICH ARE ADDED,

A History of the Restoration of Platonic Theology,
BY THE LATTER PLATONISTS:
And a Translation from the Greek of

PROCLUS’S THEOLOGICAL ELEMENTS.

IN TWO VOLUMES.

VOL I.

LONDON, PRINTED FOR THE AUTHOR:
And Sold by T. Payne and Son; B. White and Son; J. Robson; T. Cadell; Leigh and Co.; G. Nicol;
R. Faulder; and T. and J. Egerton. 1792.

[Price Two Guineas in Boards.]

Extracts from Curiosities of Literature. Second Edition.
Printed for Murray. Page 385.

Mr. T. TAYLOR, the Platonic Philosopher and the modern Plethon,
consonant to that philosophy, professes Polytheism.[1]

The Reader is requested to correct the following Errors.
Page 4. of the Dissertation, Vol. I. line 8. for admitted, read omitted. Page 16. line
8. for from, read form. Page 51. Vol. I. of the Commentaries, line 16. instead of They
are surely not the, &c. read For surely it cannot be said that there are, &c. Line 17.
for but we, read but that we. And line 19. for is by much prior to, read is by a much
greater priority.
Vol. II. page 18. line 26. for and one is, read and one part is. And line 27. for
another, read the other. Page 114. line 13. for The angle, read Let the angle; and
instead of is bisected, in the same line, read be bisected. Page 411. line 2. for is filled
with intellect, read fills intellect. And line 3. for it also participates, read also it
participates.

TO
THE SACRED

MAJESTY
OF

TRUTH.

P R E FA C E .

T he design of the present work is to bring us acquainted with the nature
and end of Mathematics in general, and of Geometry in particular: and
in the execution of this design our Author has displayed an uncommon
elegance of composition, and a most valuable store of recondite learning.
He is not content with every where unfolding the full, and most accurate
meaning of Euclid; but he continually rises in his discourse, and leads us
into the depths of the Pythagoric and Platonic philosophy. We are surprised
to find an use in Geometry, which at present it is by no means suspected to
afford. For who would conceive that it is the genuine passage to true
theology, and the vestibule of divinity? This, indeed, is by no means the
case when it is studied for lucre, and applied to mechanical purposes; for
then the soul is neither elevated nor enlightened; but degraded and filled
with material darkness. Hence these Commentaries are alone valuable to the
liberal part of mankind, who look beyond sense for certainty; and who
prefer things desirable for their own sakes, before such as minister to the
necessities of life.
The translation of this work is attended with great difficulty and labour;
not only from the sense of the philosopher, which is always profound, and
frequently obscure, but from the great incorrectness of the Greek edition, in
which, exclusive of numberless typographical errors, entire sentences,
essential to the connection, are frequently omitted; and in one place two
pages of the Latin translation are wanting in the original, as will be shewn
in our following notes. Indeed, the Latin translation of Francis Barocius the
Venetian, (Patav. 1560.) which was made from a variety of manuscripts, is
inconceivably valuable; for the diagrams, so necessary to a work of this
kind, but which are omitted in the Greek, are here inserted; and the version
is every where faithful, and sufficiently perspicuous to those who are
conversant in the ancient philosophy. Barocius justly cautions the reader not
to compare his version with the printed Greek, which he observes is rather
lacerated than printed; as indeed, without his translation, it is impossible for
any one to read the half of this invaluable work, even though he should be
as perfect in Greek as in his native tongue. If I had not, therefore,
fortunately acquired this translation, which is at present very rare, I would

have by no means engaged in this arduous undertaking. Barocius, indeed,
gives evident proofs of his possessing the philosophical genius, by the
excellence of his translation, and his preface to the reader; and it is greatly
to be lamented that he did not adorn his version with explanatory notes,
which this profound work frequently demands, and which he was doubtless
well qualified to accomplish. This defect I have endeavoured, as far as I
was able, to supply; and at the same time have been cautious neither to
weary the reader by prolixity, nor by too much brevity to leave him
destitute of proper information. In the distribution of the first book of this
work into chapters, I have followed the order of Barocius, because it is
natural and obvious; and must beg leave to solicit the reader’s indulgence
for using the words partible and impartible, differently from their common
signification. These words I have generally employed to express the
meaning of μεριστός[2] and αμεριστος[3] in the Greek, as I do not conceive
that the words divisible and indivisible always convey their full
signification. I have likewise used quadrangle instead of square, and
quinquangle for the word pentagon. For if τρίγωνος be rendered triangle,
why should not τετραγώνος be rendered quadrangle? And, as Barocius
observes, why, for a similar reason, should not πεντάγωνος and ἑξάγωνος
be rendered quinquangle and sexangle; and so of the rest? Uniformity is
always desirable when it can be obtained; and is no where so necessary as
in scientifical disquisitions.
It is likewise necessary to inform the reader, that though I have always
endeavoured to give the faithful meaning of my Author, yet I have
occasionally paraphrased his sense, when most obscure, and added such
elucidations of my own, as I either thought necessary to the full
comprehension of his matter; or which were naturally excited by the fire
and spirit of the Original. If it shall appear that I have succeeded in the
execution of this work, and rendered it intelligible to the lovers of truth, I
shall rejoice in my success, and consider my labours sufficiently rewarded.
The applause of the multitude I am neither likely nor desirous to gain; but I
am anxious to procure the approbation of the discerning few, who know that
the age of philosophy is past; and who esteem the works of her ancient
heroes as the most precious treasures which have escaped the ravages of
time.

Time, indeed, is like a deep and rapid river; whatever is trifling and
light, is precipitately borne on its surface, and what is valuable and weighty,
sinks to its bottom. Hence, the superficial observer collects nothing more
than the rubbish, which it is forever devolving into the abyss of oblivion;
while the profound and contemplative genius explores the depths of the
stream, and accounts himself happy if he can gather any of the pearl which
its bottom contains. Thus the discoveries of experimental philosophy, float
like straws on the surface, while the wisdom of Pythagoras and Plato lies
concealed in the depths of the river. I am well aware it will be said, that the
reverse of this similitude is true; that the modern philosophy is the pearl,
and the ancient the stubble; and that the former will be celebrated by
posterity, and increase in reputation when the latter shall scarcely be known.
But let us attentively examine the truth of this assertion, and shut our ears to
the unsubstantial echoes of popular applause. Is it reasonable to suppose
that men of such exalted abilities, as the Pythagoric and Platonic
philosophers possessed, even in the estimation of their opponents,
accompanied with the greatest advantages of birth and fortune, and the most
unwearied attention, have discovered nothing valuable, and have left
nothing behind them, but jargon and reveries? Is it to be supposed, that in
an age when philosophy was almost adored; when it was esteemed by
kings, cultivated by noblemen, and even reverenced by the vulgar; when
empire was relinquished for its pursuit, and every danger encountered for
its possession: is it to be supposed, that nothing but delusion was the
offspring of so glorious a period, and nothing but folly the reward of such
generous endurance? Or shall we say, that the discovery of truth was
reserved for the age of experiment; and that she is alone to be apprehended
in the infinite labyrinth of particulars? That she is to be investigated with
the corporeal senses, and not with the powers of intellect; and that the
crucible, the alembic, and the air-pump, are the only mediums of detection?
If this be the case, truth is material, and may be calcined, distilled, and
rarefied, like any other corporeal substance. It is no longer eternal and
immutable, but perishable and fluctuating; the phantastic subject of sensible
inspection, and not the steady and real object of the permanent energies of
science. Shall we call this the age of philosophy, in which talents are
prostituted for sustenance, and learning submits to the impudence of
wealth? Shall we say that we have strengthened the cause of philosophy, by
demolishing her schools; and increased her independence, by enlarging the

empire of commerce? Where shall we find the man, who is at present
reverenced for the profession of teaching speculative truth, or indeed who
teaches it at all? Or should we chance to meet with such an obsolete
character, shall we find him supported by the profession? It is a well known
fact, that men formerly lived in the highest esteem by its propagation: it is
equally as notorious, that a man at present would starve by such an attempt.
Dare we assert, that the reason of this difference must be ascribed to the
greater liberality, and more philosophical spirit of the present age? Shall we
not rather say, that the period, in which these ancient heroes lived, was the
golden age of philosophy;—a period so different from the present, as to
appear fabulous on the comparison? For mark the distinguishing
characteristics of our inferiority. The great object of ancient philosophy, was
an accurate speculation of principles and causes: but that of the modern, is a
confused investigation of effects. And if pursuits participate of the nature of
their subjects, and causes are more noble than effects, the ancient
philosophy must undoubtedly be more elevated than the modern. Again, the
object of the Pythagorean and Platonic philosophy was to make its
possessors wise and virtuous; and to elevate them above the common
frailties and imperfections of degraded humanity; and this end was happily
accomplished in its votaries, as their lives abundantly evince: but the object
of modern philosophy, is a promotion of the conveniencies and refinements
of life, by enlarging the boundaries of traffic; and the Mathematical
Sciences are studied solely with a view to this enlargement. The design of
the ancient philosophy was to remove the causes of wonder, by
contemplating effects in their causes: the grand object of the modern, is to
increase admiration, by attempting to investigate causes through the infinity
of particular effects. So that philosophy, as Mr. Harris justly observes, now
ends where it formerly began. For either there is no such thing as science,
or if its existence be admitted, it can never be obtained by experimental
enquiries; as these must be liable to all the inaccuracy and imperfection of
their material subjects.
In short, the philosophy of Pythagoras and Plato will be found, when
impartially considered, to contain every thing which can enlighten the
mind, improve the morals, and exalt the character of man. It is built on the
steady basis of truth, and will survive the wreck of ages. Its foundation is
deep, and its summit reaches the heavens. It is a mighty rock, which
modern systems may assail, like a raging sea; but, like stormy waves, they

will only be broken about its impenetrable sides. To war against wisdom is
folly; for opposition in this case is the destruction of its author. The
moderns may, indeed, expect, because their merit is raised by the present
age, above that of the ancients, to appear as giants in the eyes of posterity;
but they will only verify the elegant observation of the poet[4], that
Pygmies are Pygmies still, though perch’d on Alps,
And Pyramids are Pyramids in vales.

A

D I S S E R T AT I O N
ON THE

PLATONIC DOCTRINE of IDEAS, &c.

SECTION I.

T he Platonic doctrine of Ideas has been, in all ages, the derision of the
vulgar, and the admiration of the wise. Indeed, if we consider that ideas
are the most sublime objects of speculation, and that their nature is no less
bright in itself, than difficult to investigate, this opposition in the conduct of
mankind will be natural and necessary; for, from our connection with a
material nature, our intellectual eye, previous to the irradiations of science,
is as ill adapted to objects the most splendid of all, “as the eyes of bats to
the light of day[5].” And yet (as I presume, it will appear from the following
discourse), unless the existence of these lucid beings is admitted, there can
be no such thing as science; nor, indeed, any genuine knowledge at all.
Hence, an enquiry concerning their nature and reality, is highly proper, as
an introduction to the ensuing Commentaries, in which they are considered
as the stable pillars of all truth, and the prolific principles of the universe.
But previous to this enquiry, it is proper to observe, that Plato was not
the inventor, though he was a strenuous asserter, of ideas; for, in the
Sophista he affirms, that ideas were the discovery of men who excelled in
wisdom and piety, and who contended for an invisible essence. Diogenes
Laërtius, indeed, asserts, that Plato received the doctrine of ideas from
Epicharmus. But Epicharmus was not their inventor, because Pythagoras,
and others of still higher antiquity, were well acquainted with ideas; so that
it may be affirmed, with much greater truth, that Plato was instructed in
their nature by Philolaus his preceptor, and the disciple of Pythagoras. For
Pythagoras, after his mysterious manner, signified ideas by numbers. But,
prior to Pythagoras, Orpheus was an asserter of ideas, and called Jupiter, or
the dimiurgus of the world, “the idea of all things.” And, according to
Syrianus, the mundane sphere, celebrated by Empedocles, is no other than
the ideal world; so that the doctrine of ideas is as ancient as that of wisdom
itself.
But to begin with our enquiry: in the first place, without universals there
can be no science; for the flowing and perishing nature of particulars is
perfectly foreign from that stability and duration which is requisite to
objects of invariable truth. Neither is it possible, that infinite individuals can

exist without the subsistence of one cause endued with infinite power; for
all multitude must necessarily originate from one, and must resemble its
cause in as great a degree of perfection as its nature can admit; by a diffused
infinity, shadowing forth that infinite power which subsists in indivisible
union. Hence, if this be the case, and if infinite men, horses, and a multitude
of other univocals, are produced in an infinite time, an unity of infinite
power must be the source of each, according to which they are generated in
a terminated manner to infinity in the universe. Again, all animals are
transmuted from that which is in capacity (i. e. seed), into energy. But if this
be true, it is requisite there should be some animal in the universe,
subsisting in ever-vital energy, which may call forth that which is concealed
in dormant capacity, into perfect actuality. Thirdly, the celestial orbs would
not perpetually revolve in the same spaces, and after the same manner,
unless one and the same universal number, or idea, ruled in each. So,
likewise, there is a natural number in every animal; or those of the same
species, would not always (when perfect) be distinguished with the same
invariable organs; nor would they be subject to puberty and old age, at the
same time, unless they were detained by the same measure of nature.
Besides, the participation of universals, is evident in every sensible object.
Thus, the rational nature is united with every individual man. Thus, animal
subsists in a lion and a horse, in a man and a dog. And thus the pentad, or
number five, is participated in the five fingers, and the duad in the nostrils,
eyes, hands, and feet. But since these do not subsist without a cause, but are
perfected by certain determinate natures, it is necessary there should be an
universal animal, in the whole of nature, separate from sensibles, by means
of which this sensible animal is generated. And that there should subsist in
nature a pentad, through which the hands are always adorned with that
number of extremities; and a duad, from which the two eyes and nostrils are
derived. But if nature does not possess these numbers from herself, as she is
not the first cause of all, but derives them from another cause, in the same
manner as matter from nature, it is necessary there should be universals and
numbers prior to nature, subsisting in far greater purity and perfection.
Again, we may demonstrate the existence of ideas as follows: if the
Deity, in fabricating the universe, operated essentially (and there is no other
way in which we can conceive him to operate), he must fabricate the
universe, an image of Himself. But, if this be the case, he contains in
himself, in the manner of an exemplar, the causes of the universe; and these

causes are no other than ideas. Besides, this consideration is not to be
omitted, that the perfect must necessarily antecede and preside over the
imperfect; unity over multitude; the impartible over the partible; and that
which is perpetually the same, over that which admits of variation and
change. From whence it may be inferred, that things do not originate from
baser natures, but that their gradual processions end in these; and that they
begin from the most perfect, best, and most beautiful natures. But let us
pursue this reasoning more minutely, as it affords the strongest arguments
for the existence of ideas.
When the Deity fabricated the various species of animals, and bestowed
on them the different senses, it was doubtless with a view to the benefit of
their possessors, as he foresaw, that without these, the animal could neither
provide for its own support, nor defend itself from surrounding dangers. But
may we not enquire from whence this previous perception originated? For it
is not to be supposed, that he first made animals destitute of senses, and so,
being admonished by their sudden destruction, afterwards assigned them to
their nature. Shall we say, this foreknowledge was the result of a reasoning
process? But then, we again ask, What were the principles of this
ratiocination? For if they originated from other reasonings, it is necessary,
at length, to arrive at something prior to these discursive operations, on
which they ultimately depend; since all reasoning must be founded on
indemonstrable principles. Was sense, then, or intellect the principle of this
previous perception? But, sense, in the present instance, had not then a
being, for it could not exist prior to the animal nature: it was, therefore,
intellect. But if intellect be the repository of certain propositions, and the
conclusion be science, it must follow, that there could not then be a
consultation of any thing sensible. For the principle and the conclusion
must both depend on something intelligible. Besides, may we not ask, how
such a habit of thought arose before the existence of a sensible nature! It is
absurd in the extreme, to say from chance, and to resolve it into a sudden
volition of the Deity, is an assertion that may, indeed, satisfy vulgar minds,
but can by no means quiet the restless spirit of philosophical investigation.
Since, to suppose the cause of the universe, actuated by sudden volitions, is
to place him on a level with the vilest natures, and subject him to the
irrational impulses of the brute. Hence we infer that the formation of
animals, and by the same arguments of the world, was not the result of any
reasoning process. For, indeed, argument and foreknowledge cannot with

propriety be attributed to the Deity; but when they are ascribed to him, we
must consider it as nothing more than an indication of his constituting
particulars, in a manner somewhat similar to the providence of a wise man,
in inferior concerns. For, in subordinate natures, whose operations cannot
take effect prior to enquiry, reason is necessary, on account of the inferiority
of that power which precedes the reasoning energy. In like manner,
foreknowledge is necessary, because a power is wanting to its possessor,
which might render him superior to its use. For foreknowledge is directed to
this end, that one particular circumstance may take place in preference to
another. But if it be requisite that every energy in the Deity should be void
of defect, and if it is not lawful that any thing should be present with him,
which is not total and universal, it is necessary that all things should be
contained in every thing essential to the nature of the Deity. Hence, since
even futurity is with him present, there is nothing in him posterior; but what
is present in him becomes posterior, by its participation in another. If then
futurity be present with the Deity, it is necessary it should be so present, as
if foreknown in a posterior nature; that is, in such a manner that nothing
may be wanting to any being; and that is, lastly, so that every thing may be
complete.
Besides, reasoning cannot, by any means, belong to an eternal essence
like the deity; for if this be admitted, he must be forgetful of his former
operations. And if, in consequence of reasoning, he produces more perfect
natures afterwards, his works could not be perfectly beautiful before: but if
they were beautiful before, they must be co-existent with their cause, i.e.
they must be eternally beautiful, antecedent to the reasoning energy. Again,
if we suppose the supreme intellect, the demiurgus of the world, to operate
by enquiry, his energy could not be spontaneous, and truly his own; but his
essence would be similar to that of the artificer, who does not derive his
productions from himself, but procures them as something adventitious by
learning and enquiry. But if the universe was not formed by deliberation, it
must be co-existent with its cause, and reside in his essence; for if it be not
co-existent there must have been some particular time, in which its artificer
determined on its production; and this determination must have been the
result of a reasoning process, concluding that it would not be good to
produce it before that particular time, (from whence, by the way, we infer
the eternity of the world.) And if the universe be co-existent with its author,
it must perpetually emanate from his nature, and be dependent on it, like the

shadow on its forming substance. But in this case, its archetype must be
contained in the essence of its author; for every cause is that primarily,
which its effect is secondarily. And hence we infer, that if the sensible
universe be replete with forms of every kind, the exemplars of those forms,
must subsist in immaterial perfection, in the artificer of the world.
If this sensible world, then, be formed according to the exemplar of that
which is intelligible; may we not say, with the great Plotinus, that it is
requisite universal animal should there primarily subsist in perfect vital
energy, containing all things in its omniform essence. “Hence (says he[6])
the heavens are there a divine animal, replete with ideal stars. Earth too
does not there subsist solitary, but is much more vital than this corporeal
earth, for it is full of intellectual life. The sea too is there, and all water
subsisting in life, and an ever-abiding stream. For how is it possible that any
thing not vital, can be the progeny of life itself? He, therefore, who enquires
from whence animals originate in the intelligible world, might as well
enquire from whence all life, and soul, and universal intellect, arose. For
here there is nothing indigent nor defective, but every thing is perfect and
exuberant. Here they all flow from one fountain, not as from a certain spirit,
or heat, but as if from an universal quality, possessing and preserving in
itself, all qualities; such as sweetness, accompanied with fragrance of smell,
the vigour of wine, and the strength of all juices, bright colours, and
whatever is perceived by the taste.”
3. Such then are the arguments which the Platonic philosophy affords in
defence of ideas; the existence of which was so evident to Plato, that, in the
Sophista, he compares those who oppose the friends of ideas to the giants of
old, warring, as it were, on celestial souls, and such as are engaged in
sublime investigations. Let us now consider to what universals these lucid
beings are confined; since, according to the Pythagoreans and Platonists,
there are not ideas of all universal conceptions. “For, in the first place (says
Syrianus[7]), there are no ideas of things evil and base, because these subsist
in nature rather by a privation and absence of ideas. And, on this account,
they are said to exist contrary to nature. Nor, secondly, of negations, for
these are destructive of the bound and limitation which is attributed to every
thing from the unifying and comprehending nature of ideas; and hence,
separation is rather the result of material infinity than of that which is
formal or ideal. Nor again, are there any ideas of things which at different

times receive a variety of conditions. For these participate of transmutation
from a moveable cause, but not from the immoveable and stable illustration
of ideas. Nor again of parts, such as the hand, head, fingers, and the like.
For the causes of things existing entire, produce whole species and forms;
not divided about the parts of these, like the reasons of nature. But neither
did these wise men place in intellect the determinate causes of accidents in
bodies, such as sweetness and whiteness. For they considered that natural
reasons were sufficient for the production of accidents. Nor again, of
composites, as of a wise man. For since ideas are simple, they preside over
the simple essence of every thing. But the composition and division of
things is the business of our intellect; ideas, at the same time, and that
intellection which is co-ordinate to ideas, being exempt from all these, on
account of superlative simplicity. Neither, therefore, must we establish ideas
of things generated from dissimilars, such as mules; nor of fruit produced
by engrafting from different trees. For all these have a posterior and
adventitious generation, and are not the work of nature alone, nor of nature
proceeding according to her own reasons, but, as it were, compelled to
labour contrary to her own determinations. Hence it is manifest, that all art,
which imitates nature, and alone ministers to the use of mortal life, is
separated from the cause of ideas. But neither are the works which,
depending on the purpose of the soul, are perfected by a concourse of many
causes, and which we are accustomed to call the operations of fortune, to be
conjoined to the cause of ideas. For things which are there perfected, are
eternal, and subsist perpetually the same, free from the nature of contingent
events. It remains, therefore, that ideas must be confined to universal and
perfect essences, and to whatever confers to their natural disposition; as for
instance, to man, and every thing perfective of man, such as wisdom and
virtue. For ideas existing as the generative and energetic causes of the
perfection of every thing, distribute being to essences, and convert them to
the inexhaustible plenitude of their own omniform natures.”
4. But let us now consider the nature of numbers; for as every form is a
number, according to the Pythagoreans[8], a speculation of this kind must
afford no small light to the arduous investigation of ideas. Will it not,
therefore, be proper, in the first place, to enquire, with the great Plotinus[9],
whether multitude is not a departure and distance from one, so that infinity
itself is a separation from unity in the extreme, because it is no other than

innumerable multitude; that on this account it becomes evil; and that we
contract a similar nature when departing from intellectual unity, we are
divided by sensible multitude? For a being then properly becomes many,
when no longer able to remain collected in itself, the same, it is diffused
abroad, and thus, being dispersed, is variously extended; so that when, by
diffusion, it is absolutely deprived of unity, it becomes perfect multitude,
destitute of that universal cement, which unites one part with another. But
whenever the conciliating one is present, then that which was scattered and
diffused, becoming permanent by its bounding power, passes into
magnitude. But if any one should deny the subsistence of unity, asserting
that one is no where to be found, which is not some particular one; and
should hence affirm, that what is called one abstractedly, is only a certain
affection of the soul towards any being; we ask, what prohibits the
appellation of essence, from being nothing more than an affection of the
soul, and consequently the existence of being, a delusion? For we predicate
unity of particulars with as great propriety as being. I am well aware, that
philosophers of the present day will answer, that we have an evident proof
of the reality of being, from its agitating the soul, and becoming apparent in
the phantasy: to which we reply, that in like manner, the soul is agitated,
and the imagination influenced about the one. For every individual as much
excites the perception of one, as of being.
Besides, it is necessary to enquire whether we behold this passion and
conception of the soul, as one or multitude. And again, when we say not
one, we do not then possess one from the thing itself; for we say that one is
not contained in that individual. And hence we must possess one in our own
nature, and this must reside in the soul, separate from that which is
denominated some particular one. But here it may be objected, that the one
we possess is received from externals, and is nothing more than a
conception of the mind, produced by the thing itself. For it will be said, that
as multitude is nothing besides a number of individuals, which are called
many, so one is nothing besides one thing; and is formed by thought
separating that one particular from others. To this we reply as follows:
How can it be consonant to reason to suppose that the conception of one
arises from the sensation of some one particular subject? For one particular
man, who is discerned by sense, is by no means the same with one itself,
since, if this were the case, thought could never predicate one of that which

is not a man. Besides, as cogitation, on beholding the different positions of
things, affirms that this is here or there, so when it perceives an individual,
pronounces one; for that passion is not vain, nor does it assert one of a non-
entity. Nor must we think it predicates ones, because this individual is
different from another; for when cogitation affirms such a thing is this, and
not another, it declares, in the mean time, that the other is one. Likewise
when it affirms that any thing is this alone, it then declares, that what is
alone is one: on which account, it predicates one, prior to alone. Besides, if
there be multitude, it is necessary that one should antecede; since when it
predicates many, it pronounces more than one. And when it affirms that an
army contains a multitude of men, it conceives the soldiers reduced to one
order.
For thought, indeed, does not permit multitude to remain perfect
multitude, destitute of the conciliating power of unity; in which very
circumstance, the subsistence of one is evinced; for acutely and swiftly
perceiving the one which results from order, it reduces the nature of the
many into one. Besides, we affirm that a house and an army are each one,
but that a house is more one than an army, on account of the continuity of
its parts. If therefore, one is contained more in that which is continued than
in that which is discrete, and still more in what is perfectly indivisible, it is
evident that the one is a certain nature, and has a real being. For it is
impossible that the more and the less should take place among things which
have no subsistence. If then it be not possible to understand any thing
without one or two, or some other number, it is by no means proper to deny
existence to that, without which we cannot comprehend the existence or
properties of any being: but it is requisite that nature should antecede all
discourse, and intelligence, which is every where necessary to their
existence.
Again, if unity has no real subsistence, and is nothing more than a name
or conception of the mind, it may be destroyed without the destruction of its
subject. The unity, therefore, of a house may be taken away, without the
ruin of a house. But if a house is nothing more than certain materials,
reduced into one form, this is impossible. And, on the contrary, the
alteration of that subject, of which unity is predicated, can make no real
alteration in unity (on this hypothesis) any more than the death of a man can
affect his name. When, therefore, a body, of which one was predicated, is

divided into a multitude of parts, there is no real alteration made in the unity
of the body, because unity is nothing more than a name.
It was in consequence of this reasoning, and perceiving that unity was
participated by every being, that the Pythagoreans placed a super-essential
one at the top of the universe, intelligibly abstracted from all beings in
simplicity and excellence of nature. For they considered, that unless there
was a self-subsisting one in all things, there could neither be universals nor
particulars. Not the first, because they are by nature one and many. But it is
requisite that the one itself, should preside over that which is not one alone.
Nor again, the second, because they are many and one, (that is, they
participate more of multitude than unity, and their nature is determined
more by the many than the one.) And because of things in participation,
unless an unparticipated one is added, there can be no cause of union to
beings; in the same manner as the cause of essence to beings, is taken away
by those who deny that being itself, is the principle of all essence. For as the
good itself, is the one principle of good to the universe, and is nothing
besides good; and as a self-motive nature, which is nothing besides self-
motion, is the cause of motion to all things; so all things proceed from being
itself, and all united natures receive their union from the one, abstracted
from all things.
Hence (such is the absolute dominion of unity), continued quantities
would have no existence without its participation; for when they are
divided, so far as they lose unity, they change their being into some other
form. Hence, the bodies of plants or animals, which are each of them one,
when they fly from unity, and are dissipated into multitude, immediately
lose the essence they formerly possessed, and become something else;
which new state of being they likewise possess so far as they are one. Add
too, that health then flourishes in the corporeal frame, when the body is
conciliated into one; then beauty flourishes, when the power of one
connects the members into proportion and consent; and then virtue reigns in
the soul, when the soul is reduced into one similitude with that which is
divine.
5. But let us now investigate the nature of numbers. All number,
according to the Pythagoreans, originates from unity and the indefinite
duad; the first having the relation of form, and the second, that of matter to
all the orders of numbers. But they likewise divided number into two kinds,

essential and monadic. The essential number they considered as first
subsisting in the intelligible world, together with being, and from thence
distributed into all the various gradations of forms. But the monadic, or that
which is composed from certain units, they justly considered as nothing
more than the image of essential number. And with respect to the numbers
which the human soul participates, these from its imperfect condition have
a middle subsistence; i. e. they exist in a vital, gnostic, and speculative, but
not in an operative manner. Hence, when receiving one thing with another,
we affirm, that they are two, as a dog and a man, or two men; or when we
compute more than two, as ten, and say that there is a decad of men, this
number is not essential to the two or ten individuals, nor is it to be
conceived as subsisting in sensible natures; but it is purely quantity. But
when we distribute this ten, into units, we produce the principle of quantity,
and generate a subject in opinion[10], capable of participating the essential
decad of our soul. But when, considering man in himself, we affirm that he
is a certain number, as the duad, composed of animal and rational, we do
not observe one mode in this predication; but so far as by a discursive
operation of the soul, we numerate, we effect a particular quantum; but so
far as the subjects are two, and at the same time both one (since one fills the
essence of both, and in both unity is contained), we pronounce another, and
an essential number: and this duad is not of a posterior origin, nor alone
signifies a certain quantity, external to the subject, but a duad subsisting in
the essence of man, and containing his nature. For here we do not produce a
number by a discursive operation, while we pursue essential natures. But
when we number any ten things, which are not connected by any
conciliating unity, like a choir, or an army, then this decad, which we
predicate of the ten particulars, subsists alone in our numerating soul, which
renders the ten individuals in opinion, a definite quantum. But in a choir, or
an army, essential number is participated exclusive of that which subsists in
our soul. And if it be enquired how number subsists in the human soul, we
must say, that the soul, by her self-moving energies, procreates number,
while she numerates, and by this energy, causes the existence of quantity; in
the same manner as in walking, we give rise to a certain motion. Thus,
monadic number, or a collection of units of various kinds, subsists in
opinion, in a manner correspondent to that of geometrical figures; and by
this means participates the essential number of the soul. For as a triangular
figure in the phantasy, is the recipient of a triangular nature, or of triangle

itself; so every three units in opinion, receive the essential triad of the soul,
and, by this means, form a definite quantum.
In short, as in every being we may discern the resemblances of matter
and form, so in the pentad, or any other number, the five units, which are
the subject of participation, and the quantity of the number, originate from
the duad; but the form, that is the pentad itself, from unity. For every form
is an unity, which unites its subject quantity, and connects it with its ideal
species. It is, therefore, requisite to understand, that the two principles of
mathematical numbers are resident in our souls, with which every
mathematical number is co-existent; I mean unity, comprehending in itself
all the forms of numbers, and which corresponds to unity in intellectual
natures; and the duad, endued with a generative power, of a formless nature,
and of infinite virtue; and which is called boundless, on account of its being
the image of never-failing and intelligible duality. Hence, the unity of the
soul, with a never-ceasing energy, continually distinguishes and forms all
the orderly processions of her numbers, suffers no vacuum to intervene, and
leaves no quantity formless and innumerable. Hence too, no essential
number of the soul, as for instance, the pentad, is composed from substance
and accident, as a white man; nor from genus and difference, as man from
animal and biped; nor again, from five unities mutually touching each other,
like a bundle of wood; nor from things mixt, like water and wine, nor from
things subsisting by position, in the manner that stones compose a house;
nor lastly, does it subsist like things numerable; for it is not because they are
composed from indivisible units, that they possess any thing besides units.
For many points are indivisible, yet quantity is not produced on this
account; but because they participate of two natures, the one corresponding
to matter, and the other to form. Lastly, it is not proper to say, that the
number seven (and so of any other number), is composed from the triad and
the tetrad; for units, indeed, composed with units, form a subject adapted to
the reception of the heptad, or the ideal and essential number seven; but the
definite numerical quantity seven, is formed from so many units, and the
ideal heptad. Hence, as the soul of the shipwright gives form to the timber,
from her inherent art; so the numerative soul, from the unity endued with
the relation of a principle which she possesses, gives form and subsistence
to all her inherent numbers. But there is this difference between the two,
that the shipwright’s art is not essential to our nature, and requires manual
operation, because it is conversant with sensible matter; but the numerative

art is essentially inherent in the soul, and is therefore present with all men,
and possesses an intellectual matter, which it easily forms without the
assistance of time. And this, perhaps, is what deceives many, who think that
the heptad is nothing more than seven units. For the imagination of the
vulgar, unless it first perceives a thing destitute of ornament, and afterwards
the operations of the adorning artificer supervening its nature; and lastly,
beholds the thing perfect, and invested with form, cannot be persuaded that
it possesses two natures, the one formless, but the other endued with an
energetic and forming power.
And here it is necessary to observe, that though unity is the form of all
arithmetical forms, yet it is chiefly the form of the decad. For what unity is
simply to all the series of numbers, that the decad is to the following
hundreds, thousands, and millions; from whence, according to a secondary
progression, it is denominated unity. As intellect, therefore, is the form of
all things, but especially of the soul, so unity, though it is the idea of all
numbers, yet especially of the decad. But the reason why the Pythagoreans
extended ideal numbers no farther than ten, is because this number is the
ultimate perfection of beings, containing all things in its omniform nature.
For all proportion subsists within the number ten; the arithmetical in a
natural progression of numbers from unity; the geometrical in the numbers
1, 2, 4, and 1, 3, 9, and the harmonical in the numbers 2, 3, 6, and 3, 4, 6.
And since the causes of all things are contained in numbers, as far as to the
decad[11], it is superfluous to suppose exemplars of the following numbers.
If it should be asked in what manner we must conceive number as
subsisting in the intelligible world, we answer, with the great Plotinus, that
we must conceive it as subsisting in being itself, with a power of impelling
it to the production of multitude. “Hence (says he, Ennead vi. lib. vi.)
number is either the essence or the energy of being, and animal itself, and
intellect is number. But, perhaps, we must call being, number united
(ἀριθμὸς ηνωμένος), but beings, number evolved, or unfolded;
(ἐξεληλεγμένος ἀριθμὸς) intellect, number moving in itself; (ἀριθμὸς ἐν
ἐαυτῶ κινούμενος) and lastly, animal, number comprehending (ἀριθμὸς
περιέχων.“) It was in consequence of this reasoning, that the Pythagoreans
called ideas numbers; because the gradual evolution of these from ineffable
unity, produced all the beautiful variety of forms. Their exalted conceptions
of numbers, likewise, originated from the same sublime theory. Hence,

[12]
Pythagoras, in the sacred discourse, calls number “the ruler of forms and
ideas.” But [13]Philolaus, “the commanding and self-begotten container of
the eternal duration of mundane concerns.” And [14]Hippasus, and all those
who were called ἀκουσματικοὶ (or such as were yet under the probation of
the quinquennial silence), “the first exemplar of the mundane fabric, and
the judiciary instrument of its artificer.”
6. And here I cannot but take notice, with regret, of the very
unphilosophical mistake committed by that great mathematician Dr.
Barrow[15]: I say, with regret, on account of the extraordinary obligations I
am under to his writings, for my proficiency (whatever it may be) in
mathematical learning. But respect must yield to the truth. “Unity, says he,
is not indivisible. (For how ex. gr. can 2/6 added to 4/6 be equal to unity, if
unity be indivisible and incomposed, and represent a point) but rather only
unity is properly divisible, and numbers arise from the division of unity.”
Here the Doctor evidently confounds sensible units, which are the subjects
of vulgar practical arithmetic, with those units which are the objects of
science. Every individual sensible object, is indeed an unit, so far as it
participates the connecting and conciliating power of an immaterial one: but
the unity which stands at the top of speculative arithmetic, is perfectly
indivisible, or arithmetic would cease to be a science. The truth of this is
evident from Euclid’s definition: “Unity (says he) is that according to which
each of the things which are, is called one.” But if unity be a composite, the
definition is false; since a composite, or a certain multitude, can never be
the cause of unity, but the contrary. And that this immaterial one subsists in
sensible natures, has, I hope, been sufficiently proved in the preceding part
of this discourse. But the Platonic Theo[16] of Smyrna, fully establishes the
indivisibility of unity, as follows: “Unity is terminating quantity, the
principle and element of numbers, which remains undiminished by the most
immense multitude of subtractions, and being deprived of all number,
continues firm and fixt, because it is impossible for division to proceed
beyond the bound of unity. Thus, if we separate any one corporeal
substance into parts, the one again becomes many; and by subtracting the
several parts, we end in one part; and from this remaining part, again
divided, arises multitude; and by taking away every part, we again arrive at
one. So that one, considered as one, is incapable of diminution, and
perfectly indivisible. On the contrary, every number is diminished by

division, and is separated into parts less than itself; as the number 6 into 3
and 3, or into 4 and 2, or into 5 and 1. But unity in sensible particulars, if
divided, is diminished after the manner of body, and by section is
distributed into parts less than itself: but it receives increase after the
manner of number; for instead of the one, multitude is produced. In this
sense, therefore, is unity indivisible; for nothing is divided into parts greater
than itself. But that which is cut into parts greater than the whole, and into
parts equal to the whole, is divided as number. Thus, for instance, if any one
sensible body is divided into six parts, 1, 1, 1, 1, 1, 1, these shall be equal to
the whole; but by a section into 4 and 2, it is divided into parts greater than
the whole, considered as one; for 4 and 2 considered as numbers, exceed
unity, and the body was supposed to be one. Unity, therefore, as number is
perfectly indivisible. But unity is called by the Greek word μονάς, only, or
alone, either because it remains immoveable, and does not desert itself, nor
surpass the bounds of its nature (for it remains the same, however
multiplied into itself, through an infinite progression) or because it is placed
separate and apart from the multitude of other numbers, it is denominated
the monad, or one.”
In consequence of this very mistaken hypothesis, which opposes not
only all the wisdom of antiquity, but the sublimest truths, the Doctor asserts,
that an arithmetical cypher is the principle of numbers; and that it is
analogous to a point in geometry. Just as if a cypher, which is nothing more
than a mark expressive by its position with numbers, of a certain quantity,
had a real existence, and was productive of number: when, at the same time,
any other arbitrary character would serve the same purposes, if applied in a
similar manner. It must surely afflict every thinking mind, to see how
dreadfully the mechanical system of philosophy, which has been so long in
fashion, enslaves and perverts the minds of its votaries; for there cannot, I
think, be a more egregious instance of its fatal tendency, than the present, in
which nothing is considered as the foundation of that noble science,
arithmetic; which was deservedly placed by the ancients, in the first rank of
the mathematical disciplines. Such a foundation, indeed, may be proper to
the mechanical philosophy, but is very ill adapted to support the solid fabric
of the arithmetical science. But let us attend to the arguments of this most
learned man, in defence of so strange an assertion, “A cypher, or
arithmetical nothing (says he) is really the bound of every number coming
between it and the numbers next following, but not as a part. A cypher

being added to, or taken from a number, does neither increase nor diminish
it; from it is taken the beginning of computation, while itself is not
computed; and it bears a manifest relation to the principal properties of a
geometrical point.” But in what manner are we to conceive the nothing
which intervenes between any two numbers, to be their term or boundary?
For Euclid defines a term to be the extremity of any thing; implying by the
extremity, something belonging to that of which it is the bound. But how
can a cypher, or nothing, in any respect belong to number, or something?
For if nothing be a boundary, merely from its intervention, a point existing
between any two disjoined lines, though at the greatest distance from each,
must be their common boundary, which is evidently absurd. Besides, what
relation does it bear to a point, which is endued with a generative power, by
its flux forming the simple extension of a line, and, at the same time, every
where limiting its progression, and subsisting in infinite capacity in its
every part? Where are the real and divine properties to be found in an
arithmetical nothing, which Proclus, in the following Commentaries,
exhibits in a point? And how can computation originate from a mere non-
entity?
But a little consideration will convince us, that this Saracen, or Indian
cypher, is nothing more than an arbitrary character, invented for the purpose
of facilitating computation. For, suppose the letter (a) to be placed in its
stead, and to signify, when connected with the mark for unity, ten, or ten
multiplied by one; when connected with the mark for two, ten multiplied by
two, and so on. And again, when placed twice after unity, let it express the
second power of ten, or one hundred, in this manner, a a; when thrice
connected, one thousand, or the third power of ten, and so on: shall we say,
in consequence of this, that (a) is the bound of numbers, and the principle
of arithmetic? Or, shall we not rather say, that it is an arbitrary symbol, like
any other algebraic character, having no real connection with numbers, and
depending, for its existence and application, entirely on the will of its
inventor. But this opinion is too absurd to need any farther refutation.
7. It may here, perhaps, be expected, that I should explain how, in the
language of Syrianus[17], “divine number proceeds from the immortal
retreats of unity, until it arrives at the divine tetrad[18];” and that I should
unfold the properties of the tetractys, according to the Pythagoreans; but an
undertaking of this kind, would not only far exceed the limits of this

dissertation, but, perhaps, in the present age, might be justly deemed, by the
lovers of wisdom, a prostitution and profanation of the most exalted truths.
Enough, I hope, has been said to excite the curiosity, and rouse the attention
of the thinking and liberal part of mankind; and those who understand what
is here briefly delivered, may apply themselves, with advantage, to Proclus
on Plato’s Theology, where they will find all the mysteries of numbers
unravelled; and to the works of the great Plotinus, who will lead them into
the penetralia of the most recondite wisdom. But, in perusing the works of
these great men, the reader must not expect to find the sublimest truths
explained in a familiar manner, and adapted, like many modern
publications, to the meanest capacities. For this, indeed, is impossible to be
effected. “Mankind (says Petvin[19]), are not to be made any more truly
knowing than happy by another’s understanding.—There is no man can at
once convey light in the higher subjects, to another man’s understanding. It
must come into the mind from its own motions, within itself: and the grand
art of philosophy, is to set the mind a-going; and, even when we think
nothing of it, to assist it in its labour.” After which he observes, that “the
ancients never attempt to lead us into knowledge, by a continued chain of
reasoning; on the contrary, they write in such a manner, as to force us to
think for ourselves.” And, previous to this, he remarks, “that there are
certain truths acquired by a long exercise of reason, both in particular, and
likewise in those subjects that are most general, as much, perhaps, out of
the reach of the greatest mathematician, as Sir Isaac Newton’s speculations
are above the capacity of some that are now called mathematicians.” The
truth of this observation is sufficiently evinced, in Plato’s definition of a
philosopher (in his Sophista), “The philosopher (says he) is the man who
sufficiently sees one idea every way extended through many, every one of
them lying apart; and many ideas different from one another, externally
comprehended under one.—And farther, one idea, throughout all manys,
wrapt up in one; and many ideas, every way separate or discreet. This is to
have the knowledge to discern how ideas, as they are general, agree and
disagree.” Now, he who thinks that a perception of this kind may be
acquired by barely reading an accurate discourse on the nature of ideas,
composed in intelligible terms, without, at the same time, employing a long
course of profound meditation, and patient thought, knows but little the
difficulty of the task, and until he changes his opinion will never be the
wiser. But the folly and presumption of men, with respect to this sublime

philosophy, is really unpardonable; for there are very few who conceive that
much previous instruction is requisite to its acquisition; but almost every
man decides peremptorily on the most abstract speculations, and reckons
himself sufficient for the most profound investigations. In the sciences and
arts they are willing to proceed to perfection by gradual advances; but they
consider philosophy as easy, of instant access, and hastily approach to her
embraces with an assured confidence of success. Though, like unhappy
Ixion, through their presumption, instead of a goddess, they grasp nothing
but an empty cloud. Plato was so sensible of this truth, that, in his seventh
epistle to Dion, he expressly affirms, that he neither has written, nor ever
will write explicitly concerning these sublime speculations; “For a thing of
this kind (says he) cannot be expressed by words, like other disciplines, but
by a lasting familiarity, and conjunction of life, with this divine object, a
bright light[20] on a sudden, as it were leaping from a fire, will illuminate
the soul, and there preserve and nourish its splendor. He adds, that a
publication of such concerns, is alone useful to a few of mankind, who from
some small vestiges previously demonstrated, are sufficiently sagacious to
their invention. But it will fill others partly with a base contempt, and partly
with a rash and vain confidence, as if they had now learned some very
excellent things.” He then subjoins the following instance of the difficulty
attending such an undertaking: “There are three things (says he), from
which science must necessarily be produced; but the fourth is science itself.
And it is requisite to establish the fifth as that which is the object of
knowledge, and has a true existence. One of these is the name of a thing;
the second its definition; the third the resemblance; the fourth science. Now
take each of these, desiring to learn what we have lately asserted, and think
concerning them all, in a similar manner. A circle is called something,
whose name we have just expressed. After this follows its definition,
composed from nouns and verbs. For that which every where is equally
distant from the extremes to the middle, is the definition of that which we
signify by the name of a round, and a circumference, and a circle. But the
third is the circle which may be painted, or blotted out, which may be made
by a wheel, or destroyed. None of which affections, the circle itself, which
each of these respects, suffers, as being of a different nature. But the fourth
is science, and intellect, and true opinion about these. And this again must
be established as one whole, which neither subsists in voice, nor in
corporeal figures, but in intellect and intelligence. It is therefore manifest,

that this fourth is different from the nature itself of the circle, and again
different from the three we have previously mentioned. But among the
number of these, intellect, by its relation and similitude, proximately
adheres to the fifth, while the rest are more remote from its nature. The
same may likewise be affirmed of a straight and crooked figure, of colour,
and of the good, the beautiful, and the just. And again, of every body,
whether fashioned by the hand, or the work of nature, whether fire or water,
and the rest of this kind; likewise of every animal, and the manners of
animals; and of all actions and passions. For unless, among these, some
one, after a manner, receives that fourth, he will never perfectly participate
the science about the fifth.” He then proceeds to shew in what respect each
of the preceding four are different from the fifth. “Every circle (says he)
which by the hands of men is either painted, or fashioned by a wheel, is
plainly contrary to our fifth. For it every where participates of the right-line.
But we must affirm, that the circle itself has neither more nor less of any
thing whatever; that is, it possesses in itself, nothing of a contrary nature.
Besides, none of these are endued with any stability of name. For nothing
hinders our applying the appellation of straight to that which we now
denominate round, and calling the straight by the denomination of the
round; nor will there be any less stability in these, when their names are
changed into the contrary. The same reasoning is likewise true of definition,
since it is composed from nouns and verbs, which possess no stability. And
in a variety of ways, it may be proved, that no one of these four is certain
and firm.” Now, this fifth division of Plato’s entirely respects ideas,
considered as flourishing in intellect; by a conjunction with which, we
acquire true intelligence, and the perfection of human knowledge. The first
three of the preceding are obnoxious to various mutations; the fourth less;
but the last is perfectly stable and invariable. The three first are rather
conversant about the qualities of things, about the image and shadow; the
fourth raises us to the participation of truth; but the fifth to truth itself, and
permanent essence. In the first degrees almost all are conversant; in the
fourth a few; in the fifth, all the gods, but a very small part of mankind, as it
is asserted in the Timæus. The four first may be known, indeed, without the
fifth, confusedly; but from the knowledge of the fifth they become perfectly
manifest, as effects from the knowledge of their cause. But we cannot, by
any means, attain to the apprehension of the fifth, unless we have been first
accurately conversant with the rest; for from our imperfect condition we are

compelled to rise from difference to identity, from multitude to unity, and
from shadow to substance. While we investigate the knowledge of things, if
we are alone desirous to apprehend their resemblance (which is the case
with the multitude) we shall be placed in the third degree, and may easily
acquire the object of our pursuit. But if we should fortunately possess the
true philosophical genius, which is rare in the extreme, and aspiring to the
fifth degree, should, by a happy event, attain to its conjunction, though such
a contact is clearer and more certain than all knowledge; yet it is difficult to
express it in words, and to manifest it to others. And the reason of this is
obvious: first, because words are wanting, which exactly correspond to the
essence of a thing, since these are only the symbols of shadows. Secondly,
because we speak with those, who are alone conversant with shadows, and
are on this account derided by them, when they find that our fifth does not,
by any means, accord with material resemblances, which they consider as
the only realities.
8. And here a question very naturally presents itself for our solution,
whether the soul, while united with the body, is able to perceive ideas,
without the assistance of the phantasy, For it seems difficult to apprehend
how the soul, thus depressed and weighed down with the terrene mass,
should be able to raise herself to the supernal light of ideas, and become
united with their refulgence. The opinion of the Peripatetics is well known,
that some phantasm must always accompany intelligence; but this is denied
by the Platonists, and I think with great reason. For the operations of
intellect are not dependent on the phantasy, though the perceptions of the
latter proceed from the energies of the former. Besides, as Plotinus
beautifully observes, our most vigorous energies are accompanied with the
least animadversion; and there is no absurdity in supposing that by
increasing the force of intellectual energy, we may speculate free from all
imagination; since the phantasms attending our conceptions, became weak
in proportion as the intellectual sight increases in vigour. On this account,
the Platonists affirm, that the moral virtues free us from the vehemence of
perturbations; but the contemplative from imagination, and the senses.
Hence too, the sciences may be called living waters; in which the wings of
the soul being dipt, her feathers, which were either separated or broken by
her lapse into body, are repaired, and restored to a resemblance of their
former perfection. For the wings are the powers of the soul, leading to
intelligibles: but the feathers are as well the natural instincts to good and

truth, as reasons inserted in the soul; which either fall off, or are broken by
her descent into body, and conjunction with its ruinous bonds. But these are
repaired and invigorated by the sciences, which, like living streams, flowing
from the fountains of ideas, restore life and perfection to the soul. Hence
Plato, in the Phædrus, asserts that these wings of the soul are increased by
every thing which confers to supernal elevation; as beauty, wisdom, and the
like; and by a convenient metaphor, in the same dialogue, he considers the
chariot of the souls lives, her charioteer, and the horses by which her car is
drawn; and lastly, every thing which contributes to the elevation of the soul,
and her conjunction with intellect and ideas. We may therefore conclude,
that this conjunction is possible to be effected, though it is rarely obtained;
and that it is a flight too arduous and sacred for the groveling and sordid; a
splendor too bright for the sensible eye; and a contact too ineffable to be
described by the unstable composition of words.
But I cannot conclude this section, without soliciting the reader’s
attention to a comparison of the difference between the ancient philosophy,
and that invented by Mr. Locke, and the moderns. According to Mr. Locke’s
system ideas are formed from sensible particulars, by a kind of mechanical
operation; so that truth is something by its nature, posterior to sensation,
and entirely dependent on it for existence. According to Plato, ideas are
eternal and immaterial beings, the originals of all sensible forms, and the
fountains of all evidence and truth; so that on this system truth ranks among
the first, and not in the last of things; and would still retain its nature,
though the corporeal senses were no more. According to Mr. Locke, the
soul is a mere rasa tabula, an empty recipient, a mechanical blank.
According to Plato, she is an ever-written tablet, a plenitude of forms, a
vital and intellectual energy. On the former system, she is on a level with
the most degraded natures, the receptacle of material species, and the
spectator of delusion and non-entity[21]. Hence, her energies are nothing but
somnolent perceptions, and encumbered cogitations; for all her knowledge
terminates in sense, and her science in passion. Like a man between
sleeping and waking, her visions are turbid and confused, and the phantoms
of a material night, continually glide before her drowsy eye. But on the
latter system, the soul is the connecting medium of an intelligible and
sensible nature, the bright repository of all middle forms, and the vigilant
eye of all cogitative reasons. Hence she is capable of rousing herself from
the sleep of a corporeal life, and emerging from this dark Cimmerian land,

into the regions of light and reality. At first, indeed, before she is excited by
science, she is oppressed with lethargy, and clouded with oblivion; but in
proportion as learning and enquiry stimulate her dormant powers, she
wakens from the dreams of ignorance, and opens her eye to the irradiations
of wisdom. On Mr. Locke’s system, the principles of science and sense are
the same, for the energies of both originate from material forms, on which
they are continually employed. Hence, science is subject to the flowing and
perishable nature of particulars; and if body and its attributes were
destroyed, would be nothing but a name. But on the system of Plato, they
differ as much as delusion and reality; for here the vital, permanent, and
lucid nature of ideas is the fountain of science; and the inert, unstable, and
obscure nature of sensible objects, the source of sensation. On Mr. Locke’s
system, body may be modified into thought, and become an intelligent
creature; it may be subtilized into life, and shrink, by its exility, into
intellect. On that of Plato, body can never alter its nature by modification,
however, it may be rarefied and refined, varied by the transposition of its
parts, or tortured by the hand of experiment. In short, the two systems may
be aptly represented by the two sections of a line, in Plato’s Republic. In the
ancient, you have truth itself, and whatever participates of the brightest
evidence and reality: in the modern, ignorance, and whatever belongs to
obscurity and shadow. The former fills the soul with intelligible light,
breaks her lethargic fetters, and elevates her to the principle of things; the
latter clouds the intellectual eye of the soul, by increasing her oblivion,
strengthens her corporeal bands, and hurries her downwards into the dark
labyrinths of matter.
Nor is it wonderful there should be so great a difference between the
two systems, and so much in favour of the ancients, if we consider the great
advantages these ancients possessed over the moderns in every thing which
contributes to the advancement of philosophy. For, in the first place, they
lived in an age when abstract investigations were in the greatest request,
and the professors of such pursuits in the highest estimation. Besides this,
they united the most exalted abilities with the most unwearied attention and
obstinate perseverance; they devoted their whole lives to the search of truth;
and relinquished every thing which might be an obstacle to its acquisition.
We may add, likewise, the advantages of a language extremely
philosophical; and a freedom from the toil of learning any tongue but their
own. Now the reverse of all this is the portion of the moderns: for in the

present age, abstract speculations are ridiculed; and its professors despised.
The pursuit of truth is considered as perfectly consistent with ordinary
avocations, and is rather prosecuted as a relief from the toils of business
than as a thing desirable for its own sake, and of the greatest dignity and
worth. Hence, a few years desultory application at a college, where
language is one of the first objects of attention, qualifies a modern for
philosophy, raises him above Pythagoras and Plato, and persuades him, with
presumptuous confidence, to enter the lists against these venerable heroes.
And lastly, all modern languages are barbarous with respect to the Greek;
falling far short of its harmony and energy, its copiousness and propriety. If
such then be the true state of the case, what judgment must we form of men
who, with all these disadvantages, philosophized without the assistance of
the ancients, despising their works, and being ignorant of their contents?
Shall we call it prudence or presumption, wisdom or folly? Truth will
certainly pronounce the latter; and the general voice of posterity will
confirm her decision. There are two egregious instances in our own country
of this daring presumption; I mean Bacon and Locke. The former of these is
celebrated for having destroyed the jargon of the schoolmen, and brought
experimental enquiries into repute; and for attempting to investigate causes
through the immensity of particular effects. Hence, he fondly expected, by
experiment piled on experiment, to reach the principle of the universe; not
considering that his undertaking was as ridiculous as that of the giants of
old, who attempted to invade the heavens, by placing Ossa upon Pelion, and
Olympus upon Ossa; and ignorant that
Heaven still, with laughter, the vain toil surveys,
And buries madmen in the heaps they raise.

The latter of these, Mr. Locke, is applauded for having, without
assistance from the ancients, explained the nature, and exhibited the
genuine theory of human understanding. But that this applause is false, the
preceding comparison between his and the ancient philosophy, may evince;
and the variety of other self-taught systems which, like nocturnal meteors,
blaze for a while, and then vanish in obscurity, abundantly confirms. Had
these men, indeed, when they justly derided the barbarous writings of the
schoolmen, explored the works of antiquity, penetrated the wisdom they
contain, and enriched their native language with its illustration, they had
doubtless been celebrated by the latest posterity: but, desirous of becoming

masters in philosophy by intuition, they disdained the instruction of the
ancients, and vainly attempted to soar on artificial wings to the very summit
of science. They are, however, destined, like Icarus, to a precipitate fall; for
the influence of time, which is continually dissolving the cement of their
plumes, is likewise continually weakening their force, and will at last effect
their final separation. And thus much concerning the doctrine of ideas, and
numbers, according to Pythagoras and Plato.

S E C T I O N II.[22]

B ut let us now consider the properties of the demonstrative syllogism,
and endeavour to unravel its intricate web; appointing Aristotle for our
guide in this arduous investigation. For an enquiry of this kind is naturally
connected with the doctrine of ideas, as it enables us to gain a glimpse of
the universals participated in mathematical forms, and to rise to the
principles of science. It brings us acquainted with the laws which bind
demonstration; and teaches us that objects of intellect are alone the objects
of science, and the sources of truth.
Previous to the acquisition of all learning and ratiocinative discipline, it
is necessary we should possess certain natural principles of knowledge, as
subservient to our future progress and attainments. Thus, in every science
there are some things which require an immediate assent as soon as
proposed; whose certainty is too evident and illustrious to stand in need of
any demonstrative proof deduced from that particular science which, like
stately pillars, they equally support and adorn. Hence we are informed by
the geometrician, that a point is that which is destitute of all parts whatever;
but we must previously understand the meaning of the word part. Thus the
arithmetician defines an odd number, that which is divided according to
unequal parts; but it is necessary we should antecedently know the meaning
of the word unequal. Thus, too, art as well as science operates by
antecedent knowledge; and hence the architect, the statuary, and the
shipwright, learn the names and the use of their respective implements,
previous to the exercise of the materials themselves. This is particularly
evident in the discursive arts of rhetoric and logic; thus the logician reasons
by syllogism, the rhetorician by induction, and the sophist by digressions
and examples; while each proceeds in an orderly progression from
principles simple and evident, to the most remote and complicated
conclusions.
2. The antecedent knowledge of things may be divided into two parts:
the one a knowledge of their existence, or that they exist; the other a
knowledge of the terms expressive of their existence. Thus, previous to the
enquiry why iron is attracted by the magnet, it is necessary we should learn

the reality of this attraction, and the general mode of its operation: thus too,
in an enquiry concerning the nature of motion and time, we must be
previously convinced of their existence in the nature of things. The second
division of antecedent knowledge takes place in subjects whose very
existence admits of a dispute: thus previous to a solution of the questions,
Whether there are any gods or not? Whether there is a providence or not?
and the like, it is necessary we should first understand the meaning of the
terms; since we in vain investigate the nature of any thing while we are
ignorant of the meaning of its name; although, on the contrary, we may
have a perfect conception of the meaning of some words, and yet be totally
ignorant whether the things they express have a real, or only an imaginary
existence. Thus, the meaning of the word centaur is well understood by
every one; but its existence is questioned by most.
3. From hence it will easily appear, that no small difference subsists
between learning and knowledge. He who is about to understand the truth
of any proposition, may be said to possess a previous conception of its
truth; while, on the contrary, it may happen that he who is in the capacity of
a learner, has no antecedent knowledge of the science he is about to learn.
Thus we attain to the distinct knowledge of a thing which we formerly
knew in a general way; and frequently, things of which we were ignorant
are learned and known in the same instant.
Of this kind are the things contained under some general idea, of which
we possess a previous knowledge: thus, he who already knows that the
three interior angles of every triangle are equal to two right, and is as yet
ignorant that some particular figure delineated on paper is a triangle, is no
sooner convinced from inspection of its being a triangle, than he
immediately learns and knows: he learns it is a triangle; he knows the
equality of its angles to two right ones. That it is now a triangle he both sees
and learns; but the equality of its angles he previously knew in that general
and comprehensive idea, which embraces every particular triangle.
Indeed, a definite knowledge of this triangle requires two conditions:
the one, that it is a triangle; and the other, that it has angles equal to two
right. The first we receive from inspection; the second is the result of a
syllogistic process; an operation too refined for the energies of sense, and
alone the province of intellect and demonstration. But demonstration
without the knowledge of that which is universal, cannot subsist; and since

the proposition is universal, that in every triangle the angles are equal to
two right; as soon as any figure is acknowledged to be a triangle, it must
necessarily possess this general property.
Hence we infer, that of the triangle delineated on paper, and concealed,
we are partly ignorant of this general property, the equality of its angles
(because we are ignorant of its existence); and we partly understand it as
included in that universal idea we previously possessed. Hence too, it is
evident that actual science arises from a medium between absolute
ignorance and perfect knowledge; and that he who possesses the principles
of demonstration, possesses in capacity the conclusions also, however
complicated and remote; and that by an evocation of these principles from
dormant power into energy, we advance from general and abstracted
knowledge to that which is sensible and particular.
4. Two acceptations of knowledge may be admitted; the one common
and without any restriction; the other limited and peculiar. Since all
knowledge, whether arising from accidents, or supported by necessary
principles, is called science. Knowledge, properly so called, arises from a
possession of that cause from which a thing derives its existence, and by
which we infer the necessity of its existence; and this constitutes simple and
absolute science. Thus too, the definitions of those general conceptions and
suppositions, which from their primary nature are incapable of
demonstration, are called science. But the science which treats of the
method of arriving at knowledge, is called demonstration; for every
demonstration is a syllogism producing science. Hence, if in every
syllogism it is necessary that the propositions should be the cause of the
conclusion; and to know any thing properly, a knowledge of its cause is
requisite; in the propositions of demonstration, both these conditions are
required: that they should be effective of the conclusion; and the causes of
the thing demonstrated.
Thus, from the ruins of a stately edifice, we may justly infer, that the
building was beautiful when entire; and from the smoke we may collect the
existence of the fire, though concealed: but the ruins of the edifice are not
the cause of its beauty; nor does fire originate from smoke, but, on the
contrary, smoke is the natural result of fire: the inference, therefore, is in
neither case a demonstrative one. Again, since every cause is both prior to,
and more excellent than its effect, it is necessary that the propositions

should be more peculiar, primary, and excellent than the conclusions. And
because we then know a thing properly when we believe it to have a
necessary existence, hence it is requisite that the propositions should be
true; for if false, a false conclusion may ensue, such as, that the diameter of
a square is commensurable with its side. But if every science arises from
antecedent knowledge, demonstration must be founded on something
previous; and on this account it is requisite that the propositions should be
more known than the conclusions. The necessary properties, then, of all
demonstrative propositions, are these; that they exist as causes, are primary,
more excellent, peculiar, true, and known, than the conclusions. Indeed,
every demonstration not only consists of principles prior to others, but of
such as are eminently first; for if the assumed propositions may be
demonstrated by other assumptions, such propositions may, indeed, appear
prior to the conclusions, but are by no means entitled to the appellation of
first. But others, on the contrary, which require no demonstration, but are of
themselves probable or manifest, are deservedly esteemed the first, the
truest, and the best. Such indemonstrable truths were called by the ancients,
axioms, from their majesty and authority; as the assumptions which
constitute the best syllogisms derive all their force and efficacy from these.
And on this account, above all others, they merit the title of the
principles of demonstration. But here it is worth observing, that these
primary propositions are not the first in the order of our conceptions; but
first to nature, or in the nature of things. To us, that which is first is
particular, and subject to sensible inspection; to nature, that which is
universal, and far remote from the apprehension of sense. Demonstration
does not submit itself to the measure of our ingenuity, but, with invariable
rectitude, tends to truth as its ultimate aim; and without stopping to consider
what our limited powers can attain, it alone explores and traces out the
nature of a thing, though to us unperceived and unknown.
This demonstrative syllogism differs not a little from others, by the
above property; the rest can as well educe a true conclusion from false
premises, which is frequent among the rhetoricians, as that which is prior
from that which is posterior; such as, Is every syllogism derived from
conjecture?
With respect to the rest, as we have already confessed, they may be
formed from principles that are true, but not from such as are proper and

peculiar; as if a physician should endeavour to prove an orbicular wound
the most difficult to coalesce and heal, because its figure is of all others, the
most capacious; since the demonstration of this is not the province of the
physician, but of the geometrician alone.
5. That proposition is called immediate, which has none superior to
itself, and which no demonstration whatever can confirm: such as these are
held together by the embraces of universals. There are some, indeed, united
from that which is sensible and particular: thus, that the garment is white, is
an immediate proposition, but not of that kind whose principles require to
be demonstrative ones; the cause of which we shall hereafter investigate. Of
immediate propositions subservient to the purposes of demonstration, some
are of such a superior nature, that all men possess a knowledge of them
without any previous instruction; and these are called axioms, or general
notions; for without these all knowledge and enquiry is vain. Another
species of immediate propositions is position; incapable of being
strengthened by demonstration, yet not necessarily foreknown by the
learner, but received from the teacher. With respect to the genus of position,
one of its species is definition, and another hypothesis. Definition is an
oration, in which we neither speak of the existence, nor non-existence of a
thing; but alone determine its nature and essence. It is common to every
hypothesis, not to be derived from nature, but to be the entire result of the
art of the preceptor.
It likewise always affirms the existence or non-existence of its subject:
such as, that motion is, and that from nothing nothing is produced. Those
which are not so perspicuous are called postulates, or petitions; as that a
circle may be described from any centre, and with any radius; and such as
these are properly hypotheses and postulates.
6. We have now seen the privilege assigned to the principles of
demonstration:—whether or no our decision has been just, the ensuing
considerations will evince. We said that the assumptions in demonstration
were more known than the conclusions,—not indeed without reason, since
through these our knowledge and belief of the conclusion arises. For
universally, that quality which is attributed to many different things so as to
be assigned to one through the medium of another, abounds most in that
medium by which it is transmitted to the rest.

Thus the sun, through the medium of the moon, illuminates the earth by
night; thus the father loves the preceptor through the medium of his child.
And in the first instance the moon is more lucid than any object it
enlightens: in the second, the child possesses more of the father’s regard
than his preceptor. If then we assent to the conclusions through our belief of
the principles alone, it is necessary that the principles should be more
known, and inherit a greater degree of our assent. Hence, if it be true that
the principles are more known than the conclusions, it follows, that either
our knowledge of them is derived from demonstration, or that it is superior
to any demonstrative proof; and after this manner we must conceive of
those general self-evident notions which, on account of their
indemonstrable certainty, are deservedly placed at the top of all human
science.
These propositions not only possess greater credibility than their
conclusions, they likewise inherit this property as an accession to their
dignity and importance; that no contrary propositions deserve greater belief;
for if you give no more assent to any principle than to its contrary, neither
can you give more credit to the conclusion deduced from that principle than
to its opposite. Were this the case, the doctrine of these propositions would
immediately lose its invariable certainty.
7. There are, indeed, some who, from erroneously applying what we
have rightly determined, endeavour to take away the possibility of
demonstration. From the preceding doctrine it appears that the principles
are more aptly known than the conclusions. This is not evident to some,
who think nothing can be known by us without a demonstrative process;
and consequently believe that the most simple principles must derive all
their credit from the light of demonstration.
But if it be necessary that all assumptions should be demonstrated by
others, and these again by others; either the enquiry must be continued to
infinity, (but infinity can never be absolved), or if, wearied by the immense
process, you at length stop, you must doubtless leave those propositions
unknown, whose demonstration was declined through the fatigue of
investigation. But how can science be derived from unknown principles?
For he who is ignorant of the principles, cannot understand the conclusions
which flow from these as their proper source, unless from an hypothesis or
supposition of their reality.

This argument of the sophists is, indeed, so far true, that he who does
not understand that which is first in the order of demonstration, must remain
ignorant of that which is last:—But in this it fails, that all knowledge is
demonstrative; since this is an assertion no less ridiculous than to maintain
that nothing can be known. For as it is manifest that some things derive
their credit and support from others, it is equally obvious that many, by their
intrinsic excellence, possess indubitable certainty and truth; and command
our immediate assent as soon as proposed. They inherit, indeed, a higher
degree of evidence than those we assent to by the confirmation of others;
and these are the first principles of demonstration: propositions
indisputable, immediate, and perspicuous by that native lustre they always
possess. By means of these, we advance from proposition to proposition,
and from syllogism to syllogism, till we arrive at the most complicated and
important conclusions. Others, willing to decline this infinite progression,
defend the necessity of a circular or reciprocal demonstration. But this is
nothing more than to build error upon error, in order to attain the truth; an
attempt no less ridiculous than that of the giants of old. For since, as we
shall hereafter accurately prove, demonstration ought to consist from that
which is first, and most known; and since it is impossible that the same
thing should be to itself both prior and posterior: hence we infer the
absurdity of circular demonstration; or those syllogisms in which the
conclusions are alternately substituted as principles, and the principles as
conclusions. It may, indeed, happen, that the same thing may be both prior
and posterior to the same; but not at one and the same time, nor according
to the same mode of existence. Thus, what is prior in the order of our
conceptions, is posterior in the order of nature; and what is first in the
arrangement of things, is last in the progressions of human understanding.
But demonstration always desires that first which is prior in the order and
constitution of nature. But the folly of such a method will more plainly
appear from considering its result: let us suppose every a is b, and every b
is c; hence we justly infer, that every a is c. In like manner, if we prove that
every a is b, and by a circular demonstration, that every b is a, the
consequence from the preceding is no other than that every a is a; and thus
the conclusion terminates in that from which it first began; a deduction
equally useless and ridiculous. However, admitting that, in the first figure,
circular demonstration may be in some cases adopted, yet this can but
seldom happen from the paucity of reciprocal terms.

But that reciprocal terms are very few, is plain from hence: let any
species be assumed, as man; whatever is the predicate of man, is either
constitutive of his essence, or expressive of some accident belonging to his
nature. The superior genera and differences compose his essence, among
which no equal predicate can be assigned reciprocable with man, except the
ultimate differences which cannot be otherwise than one, i. e., risibility,
which mutually reciprocates with its subject; since every man is risible, and
whatever is risible is man. Of accidents some are common, others peculiar;
and the common are far more in number than the peculiar; consequently the
predicates which reciprocate with man, are much fewer than those which do
not reciprocate.
8. It is now necessary to enumerate the questions pertaining to
demonstration; and for this purpose, we shall begin with propositions, since
from these, syllogisms are formed; and since every proposition consists of a
subject and predicate, the modes of predication must be considered, and
these are three which I call total, essential, and universal; a total
predication takes place when that which is affirmed or denied of one
individual is affirmed or denied of every individual comprehended under
the same common species.
Thus, animal is predicated of every man, and it has this farther property
besides, that of whatever subject it is true to affirm man, it is at the same
time true to affirm animal.
Those things are said to be essentially predicated; first, when the
predicate is not only total, but constitutes the essence of the subject;
instances of this kind are, animal of man; tree of the plantain; a line of a
triangle; for a triangle is that which is contained under three right-lines. But
here we must observe, that not every total predicate is an essential one;
thus, whiteness is predicated of every swan, because it is inherent in every
swan, and at every instant of time; but because whiteness does not
constitute the essence of a swan, it is not essentially predicated; and this,
first, is one of the modes of essential predication of the greatest importance
in demonstration. The second mode is of accidents, in the definition of
which their common subject is applied: thus, a line is essentially inherent in
rectitude, because in its geometrical definition, a line is adopted; for
rectitude is no other than a measure, equally extended between the points of
a line. In the same manner, imparity is contained in number; for what is that

which is odd, but a number divided into unequal parts? Thus, virtues are
resident in the soul, because, in their definition, either some part of the soul,
or some one of its powers, is always applied. The third mode of essential
predicates pertains to accidents which are inseparably contained in some
particular subject, so as to exclude a prior existence in any other subject;
such as colour in superficies. The fourth mode is of things neither contained
in another, nor predicated of others; and such are all individuals, as Callias,
Socrates, Plato. Causes are likewise said to exist substantially, which
operate neither from accident nor fortune.
Thus, digging up the ground for the purposes of agriculture, may be the
cause of discovering a treasure, but it is only an accidental one. But the
death of Socrates, in despite of vigilance, is not the result of a fortuitous
cause, but of an essential one, viz. the operation of poison.
9. These posterior significations of essential predicates are added more
for the sake of ornament than use; but the two former have a necessary
existence, since they cannot but exist in the definition of names which
predicate the essence of a thing, and in subjects which are so entirely the
support of accidents, that they are always applied in their definition. But it
is a doubt with some, whether those accidents are necessary, which cannot
be defined independent of their common subject? To this we answer, that no
such accident can, from its nature, be contained in every individual of any
species; for curvature is not contained in every line; nor imparity in every
number; from whence we infer, that neither is curvature necessarily existent
in a line, nor parity in number. The truth of this is evident from considering
these accidents abstracted from their subjects; for then we shall perceive
that a line may exist without curvature, and number without imparity.
Again, I call that an universal predicate, which is predicated of a subject
totally and essentially, and considered as primarily and inseparably inherent
in that subject: for it does not follow that a predicate, which is total, should
be immediately universal; for whiteness is affirmed of every swan, and
blackness of every crow, yet neither universally. In like manner, a
substantial predicate is not consequently an universal one; for the third
mode of essential predicates, and the two following (instanced before)
cannot be universal. Thus, colour, although inherent in superficies
essentially, is not inherent in every superficies, and consequently not
universally. Thus again, Socrates, Callias, and Plato, though they exist

essentially, are not universals, but particulars; and thus, lastly, the drinking
of poison was an essential cause of the death of Socrates, but not an
universal one, because Socrates might have died by other means than
poison. If then, we wish to render an accurate definition of an universal
predicate, we must not only say it is total and essential, but that it is
primarily present to its subject and no other. Thus, the possession of angles
equal to two right, primarily belongs to a triangle; for this assertion is
essentially predicated of triangle, and is inherent in every triangle. This
property, therefore, is not universally in figure, because it is not the property
of every figure, not of a square, for instance; nor as universal in a scalene
triangle: for although it is contained in every scalene, and in every
equilateral, and isosceles triangle, yet it is not primarily contained in them,
but in triangle itself; because these several figures inherit this property, not
from the particular species to which they belong, but from the common
genus triangle. And thus much concerning total, essential, and universal
predicates.
10. Concerning that which is universal, we are frequently liable to err;
often from a belief that our demonstration is universal, when it is only
particular; and frequently from supposing it particular when it is, on the
contrary, universal. There are three causes of this mistake; the first, when
we demonstrate any particular property of that which is singular and
individual, as the sun, the earth, or the world. For since there is but one sun,
one earth, and one world, when we demonstrate that the orb of the earth
possesses the middle place, or that the heavens revolve, we do not then
appear to demonstrate that which is universal.
To this we answer: when we demonstrate an eclipse of the sun to arise
from the opposition of the moon, we do not consider the sun as one
particular luminary, but we deduce this consequence as if many other suns
existed besides the present.
Just as if there were but one species of triangles existed; for instance,
the isosceles; the equality of its angles at the base would not be considered
in the demonstration of the equality of all its angles to two right ones: but
its triangularity would be essential, supposing every species of triangles but
the isosceles extinct, and no other the subject of this affection. So when we
prove that the sun is greater than the earth, our proof does not arise from
considering it as this particular sun alone, but as sun in general; and by

applying our reasoning to every sun, if thousands besides the present should
enlighten the world. This will appear still more evident, if we consider that
such conclusions must be universal, as they are the result of an induction of
particulars: thus, he who demonstrates that an eclipse of the sun arises from
the opposition of the moon between the sun and earth, must previously
collect, by induction, that when any luminous body is placed in a right-line
with any two others opaque, the lucid body shall be prevented, in a greater
or less degree, from enlightening the last of these bodies, by the
intervention of the second; and by extending this reasoning to the sun and
earth, the syllogism will run thus:
Every lucid body placed in a right-line with two others opaque, will be eclipsed in
respect of the last by the intervention of the second;
The sun, or every sun, is a luminous body with these conditions;
And consequently the sun, and so every sun, will be eclipsed to the earth by the
opposition of the moon.

Hence, in cases of this kind, we must ever remember, that we
demonstrate no property of them as singulars, but as that universal
conceived by the abstraction of the mind.
Another cause of deception arises, when many different species agree in
one ratio or analogy, yet that in which they agree is nameless. Thus number,
magnitude, and time, differ by the diversities of species; but agree in this,
that as any four comparable numbers correspond in their proportions to
each other, so that as the first is to the second, so is the third to the fourth;
or alternately, as the first to the third; so is the second to the fourth: in a
similar manner, four magnitudes, or four times, accord in their mutual
analogies and proportions. Hence, alternate proportion may be attributed to
lines as they are lines, to numbers as they are numbers, and afterwards to
times and to bodies, as the demonstration of these is usually separate and
singular; when the same property might be proved of all these by one
comprehensive demonstration, if the common name of their genus could be
obtained: but since this is wanting, and the species are different, we are
obliged to consider them separately and apart; and as we are now speaking
of that universal demonstration which is properly one, as arising from one
first subject; hence none of these obtain an universal demonstration,
because this affection of alternate proportion is not restricted to numbers or
lines, considered in themselves, but to that common something which is

supposed to embrace all these, and is destitute of a proper name. Thus too
we may happen to be deceived, should we attempt to prove the equality of
three angles to two right, separately, of a scalene, an isosceles, and an
equilateral triangle, only with this difference, that in the latter case the
deception is not so easy as in the former; since here the name triangle,
expressive of their common genus, is assigned. A third cause of error arises
from believing that to demonstrate any property inherent after some
particular manner in the whole of a thing, is to demonstrate that property
universally inherent. Thus, geometry proves[23] that if a right-line falling
upon two right-lines makes the outward angle with the one line a right-
angle, and the inward and opposite angle with the other a right one, those
two right-lines shall be parallel, or never meet, though infinitely extended.
This property agrees to all lines which make right-angles: but they are not
primarily equidistant on this account, since, if they do not each make a
right-angle, but the two conjointly are equal to two right, they may still be
proved equidistant. This latter demonstration, then, is primarily and
universally conceived; the other, which always supposes the opposite angles
right ones, does not conclude universally; though it concludes totally of all
lines with such conditions: the one may be said to conclude of a greater all;
the other of a lesser. It is this greater all which the mind embraces when it
assents to any self-evident truth; or to any of the propositions of Euclid. But
by what method may we discover whether our demonstration is of this
greater or lesser all? We answer, that general affection which constitutes
universal demonstration is always present to that subject, which when taken
away, the predicate is immediately destroyed, because the first of all its
inherent properties.
Thus, for instance, some particular sensible triangle possesses these
properties:—it consists of brass; it is scalene; it is a triangle. The query is,
by which of these we have just now enumerated, this affection of
possessing angles equal to two right is predicated of the triangle? Take
away the brass, do you by this means destroy the equality of its angles to
two right ones? Certainly not:—take away its scalenity, yet this general
affection remains: lastly, take away its triangularity, and then you
necessarily destroy the predicate; for no longer can this property remain, if
it ceases to be a triangle.

But perhaps some may object from this reasoning, such a general
affection extends to figure, superficies, and extremities, since, if any of
these are taken away, the equality of its angles to two right can no longer
remain. It is true, indeed, that by a separation of figure, superficies, and
terms, from a body, you destroy all the modes and circumstances of its
being; yet not because these are taken away, but because the triangle, by the
separation of these, is necessarily destroyed; for if the triangle could still be
preserved without figure, superficies, and terms, though these were taken
away it would still retain angles equal to two right; but this is impossible.
And if all these remain, and triangle is taken away, this affection no longer
remains. Hence the possession of this equality of three angles to two right,
is primarily and universally inherent in triangle, since it is not abolished by
the abolition of the rest:——such as to consist of brass; to be scalene, or the
like. Neither does it derive its being from the existence of the rest alone; as
figure, superficies, terms; since it is not every figure which possesses this
property, as is evident in such as are quadrangular, or multangular. And thus
it is preserved by the preservation of triangle, it is destroyed by its
destruction.
11. From the principles already established, it is plain that
demonstration must consist of such propositions as are universal and
necessary. That they must be universal, is evident from the preceding; and
that they must be necessary, we gather probably from hence; that in the
subversion of any demonstration we use no other arguments than the want
of necessary existence in the principles.
We collect their necessity demonstratively, thus; he who does not know
a thing by the proper cause of its existence, cannot possess science of that
thing; but he who collects a necessary conclusion from a medium not
necessary, does not know it by the proper cause of its existence, and
therefore he has no proper science concerning it. Thus, if the necessary
conclusion c is a, be demonstrated by the medium B, not necessary; such a
medium is not the cause of the conclusion; for since the medium does not
exist necessarily, it may be supposed not to exist; and at the time when it no
longer exists, the conclusion remains in full force; because, since necessary,
it is eternal. But an effect cannot exist without a cause of its existence; and
hence such a medium can never be the cause of such a conclusion. Again,
since in all science there are three things, with whose preservation the

duration of knowledge is connected; and these are, first, he who possesses
science; secondly, the thing known; and thirdly, the reason by which it is
known; while these endure, science can never be blotted from the mind, but
on the contrary, if science be ever lost, it is necessary some of these three
must be destroyed.
If then you infer that the science of a necessary conclusion may be
obtained from a medium not necessary, suppose this medium, since capable
of extinction, to be destroyed; then the conclusion, since necessary, shall
remain; but will be no longer the object of knowledge, since it is supposed
to be known by that medium which is now extinct. Hence, science is lost,
though none of the preceding three are taken away; but this is absurd, and
contrary to the principles we have just established. The thing known
remains; for the conclusion, since necessary, cannot be destroyed;—he who
knows still remains, since neither dead, nor forgetful of the conclusion:—
lastly, the demonstration by which it was known, still survives in the mind;
and hence we collect, that if science be no more after the corruption of the
medium, neither was it science by that medium before its corruption; for if
science was ever obtained through such a medium, it could not be lost while
these three are preserved. The science, therefore, of a necessary conclusion
can never be obtained by a medium which is not necessary.
12. From hence it is manifest, that demonstrations cannot emigrate from
one genus to another; or by such a translation be compared with one
another. Such as, for instance, the demonstrations of geometry with those of
arithmetic. To be convinced of this, we must rise a little higher in our
speculations, and attentively consider the properties of demonstration: one
of which is, that predicate which is always found in the conclusion, and
which affirms or denies the existence of its subject: another is, those axioms
or first principles by whose universal embrace demonstration is fortified;
and from whose original light it derives all its lustre. The third is the subject
genus, and that nature of which the affections and essential properties are
predicated; such as magnitude and number. In these subjects we must
examine when, and in what manner a transition in demonstrations from
genus to genus may be allowed. First, it is evident, that when the genera are
altogether separate and discordant, as in arithmetic and geometry, then the
demonstrations of the one cannot be referred to the other. Thus, it is
impossible that arithmetical proofs can ever be accommodated with

propriety to the accidents of magnitudes: but when the genera, as it were,
communicate, and the one is contained under the other, then the one may
transfer the principles of the other to its own convenience. Thus, optics
unites in amicable compact with geometry, which defines all its
suppositions; such as lines that are right, angles acute, lines equilateral, and
the like. The same order may be perceived between arithmetic and music:
thus, the double, sesquialter, and the like, are transferred from arithmetic,
from which they take their rise, and are applied to the measures of harmony.
Thus, medicine frequently derives its proofs from nature, because the
human body, with which it is conversant, is comprehended under natural
body. From hence it follows, that the geometrician cannot, by any
geometrical reasons demonstrate any truth, abstracted from lines,
superficies, and solids; such as, that of contraries there is the same science;
or that contraries follow each other; nor yet such as have an existence in
lines and superficies, but not an essential one, in the sense previously
explained.
Of this kind is the question, whether a right-line is the most beautiful of
lines? or whether it is more opposed to a line perfectly orbicular, or to an
arch only. For the consideration of beauty, and the opposition of contraries,
does not belong to geometry, but is alone the province of metaphysics, or
the first philosophy.
But a question here occurs, If it be requisite that the propositions which
constitute demonstration should be peculiar to the science they establish,
after what manner are we to admit in demonstration those axioms which are
conceived in the most common and general terms; such as, if from equal
things you take away equals, the remainders shall be equal:——as likewise,
of every thing that exists, either affirmation or negation is true? The
solution is this: such principles, though common, yet when applied to any
particular science for the purposes of demonstration, must be used with a
certain limitation. Thus the geometrician applies that general principle, if
from equal things, &c. not simply, but with a restriction to magnitudes; and
the arithmetician universally to numbers.
Thus too, that other general proposition:——of every thing, affirmation
or negation is true; is subservient to every art, but not without
accommodation to the particular science it is used by. Thus number is or is

not, and so of others. It is not then alone sufficient in demonstration that its
propositions are true, nor that they are immediate, or such as inherit an
evidence more illustrious than the certainty of proof; but, besides all these,
it is necessary they should be made peculiar by a limitation of their
comprehensive nature to some particular subject. It is on this account that
no one esteems the quadrature of Bryso[24], a geometrical demonstration,
since he uses a principle which, although true, is entirely common. Previous
to his demonstration he supposes two squares described, the one
circumscribing the circle, which will be consequently greater; the other
inscribed, which will be consequently less than the given circle. Hence,
because the circle is a medium between the two given squares, let a mean
square be found between them, which is easily done from the principles of
geometry; this mean square, Bryso affirms, shall be equal to the given
circle. In order to prove this, he reasons after the following manner: those
things which compared with others without any respect, are either at the
same time greater, or at the same time less, are equal among themselves: the
circle and the mean square are, at the same time, greater than the internal,
and at the same time less than the external square; therefore they are equal
among themselves. This demonstration can never produce science, because
it is built only on one common principle, which may with equal propriety
be applied to numbers in arithmetic, and to times in natural science. It is
defective, therefore, because it assumes no principle peculiar to the nature
of the circle alone, but such a one as is common to quantity in general.
13. It is likewise evident, that if the propositions be universal, from
which the demonstrative syllogism consists, the conclusion must
necessarily be eternal. For necessary propositions are eternal; but from
things necessary and eternal, necessary and eternal truth must arise. There is
no demonstration, therefore, of corruptible natures, nor any science
absolutely, but only by accident; because it is not founded on that which is
universal. For what confirmation can there be of a conclusion, whose
subject is dissoluble, and whose predicate is neither always, nor simply, but
only partially inherent? But as there can be no demonstration, so likewise
there can be no definition of corruptible natures; because definition is either
the principle of demonstration, or demonstration differing in the position of
terms, or it is a certain conclusion of demonstration. It is the beginning of
demonstration, when it is either assumed for an immediate proposition, or
for a term in the proposition; as if any one should prove that man is risible,

because he is a rational animal. And it alone differs in position from
demonstration, as often as the definition is such as contains the cause of its
subjects existence. As the following: an eclipse of the sun is a concealment
of its light, through the interposition of the moon between that luminary and
the earth. For the order of this definition being a little changed, passes into a
demonstration; thus,
The moon is subjected and opposed to the sun:
That which is subjected and opposed, conceals:
The moon, therefore, being subjected and opposed, conceals the sun.

But that definition is the conclusion of demonstration, which extends to
the material cause; as in the preceding instance, the conclusion affirming
that the subjection and opposition of the moon conceals the sun, is a
definition of an eclipse including the material cause.
Again, we have already proved that all demonstration consists of such
principles as are prior in the nature of things; and from hence we infer, that
it is the business of no science to prove its own principles, since they can no
longer be called principles if they require confirmation from any thing prior
to themselves; for, admitting this as necessary, an infinite series of proofs
must ensue. On the contrary, if this be not necessary, but things most known
and evident are admitted, these must be constituted the principles of
science. He who possesses a knowledge of these, and applies them as
mediums of demonstration, is better skilled in science, than he who knows
only posterior or mediate propositions, and demonstrates from posterior
principles. But here a doubt arises whether the first principles of geometry,
arithmetic, music, and of other arts, can ever be demonstrated? Or shall we
allow they are capable of proof, not by that particular science which applies
them as principles or causes of its conclusions? If so, this will be the office
of some superior science,—which can be no other than the first philosophy,
to whose charge the task is committed; and whose universal embrace
circumscribes the whole circle of science, in the same manner as arithmetic
comprehends music, or geometry optics.—This is no other than that
celebrated wisdom which merits the appellation of science in a more simple,
as well as in a more eminent degree than others: not, indeed, that all causes
are within its reach, but such only as are the principal and the best, because
no cause superior to them can ever be found. Hence the difficulty of
knowing whether we possess science or not, from the difficulty of

understanding whether it is founded on peculiar or common principles;
since it is necessary that both these should be applied in the constitution of
all real knowledge and science.
[25]
Again, axioms differ from postulates in this:—they demand our
assent without any previous solicitation, from the illustrious certainty they
possess. Their truth may, indeed, be denied by external speech, but never
from internal connection. He who denies that equal things shall remain
from the subtraction of equal, dissents, as Euripides says, with his tongue,
and not with his heart. But demonstration depends not on external speech,
but on intellectual and internal conviction; and hence, axioms derive all
their authority from intrinsic approbation, and not from public proclaim.
For the prompt decisions of the tongue are frequently dissonant from the
sentiments concealed in the secret recesses of the heart. Thus the
[26]
geometrician does not speculate those lines which are the objects of
corporeal sight, but such as are exhibited by mental conception, and of
which the delineations on paper, or in the dust, are no more than imperfect
copies, notes, and resemblances. Thus, when he draws a pedal line which is
not pedal, or an equilateral triangle which is not equilateral, we must pay no
regard to the designations of the pen, but solely attend to the intellection of
the mind; for the property demonstrated of some particular line, is in the
conclusion applied to one that is universal, and this true line could be no
otherwise signified to the learner than by a material description.
The certainty of axioms is, indeed, in a measure obvious to every one.
For what more evident than that nothing exists of which it is possible, at the
same time, to affirm and deny any circumstance of being? Indeed, so
illustrious and indubitable is the light of this axiom, that in any
demonstration we are ashamed to assign it the place of an assumption. It
would almost seem prolix and superfluous, since there is nothing more
manifest and certain; and yet there are cases in which it is necessary to rank
it among assumptions. And these take place whenever the intention is to
conclude the existence of something as true, and of its opposite as false.
Thus, for instance, in the demonstration that the world is finite, we assume
this principle, and then reason as follows:
Bound and infinite cannot be at the same time affirmed and denied of any body:
The world is a body:
Therefore the world is not at the same time finite and infinite.

And in this genus of demonstration, the major proposition ought always
to assimilate with the conclusion. But the above axiom is not the only one
obvious, for the following possesses equal certainty; that of every thing
which exists, either affirmation or negation is true. This axiom is of great
use in demonstrations leading to an absurdity; for he who demonstrates the
impossibility of any opposite assertion, necessarily establishes his own.
Hence it is we affirm that the diameter of a square is either commensurable
or incommensurable with its side; and this general principle is
accommodated, and, as it were, descends into its proper matter as often as
that which it possesses of universal is contracted to a certain genus; for, as
we have previously observed, common principles are not admitted in
demonstration without any restriction; but then only when their general
nature is limited to some particular subject, by which they become peculiar
and apposite.
14. [27]Wisdom, or the first philosophy and logic, agree in not using
axioms after the same manner as other arts; but on the contrary, they
confirm and establish their certainty, though with this difference, that the
logician reasons only from probabilities, but the metaphysician from the
highest certainty and evidence. Besides, we do not rank logic in the order of
the sciences, because it is destitute of some determinate genus or subject, as
it is neither conversant about lines, nor numbers, nor proportions. And its
chief concern is about apparent properties, and not such as are essential to a
subject.
Hence, in logical disquisitions, we confidently employ interrogations, as
equally subservient to the affirmation or negation of an opinion:—a method
utterly impracticable, if we only employed those principles which are
universally acknowledged; since it is impossible of the same thing to prove
contrary properties,—as of the soul, that it is mortal and immortal; but he
who demonstrates, assumes one definite part of a question, because his
purpose is not to interrogate, but to trace out the latent paths of truth. And
hence, if any one affirms that the soul is moved, and immediately after
denies it, he is no longer a subject worthy the exercise of our discursive and
reasoning powers.
Again, it may so happen, that the same science at one time considers
why a thing is, at another only explains its existence, or that it exists,
without considering the cause. Thus, the syllogism which concludes by

mediate propositions, demonstrates without assigning the proper cause: but
that which determines by immediate ones, in a great measure explains the
cause or reason of existence. Thus, he who infers that trees do not breathe
because they are not animals, reasons from a mediate and secondary cause,
because there are many animals, such as insects, which exist without
breathing: but he who infers this from their want of lungs, demonstrates
from the immediate and primary cause.
Thus, the following syllogism is a mediate one, or such as requires one
or more mediums to establish its certainty:
Every thing that is not an animal does not breathe;
A tree is not an animal;
Therefore a tree does not breathe.

Here the major proposition is evidently mediate, because we are still to seek
why that which is not an animal does not breathe, which the following
immediate syllogism solves.
Every thing that is not endued with lungs does not breathe;
Every thing that is not an animal is not endued with lungs; ergo,
Every thing that is not an animal does not breathe.

Again, the same science may demonstrate the existence of a thing, or
that it exists, and the cause of such existence as often as it assigns two
immediate reasons; but the one from the proper cause, the other only from a
sign. Thus, he who demonstrates the increase of the moon, from the
plenitude of her orb, infers the cause of such increase; but on the contrary,
he who collects the plenitude of her orb from her increase, reasons only
from a sign, and can alone declare its existence. And, indeed, it often
happens that the cause and sign reciprocate, so that as from the sign we
advance to the cause, demonstration from the cause frequently recurs to the
sign. Thus, from the breadth and firmness of the basis, we collect the
permanent duration of the pyramid; and from its extended existence we
infer the strength of its support. Whenever, then, the argument originates
from a sign, it gives evidence to the conclusion, as from something more
known than its cause. When it begins from the cause, it proceeds from that
which is first in the order of nature, to that which is last, and reasons as
from the proper principle of the thing.

Sometimes the cause and sign do not reciprocate. Thus, although
wherever there is smoke, we infer the existence of fire; yet we cannot infer,
that wherever there is fire smoke exists. Thus, from the palace and the
picture we collect the existence of the architect and painter; but the last may
exist without the first;—the living architect without the actual palace; and
the living painter without the energies of his art. And thus it is that the cause
is illustrated by its sign; but not always the sign by its cause.
Hence then, as all causes do not reciprocate with their effects; so neither
is it always causes and effects which do reciprocate: because a multitude of
signs, mutually inferring each other, may accompany a certain cause. Thus,
the signs which attend the causes of a fever, are a quick pulsation of the
artery, and an intense heat: and these signs mutually assert each other; but
no syllogism can be composed from either expressing the why, but only
simply that the other exists.
15. We now propose to consider the mode in which the two preceding
demonstrations are distributed in different sciences. When sciences then are
so related, that the one is dependent on the other, as optics on geometry,
navigation on astronomy, and music composed by the arbitration of the ear,
on that which consists in the knowledge of mathematical proportions: in
this case, the demonstration of simple existence, or that they exist, pertains
to the science of sensibles; but the demonstration why they exist to the
science which is speculative and mathematical.
Thus the mathematician speculates the causes of a certain sensible
effect, without considering its actual existence; for the contemplation of
universals excludes the knowledge of particulars; and he whose intellectual
eye is fixed on that which is general and comprehensive, will think but little
of that which is sensible and singular. Thus, by mathematics we may learn
the responsive harmony of the last chord, and its consonance with the
mean; but we cannot perceive this concord, if unaccustomed to the practice
of the musical art. In fine, those sciences which are more of a mathematical
nature, I mean such as are more amply conversant with the inspection of
things, considering their forms abstracted from every material subject,
always demonstrate the why; and such is geometry in respect of optics.
Thus geometry considers only such things as are peculiar to right-lines,
independent of every sensible connection. For the geometrician does not

investigate a right-line as contained in stone or brass; but considers it as
entirely detached and unconnected with any object of sense.
On the contrary, optics receives a right-line just as it is perceived in a
rule, or engraved in brass. And, indeed, in treating of some particulars,
natural science has the same relation to optics, as optics to geometry. Thus,
in considering the reason of the appearance of the rainbow, the natural
philosopher defines the bow to be an image refracted from a certain cloud
against the sun; but why it is endued with such a form, and seen with such a
colour, must be assigned by him who is skilled in optics. There are, again,
sciences, one of which is not subordinate to the other, because founded on
principles totally different; yet, in some particulars they agree with the
preceding. Thus, to know that an orbicular wound is the most difficult of
cure, belongs to the physician; but to know why, to the geometrician.
16. Of all syllogistic figures, the first is the best adapted to science,
since the arithmetician, geometrician, and lastly all those who demonstrate
any effect from its proper cause, fabricate their reasonings according to this
figure. For the middle figure is seldom used, because only adapted to a few
occasions: and since the knowledge of the why is of all others the most
important, which is alone obtained by this figure: hence, in the pursuit of
science, it is always preferred before the rest. Besides, it is equally
accommodated to the knowledge of final causes; to which it alone tends: for
it composes definitions from words universal, and affirmative. In the second
figure, a complex negative is conceived; and in the last, a particular one.
Add to this, that mediate propositions are no other ways reducible to
immediate ones than by this figure, in which the mediate proposition tends,
by a continued series, to that which is immediate. But the second does not
conclude affirmatively, nor the last universally; from whence it appears, that
a mediate proposition can never become immediate by these figures: not
that all affirmative propositions are immediate ones, since some negatives
are of this kind; for all propositions are equally immediate, which cannot be
confirmed by syllogism; and such are those negatives, of whose terms it is
impossible any genus can be affirmed. Thus the proposition, no substance is
quality, is an immediate negative of this kind, whose terms are two of the
most universal genera of things.
Again, as we have frequently affirmed that he who demonstrates,
always assumes such things as are essentially predicated; but that he who

argues dialectically or topically, not always, but generally assumes such as
are accidentally predicated, and which appear more probable and known
than such as are essentially inherent; it is proper we should define what is
meant by accidental predication; or something predicated by means of
another. Indeed, the term has a diffuse signification: for, first, a body is said
to be white by something else, because by its superficies; and in this
manner vines are white, because their branches are white. Thus, if accident
be predicated of accident, it is by means of another; as when we say the
musician is fair; for the being a musician is an accident of man, and the
being fair of the musician: and man is the subject of each. The predicate of
substance is equally accidental, when not included in the number of things
substantially inherent; as when we affirm of any particular man that he is
red, or black. But the predication is especially accidental, as often as, by
perverting the order of nature, substance is predicated of accident; as when
we say something white is an animal: for this assertion differs from that
other, animal is white. In the latter, the subject animal is neither inherent in
another, nor subsists by another, but has an essential existence. In the
former, what is assumed as a subject derives its existence from that of
which it is the accident. It is only dialectically, therefore, that we can argue
from predicates as probable and known without any distinction: but in
demonstration, all that are preposterous and accidental must be carefully
avoided, excepting such accidents as being essentially in a subject, admit of
an essential predication; and some of these we have enumerated before.
17. We are now entering on a disquisition neither ignoble nor useless: it
is this, whether the number of things predicated essentially of a subject is
finite, or whether things in a continued series run on to infinity. For
instance, let us suppose some ultimate subject, which is not the predicate of
any thing besides; and let c represent such a subject, of which b is the first
and immediate predicate; and in the same manner d of b, and e of d: the
query is, Whether or not this extraction must necessarily stop, or will admit
of an immense progression, so that f may be predicated of e, and g of f, and
so on infinitely; the power of the predicates, which supplies the common
identity, still remaining inexhaustible and undiminished? The second query
is this, Supposing some general subject, which we call a, of such a nature as
to be no longer the subject of any farther predication, but to be itself the
supreme and primary predicate; and supposing that it is immediately
inherent in f, and f in e, and e in g, whether or not the process must stop, or

extend to infinity, and no subject be found which is not directly predicable
of another? There is a remarkable difference in the two considerations; for,
in the former we enquire whether any ultimate subject can supply an
infinite ascent of predicates; in the latter, whether any first predicate can
exist in an infinite descending series of subjects. The third question is,
supposing two extremes constituted from a first predicate and last subject,
whether it is possible an infinite number of mediums can intervene? And
this is no other than to enquire whether demonstrations admit of an infinite
progression, so that whatever is assumed in proof of another, must be
proved itself? Or whether it is not more agreeable to truth, that there should
be some immediate propositions and ultimate terms, whose discovery may
give respite to enquiry, and stay the elaborate process of demonstration?
The same question occurs in negatives. But that some of these are
immediate, the instance lately alledged sufficiently evinces. The solution of
this enquiry is not so difficult in subjects which mutually reciprocate; for in
these, when the ultimate subject is given, no one can doubt the existence of
their primary predicate; nor when the primary predicate is admitted, can
there be any doubt of the existence of some ultimate subject. For, in things
which mutually reciprocate, whatever is enquired of the one, is immediately
questioned of the other; and wherever there is a last subject, there must be a
first predicate; for by the conversion of the ultimate subject you effect the
primary predicate.
Previous to the discussion of the first question, it is necessary to know
that infinite intermediates cannot intervene between two finite terms in an
ascending and descending series of predications. I call the series ascending
which rises to universals; but descending, which, by a contrary process,
stops at particulars. Thus, if any one admits that a is some first predicate,
and g some ultimate subject, and should contend, that between these terms
there may be infinite mediums, he contradicts himself; since he who begins
from a in a descending progression, will never, by this means, arrive at g;
and he who departs from g in an ascending series, can never finally rise to
a. So that the extremes can be no longer finite, as the hypothesis admitted.
Indeed, the absurdity of such a supposition is the same as to contend that
between one and ten, an infinity of numbers may exist; which is evidently
impossible, because the discrete nature of numbers excludes their actual
existence in infinitum, between any finite limits; since they can only
become infinite from their actual existence and precedence, and not from

any dormant power or capacity they possess: for between any two given
numbers there is nothing similar to number in capacity, which can ever
become number in energy; as in quantity continuous between any two
points there are always parts in capacity, which, whenever a proper agent is
at hand, become immediately actual. In like manner, he who admits the
terms finite, but believes that the mediums are infinite, asserts what is
impossible, since these logical predications are of the same discrete nature
with numbers themselves. Thus all the predicates which can exist between
Socrates and substance, must exist actually, or not at all; for surely between
these two terms, or periods, no predicate in capacity can ever be supposed
to subsist. If it be urged, that the capacity of receiving these predicates
exists between Socrates and substance, still we reply, it is not that kind of
capacity in which these predicates can retain the most shadowy existence;
out of which they can ever be called forth into energy, as from some latent
retreat; or into which they can finally retire, when energy is no more. And
hence we conclude it impossible that infinite mediums can exist between
any finite terms.
18. It now remains that we prove, first, by probable arguments, and then
by such as are demonstrative, that the extremes in any series of predications
are finite; and that an infinite progression is impossible, not only in
substantial predicates, but in such as are accidental. For every thing
predicated of another is either essentially or accidentally inherent; and is
predicated in a natural or preposterous order. It is predicated according to
nature, when accident is declared of substance; contrary to nature, when
substance of accident. That essential predicates are finite, appears from
hence, because a contrary hypothesis excludes the existence of definition,
by admitting that all things are contained in some superior genus, and
acknowledge some farther definition; since it is impossible that the
definitions of genus can ever be circumscribed, while there is a continual
supply of other genera, which can never be known without definition; for
thus we shall never obtain either a beginning or an end. But to define all
things is not possible, because infinity can never be absolved by the most
unwearied progression. Predictions then, of this kind, are always
circumscribed by a certain number of terms, which prevent their infinite
process, and cause all the strength of demonstration, and all the certainty of
human knowledge. The same may be proved in accidents; for such as are
predicated of substance, are either predicated as qualities or quantities, as

relatives, or as actions and passions; as expressive of some habit, or
significant of some place; or as connected with some time. Thus we say the
wood is white, the triangle is scalene; whiteness being accidental to the
wood, and scalenity to the triangle. It is therefore certain, that every
accident is predicated of substance; and it is no less certain that the
predicates of substance are finite, since they are all included in the ten
universal genera of things.
19. We have hitherto defended the impossibility of an infinite
progression of logical predicates and subjects, in a demonstrative process,
by such arguments as are dialectical and common: it now remains that we
adopt such as are peculiar and certain. Demonstrations, then, are derived
from affections essentially inherent in a subject; and these are either such as
take place in definitions of a subject, as multitude and quantity, are
essentially predicated of number; or, secondly, accidents which are defined
from their subjects, as imparity by number. But the predication cannot, in
either case, be extended to infinity. For it is not necessary that in the same
manner that imparity is predicated of number, something else, suppose c,
should be predicated of imparity; and so imparity be contained in its
definition, similar to number in the definition of imparity. For in
predications of this kind, the terms are always assumed more contracted
than their subject; and at length, by a continued procession, must terminate
in an indivisible. Thus, as imparity is more contracted than number, c must
be more contracted than imparity. Hence, these predications either finally
stop, for the reasons we have assigned; or because whatever is predicated of
imparity, is necessarily predicated of number; so that one thing as number
would be actually contained in the definition of an infinity of things; and so
actual infinity must ensue, which is absurd. Lastly, whatever is said to
reside in the terms, must be allowed to reside in the subject; so number
must be applied in the definition of every affection; and an infinite number
of properties will be essentially inherent in number; and number will inherit
infinite definitions. But affections essentially resident in a subject cannot be
infinite, because it is necessary they should exist in energy. Thus, imparity
cannot exist potentially in number; nor reason in man; nor rotundity in a
circle, because wherever these subjects have an actual being, it is necessary
these essential attributes should be actually inherent. Again, in the
definitions of a subject, an infinite process is impossible, because from such
an hypothesis nothing could ever be defined; and thus it appears that neither

can demonstrations be infinitely extended, nor every thing admit of
demonstration, an opinion we have already noticed in the beginning of this
section: for if neither universally, nor in every proposition a middle term
can be assumed, but as soon as we arrive at immediate propositions, the
labour of investigation is finished, the possibility of demonstrating every
thing can no longer be defended; since it is proved above, that by limiting
the extremes, an infinite number of mediums is necessarily excluded.
And thus, by taking away infinity from the reasoning art, we have given
a support to science, which the most vigorous efforts of subtle sophistry can
never finally subvert. We have set bounds to that restless spirit of enquiry
which wanders uncontrouled in the mind unenlightened by science, by
every where circumscribing its progress within the limits of that which is
most particular, and most universal, a first predicate, and an ultimate
subject: and finally, by asserting that all the evidence of human knowledge
results from the lustre of primary and immediate principles, we have held
up a steady and permanent light, ever sufficient to direct our steps through
the dark mazes of ignorance and error, into the bright paths of certainty and
truth.
20. Let us next consider whether universal demonstration is preferable
to particular, or not. And first, in favour of particulars we may say that their
evidence is more exquisite and certain than that of universals. Thus, the
knowledge, from inspection, that Callias is a rational animal, is superior to
that acquired by a reasoning process which infers his rationality, because
every man is a rational animal. By particular demonstration a thing is
known as it is, by universal only in common. Besides, particulars possess
some solidity, universals none: and the demonstration of things which have
a real existence, is more excellent than that of things which have none. And
there are no errors more frequent than those about universals;
demonstration considering them as things entirely abstracted from
singulars. On the contrary, particulars are usurped by the sight, grasped, as
it were, by the hand, and the general subject of every sense; so that
concerning these, demonstration affirms nothing false or inconstant. But
these reasons, however plausible, are easily confused. And, first, the term
essential is more closely connected with universals than particulars. Thus
the possession of three angles equal to two right, is an affection more
essential to the triangle itself, than to one equilateral or scalene. Add too,

that in the demonstration of universals we always infer some property of a
subject from its simple existence, or because it is such a subject. Again,
many affections are contained in singulars assumed from no particular
nature, but from that which is universal; as rationality in Socrates, which is
not inferred from his existence as Socrates, but from his existence as man.
Farther, that demonstration is the more excellent which is derived from the
better cause: but an universal cause is more extended and excellent than a
particular one; since the arduous investigation of the why in any subject is
stopt by the arrival at universals. Thus, if we desire to know why the
exterior angles of a triangle are equal to four right ones, and it is answered,
because the triangle is isosceles; we again ask, But why because isosceles?
And if it be replied, because it is a triangle, we may again enquire, But why
because a triangle? To which we finally answer, because a triangle is a
right-lined figure; and here our enquiry rests at that universal idea which
embraces every preceding particular one, and is contained in no other more
general and comprehensive than itself. Add too, that the demonstration of
particulars is almost the demonstration of infinites; of universals, the
demonstration of finites.—We add farther, that demonstration is the best,
which furnishes the mind with the most ample knowledge; and this is alone
the province of universals. Again, the principles of science become
immediate only in proportion as the demonstration becomes universal; and
he who knows universals, knows particulars in capacity: but we cannot
infer, that he who has the best knowledge of particulars, knows any thing of
universals. Lastly, that which is universal, is the province of intellect and
reason, particulars are the offspring of sense; and hence we conclude that
universal demonstration exceeds particular both in dignity and excellence,
and is first in the nature of things, although last in the progressions of the
reasoning power.
Again, That affirmative demonstration is superior to negative, appears
from hence: the affirmative does not require the assistance of the negative;
but the negative cannot exist without the affirmative; on which account, the
demonstration composed from negatives alone, is incapable of producing
real evidence and conviction. Besides, affirmation exceeds negation both in
priority and simplicity of existence.
Again, the demonstration which concludes directly, is better than that
which confirms a proposition by evincing the absurdity of its contrary. The

first proceeding in a regular order, establishes, by a natural deduction, the
truth which was first advanced. The second taking a wider circuit, yet with
the same intentions produces a conclusion quite opposite to its apparent
design. The one may be compared to the open attack of a valiant and skilful
soldier, who expects the conquest of his enemy from strength and courage
alone: the progress of the other resembles the same soldier, uniting force
with stratagem, and advancing, by an irregular march, which his foe
mistakes for a retreat, but finds the secret cause of his destruction. The first
is simple and impromiscuous, as composed from propositions alone: the
second is compound and miscellaneous, calling in hypothesis to its
assistance.
21. One science is said to be prior to, and more certain than another in
many respects;—when the one reasons from primary causes, but the other
from such as are secondary:—when the one may be ranked in the genera of
intelligibles and universals; but the other in the genera of sensibles and
particulars. And such is the relation of arithmetic to music; of geometry to
optics; and lastly, of every superior to every subordinate science. Again, this
happens when the one reasons from simple principles, the other from such
as are complex and connected; on which account arithmetic seems to
possess greater certainty than geometry. For the principle of arithmetic is
unity; but of geometry a point; and unity is without position, with which a
point is always connected. And in this manner geometry inherits greater
evidence than astronomy; for the one considers body simply, the other as
connected with a circular motion. The science is called one which
contemplates actions belonging to one genus: the genus is one which
possesses the same first principles; and hence geometry and stereometry
form one science. On the contrary, the sciences are called different which
have different principles, such as geometry and optics; the latter of which
does not originate from the principles of the former.
Again, the same thing may admit of many demonstrations, and may be
known from many mediums: at one time from the application of such as are
congenial: at another, from those of a different order or genus. From
congenials, as when we demonstrate that the plantain is a substance, first,
by the medium of a tree, and then by the medium of a plant, thus:
Every tree is a substance;
The plantain is a tree:

Therefore the plantain is a substance. And again,
Every plant is a substance:
The plantain is a plant:
Therefore the plantain is a substance.

We demonstrate, from mediums, of a following order or genus, as when
we prove man to be a substance, at one one time from his being rational, at
another from his being a biped; and these mediums, in part, mutually
contain each other.
22. Fortuitous events can never, in any science, become the subject of
demonstration; since they are neither limited by necessity, nor admit the
arrangement of syllogism. Indeed, so far from obtaining a necessary, they
do not possess a frequent existence, but every syllogism is composed from
one or other of these.
Again, science is not the business of sense, since that which is universal
is the object of perception in particulars themselves. For the object of sight
is colour in general, and not this particular colour: the object of hearing is
sound in general, and not any particular sound; and, on this account we see
or hear not only this or that colour or sound, but likewise every other which
falls under the cognizance of these senses. Hence, if it were possible for any
one to discern by his sight, the equality of the three angles of some
particular triangle to two right, he would not by this means possess a
demonstration of the conclusion which affirms this to be the property of
every triangle; but his knowledge would extend no farther than the triangle
he inspects. Thus too, if we could perceive an eclipse of the moon to arise
from the interposition of the earth, we could not universally conclude that
this is the cause of every eclipse, but only of the particular one we behold.
For the explication of causes extends to universals; and comprehends not
only the knowledge of one particular defect of the moon, but simply of
every eclipse; since the interposition of the earth is not so much the cause of
any present eclipse, as of all which can possibly exist in every age.
Whenever, then, the cause is universal, the knowledge of any effect
deduced from such a cause is, in every respect, superior to the evidence
arising from the perceptions of sense. It is likewise more excellent than the
apprehension which subsists independent of the proper cause; as if any one
should give absolute credit to the proposition, that the three angles of a
triangle are equal to two right, without a previous conviction that the

external angle of a triangle, is equal to the two interior opposite ones; and
without applying this last proposition as the cause of the first. The
comprehension, then, which is conjoined with the proper cause, far exceeds
the strongest evidence of sense.
But perhaps it may be said that science consists in sense, because the
science of any particular, fails from a defect of the sense by which it is
apprehended. To this we reply, that science, indeed, is not acquired without
the assistance of sense, but it does not follow from hence, that to perceive is
to know; because the object of science is that which is universal; but of
sense, that which is particular. Thus, if we could see light penetrating the
pores of glass (on the atomical hypothesis) the cause why it illuminates
would be manifest from sensible inspection as the means, and from the
universal apprehension of science, by which we should understand this to
be universally true.
Again, the principles of all sciences cannot be the same neither
considered as remote or proximate. Not considered as proximate, because
the principles always correspond to the demonstrated conclusions; but these
are not the same, since they are often generically different; and
consequently the propositions from which they result must be derived from
discordant genera. But propositions consist of such things as essentially
exist; and hence we infer, that the principles of geometry are essentially
distinguished from those of arithmetic, that they cannot admit of reciprocal
accommodation, so that the one may be predicated, or become the subject
of the other, and that the one can never be subservient as a medium to the
other. Again, common and first principles are not applied in every science;
such as this, that every thing must either be affirmed or denied. Nor can any
thing be proved by their assistance alone, but as often as these are required
in demonstration, other principles more proximate and peculiar to the given
proposition, must always be adopted. Again, axioms universally conceived,
cannot be assumed in syllogism, but they must be contracted, as it were, to
some subject genus. Of this kind is that common axiom, that as often as any
four quantities are proportionable, by permutation, or changing the order of
the terms, the same ratio will result. For the arts apply this axiom in a
restricted sense; geometry, by considering the relatives as four magnitudes,
and arithmetic as four numbers; but the natural philosopher, by adapting the
comparison to four motions, or four times. Besides, if the principles of all

sciences were the same, it is necessary they should be comprehended by
some certain number, similar to the limitation of the elements: but every
science is capable of immense increase from the many different modes of
amplification the conclusions will admit; and consequently it is requisite to
establish a correspondent number of proper principles; for such as are
common cannot be alone sufficient. Lastly, if the same principles accord
with every science, it follows, that any thing may be demonstrated from
such principles: but the certainty of geometrical conclusions cannot be
established from the principles of music; and from hence it follows, that
although the principles of every science are not the same, they do not
possess an entire diversity, nor yet an absolute affinity of nature.
23. There is a remarkable difference between science and opinion.
Whatever is the subject of science must have a necessary existence; on the
contrary, opinion is conversant with things liable to mutation and decay.
Again, as science depends on necessary propositions for support; so opinion
on such as possess only a possibility of existence; and so there is one mode
of approbation in subjects of opinion, and another in those of science.
Hence science is distinguished from opinion by two discriminations, the
one arising from their subjects, the other from the mode of approbation.
That opinion is conversant with things possible or contingent, we may learn
from hence; contingencies cannot belong to science, because their existence
is not necessary; nor to intellect, or that principle of science by which its
terms are known; nor to the apprehension or belief of immediate
propositions, called indemonstrable science. Hence, if every habit by which
truth is known, is either science, or intellect, or opinion, it remains that
opinion alone consists of things which are, indeed, true; but not necessary.
It is, therefore, inconstant and unstable, from the mutable nature of its
subjects. Besides, no one thinks he possesses an opinion of things which he
believes to have a necessary existence, so that they cannot be otherwise
than they are; but to such conviction he properly gives the name of
knowledge, and to its contrary the name of opinion.
Again, the same thing from the same propositions may at one time
become the subject of knowledge, at another, of opinion; and this happens
according to the different formation of the syllogism which the propositions
compose; whether reasoning from the proper cause it explains the why, or
only simply declares a thing exists. Hence a doubt arises why opinions of

this kind may not be called science, since both the subjects and propositions
are the same? The solution is obvious. If it is believed that the propositions
cannot be otherwise than they are, or that they have a necessary existence,
such an assent of the mind is not opinion, but science; because things which
inherit an essential existence are the ornaments of science alone. On the
contrary, if we are convinced that the propositions are true, but at the same
time not necessary, such conviction is not science, but opinion. Hence, it is
impossible that science and opinion can be the same, since they vary in
their definition and mode of approbation, and in a different manner demand
our belief. Similar to this, although it may happen that of the same thing a
true and a false opinion may arise, it will not therefore follow, that true and
false opinions are the same. For that which is firm and constant can never
be the same with that which is mutable and frail; and that which is always
true must be essentially different from that which may be changed into
false. By the power of habit indeed in different men, the same thing may be
comprehended by opinion and science. Thus it was opinion in Epicurus
when he said that the sun was eclipsed by the moon passing under its orb,
because he thought it might otherwise happen, and that the moon might be
interposed without obscuring the light of the sun. It was science in
Hipparchus, because he knew it as a necessary event. But in the same mind,
at the same time, and of the same thing, it is impossible that science and
opinion can exist. And thus much concerning the difference of the two.
24. Lastly, sagacity is an acute and sudden apprehension of the medium,
or proper cause of a certain effect: as if any one, beholding the moon,
should in a moment conjecture the cause of the part opposite to the sun
being lucid, and the other parts obscure, because she derives her splendor
from the sun. Hence he is universally called acute and sagacious, who, from
the aspect or hearing of the extremes, can readily perceive the medium
which exists between them: as the term imports a certain revolution of the
conclusion into its first propositions, and, as it were, a swift comprehension
and continuation of the medium.

S E C T I O N III.

I n the ensuing Commentaries, the soul is considered as immaterial; and as
possessing a middle nature between intelligibles and sensibles; but that
this important assertion may not remain without proof, the following
demonstrative arguments, derived from the Platonic philosophy, are offered
to the reader’s consideration. And first, that the soul is an unextended, and
consequently immaterial essence, may be thus proved. If, after the manner
of magnitude, it consisted of continuous parts, it would be impossible that
any one part could be sentient from the passion of another; but the soul, for
instance, which is situated in the finger, would be sensible of passion, as if
detached from soul in the other parts of the body, and existing by itself; for
the soul, from this hypothesis, would be sentient by parts, and not
considered as a whole. Besides, there must be many souls governing each
part of us, different among themselves, and endued with their own peculiar
energies. For whatever may be said of continuity, is to no purpose, unless it
conduces to unity of sensation; so that the hypothesis which supposes that
the sensations gradually arrive at the principal part of the soul by a certain
continued succession, is not to be admitted, since it may be reasonably
asked, How is the principal part to be peculiarly distinguished? By what
rule of quantity can the parts be discerned, by what difference are they to be
distinguished, where the quantity is one, and the bulk continuous? Besides,
is the principal part alone, or are the other parts, sentient? If the principal
part alone be perceptive, the soul can then alone be sentient when a sensible
passion meets with this principal part, situated in its peculiar seat; but if a
sensible passion falls upon any other part of the soul destitute of sense, it is
impossible that such a part should be able to transmit the same passion to
the principal, or be at all sentient. For how can that which is void of sense,
receive passion, and convey it entire to a sentient part? Besides, if passion
accedes to the principal, it either falls on one of its parts, and so either one
part alone will be sentient, and the rest without sensation, and consequently
superfluous, or there must be innumerable and dissimilar sensations; for if
the sensation of each of the parts singly, is the same with the aggregate of
them all, of what use is a multitude of parts? But if the sensations are
various, a man may say, as it were, I am primarily sentient in this place, and
secondarily in another; and every sentient part besides the first, will be

ignorant where sensation is most powerful: or perhaps (from such an
hypothesis) every part of the soul will be equally deceived, each part
thinking the passion to arise in the place where it is situated. But if not the
principal part alone, but every part of the soul be sentient, a principal part is
superfluous for the purpose of sensation; and if the soul be divisible like
magnitude, how is it able to recognize, as belonging to one subject, the
qualities which flow, as it were, through many senses, as through the eyes
and ears? For that part of the soul which is perceptive through the eyes,
distinguishes nothing but colours; that which energises through the ears,
nothing but sounds; and that which acts through the medium of the touch,
nothing but the surfaces of bodies: what is it then which perceives all these
properties of bodies united in one subject, or what is that which perceives
any thing as a certain one? For unless the intentions of the senses, and of
sensible objects, were collected together in one, the soul could never be
able to judge of the peculiar and different properties of bodies: Hence it is
necessary that the soul should be, as it were, a general centre; that the
several senses should be extended on all sides to this, similar to lines
verging from the circumference of a circle to the centre; and that a power of
this kind, comprehending all things; should be truly one. For if the soul was
any thing divisible, and the intentions of the senses reached the soul, and
ended in its essence like the extremities of various lines, they must either
again concur in one and the same as a medium, or have different situations,
in such a manner that each sense may perceive different from one another:
as if, for instance, the sense of sight should perceive the form of Socrates,
and the sense of hearing recognize his voice; that essence which pronounces
the whole to be one person, that of Socrates, must be something different
from each of the senses. Hence it is necessary that the soul should be an
indivisible essence; for if she possessed magnitude, she must be divided
along with every sensible object she perceives; so that one part of the soul
would perceive a certain part of a sensible object, and we should possess no
sentient power capable of perceiving the whole, or of pronouncing any
thing one. Thus, in the perception of a man, considered as one, how is it
possible that the soul can be divided, so that the perception of a part shall be
the same with the perception of the whole. But if we suppose the soul
divisible in all her perceptions, since it is impossible she should be co-
extended with every sentient object, in how many parts is the division to be
made? Is the soul to be distributed into the same number of parts as the

sensible object she perceives, so that every part of the soul may perceive the
same part of the object? Or shall we say that the parts of the soul have no
sensation of the parts of the object? But this would be absurd and
impossible. If every part then of the soul perceived every part of a
magnitude as a whole, since magnitude is divisible to infinity, and since, on
this hypothesis, there must be innumerable sensations of every sensible
object, there must be innumerable images, as it were, of the same thing in
our principal part. Besides, if that which perceives is corporeal, it will not
be possible for it to perceive in any other manner than as if certain images
were impressed from a seal in wax, or in brass, or in any other sensible
substance. But if the images of sensation exist as in humid bodies (which is
most probable), they will certainly be confounded like images in water; nor
can there be any memory, the image departing with its forming substance.
And if we suppose the figures to remain like impressions in solid bodies,
either it will not be possible for others to succeed while the former endure;
and thus, sensations of other things cannot take place; or if others succeed,
the former images must be immediately destroyed, and memory be no
more. So that if we allow it possible to remember, and, besides this, to
perceive other things, without any hindrance from former impressions, it is
impossible that the soul should be corporeal. Since the soul, therefore, is an
unextended, indivisible, and immaterial substance, it is consequently
incorruptible and immortal; for every thing capable of dissolution and
dispersion is either corporeal and composite, or exists in some subject from
which it is inseparable. And indeed, whatever may be dissolved is
corruptible, as being compounded from many. But whatever naturally
subsists in something different from itself, when it is separated from its
subject, immediately vanishes into non-entity. But the soul, as we have
proved, is incorporeal; it is likewise removed from every subject, and
naturally reverts to itself, and is therefore immortal and incorruptible.
2. Let us now consider how, and on what account the soul is said to be
of a middle nature, and to be the receptacle of all middle energies, both vital
and gnostic. Since, then, there is a long gradation of beings, proceeding
from the first being, even to formless matter, which is nothing more than
the dark shadow of essence, it is requisite to enquire what the properties are
of the first and last beings, and what the condition is of the middle orders;
for thus we shall know where the essence of the soul ought to be placed.
The properties of intelligible natures, therefore, are as follows: true being,

eternal, indivisible, immoveable, total, perfect, full of essence, replete with
life, free, moving all things, similitude, presiding over all things, and at the
same time separated from all; for each of these properties appears in
intelligibles, according to the processions of being. But the properties of
sensible natures, different from these by the greatest interval, are such as,
not-true-being, temporal according to essence, partible, moveable,
particular, indigent of another, always replete with subsistence, living by
participation, moved by another, dissimilitude, and occupying place by
parts. But the middle properties of these are, not-true-being, an essence
better than non-being, and inferior to true-being, according to essence
eternal, but according to its energies extended with time, indivisible
according to its divine part, but divisible according to the various
processions of reasons, self-motive, governing things moved by another, but
subordinate to such as are immoveable, bearing before itself a particular
nature, together with its totality; (for, because it contains in itself all
reasons, it is after a manner a whole, but because it is diminished and fallen,
ends in parts, and suffers a transition of its energy, it must be esteemed a
particular nature): and again, perfecting itself, yet, nevertheless, perfected
by natures prior to its own; filling itself with power and strength, and at the
same time filled by others: living from itself, and receiving life from others,
being more divine, indeed, than things which live only by participation, but
inferior to things primarily vital; moving other things, and itself moved by
others; at the same time similar and dissimilar; and separated, at the same
time, from last natures, and co-ordinated with them. Such then, being the
properties of the first, middle, and last orders, let us consider where we
ought to place the soul, whether in the first order, or in those which retain
the last place: but if we establish it among the first, it must be true being,
every way eternal and immoveable, and it must consequently possess every
thing which we have attributed to the first beings; add too, that on this
hypothesis we can no longer attribute to the soul a power of self-motion,
nor the discursive processions of reasoning, nor a variety of other
particulars, which manifestly belong to the soul. But can we place it with
propriety among the last of beings? The least of all: for on this hypothesis
we shall make it alone moved by others, divisible, composite, and alone
possessing perfection from others, the opposite of which is evident in all
our souls; since they move and perfect themselves, and are led wherever
they please. Since then it is not possible to place the soul either in the first,

or last order of beings, it is requisite to assign it a middle place, in imitation
of its divine cause Rhea (according to the theologists), who is the
conciliating band of the two parents Saturn and Jupiter, and is reported,
from her prolific bosom to produce the life of the soul. But though the soul
is thus the extremity of intelligible, and the principle of sensible natures, we
must not conceive it to be such a principle or extremity as a point in a line,
for it is not in both the natures it terminates, like a point in both the sections
of a line; but it is to be called the extremity of intelligibles, because it
appears after an intelligible essence, and the principle of sensibles, as being
abstracted from them, and the source of their motion. And thus it will
preserve to us a certain proportion, that as the natures which are moved by
others, are to those which are moved by themselves, so are these last to
immoveable natures; and hence it will obtain the condition of a bond, on
account of its peculiar mediocrity, unfolding, indeed, united causes, but
reducing the dissipated powers of sensibles into one, and being contained
by an immoveable and perpetually abiding cause; but containing itself the
generation, which, moved by another, is subject to continual mutation. It is
likewise intelligible, if we regard generated natures; but generated, if we
compare it with intelligibles; and thus it exhibits in its middle nature both
extremes, imitating also, by this means, (according to the Greek theologists)
its divine cause, for it is said to be on both sides refulgent, ἀμφιφαὴς, and to
be endued with two faces ἀμφιπρόσωπος, and to receive in its bosom the
processions of intelligible natures. It is likewise said to be replenished with
intellectual life, and to be the fountain of the ever-running streams of
corporeal life, and to contain in itself the centre of the processions of all
beings. On this account it is, with great propriety, affirmed to be generated,
and at the same time without generation. For true being, according to the
Platonists, is without generation, because it has an infinite power of being
totally present at the same time: and body is said to be generated, because it
always possesses in itself an infinite flowing power, which it cannot at once
totally receive. The soul, therefore, because it is incorporeal, abiding in
itself, has an infinite power of being, and this total with respect to its
essence, and immortal without generation; but according to parts it may be
considered in infinite production. For it has not the same total infinity ever
present, or there would be the same infinity of the whole and part, of the
perfect and imperfect, of the contained and containing, which is impossible.
But neither is it possible that the whole of its essence should be in the act of

perpetual production, any more than that a part of it should be eternal being,
lest the part should be more worthy and better than the whole. Hence the
ὑπόϛασις, or subsistence of the soul, is at the same time of infinite power,
and is generated in infinitum; for by this means it participates of being, and
obtains the first place among generated natures; while body alone, both
with respect to its whole and parts, is obnoxious to a perpetual generation.
3. But let us now enquire from what genera Plato composes this nature
of the soul, which contains in itself the bond of all beings; previous to
which it will be requisite to explain what these genera are, and from whence
they originate. Of the species, then, existing in the intelligible world, or the
divine intellect, which contains in itself the causes of all posterior natures,
some are most general, extending themselves to the universality of things;
but others are more particular, like the most special species, and others
subsisting between these, expand themselves, indeed, to a multitude of
things, but not to all, according to the division, of the Elean guest in the
Sophista. For man is produced from the ideal man, and horse from the ideal
horse, in the intelligible world; but the similitude which is found in man and
horse, and other animals, is produced from likeness itself, or the ideal
similitude, as dissimilitude from unlikeness itself; but the sameness and
difference which are found in all beings, proceed from the sameness and
difference which subsist in ever-vital energy and perfection, in the supreme
intellect, or the ideal world. Now, as among the sciences some are
especially universal, so in intelligible causes some are perfectly particular,
presiding alone over the proper and peculiar number of one species; but
others extend themselves to a multitude, such as equality, likeness, totality,
(for the whole considered as a whole is not common to all things, since the
part is not a whole); but others, again, expand themselves to all things, as all
beings participate of these, considered as beings, and not considered as vital
or animated, or possessing any other property exclusive of the
denomination of being. Because, therefore, being is the first, the causes of
being obtain the most universal order among genera; and these are five in
number, as follows, essence, sameness, difference, motion, and station. For
every being is endued with essence; is united to itself; is by itself, or its own
sameness, separated from others; proceeds from itself, and its own state and
principle, and no less appears to participate of a certain abiding, in
preserving its own proper species. All things, therefore, whether
intelligibles or sensibles, or subsisting between both, depend on these

genera for their existence. For without the being of essence, nothing could
subsist; in like manner, without sameness every whole would be dissipated,
and divided from itself; and difference being taken away, all things would
be one alone, and multitude be destroyed. But without motion and station,
all things would either be inefficacious and dead, or, losing their proper
state and stability, would end in non-entity.
4. Such then being the middle nature of the soul, Plato, with great
propriety, in the Phædrus, and in his tenth book of laws, defines it to be
number moving itself; which definition he received from Philolaus, and
Philolaus from Pythagoras. For since mathematical species have a middle
subsistence, as is proved in the following Commentaries, they are of all
things most accommodated to the nature of the soul. Hence Plato, in
imitation of Pythagoras, sometimes explains the soul by number, as in the
present instance; and sometimes by figure and magnitude, as in the Timæus;
while he considers in the soul the intersection of lines, and a twofold circle.
For since mathematical forms are separated from the flux and inconstancy
of matter, they participate of a certain, exact, sure, and exquisite condition,
by means of which they eminently confer to the elevation of our ingenuity,
and the explication of latent concerns; and, on this account, as they pertain
to numbers, we may say, preserving the analogy, that there are five orders of
numbers, the divine, the essential, the animative, the natural, and the
mathematic. The first of these is uniform, the second immoveable, the third
self-motive, the fourth moved by another, and the last the image of the
others, and their external measure. The divine number is considered
eminently in the deity, as in the principle of all things; the essential belongs
to intellect through ideas, and is called essence, unity, and the first being;
the animative number belongs to the soul, through the medium of her
inherent reasons; the natural to physical concerns, through the seeds of
nature; and lastly, the mathematical belongs to opinion, as it is nothing
more than the image of essential number, formed by the energies of the
rational soul. The soul, therefore, is number, not limited by quantity, and
mathematical, but animative; it is number, not indeed numbering, but
numbered, generating and converted into itself. Hence too, because
harmony arises from number, the soul is called harmony; not, indeed, a
harmony of the parts of the body, nor the harmonic quantity which subsists
in sound or in voice; but a harmony arising from its essential numbers,
placed in its inherent reasons, and in the genera which constitute its nature.

It is this harmony which produces, as from its proper cause, the harmony of
the corporeal parts, the rhythm of motions, and the melody of voices and
sounds. It is this which produces that delight in the soul from sensible
harmony, which sufficiently indicates it to be something familiar and
domestic to her nature. From hence it may be inferred, that Plato is not
inconsistent with himself when, in the Phædo, he denies, and in the Timæus
affirms, the soul to be harmony; for he denies that it is a harmony of a
definite quantity, or such as arises from the parts of the body; but he asserts
it to be a harmony in the manner already explained. It may likewise be
inferred, that Plato is ignorantly accused by a many, for affirming that the
soul is harmony, or number; for they only regard vulgar mathematical
number, and sensible harmony; while Plato, far more elevated, discourses of
intelligible numbers, and ideal harmony, subsisting in immaterial energy
and perfection.
5. And here it is necessary to consider what number, in a particular
manner belongs to the soul; for various numbers, differently considered,
accord with her self-motive nature. In the first place, union and unity may
be considered in the soul, as in her proper degree she participates of divine
unity; and likewise with relation to her totality, for she is one certain whole.
And because a whole may be considered in a triple respect, one before the
parts, another rising from the aggregate of parts, and a third subsisting in
the single parts; the soul is a whole in each of these respects. Thus she is a
whole prior to the parts, while she is considered as divisible into them, in an
incorporeal manner; she is a whole rising from parts, while assuming the
parts in the first place, we consider how her nature is fabricated from their
conjunction; and she is a whole in the single parts, since she is total in the
whole, and in every part. Besides, the duad belongs to the soul, because she
contains in her nature bound and infinite, sameness and difference; and
lastly, a conversion to intelligibles and sensibles. And, indeed, the duad
conjoined with unity, very properly accords with the soul; for to intellect
above soul, unity particularly belongs; to body beneath soul, the infinite
alone; and to soul situated in the middle, duality properly agrees, being, as
it were, infinite, connected with unity. Again, the ternary number is
attributed to the soul, as well on account of her beginning, middle, and end,
as because she abides in herself, proceeds to inferiors, and returns to
supernal natures. Besides, she flows from the one, recedes from him, and is
reflected into the one when she acquires her proper perfection. Lastly, as

Proclus observes, the nature of the soul is divided into essence, power, and
energy; so that she may be said to rejoice in the ternary number, and to be
replete with its perfection. But the quaternary number belongs to the soul so
far as she is connected with matter, which is tempered with four qualities,
and four elements; and she is endued with four principal faculties, nutrition,
sensation, local motion, and intellection. But to omit other numbers, and
their conformity with the soul, the quinary, and septenary numbers are
especially attributed to the soul. The quinary, because the soul is composed
from the five genera of things, we have previously explained; and because
five particulars merit a principal consideration in the soul; first, her essence;
secondly, the harmony of her reasons; thirdly, the species arising from the
concord of her parts or reasons; fourthly, her virtue; and, lastly, her
energies: and on this account, Proclus observes, the consideration of the
soul ought to receive a quintuple distribution. Besides, as the soul consists
from a divisible and indivisible nature; so the quinary is composed from the
first even and the first odd number. Lastly, as the soul is the connecting
medium of the universe; so the quinary obtains the middle situation in
universal number, that is in the decad. But the septenary number belongs to
the soul, because, as Plato shews, in his Timæus, all harmonical reasons are
contained in the seven numbers, 1, 2, 3, 4, 9, 8, 27; and from these the soul
is composed. Hence Proclus elegantly observes, that the septenary number
is dedicated to Apollo, the parent of all harmony; because in one, two, and
four, from which the septenary results, the first bisdiapason is found.
Besides, the writers on harmony affirm that all the difference of voices
proceeds as far as to the seventh degree. So that Plato uses, with great
propriety, the septenary number for the composition of the soul. Again, in
these numbers of the soul, every medium is found in a convenient
proportion; in these the geometric medium is contained, corresponding to
the right institution of laws; of which Plato, in his Republic, says, that by
this cities are properly governed: there is found too, an harmonical medium,
which is the similitude of justice: and lastly, we may discover an
arithmetical medium, which is called the illustrious symbol of peace. After
this manner, therefore, Plato, with a wonderful fecundity of significations,
affirms that the soul is composed from numbers. He likewise considers the
figures she contains, I mean the circle and triangle; because as the soul is
the first nature which verges to body, so these are the first of all figures, as
well rectilinear as curvilinear. Besides, an orbiculation agrees to the soul,

through intellect; but progression and rectitude according to her own proper
nature; and on this account she contains both a straight and circular figure. I
omit other correspondencies of a circle and triangle with the soul, as they
are exhibited in the following Commentaries; and particularly by Proclus, in
the third book of his Commentaries on the Timæus: it is sufficient to the
design of this Dissertation, just to have mentioned this analogy, that the
nature of the numbers and figures may appear, which are considered by
Plato in the composition of the soul.
6. Let us now pass from contemplating the nature of the soul, to a
survey of its various gradations of knowledge, and the means by which it
acquires the illuminations of science; as this is a speculation perfectly
essential to a full comprehension of the ensuing Commentaries. According
to Plato, then, in the sixth and seventh books of his Republic[28], there are
four degrees of the internal cognitions of our soul; imagination, or
assimilation; faith, cogitation (διάνοια); and lastly, science or wisdom. The
two first degrees conjoined constitute opinion; but the two last equally
joined produce intelligence in its large acceptation. I say in its large
acceptation, because the word intelligence is considered by the Platonists in
a triple respect. First, as it rises from opinion and science; as Plato asserts,
in the seventh book of his Republic. Secondly, as it passes into the same
with science; for thus, in the end of the sixth book, he considers intelligence
and science as the same. Lastly, as it is distinguished from science, and
intelligible from that which is the object of cognition: thus, science regards
the essential reasons of the soul; but intelligence elevates us to ideas, and
this is intelligence in its proper acceptation. The distinction of these four
degrees, especially depends on the distinction of things with which the soul
is conversant. For these four degrees of things are, the image of a sensible
object, the sensible object itself, the image of an intelligible object, and the
intelligible itself. Imagination or assimilation regards the image of the
sensible object; which image is nothing more than the shadow or
resemblance appearing in water, or other lucid and polished bodies. Faith is
conversant with sensible objects; and these are animals, plants, and every
thing subject to the energies of sense. From the junction of imagination and
faith, opinion is produced. Cogitation is conversant with the image of an
intelligible object; which is nothing more than a certain universal, collected
from sensibles, related to the reasons of things existing in the soul, and
constituted by their assistance; but not elevated to ideas, and resolved into

their lucid nature. Those who are conversant with this image of an
intelligible object, use hypotheses; which do not elevate us to principles,
which are reasons and ideas, but bring us down to subordinate objects.
Lastly, science, considered as the same with intelligence, is conversant with
that which is intelligible, or the essence of things; and of this kind are
reasons pertaining to the soul, and ideas to intellect. Hence, as intelligible is
to sensible in splendor and truth, and form to its image; such is the relation
of the superior to the inferior degrees of cognition. And as these four
degrees may be resolved into five, by separating intelligence from science;
so they are reduced by Plato into two principal degrees; so far as the two
first are conversant about generation; but the two last about essence. And
thus much for the first particular proposed.
It now remains that we investigate the mode in which science is
produced in the soul, according to the doctrine of Plato. For this purpose, I
think it will be necessary to consider the rational soul, in whose nature all
reasons corresponding to ideas, have been inserted from eternity, received
into the human body, as into the plain of oblivion; bordering on the river of
negligence, that is placed near to the flux of humours; which producing in
its nature various kinds of perturbations, are the causes of its self-oblivion
and neglect. Hence the soul, thus constituted, as it were, sleeping and
intoxicated; sleeping before it is roused; intoxicated before it is purified;
begins from external sensible objects to be moved and excited, and to seek
with avidity, the knowledge congenial to its nature. The soul now, enquiring
after truth and the sciences, is first conversant with the external images of
things, in which the glimmering light of similitude to truth, presents itself to
the view; then it distinguishes these images among themselves by its
reasoning power; and if they agree in any particular, collects them into one.
Thus, being employed in separating into many, things united, and reducing
many into one, it advances from shade to substance, and is elevated from
similitude to truth itself; and thus apprehends the essence of a thing free
from every foreign or contrary quality, shining in reason and idea.
Afterwards, from this contact, especially salutary to itself, it experiences an
ineffable joy, as from a return into its proper nature, and best disposition;
and so great is its exultation, that it neglects and despises the shadows
which it formerly pursued. Then the soul truly knows, that while a man
regards corporeal natures, he is employed in resemblances; and that though
he may esteem himself knowing in many things, he knows nothing in

reality; but is then alone elevated to the sublime degree of science, when he
arrives at ideas. Hence it appears, that there are four subordinate
dispositions of mankind in order to science. For, in the first place, children,
as new guests of this world, are ignorant of every thing, without being
conscious of their ignorance. But as they advance in years, they are
employed in the shadows and images of natural concerns; and not being yet
converted to the essences of things, though they possess no real knowledge,
yet they conceive themselves to abound in knowledge. But in the third
place, being elevated to supernal natures, and judging these to be alone true,
they affirm themselves to be ignorant of all they formerly imagined
themselves to know; in which degree Socrates professed to find himself,
when he said, this one thing I know, that I know nothing; an ignorance
preferable to all the knowledge gained by the most unwearied experimental
enquiries. Indeed, this order is indicated by Socrates himself, in the Phædo,
when he says, that on his first acquaintance with natural concerns, he
thought he abounded in knowledge, but as he advanced in these pursuits, he
perceived that he knew nothing. Lastly, when, through the piercing sight of
the soul, men are elevated to ideas, they become illustrated with the
splendors of true science, and pass into the regions of perfect reality. And
hence it appears how true science, which is the same with intelligence and
wisdom, produces piety and religion: for science elevates us to intellect and
divinity; and copulates the soul with natures of the greatest purity and
perfection; so that an union of this kind cannot take place without piety,
sanctity, and religion; as dissimilars can never be blended in amicable
conjunction.

S E C T I O N IV.

W e are informed by Proclus, in the ensuing Commentaries, that the end
of geometry, and, indeed, of mathematics in general, is to be referred
to the energies of intellect; and that it is degraded when made subservient to
the common utilities of a mere animal life. But as the very opposite to this
is the prevailing opinion of the present age, let us examine the truth of this
doctrine, and attend to the arguments which the Platonic philosophy affords
in its defence. For if we can prove that this assertion of Proclus is supported
by the strongest evidence, we shall vindicate the dignity of true geometry,
restore it to its ancient esteem in the minds of the liberal, and shew how
much it is perverted by applying it to contrary purposes.
In order to this, I shall endeavour to prove the following position, that
things valuable for their own sakes, are preferable to such as refer to
something else. Now, this may be demonstrated, by considering that every
natural production was made with reference to some end, as is evident from
an induction of particulars; and if this be the case, it may be safely inferred,
that every thing exists for the sake of the end. But that for the sake of which
any being subsists is the best of all; and the end, according to nature, is that
which is perfected the last of all, from the birth of any being. Hence the
human body receives its end or perfection first, but the soul last. And hence
the soul is posterior to the body, in the accomplishment of its nature; and its
ultimate perfection is wisdom. It is on this account that old age alone
pursues and desires the goods of prudence and wisdom. Hence, wisdom is a
certain end to us according to nature; and to be wise, is the extreme or final
cause for which we were produced. It was, therefore, beautifully said by
Pythagoras, that man was constituted by divinity, that he might know, and
contemplate. If then wisdom be the end of our nature, to be endued with
wisdom must be the best of all. So that other things are to be performed for
the sake of the good which this contains. But to enquire in every science
something besides this, and to require that it should be useful, is alone the
employment of one ignorant of the great difference between the most
illustrious goods, and things necessary. For they differ, indeed, widely;
since things are to be called necessaries, which are the objects of desire for
the sake of others, and without which it is impossible to live. But those

concerns alone are properly good, which are loved by themselves, though
nothing else should fall to the lot of their possessor; for one thing is not to
be desired for the sake of another infinitely, but it is requisite to stop at
some limited object of desire, of which it would be ridiculous to require any
utility abstracted from itself. But you will ask, What is the emolument of
contemplative wisdom, what the good it confers on its possessor? What if
we should say (for such is the truth of the case) that it transports us by
intellect and cogitation, to regions similar to the fortunate islands; for utility
and necessity are strangers to those happy and liberal realms. And if this be
admitted, ought we not to blush, that having it in our power to become
inhabitants of the fortunate islands, we neglect the pursuit, through a sordid
enquiry after what is useful and profitable according to vulgar estimation?
The rewards of science, therefore, are not to be reprehended, nor is it a
trifling good which results from its acquisition. Besides, as men travel to
the mountain Olympus for its spectacle alone, preferring a view of its lofty
summit to much wealth; and as many other spectacles are desired for their
own sakes, and valued beyond gold, in like manner the speculation of the
universe is to be prized above every thing which appears useful to the
purposes of life: for it is surely shameful that we should eagerly frequent
the theatre, and the race, for the sake of the delight afforded to our corporeal
sight, and should look for no farther utility in these than the pleasure they
produce; and yet should be so sordidly stupid as to think that the nature of
things, and truth itself are not to be speculated without some farther reward
than the sincere delight their contemplation affords.
It is on this account that the apprehension of truth is compared to
corporeal vision; for the sight is the most liberal of all the senses, as is
confirmed by the general testimony of mankind. Hence, the sight of the sun
and moon, and the glorious spectacle of the stars is desired by the most
illiterate as well as the most knowing, for the delight such visions afford;
while, on the contrary, the desires of the other senses are for the most part
directed to something farther than the mere objects of their energy. Thus,
even the sense of hearing, which is the next in dignity to the sight, is not
always desirable for its own sake; for light is the general object of sight, and
sound that of hearing; but it is evident that light is more universally desired
than sound, since all light, when not excessive, is always pleasing, but this
is by no means the case with every kind of sound. Hence it is, that all
contemplation is so delightful, and this in proportion as it becomes

abstracted from sensible objects; for the most beautiful forms do not
produce genuine delight, until they are strongly represented in the phantasy,
as is evident in the passion of love; since the fairest face then alone causes
love when it presents itself clearly to the inward eye of thought, in the
mirror of imagination, accompanied with living elegance, and a resistless
energy of form.
Indeed, so liberal and so exalted an employment is contemplation, that
Plotinus, with his usual profundity, proves that the universe subsists for its
sake; that all the productions of nature originate from this; and that even
actions themselves are undertaken with a view to the enjoyment of after-
speculation. May we not, therefore, say that the sportsman follows the
chace for the sake of a subsequent review of his favourite pursuit? That the
glutton for this rejoices in the meal; and even the miser in his wealth? And
that conversation is alone sollicited, that it may recal past images to the
soul? In short, contemplation is the first spring of action, and its only end;
since we are first incited to any external object by speculating its image in
the phantasy: and our subsequent conduct tends, without ceasing, to the
energy of reflection; for destroy prior and posterior contemplation, and
action is no more.
Now if this be the case, and if geometry is a speculative science (I mean
the geometry of the ancients), it is both desirable for its own sake, and for
still higher contemplations, the visions of intellect, to which it is ultimately
subservient. For, when studied with this view, it opens the eye of the soul to
spectacles of perfect reality, and purifies it from the darkness of material
oblivion. Away then, ye sordid vulgar, who are perpetually demanding the
utility of abstract speculations, and who are impatient to bring down and
debase the noblest energies, to the most groveling purposes; ignorant of that
mighty principle of action, which influences every part of the universe, and
through which even division and discord tend as much as possible to union
and consent; ignorant that from the depravity of your nature, and the
blindness of your inward eye, you are incapable of speculating the
substance of reality, and are therefore eagerly gazing on its shadow: and
lastly, unconscious that this is the point about which you are continually
making excentric revolutions, mistaking the circumference for the centre,
motion for rest, and a departure from good for a tendency to felicity.

It was for the sake of this most exalted and liberal contemplation that
Heraclitus yielded his right of succession to a throne, to his brother; and
that Anaxagoras neglected his patrimony, esteeming one drop of genuine
wisdom preferable to whole tuns of riches. Led by a desire of this, as by
some guiding star, Pythagoras travelled into Egypt, and cheerfully
encountered the greatest difficulties, and maintained the most obstinate
perseverance, until at length he happily penetrated the depths of Egyptian
wisdom, and brought into Greece a treasury of truth for future speculation.
But these were happy days; this was the period destined to the reign of true
philosophy, and to the advancement of the human soul to the greatest
perfection its union with this terrene body can admit. For in our times, the
voice of wisdom is no longer heard in the silence of sacred solitude; but
folly usurping her place, has filled every quarter with the barbarous and
deafning clamours of despicable sectaries; while the brutal hand of
commerce has blinded the liberal eye of divine contemplation. For
unfortunately, the circle of time, as it produces continual variations, at
length reverses the objects of pursuit; and hence, that which was once
deservedly first, becomes at length, by a degraded revolution, the last in the
general esteem.
2. If geometry, therefore, be both valuable for its own sake, and for its
subserviency to the most exalted contemplations, there can be no doubt but
that the great perfection to which this science was brought by the Greeks,
was entirely owing to their deep conviction of this important truth. Euclid,
we are informed by Proclus, in this work, was of the Platonic sect; and
Archimedes is reported, by Plutarch, in his Life of Marcellus, to have
possessed such elevated sentiments of the intrinsic dignity of geometry, that
he considered it perverted and degraded, when subservient to mechanical
operations; though, at the request of king Hiero, he fabricated such
admirable engines for the defence of Syracuse. From this source alone, the
great accuracy and elegance of their demonstrations was derived, which
have been so deservedly applauded by the greatest modern mathematicians,
and the warmest advocates for the farrago of algebraic calculation. Algebra,
indeed, or as it is called, specious analysis, is the modern substitute for the
perfect method adopted by the ancients in geometrical demonstrations; and
this solely, because it is capable of being applied with greater facility to the
common purposes of life. Hence, hypotheses have been eagerly admitted in
geometry, which the ancients would have blushed to own: I mean the

multiplications and divisions of lines and spaces as if they were numbers,
and considering geometry and arithmetic as sciences perfectly the same.
But we have fortunately the testimony of the first mathematicians among
the moderns against the unlawfulness of this ungeometrical invasion. And
to begin with the great sir Isaac Newton, in his Universal Arithmetic[29]:
“Equations (says he) are expressions of arithmetical computation, and
properly have no place in geometry, except so far as quantities truly
geometrical (that is, lines, surfaces, solids, and proportions), may be said to
be some equal to others. Multiplications, divisions, and such sort of
computations, are newly received into geometry, and that unwarily, and
contrary to the first design of this science. For whoever considers the
construction of problems by a right-line and a circle, found out by the first
geometricians, will easily perceive that geometry was invented that we
might expeditiously avoid, by drawing lines, the tediousness of
computation. Therefore, these two sciences ought not to be confounded. The
ancients so industriously distinguished them from one another, that they
never introduced arithmetical terms into geometry. And the moderns, by
confounding both, have lost the simplicity in which all the elegancy of
geometry consists.” And in another part[30] of the same work he observes,
that “the modern geometers are too fond of the speculation of equations.”
To this very high authority we may add that of Dr. Halley, in the preface to
his translation of Apollonius de Sectione Rationis; for which work he
conceived so great an esteem, that he was at the pains to learn Arabic in
order to accomplish its translation into Latin[31]: “This method, says he, (of
Apollonius) contends with specious algebra in facility, but far excels it in
evidence and elegance of demonstrations; as will be abundantly manifest if
any one compares this doctrine of Apollonius de Sectione Rationis, with the
algebraic analysis of the same problem, which the most illustrious Wallis
exhibits in the second volume of his mathematical works, cap. liv. p. 220.”
And in the conclusion of his preface, he observes[32], “that it is one thing to
give the resolution of a problem some how or other, which may be
accomplished by various ways, but another to effect this by the most
elegant method; by an analysis the shortest, and at the same time
perspicuous; by a synthesis elegant, and by no means operose.” And Dr.
Barrow, notwithstanding he was so great an advocate for the identity of
arithmetic and geometry, expressly asserts[33], that algebra is no science. To

these authorities we may add Simson and Lawson, who, sensible of the
superior skill of the ancients, both in analysis and synthesis, have made
laudable attempts to restore the Greek geometry to its pristine purity and
perfection.
Again, the greatest men of the present times have been of opinion, that
algebra was not unknown to the ancients; and if this be true, their silence
respecting it is a sufficient proof of their disapprobation. Indeed, if we
consider it when applied to geometry, as an art alone subservient to the
facility of practice, as conveying no evidence, and possessing no elegance
of demonstration, we shall not wonder at its being unnoticed by the
ancients, with whom practice was ever considered as subservient to
speculation; and in whose writings elegance of theory and accuracy of
reasoning are found perpetually united.
3. But the lives of the first cultivators of this science (I mean the
Egyptian priests) as well as of the Pythagoreans and Platonists, by whom it
afterwards received such improvements, sufficiently evince that this science
advanced to perfection from an intellectual theory as its source, and from
being referred to contemplation as its end; and this will be evident, by
attending to the following history of the Egyptian priests, as preserved to us
by Porphyry, in his excellent work on abstinence[34]; a translation of which
will not, I presume, be unacceptable to the philosophical reader,
“Chæremon, the Stoic (says he) explaining the rites of the Egyptian priests,
who, he says, are accounted philosophers by the Egyptians, relates, that
they choose a place best adapted to the study and performance of sacred
rites; so that a desire of contemplation is excited by only frequenting those
recesses which are dedicated to their use, and which procure safety to the
priests, on account of that reverence of the divinity, whose sacred mysteries
they perform; so that all possible honour is paid to these philosophers, in
the same manner as to some sacred animals. But he says they live entirely
solitary, except at particular times, when they mix with others in such
assemblies as are usually held, and in public feasts; and that on all other
occasions they are scarcely to be approached. For he who desires to
converse with them must first purify himself, and abstain from a multitude
of things after the manner of these Egyptian priests. He adds, that these
men, renouncing every other occupation, and all human affairs, give
themselves entirety, through the whole of life, to the contemplation of

divine concerns, and to enquiring into the divine will: by the latter of these
employments procuring to themselves honour, security, and the estimation
of piety; by contemplation, tracing out the latent paths of science; and by
both these occupations united, accustoming themselves to manners truly
occult, and worthy of antiquity. For to dwell always on divine knowledge,
and be disposed for divine inspiration, removes a man beyond all
immoderate desires, calms the passions of the soul, and raises her
intellectual eye to the perception of that which is real and true. But they
studied tenuity of aliment, and frugality in their apparel, and cultivated
temperance and patience, together with justice and equity, in all their
concerns. Indeed, a solitary life rendered them perfectly venerable; for
during that period which they call the time of purification, they scarcely
mixed with the associates of their own order, or saw any one of them,
except him who was conversant with them in that exercise of purity, on
account of necessary uses. But they by no means concerned themselves
with those who were unemployed in the business of purification. The
remaining part of their time they conversed familiarly with those similar to
themselves; but they lived separate and apart from those who were
estranged from their ceremonies and manner of living. He adds, they are
always seen employed among the resemblances of the gods, either carrying
their images, or preceding them in their accustomed processions, and
disposing them with gravity of deportment, and in a graceful order. In all
which operations they did not indicate any pride of disposition; but
exhibited some particular natural reason. But their gravity was conspicuous
from their habit; for when they walked, their pace was equable, and their
aspect so perfectly steady, that they refrained from winking whenever they
pleased. Their risibility too, extended no farther than to a smile. But their
hands were always contained within their garments; and as there were many
orders of priests, every one carried about him some remarkable symbol of
the order he was allotted in sacred concerns. Their sustenance too was
slender and simple; and with respect to wine, some of them entirely
refrained from it; and others drank it very sparingly, affirming that it hurt
the nerves, was an impediment to the invention of things, and an incentive
to venereal desires. They also abstained from many other things, never
using bread in exercises of purity; and if they ate it at other times, it was
first cut in pieces, and mingled with hyssop. But they abstained, for the
most part, from oil, and when they used it mixt with olives, it was only in

small quantities, and as much as was sufficient to mitigate the taste of the
herbs.
In the mean time, it was not lawful for any one to taste of the aliment,
whether solid or fluid, which was brought into Egypt from foreign parts.
They likewise abstained from the fish which Egypt produced; and from all
quadrupeds having solid or many fissured hoofs; from such as were without
horns; and from all carniverous birds: but many of them abstained entirely
from animal food. And at those times when they all rendered themselves
pure, they did not even eat an egg. But when the time drew near in which
they were to celebrate some sacred rites, or festival, they employed many
days in previous preparation, some of them setting apart forty-two days,
others a greater length of time than this; and others again a shorter; but
never less than seven days; abstaining, during this period, from all animals,
and from all leguminous and oily nutriment, but especially from venereal
congress. Every day, they washed themselves three times in cold water;
after rising from bed, before dinner, and when they betook themselves to
rest. And if they happened to be polluted in their sleep, they immediately
purified their bodies in a bath. They made cold water too subservient to the
purposes of purification at other times, but not so often as the bath. Their
beds were composed from the branches of palm, which they called βαίς,
bais. A piece of wood, of a semi-cylindrical form, and well planed, served
them for a pillow. But through the whole of life, they were exercised in the
endurance of hunger and thirst, and accustomed to a paucity and simplicity
of nutriment.
But as a testimony of their temperance, though they neither used the
exercise of walking nor riding, yet they lived free from disease, and were
moderately strong. For, indeed, they endured great labour in their sacred
ceremonies, and performed many services exceeding the common strength
of men. They divided the night between observations of the celestial bodies,
and offices of purity; but the day was destined by them to the cultivation of
the divinities, whom they worshiped with hymns each day three or four
times; in the morning and evening, when the sun is at his meridian, and
when he is setting. But the rest of their time they were occupied in
arithmetical and geometrical speculations, always laborious and inventing,
and continually employed in the investigation of things. In winter nights
also, they were diligent in the same employments, and were ever vigilant to

literary studies; since they were not solicitous about external concerns, and
were freed from the base dominion of intemperate desires. Their unwearied
and assiduous labour, therefore, is an argument of their great patience; and
their continence is sufficiently indicated by their privation of desire.
Besides this, it was esteemed very impious to sail from Egypt, as they were
particularly careful in abstaining from the manners and luxuries of foreign
nations; so that to leave Egypt was alone lawful to those who were
compelled to it by state necessities. But they discoursed much concerning a
retention of their native manners; and if any priest was judged to have
transgressed the laws in the least particular, he was expelled from the
college. Besides, the true method of philosophizing was preserved in
Commentaries and Diaries, by the prophets and ministers of sacred
concerns: but the remaining multitude of priests, Pastophori, or priests of
Isis and Osiris, governors of temples, and servants of the gods, studied
purity, yet not so exactly, nor with so great continence as those we have
mentioned. And thus much is related of the Egyptians, by a man who is
equally a lover of truth, and of accurate diligence, and who is deeply skilled
in the Stoic philosophy.”
4. But the lives of the Pythagoreans and Platonists, who carried this
divine science to its ultimate perfection, no less eminently evince the truth
of our position. For, as Porphyry informs us, in the same invaluable
treatise[35], “some of the ancient Pythagoreans, and wise men, inhabited the
most desert places; and others retired into temples, from which the
multitude and every tumult were expelled. But Plato was willing to fix his
academy in a place not only solitary, and remote from the city, but, as they
report, insalubrious. Others, again, have not spared their eyes, through a
desire of more perfectly enjoying that blissful contemplation, from which
they wished never to be separated.” After this, he presents us with a
description from Plato[36] of those intellectual men, by whom the world has
been enlightened with the sublimest wisdom and truth: “For it was not
falsly, or in vain (says he), that a certain philosopher, speaking of
contemplative men, affirms, that such as these are ignorant, from their early
youth, of the way which leads to the forum, or in what place the court or
senate-house is situated, or any public council of the state. They neither see
nor hear the laws, whether decreed or promulgated, or written; and with
respect to the factions and contentions of their companions for magistracy,

for assemblies and splendid entertainments, luxurious eating and minstrels,
they do not even think of these as in a dream. Such an one knows no more
of the evil which has happened to some one of his ancestors, whether male
or female, or any thing belonging to them, than how many pitchers of water
are contained in the sea. Nor does he abstain from things of this nature for
the sake of acquiring fame; but in reality, his body alone abides in the city,
and wanders about from place to place, but his intellect esteeming all these
as of small importance, or rather as non-entities, he despises them, and,
according to Pindar, “from these on every side he soars:” by no means
applying himself to things which are near him, and to sensible concerns.”
If such then were the lives of the men who brought this contemplative
science to its present perfection, and who are to this day our masters in
geometry; if such were the exalted sentiments they entertained of its dignity
and worth, what greater proof can we require of its being valuable for its
own sake, and as subservient to the energies of intellect? We have ample
evidence too, of its being degraded when brought down to the common
purposes of life, in the example of those who, with this view, have
disguised it with the dark and sordid involutions of algebraic calculation;
for it was solely to facilitate practice, that this barbarous invasion has been
admitted by the moderns. Let me then be permitted to persuade the few who
study geometry in its ancient purity, and who consider the ruins of Grecian
literature on this, as well as on every other science, the models of
perfection, to enter with avidity on the study of the ensuing Commentaries,
and endeavour to fathom the depth of our profound and elegant
philosopher: for by this means they may happily obtain the end of all true
science, the purification of the soul; and be able to draw the light of perfect
wisdom, from the undecaying and inexhaustible fountain of good.
But if it should be asked in what these energies of intellect consist, to
which all science ultimately refers? I answer, in the contemplation of true
being, or those ideal and divine forms, with which the intelligible world is
replete. Now this great end is not to be accomplished without previous
discipline, a long exercise of the reasoning power, and a continued series of
philosophic endurance. For this end, when attained, is no other than the
enjoyment of that felicity congenial to the soul previous to her immersion in
body. But, for the further information of the liberal reader on this important
subject, the following paraphrases from Porphyry and Proclus are

subjoined; the former instructing us in the various purifications necessary to
this end; and the latter exhibiting the gradations by which we may rise to
the speculation of reality, and (leaving all multitude behind) ascend to the
divinely solitary principle of things, the ineffable One.
5. “In the first place, then (says Porphyry[37]) my reasons are not
addressed to those who are occupied in illiberal arts, nor to those engaged
in corporeal exercises, neither to soldiers nor sailors, neither to rhetoricians
nor to those who have undertaken the duties of an active life. But I write to
the man continually employed in thinking what he is, from whence he
comes, and whither he ought to tend: and who, with respect to every thing
pertaining to food, and other offices of life, is entirely changed from those
who propose to themselves a different manner of living; for to a man of this
kind alone is my present discourse addressed. Indeed, in this common state
of existence, one and the same mode of persuasion cannot be addressed to
the sleeper, who, if it was possible, would conciliate to himself perpetual
sleep, and who, for this purpose, seeks on every side for soporiferous
incentives, as to him who studies continually to drive away sleep, and to
dispose every thing about him to vigilance and intellectual activity. But to
the former, it is necessary to advise intoxication, surfeiting, and satiety, and
to recommend a dark house; and, as the poets say, a bed luxurious, broad,
and soft. Such a one should chuse whatever tends to produce stupor, and
give birth to indolence and oblivion, whether consisting of odours,
ointments, or medicaments which are accustomed to be eat or drank. But it
is necessary that the intellectual man should use sober drink, unmixed with
the lethargic fumes of wine; nutriment slender, and almost approaching to
fasting; a lucid house, receiving a subtle air and wind; that he should be
continually agitated with cares and griefs; and lastly, that he prepares for
himself a small and hard bed, while thus employed in purifying his soul
from the stains contracted by corporeal involution. But whether we are born
for this exalted purpose, I mean for vigilant intellectual energies, allowing
as small a part of our life as possible to sleep; (since we do not exist in a
place where souls perpetually vigilant abide), or whether we are destined to
a contrary purpose, I mean, to sleep and oblivion, would be foreign from
our design to explain; and would require a longer demonstration than the
limits of our work will admit.

But whoever once cautiously surmizes the delusions of our life in the
present world, and the inchantments of this material house in which we are
employed, and who perceives himself naturally adapted to vigilant energies;
lastly, who apprehends the soporiferous nature of the place in which he acts,
to such a one we would prescribe a diet congruous to his suspicion of this
fallacious abode, and to the knowledge he possesses of himself; in the mean
time, advising him to bid a long farewel to the sleeper, stretched on his
couch, as on the lap of oblivion. Nevertheless, we should be careful lest, as
those who behold the bleer-eyed, contract a similar defect, and as we gape
when present with those who are gaping, so we should be filled with
drowsiness and sleep, when the place in which we reside is cold, and
adapted to fill the eyes with watery humours, from its abounding with
marshes and vapours, which incline their inhabitants to heaviness and sleep.
If then, legislators had composed the laws with a view to the utility of the
state, and had referred these to a contemplative and intellectual life as their
end, we ought to submit to their institutions, and acquiesce in the diet they
have prescribed for our subsistence. But if they, only regarding that life
which is according to nature, and is called of the middle kind, ordain such
things as the vulgar admit, who only estimate good and evil as they respect
the body, why should any one, adducing these laws, weary himself in
endeavouring to subvert a life which is far more excellent than every law
written and composed for the sake of the vulgar, and which follows a law
not written, but divinely delivered? For such is the truth of the case.
That contemplation which procures us felicity, is not a mass of
discourses, and a multitude of disciplines; or, as some may think, consisting
from hence; nor does it receive any increase from a quantity of words. For
if this was the case, nothing could hinder those from being happy, who
comprehend all disciplines, and are accurately skilled in a variety of
languages. But the whole circle of the sciences cannot by any means
accomplish this blissful contemplation, nor even those disciplines which are
conversant with true and substantial being, unless there is also a
conformation of our nature and life to this divine end. For since there are, as
they say, three ends of living, if we regard the particular objects to which
mankind tend, the end with us is to follow the contemplation of true being,
promoting, as much as possible, by an acquisition of this kind, an intimate
union of the contemplating individual with the object of contemplation. For,
in nothing else besides true being, is it possible for the soul to return to its

pristine felicity; nor can this be effected by any other conjunction. But
intellect is true being itself: so that the proper end is to live according to
intellect. And on this account, exoteric discourses and disciplines, retarding
the purgation of the soul, are far from filling up the measure of our felicity.
If then, felicity was defined by the comprehension of words or sciences,
they who do not pay a proper attention to the kind and quantity of their
food, nor to any thing else pertaining to their present existence, might
obtain this end: but since it is requisite to change our life, and to be pure
both in speech and action, let us consider what discourses and what works
may render us partakers of this most necessary means of acquiring
substantial felicity.
Are, then, those things which separate us from sensible objects, and
from the affections which they excite, and which lead to a life intellectual,
and void of imagination and passion, are these the means we are in pursuit
of? So that every thing contrary is foreign from our purpose, and worthy to
be rejected? And in such proportion as it draws us aside from intellect?
Indeed, I think it is consonant to truth, that we should eagerly contend
where intellect leads; for in this material abode, we are similar to those who
enter or depart from a foreign region, not only in casting aside our native
manners and customs, but from the long use of a strange country, we are
imbued with affections, manners, and laws foreign from our natural and
true region, and with a strong propensity to these unnatural habits. Such an
one, therefore, should not only think earnestly of the way, however long and
laborious, by which he may return to his own, but that he may meet with a
more favourable reception from his proper kindred, should also meditate by
what means he may divest himself of every thing alien from his true
country, which he has contracted; and in what manner he may best recal to
his memory, those habits and dispositions without which he cannot be
admitted by his own, and which, from long disuse, have departed from his
soul. In like manner, it is requisite, if we wish to return to such things as are
truly our own, and proper to man considered as a rational soul, to lay aside
whatever we have associated to ourselves from a mortal nature, together
with all that propensity to material connections, by which the soul is
allured, and descends into the obscure regions of sense; but to be mindful of
that blessed and eternal essence intellect, our true father, and hastening our
return to the contemplation of the uncoloured light of good, to take especial
care of these two things; one, that we divest ourselves (as of foreign

garments) of every thing mortal and material; the other how we may return
with safety, since thus, ascending to our native land, we are different from
ourselves before we descended into mortality. For we were formerly
intellectual natures; and even now we are essences purified from every stain
contracted by sense, and from that part which is destitute of reason: but we
are complicated with sensible connections, on account of our impotence and
infirmity, which is the cause that we cannot always be conversant with
intellectual concerns; but with mundane affairs we can be present with
frequency and ease; for all our energetic powers are stupified and clouded
with oblivion, through body and sense; the soul not remaining in an
intellectual state; (as the earth when badly affected, though good fruit is
deposited in its bosom, produces nothing but weeds); and this, through the
improbity of the soul, which does not, indeed, destroy its essence, while it
acquires brutality; but by such an accession it becomes complicated with a
perishing nature, is bound in the dark folds of matter, and is drawn aside
from its proper state, into one that is foreign and base.
So that it is highly requisite to study, if we are solicitous of returning to
our pristine state of felicity, how to depart from sense and imagination, and
her attendant brutality, and from those passions which are raised by her
phantastic eye, as much as the necessity of our nature will permit. For the
intellect must be accurately composed; and it is proper it should obtain a
peace and tranquility free from the contentions of that part which is
destitute of reason, that we may not only hear with attention concerning
intellect and intelligible objects, but to the utmost of our ability, may enjoy
their contemplation; and thus, being reduced into an incorporeal nature,
may truly lead an intellectual life, and not in a false delusive manner, like
these who are at the same time entangled with corporeal concerns. We must,
therefore, divest ourselves of the various garments of mortality by which
our vigour is impeded; as well this visible and fleshly garment, as that more
interior one with which we are invested contiguous to the skin. We must
enter the place of contest naked, and without the incumbrance of dress,
striving for the most glorious of all prizes, the Olympiad of the soul. But the
first requisite, and without which it is not lawful to contend, is, that we strip
off our garments. And since our vestments are some of them exterior, and
some interior, so with respect to the denudation of the soul, one process is
by things more open, another by such as are more occult. For instance, not
to eat, or not to accept what is offered, is among things obvious and open;

but not to desire is more obscure; so that it is here requisite not only to
abstain from things improper in deeds, but likewise in desire. For what does
it profit to abstain in actions from what is base, in the mean time adhering
to the causes which produce such actions, as if bound in indissoluble
chains?
But this receding from material affections is brought about partly by
force, and partly by persuasion; and by the assistance of reason the
affections languish, and are, as it were, buried in oblivion, or in a certain
philosophical death; which is, indeed, the best mode of desertion, without
oppressing the terrene bandage from which the soul departs. For in things
which are the objects of sense, a violent devulsion cannot take place
without either a laceration of some part, or at least a vestige of separation.
But vice steals in upon the soul through continual negligence: and
carelessness is produced by not sufficiently attending to intelligible objects;
the affections in the mean time being excited by the drowsy perceptions of
sense, among which must be also reckoned the sensations arising from
food. We must therefore abstain, not less than from other things, from such
food as usually excites the passions of our soul. Let us then in this particular
enquire a little farther.
There are two fountains, whose noxious streams detain the soul in
matter, and with which, as if saturated with lethargic potions, she forgets
her own proper speculations: I mean pleasure and grief, the artificer of
which is sense and its perceptions, together with the operations attendant on
the senses, imaginations, opinions, and memory. The passions, roused by
the energies of these, and the irrational part, now fattened with noxious
nutriment, draw down the soul, and avert her inclinations from her native
love of true being. It is requisite, therefore, that we revolt from these to the
utmost of our ability. But true defections can alone take place by avoiding
the passions and rash motions produced by the senses. But, sensation
respects whatever moves the sight, or the hearing, or the taste, or the smell.
And sense is, as it were, the metropolis of that foreign colony of passions
which reside in the soul, and which must be expelled by him who wishes,
while connected with body, to become an inhabitant of the royal regions of
intellect. Let us then enquire how much fuel of the passions enters into us
through each of the senses; and this either when we behold the spectacles of
horses in the race, and the labours of the athletic, or the contests of those

who twist and bend their bodies in leaping, or when we survey beautiful
women. For all these insnare us, unconscious of the danger, and subject to
their dominion the irrational appetite, by proffered inchantments of every
kind.
For by all such inchantments the soul, as if driven into fury, compels the
compound man to leap rashly, and without reason, and full of the brutal
nature to bellow and exclaim. In the mean time, the perturbation appearing
from without, being inflamed by the internal, which was first of all roused
by sense. But the vehement motions excited by the hearing, arise from
certain noises and sounds, from base discourse, and mixed assemblies; so
that some, exiled from reason, behave as if struck mad; and others,
enervated by effeminate softness, agitate themselves by a multitude of
trifling gesticulations. And who is ignorant how much the soul is fattened,
and infested with material grossness, by the ointments and perfumes which
commend lovers to each other? But why is it necessary to speak of the
passions originating from the taste: in this respect especially, binding the
soul in a double band; one of which is thickened by the passions excited by
the taste; the other becomes strong and powerful by the different bodies
which we receive in food. For as a certain physician observed, those are not
the only poisons which are prepared by the medical art, but such things as
we daily receive for food, as well liquid as solid, are to be reckoned among
this number; and much greater danger arises to our life from these, than to
our bodies from poisons. But the touch does all but transmute the soul into
body, and excites in it, as in a dissonant body, certain broken and enervated
sounds. The remembrance, imagination, and cogitation of all these raise a
collected swarm of passions, i. e. of fear, desire, anger, love, emulation,
cares, and griefs, they fill the soul with perturbations of this kind, cloud its
intellectual eye with oblivion, and bury its divine light in material darkness.
On which account it is a great undertaking to be purified from all this
rout of pollutions; and to bestow much labour in meditating day and night,
what measures we shall adopt to be freed from these bonds, and this
because we are complicated with sense, from a certain necessity. From
whence, as much as our ability will permit, we ought to recede from those
places in which we may (perhaps unwillingly), meet with this hostile rout;
and it is requisite we should be solicitous not to engage in combat with
these dangerous foes, lest, through too great a confidence of victory and

success, instead of vigorous contention, we produce only unskilfulness and
indolence.”
And in the conclusion of the first book, he adds, “For, indeed, if it be
lawful to speak freely, and without fear, we can by no other means obtain
the true end of a contemplative, intellectual life, but by adhering to the
Deity (if I may be allowed the expression), as if fastened by a nail, at the
same time being torn away and separated from body and corporeal delights;
having procured safety from our deeds, and not from the mere attention to
words. But if friendship is not to be conciliated with a divinity, who is only
the governor of some particular region, with any kind of food, or by the use
of animal nutriment, much less can a gross diet effect an union with that
God who is exalted above all things, and who is superior to a nature simply
incorporeal; but after every mode of purgation, and the greatest chastity of
body, and purity of soul, we shall scarcely be thought worthy to obtain the
vision of his ineffable beauty; though this is sometimes permitted to him
whose soul is well disposed, and who has passed through life with the
greatest sanctity and purity of manners. So that, by how much the Father of
all exceeds every nature in simplicity, purity, and self-sufficiency, as being
infinitely remote from all suspicion of material contagion, by so much the
more ought he who approaches to the Deity, to be entirely pure and holy,
first in his body, and afterwards in the most secret recesses of his soul;
having distributed a purgation adapted to every part, and being completely
invested with purity, as with a transparent garment, fit for the intimate
reception of divine illumination.” Thus far Porphyry, whose excellent
sentiments on this subject are a lasting monument of the elevation and
purity of soul which the Platonic philosophy affords; and at the same time
sufficiently prove the arrogance and ignorance of those who depreciate the
wisdom of the ancients, and consider their greatest philosophers as involved
in mental darkness and delusion. But presumption of this kind is continually
increased by indolence, and strengthened by interest; and it is common to
find scribblers of every kind, laughing at Plato and his philosophy, who are
too mean for criticism, and even too insignificant for contempt. Let us,
therefore, leave such in their native inanity, and listen to the instructions of
the divinely elegant Proclus, by which we may ascend to the contemplation
of true being, and the ineffable principle of things.

6. [38]“Pythagoras and Plato command us to fly from the multitude, that
we may pursue the most simple truth, and apply ourselves wholly to the
contemplation of real being. From the multitude of exterior people drawing
us aside in various ways, and deceiving us by fallacious appearances. But
much more to shun the multitude of interior people; for this much more
distracts and deceives. We must, therefore, fly from the various multitude of
affections, the obscure informations of sense, the shadowy objects of
imagination, and the dusky light of opinion. For every multitude of this
kind is so different in itself, that its parts are contrary to one another; from
whence it is necessary to betake ourselves to the sciences, in which
multitude has no contrariety. For though affections are contrary to
affections, one perception of sense to another, imaginations to imaginations,
and opinions to opinions, yet no one science is found contrary to another. In
this multitude, therefore, of propositions and notions, we may collect into
one the number of sciences binding them in one according bond. For they
are so remote from contrariety to each other, that notion is subservient to
notion, and inferior sciences minister to superior, depending on them for
their origin. Above all, it is here necessary, from many sciences which pre-
suppose one, to betake ourselves to one science itself, no longer supposing
another, and in an orderly series to refer them all to this original one. But
after science, and its study, it will be necessary to lay aside compositions,
divisions, and multiform discourses, and from thence to ascend to
intellectual life, to its simple vision, and intimate perception. For science is
not the summit of knowledge, but beyond it is intellect; not that intellect
only which is separated from soul, but the illustration infused from thence
into the soul, which Aristotle affirms to be the intellect by which we
acknowledge the principles of science; and Timæus says, that this exists in
no place but the soul. Ascending, therefore, to this intellect, we must
contemplate together with it intelligible essence, by indivisible and simple
perceptions, speculating the simple genera of beings. But after venerable
intellect itself, it will be proper to contemplate that summit of the soul, by
which we are one, and under whose influence our multitude is united. For
as by our intellect we touch the divine intellect; so by our unity, and as it
were the flower of our essence, it will be lawful to touch that first one, from
whom all subordinate unities proceed. And by this our one, we are
especially conjoined with divinity. For similitude may be every where
comprehended by that which is similar; the objects of knowledge by

science; things intelligible by intellect; and the most unifying measures of
being, by the unity of the soul. But this unity and its energy is the summit of
our actions; for by this we become divine, when, flying from all multitude,
we retire into the depths of our unity, and, being collected into one,
uniformly energize. Thus far we admonish to shun the multitude, by steps
proceeding from the order of knowledge: in the next place, we shall proceed
in the same design by the series of knowable objects. Fly then every
sensible species, for they are heaped together, are divisible, and perfectly
mutable, and incapable of affording sincere and genuine knowledge. From
these dark informations, therefore, betake yourself to incorporeal essence;
since every sensible object possesses adventitious unity, is by itself
scattered and confused, and full of formless infinity. Hence its good is
divisible, and adventitious, distant and separated from itself, and residing in
a foreign seat. When you have ascended thither, and are placed among
incorporeal beings, you will behold above the fluctuating empire of bodies,
the sublime animal order, self-moving, spontaneously energizing in itself,
and from itself possessing its own essence, yet multiplied, and anticipating
in itself a certain apparition or image of the essence divisible about the
unstable order of bodies. You will there perceive many habitudes of
reasons, various proportions, and according bonds. Likewise the whole and
parts, vivid circles, and a multiform variety of powers; together with a
perfection of souls not-eternal, not subsisting, together as a whole, but,
unfolded by time, gradually departing from their integrity, and conversant
with continual circulations. For such is the nature of the soul.
But after the multitude belonging to souls, betake yourself to intellect,
and the intellectual kingdoms, that you may possess the unity of things.
There remain in contemplation of a nature ever abiding in eternity, of life
ever flourishing, intelligence ever vigilant, to which no perfection of being
is wanting, and which does not desire the chariot of time, for the full energy
of its essence. When you have beheld natures of this exalted kind, and have
seen by how great an interval they are superior to souls; in the next place
enquire whether any multitude is there, and if intellect, since it is one, is
also universal; and again, since it is uniform, if not also multiform: for you
will find it subsists after this manner. When, therefore, you have intimately
beheld this intellectual multitude, though profoundly indivisible and united,
transport yourself again to another principle, and having considered, as in a
more exalted rank, the unities of intellectual essences, in the last place

proceed to unity perfectly separate and free from all things. And when
advanced thus far, lay aside all multitude, and you will at length arrive at
the ineffable fountain of good. And since it appears, from these various
gradations, that the soul then properly obtains perfection, when she flies
from all external and internal multitude, and the boundless variety of the
universe, we may likewise conclude from hence, that our souls do not alone
collect their knowledge from the obscure objects of sense, nor from things
particular and divisible discover a perfect whole, and a perfect one, but
draw forth science from their inmost recesses, and produce accuracy and
perfection from whatever in appearances is inaccurate and imperfect. For it
is not proper to suppose that things false and obscure, should be the
principal sources of knowledge to the soul; and that things discordant
among themselves, which require the reasonings and arguments of the soul,
and which are ambiguous and confused, should precede science which is
immutable; nor that things variously changed, should generate reasons
abiding in one; nor that indeterminate beings should exist as the causes of
determinate intelligence. It is not, therefore, fit to receive the truth of eternal
entities from boundless multitude; nor from sensible objects the judgment
of universals; nor from things destitute of reason, accurate discrimination of
that which is good: but it is proper that the soul, retiring into her immortal
essence, should there scrutinize the good and the true, and the immutable
reasons of all things: for the essence of the soul is full of these, though they
are clouded by oblivion. The soul, therefore, beholding exteriors, enquires
after truth, in the mean time possessing it in the depths of her essence, and
deserting herself, explores the good in the dark regions of matter. Hence,
every one in the pursuit of reality ought to begin with the knowledge of
himself. For, if we constantly extend our views among the multitude of
men, we shall never discern the one species man, obscured by the
multitude, and distracted by the division and discord, and the various
mutations of those who participate the species. But if we turn our eye
inwards, there, remote from perturbation, we shall behold one reason and
nature of men; since multitude is an impediment to the conversion of the
soul into herself. For here variety darkens unity, difference obscures
identity, and dissimilitude clouds similitude; since species are confused in
the folds of matter; and every where that which is excellent is mixed with
the base.” Thus far Proclus; and thus much for our intended Dissertation.

L I F E

AND

COMMENTARIES
OF

PROCLUS.

THE

LIFE of PROCLUS,
BY MARINUS[39];
OR,

CONCERNING FELICITY.

W hen I consider the magnitude of mind, and dignity of character
belonging to Proclus, a philosopher of our time, and attend to those
requisites, and that power of composition which those ought to possess who
undertake a description of his life; and lastly, when I regard my own
poverty of diction, I am inclined to believe it more proper to refrain from
such an undertaking, not to leap over the fosse (according to the proverb),
and to decline a discourse involved in so much difficulty and danger. But
my scruples are something diminished when I consider, on the other hand,
that even in temples, those who approach to the altars do not all sacrifice
alike; but that some are solicitously employed in preparing bulls, goats, and
other things of a similar kind, as not unworthy the beneficence of the Gods
to whom those altars belong: likewise that they compose hymns, some of
which are more elegant in verse, but others in prose; while some, who are
destitute of all such gifts, and sacrifice with nothing more perhaps than a

cake and a small quantity of bread, with frankincense, and who finish their
invocations with a short address to the particular divinity they adore, are not
less heard than others. While I thus think with myself, I am afraid,
according to Ibycus[40], lest I should not offend against the Gods (for these
are his words) but against a wise man, and thus obtain the praise of men.
For I do not think it lawful, that I who was one of his familiars, should
be silent concerning his life; and should not, according to my utmost ability,
relate such particulars concerning him as are true, and which perhaps ought
to be published in preference to others. And indeed by such a neglect I shall
not perhaps obtain the esteem and honour of mankind, who will not entirely
ascribe my conduct to the desire of avoiding ostentation, but will suppose I
avoided such a design from indolence, or some, more dreadful disease of
the soul. Incited, therefore, by all these considerations, I have taken upon
me to relate some illustrious particulars of this philosopher, since they are
almost infinite, and may be depended on for their undoubted reality.
I shall begin therefore not according to the usual manner of writers, who
are accustomed to distribute their discourse into chapters; but I consider that
the felicity of this blessed man ought, with the greatest propriety, to be
placed as the foundation of this treatise. For I regard him as the most happy
of those men who were celebrated in former ages; I do not say happy only
from the felicity of wisdom, though he possessed this in the highest degree
of all men; nor because he abundantly enjoyed the goods of an animal life;
nor again on account of his fortune, though this belonged to him in a most
eminent degree, for he was supplied with a great abundance of all such
things as are called external goods: but I call him happy, because his felicity
was perfect, complete in all parts, and composed from each of the preceding
particulars. Having then in the first place distributed[41] virtues according to
their kinds into natural, moral, and political, and also into those of a
sublimer rank, which are wholly conversant with purification and
contemplation, and are therefore called Cathartic and Theoretic, and also
such as are denominated Theurgic, by which we acquire a similitude with
some particular divinity; but omitting such as are superior to these, as
beyond the reach of man, we shall begin from such as are more natural, and
which are first in the progressions of the human soul, though not first in the
nature of things.

This blessed man, then, whose praise is the subject of this treatise,
naturally possessed, from the hour of his birth, all those physical virtues
which fall to the lot of mankind; the traces of which were manifest in the
latest period of his life, and appeared to surround and invest his body after
the manner of a tenacious shell. In the first place, he was endued with a
singular perfection of sensation, which they denominate corporeal
prudence; and this was particularly evident in the nobler senses of seeing
and hearing, which are indeed given by the gods to men for the purpose of
philosophizing, and for the greater convenience of the animal life; and
which remained entire to this divine man through the whole of his life.
Secondly, he possessed a strength of body which was not affected by cold,
and which was neither weakened nor disturbed by any vicious or negligent
diet, nor by any endurance of labours, though it was exhausted day and
night, while he was employed in prayer, in perusing the works of others, in
writing books himself, and in conversing with his familiars; all which he
performed with such expedition, that he appeared to study but one thing
alone. But a power of this kind may with propriety be called fortitude of
body, from the singular strength employed in such exertions. The third
corporeal virtue with which he was endued was beauty, which, when
compared with temperance, the authors of these appellations have very
properly considered as possessing a similitude of nature. For as we consider
temperance as consisting in a certain symphony and consent of the powers
of the soul, so corporeal beauty is understood to consist in a certain
agreement of the organical parts. He was indeed of a most pleasing aspect,
not only because he was endued with this excellent proportion of body, but
because the flourishing condition of his soul beamed through his corporeal
frame like a living light, with splendors too wonderful for language to
explain. And indeed he was so beautiful that no painter could accurately
describe his resemblance; and all the pictures of him which were circulated,
although very beautiful, were far short of the true beauty of the original. But
the fourth corporeal virtue which he possessed was health, which they
affirm corresponds to justice in the soul; and that this is a certain justice in
the disposition of the corporeal parts, as the other in those of the soul. For
justice is nothing more than a certain habit, containing the parts of the soul
in their proper duty. Hence, that is called health by physicians, which
conciliates the jarring elements of the body into union and consent; and
which Proclus possessed in such perfection, that he affirmed he was not ill

above twice or thrice, in the course of so long a life as seventy-five years.
But a sufficient proof of this is evident from hence, that, in his last illness,
he was entirely ignorant what the disorders were which invaded his body,
on account of the great rarity of their incursions.
Such then were the corporeal goods which Proclus possessed, and
which may be called the forerunners, and as it were messengers, of those
forms into which we have distributed perfect virtue. But the first powers
and progeny of his soul, which he naturally possessed, previous to
instruction, and those parts of virtue with which he was adorned, and which
Plato reckons the elements of a philosophic nature[42], must excite the
wonder of any one who considers their excellent quality. For he was
remarkable for his memory and ingenuity; he was of a disposition
magnificent, gentle, and friendly; and a companion, as it were, of truth,
justice, fortitude and temperance; and his love of truth was so great, that he
never admitted any prudent dissimulation, but hated falsehood vehemently.
Indeed it is necessary that he who prosecutes truth, with so much
earnestness and sincerity, should be extremely desirous of it from his
infancy, since truth is the source of every good, both to gods and men. But
that he despised corporeal pleasures, and was an eminent lover of
temperance, is sufficiently evident from his great propensity to disciplines,
and his desire of every kind of studies; for dispositions of this kind never
suffer beastly and illiberal pleasure to dwell it the mind, but are able to
excite in the soul, from her own internal operations, sincere pleasure and
delight. But it is impossible to say how foreign he was from avarice, so that
when a boy he despised the wealth of his parents, though very rich, on
account of his incredible love towards philosophy. Hence he was far
removed from illiberality, and from the care of lesser concerns, as he was
most studious of the universe, and of every thing divine and human. But
from such a disposition of the rational soul, having acquired true
magnanimity, he considered human life as of no account, and, unlike the
multitude, viewed nothing dreadful in death. So that he by no means feared
all that rout of molestations which appear terrible to others, and this in
consequence of that natural affection which it is proper to call by no other
name than that of Fortitude alone. But, from all these virtues, I think it must
be evident to those who have not experienced his best of dispositions, that
he loved equity from a boy; that he was just and mild, and by no means
difficult or unjust in his associations or contracts. To us indeed he certainly

appeared modest and elegant, neither avaritious nor illiberal, neither
arrogant nor timid.
But will it not be superfluous to mention the goodness and fertility of
his ingenuity? Especially among those who know and who have heard, that
he was full of the most beautiful disciplines, and who are acquainted with
the multitude he produced and published to the world, so that he alone
seemed to have drank nothing of the cup of oblivion, as he was endued with
a power of memory which was never disturbed, and that which belongs to
the oblivious, never happened to him. Besides, he never neglected fresh
acquisitions, as if possessing a sufficiency of disciplines; and as one who is
merely delighted with their study. But he was most remote from a nature
rustic and horrid, and averse from the Muses, and particularly propense to
more cultivated endowments: for on account of his singular urbanity and
festivity (without transgressing the bounds of true honesty) in his common
associations, sacred feasts, and other actions, he allured and charmed his
familiars, and always dismissed them more cheerful and pleased.
His mother, therefore, Marcella, lawfully united to his father Patricius,
both of the Lycian nation, and excelling in birth and virtue, produced our
philosopher, thus endued from the beginning with all these, and other gifts
of nature. And[43] Minerva, the tutelar goddess of Byzantium, received him
when born, and took care of him as a midwife, she being the cause of his
birth in that city: but afterwards she provided for his well-being, when he
was numbered among boys and young men; for she appeared once to him in
a dream, and exhorted him to the study of philosophy, from whence arose
his great propensity to this goddess, as he particularly performed her sacred
rites, and cultivated with a greater fury (as I may say) her institutions.
Lastly, his parents brought him, when born, into their native country
Zanthus, consecrated to Apollo: and I cannot but think that this country
happened to him by a certain divine providence; as it was requisite that he,
who was to be the prince of all sciences, should be educated under the
presiding deity of the Muses. Here, being instituted in the most elegant
manners, he pursued moral virtues, and was accustomed to right conduct,
and to a declination of its contrary, that which is base.
But at that time the love of the gods, who had attended him from his
nativity, manifestly appeared; for being once detained by some disease of
body, and it appearing very difficult, and scarcely possible to cure him,

there stood at his bed a youth of a more than ordinary appearance, so that
even previous to the declaration of his name, he might be considered as[44]
Telesphorus or Apollo: but the god proclaiming who he was, and
pronouncing his name, touched the head of Proclus (for he stood reclining
his head on Proclus’ pillow) and having immediately restored him to health,
vanished from his sight. And such was the[45] divine vision, and the divine
benevolence at that time exhibited to our youth.
But having, for a short space of time, in Lycia, applied himself to
grammar, he went to Alexandria in Egypt, bringing with him very singular
moral virtues, by which he excited towards himself the love of the masters
resident in that place. Hence Leonas the rhetorician, who derived (as I
think) his lineage from Isaurus, and was illustrious among many of that
profession, who were then at Alexandria, not only made him a partaker of
his studies, but thought him worthy to become his domestic, and ordered
that he should be supplied with food, together with his wife and children, no
otherwise than if he had been his[46] true son. He likewise took care to
procure him the notice of the principal men in Egypt, who being
wonderfully delighted with the acumen of the youth’s ingenuity, and with
the elegance and integrity of his morals, reckoned him among their greatest
friends. But he was also instructed by Orion the grammarian, whose
ancestors discharged the sacerdotal office among the Egyptians, and who
had made such a progress in the knowledge of his art, that he composed
elaborate books on this subject, which he left not without advantage to
posterity. He also went to the schools of the Roman preceptors, and made a
great progress in that language; for he was at first led to the study of his
father’s profession, in which he was illustrious, his employment being the
study of law in the royal city. But when it appeared how vehemently the
young man was delighted with the study of rhetoric, as he had not yet
touched the writings of the philosophers, he both acquired great glory from
his acquisitions, and became the admiration of his fellow pupils and
masters, on account of the elegance of his discourse, and his celerity in
perceiving; and from his exhibiting more the habit and industry of the
master, than that of the scholar.
But while he yet frequented the rhetorical school, the sophist Leonas,
made him the companion of his journey to Byzantium: which he undertook
for the purpose of gratifying his friend Theodorus, who was at that time

præfect of Alexandria, and who was a man both polite and magnificent, and
a lover of philosophy. But Proclus, though a youth, followed his master
more cheerfully in this journey, lest he should be compelled to interrupt his
studies. However, that I may speak more truly, a certain good fortune
brought him back to the source of his nativity. For, on his return, his tutelar
goddess exhorted him to philosophy, and to visit the Athenian schools. But
having first returned to Alexandria, and bid farewel to rhetoric, and the
other arts which he had formerly studied, he gave himself up to the
discourses of the philosophers then resident at Alexandria. But he
frequented[47] Olympiodorus, the most illustrious of philosophers, for the
sake of imbibing the doctrine of Aristotle; and in mathematical disciplines
gave himself to Hero[48], a religious man, and one who was eminently
skilful in the proper methods of institution. But these men were so delighted
with the manners of the youth, that Olympiodorus wished him to espouse
his daughter, whom he had taken care to instruct in philosophy, and Hero
committed to him all his religion, and made him his constant companion.
But having, on a certain time, heard Olympiodorus, a man who was endued
with a great power of speaking, and on account of the celerity of his speech,
and the gravity of his subjects, was understood by very few of his auditors,
as he was departing with the dismissed multitude, he repeated to his
companions all that was said, and almost verbatim, though the discourse
was copious; as Ulpianus Gazæus, one of his fellow-disciples informed me,
who also consumed not the least part of his life in the study of philosophy.
But he likewise learned, with great facility, the writings of Aristotle,
pertaining to rational philosophy[49], though the bare reading of them is
difficult to those who are engaged in the attempt.
Having therefore, at Alexandria, applied himself to these masters, and
enjoyed their confidence in such instruction as they were able to afford,
when upon reading together with them a certain author, they appeared to
him not to interpret the mind of the philosopher as they ought; conceiving a
contempt for these schools, and at the same time being mindful of the
exhortation which had been divinely sent to him at Byzantium, he went to
Athens, attended by the presiding deities of eloquence and philosophy, and
by beneficent dæmons. For that he might preserve the genuine and entire
succession[50] of Plato, he was brought by the gods to the guardian city of
philosophy, as the circumstances which happened on his first entrance into

the city, and all the divine excitations manifestly evince: for they openly
presaged, that this gift was sent from the father Apollo, and was a future
suffrage of his succession confirmed by divine events. For when his vessel
drove to the Pyræum, and it was told to the citizens, Nicolaus, who
afterwards flourished in the rhetorical art, but at that time studied under the
masters of Athens, descended to the shore as if to an acquaintance, received
him for his guest as if he had been a citizen, and brought him to the city; for
Nicolaus was also a Lycian. But Proclus, who perceived himself weary
from his journey, sat down at the temple of Socrates, though he did not yet
know, nor had heard that Socrates was worshipped in that place, and
requested Nicolaus that he would stay there for a short time, and, if
possible, procure him some water, as he said that he was exceeding thirsty.
Immediately Nicolaus, from that very consecrated place, brought him some
water; for the fountain belonging to the statue of Socrates was not far
distant. But while he was drinking, Nicolaus, for the first time, considering
the circumstance: This is an omen, says he, because you have sate in the
temple of Socrates, and have there first drank the Attic water. Then Proclus
rising, and having paid due reverence to the place, proceeded to the city.
But when he came to the tower, the porter who was present at his entrance,
and was about to close the gates with bars, said to him, (that I may repeat
the words of the man,) “Certainly unless you had come, I should have
closed the gates.” And what omen could be more manifest, or could less
require the interpretations of[51] Polletes or Melampodes, or such like
diviners?
But Proclus, despising the schools of the rhetoricians, though they very
much desired his association, as if he had come for that very purpose, met
with the prince of philosophers Syrianus[52], the son of Philoxenus. At that
time too, Lachares was in the same company, a man much conversant in
philosophy, and the companion of Syrianus in his study, but in eloquence he
was in as great admiration as Homer in the poetic art. He then was, as I
have said, present at the same time. But it was now the evening twilight,
and while they were engaged in mutual converse, the sun sate, and the
moon made her first appearance after her change: wherefore, having saluted
the stranger, they endeavoured to dismiss him, as being a young man, from
their company, that they might adore the goddess apart. But he not having
proceeded far, beheld also the moon appearing from the same house, and

laying aside his sandals, in their presence saluted the goddess. Here
Lachares, admiring the confidence of the youth, said, speaking to Syrianus,
“This is what Plato[53] divinely affirms of great geniuses; that they either
produce great good, or its contrary.” And such, that I may mention a few out
of many, were the signatures of divine original, which happened to our
philosopher, on his first arrival at Athens.
But Syrianus brought him to the great Plutarch[54], the son of Nestor,
who, when he saw the young man, not yet twenty years old, and heard of
his love and desire of a philosophic life, he was much delighted, and
immediately made him a partaker of his philosophic study, though his age
almost forbade such an attempt; for he was then very old. He therefore read
to Proclus his commentary on Aristotle’s books on the soul, and on the
Phædo of Plato: and this great man exhorted him to commit to writing what
he heard, employing the ambition of the youth as an instrument, by telling
him, that if he completed those scholia, they would be reported as the
commentaries of Proclus. And as he loved the youth very much on account
of his inclination to the best studies, he called him his son, and caused him
to reside with him as a domestic. But after he saw his temperance, with
regard to animal food, he exhorted him not to abstain from animals entirely,
but to use them so far as was necessary to the vital energies of the corporeal
part. He likewise gave the same advice to Syrianus, concerning the youth’s
diet. But he answered the old man, as that divine head (Proclus) informed
us: “Suffer him with that frugality to learn what I wish, and then, if he
pleases, he may die.” Such was the care of his masters respecting him, in
every concern. But the old man lived about two years after the arrival of
Proclus; and dying, commended the youth to his successor Syrianus, as also
his grandson Archiadas. But Syrianus, when he had received Proclus as his
pupil, not only much assisted him in learning, but made him his domestic as
to other concerns, and the companion of his philosophic life, having found
him such an auditor and successor as he had for a long time sought for; and
one who was adapted for the reception of a multitude of disciplines, and
divine dogmata.
In a shorter space, therefore, than two years, he read, together with
Syrianus, all the works of Aristotle, i. e. his logic, ethics, politics, physics,
and theological science. But being sufficiently instructed in these, as in
certain[55] proteleia, and small mysteries, Syrianus led him to the sacred

discipline of Plato, and this by an orderly progression, and not[56] according
to the oracle, with a transcendent foot. And he was careful that he might
survey with him true mysteries, with the eyes of his soul, free from material
darkness, and with a speculation of intellect refined and pure. Hence
Proclus was employed night and day in vigilant energies, and in writing
compendiously what he had heard, employing his own judgment in the
selection and order. And in consequence of this unwearied assiduity, he
made so great a progress in a short time, that by then he was twenty-eight
years of age, he composed a multitude of works; and among the rest his
very learned and elegant commentaries on the Timæus. But from an
institution of this kind, his manners also received a greater ornament, since
as he advanced in science he accumulated virtue.
In the original κατὰ τὸ λόγιον, which I wonder Fabricius should
translate, quod aiunt, as it is usual with the Platonists, to cite the
Zoroastrian oracles exactly in these words, instances of which may be
found in Proclus on Plato’s theology; and the very words prove themselves
to be a part of an oracle, when attentively considered.]
But he likewise pursued the civil virtues from Aristotle’s political
writings, and Plato’s books concerning laws and a republic. However, lest
he should be conversant only in the contemplation of these, and should
attempt nothing actual, since he was hindered from engaging in public
affairs himself, from his being employed in more important concerns, he
exhorted Archiadas, who was a religious man, to a political life, at the same
time instructing him in its nature and virtues, and explaining to him the
methods proper to be observed in the discharge of its duties. And in the
same manner as they incite those who run in the race, so he exhorted him to
preside in common over the whole of his city, and to confer benefits
privately upon each according to all the virtues, but particularly in
conformity with the laws of justice. But he excited in reality the emulation
of himself in Archiadas, when he exhibited to him his own liberality and
magnificence with respect to riches, by bestowing gifts at one time on his
friends, and at another time on his kindred, guests, and citizens; proving
himself by this means not in the least affected with avarice, and conferring
not a little to the good of public concerns. But when dying, he bequeathed
that part of his possessions which he did not leave to Archiadas, to his own
country, and to Athens. But Archiadas became so great a lover of truth, as

well from the company of his own associates as from the friendship of
Proclus, that he is never mentioned by our acquaintance, without being at
the same time celebrated as the religious Archiadas.
But the philosopher also employed himself in civil consultations among
public assemblies, where things pertaining to the republic were discussed;
giving the most prudent counsels, conferring with governors concerning
equity, and not only exhorting them to an impartial distribution of justice,
but in a manner compelling them by philosophical authority. For he had a
certain public care of the morals of princes, and not only instructed them in
the art of temperate government by his discourse, but also by his own
example through the whole of his life; since he was, as it were, the
exemplar of temperance to the rest. But he gave a specimen of civil
fortitude perfectly Herculean: for since at that time there was, as it were, a
sea of troubles upon him, and mighty waves of stormy employments were
roused by adverse winds against his upright life, he conducted himself,
though in danger, with gravity, and an unshaken constancy. And when he
was once very much molested by the improbity of some violent men, which
was both pernicious and dangerous to himself, he undertook a journey into
Asia, which contributed greatly to his own advantage: for as he was not
unskilful in the more ancient rites of that place, which he yet preserved, a
divine power afforded him this occasion of departure. Hence, as he well
knew the whole of their concerns, he taught them more accurately in things
pertaining to the gods, if they happened to have neglected any thing through
a long interval of time. And while he was engaged in all these
employments, and lived in a correspondent manner, he was so concealed
from the multitude that he even excelled the Pythagoreans, who keep with
unshaken constancy this precept of their founder, λάθε βιώσας, live
concealed. But having passed a year only in the parts of Lydia, he returned
to Athens, under the providential protection of the presiding goddess of
philosophy. And thus fortitude was perfected in our philosopher, first by
nature, then by custom, and afterwards by science and the consideration of
causes. Besides this, he exhibited in another manner his politic habit
practically, by composing letters for noblemen; and by this means procuring
good to entire cities. But of this I have a sufficient testimony from those on
whom they were bestowed, as well Athenians as Argives, and others of
different nations.

But he likewise much promoted and increased literary studies,
demanding of princes rewards for the preceptors, according to their several
deserts. Nor did he undertake this rashly, nor with any interested views, but
he compelled them (as he considered it a matter of great moment) to be
diligent in their profession, interrogating and discoursing with them
respecting every particular: for he was a judge sufficiently instructed in the
employments of them all. And if he ever found any one negligent in his
profession, he sharply reproved him; so that he appeared very vehement and
ambitious, because he was both willing and able to give a just determination
on every subject: and he was indeed a lover of glory. But this was not a
fault in him, as in most, because it alone regarded virtue and goodness. And,
perhaps, without an energy of this kind, nothing great and excellent would
ever subsist in the human mind. But he was in this respect vehement: this I
will not deny. Yet, at the same time, he was gentle; for he was easily
pleased, and demonstrated in a moment that his anger was as pliable as
wax. For, almost at the same time, he was (as I may say) wholly transported
in reprehension, and with a desire of becoming subservient to their interest,
and that he might intercede with princes in their names; being moved with a
certain natural conjunction of soul, and, as it were, sympathy of grief.
And here, I very opportunely recollect a peculiar example of his natural
sympathy of soul with others: nor do I think the like was ever related of any
other man. For, notwithstanding he was unmarried, and had no children
(because he was not desirous of such connections, but remained free from
them all, though many noble and rich alliances were offered him), yet his
care of all his familiars and friends, and of their wives and children, was as
great as if he had been some common father, and the cause of their birth; for
he bestowed a singular attention on the life of each. And whenever any one
of them was detained by any disease, he first earnestly supplicated the gods
on their behalf with sacrifices and hymns; afterwards he gave a prompt
attendance on the sick person himself, convened the physicians, and urged
them to make an immediate trial, if they knew of any thing in their art
advantageous to the condition of the diseased; and sometimes he produced
some singular advice himself, among the physicians; and thus delivered
many from imminent dangers. And the greatness of this blessed man’s
humanity towards his servants, may be understood by those who desire it,
from his will. But of all his familiars, he loved Archiadas and his kindred
the most; because, in the first place, their succession was derived from the

genus of Plutarch the philosopher; and afterwards on account of that[57]
Pythagoric friendship which he maintained with Archiadas, as he was both
the companion and preceptor of his studies. And this other kind of
friendship, differing from the two already mentioned, appears to have been
the most firm and excellent, among these illustrious men. For Archiadas
desired nothing, which was not also the wish of Proclus; and on the
contrary, the desires of Proclus were the constant wishes of Archiadas.
But having now brought the political virtues, which are inferior to the
true ones, to an end, and terminating them in[58] friendship, as their proper
bound, we shall now pass to the Cathartic differing from the politic virtues.
Indeed, the employment of these last consists in purifying the soul, that so
being liberated from the body as much as they are able to effect, it may
regard human concerns, and possess a certain similitude with divinity;
which is the soul’s best and most exalted end. Yet they do not all liberate
after the same manner, but some more, and others less. Since there are
certain political purgations which adorn their possessors, even while
connected with body, and reduce them to a better condition; bringing under
the dominion of reason, anger and desire, and entirely destroying passion
and every false opinion: but the Cathartic virtues, which are superior to
these, separate entirely from this truly leaden weight of body, and procure
an easy flight from mundane concerns. And in these, indeed, our
philosopher was studiously employed during the whole of his life, which
was devoted to philosophy; since he both taught by his discourses what they
were, and after what manner they were preparatory to felicity, and in a
particular manner conformed his life to their institutions; performing every
thing which could contribute to the separation of his soul, using both night
and day prayers, lustrations, and other purifications, as well according to
the Orphic as the Chaldaic institutions: and every month he descended, with
great diligence, to the sea; and this sometimes twice or thrice. But he was
exercised in these, not only in the vigour of his age, but also towards the
close of his life; and these customs he observed perpetually, as if they were
certain invariable statutes.
But he used meat and drink, and other necessary pleasures, only so far
as was necessary to avoid the molestations of disease; for he was in these
by much the most frugal, and particularly loved abstinence from animal
food. And if at any time he was invited to eat it more vehemently, he was so

cautious in its use, that he ate it merely after the manner of a taster. But he
purified himself every month by the sacred rites, in honour of the mother of
the gods, celebrated by the Romans, and prior to them by the Phrygians: he
likewise more diligently observed the unfortunate days of the Egyptians
than they themselves; and, besides this, fasted, on certain days in a peculiar
manner, on account of the lunar appearances[59]. He likewise instituted a
fast on the last day of the month, not having supped the day before. But in
what a splendid manner, and with what piety, he celebrated the new moon,
and properly observed, with sacrifices, the more illustrious feasts of almost
all nations, according to the manner of each country; and how from these he
did not, according to the custom of many, take occasion of becoming idle
and intemperate, but employed himself in continual prayers, hymns, and the
like, his hymns sufficiently evince, which not only celebrate the divinities
of the Greeks, but likewise Marna Gazæus, Esculapius Leonteuchus,
Ascalonites, and Theandrites, another god much venerated by the Arabians;
together with Isis, worshipped by the Philians; and lastly, all the rest which
were the subjects of his devotion. For this sentiment was very familiar to
this most religious man, that it was proper a philosopher should not be
careful in the observance of the rites and institutions of one particular city,
nor of certain nations only, but that he should be the general priest of the
universe. And thus was he pure and holy, so far as pertains to the virtue of
temperance.
But he declined, as much as possible, pain: and if it ever happened to
him, he bore it with gentleness, and diminished it with this view, that his
best part might not at the same time be affected with its molestations. And
the fortitude of his soul in this respect, was sufficiently evinced in his last
illness; for when, at that time, he was oppressed and tormented with the
most excruciating pains, he endeavoured to the utmost to mitigate and expel
their afflictive invasions. Hence, on such occasions, he often commanded us
to repeat certain hymns, which when repeated procured him a remission and
cessation of pain. And what is more wonderful, he remembered what he
heard of these, though forgetful of almost all human concerns, from the
dissolution of his corporeal part continually increasing. For when we began
to repeat, he supplied what was unfinished of the hymns, together with
many of the Orphic verses; for it was these we were then reciting. Nor was
he only thus constant in enduring corporeal evils, but much more so in
external unfortunate events, and such as appeared to happen contrary to

expectation. So that he would say, concerning particulars of this kind, So it
is, such things are usual; which seemed to me, or rather at that time
appeared to be, worthy of remembrance, and an evident argument of our
philosopher’s magnanimity. But besides this, he restrained anger as much as
possible, so that it might either remain free from all excitation, or that at
least reason might not consent to its indulgence, but the irrational faculty
alone, contrary to his will, might be moderately and lightly excited. And
with respect to venereal concerns, he used them in the natural way, but so as
that he might not proceed beyond a very moderate and light phantasy, in
their indulgence.
And thus the soul of this blessed man, having collected itself from all
parts, and retiring into the depths of its essence, departed after a manner
from body; while it yet appeared to be contained in its dark receptacle. For
he possessed a prudence, not like that of a civil nature, which is conversant
in the administration of fluctuating particulars, but prudence itself, by itself
sincere, which is engaged in contemplating, and converting itself into itself,
without any longer consenting to a corporeal nature. He likewise possessed
a temperance free from evil; and which is not even moderately influenced
by perturbations, but is abstracted from all affections. And lastly, he
acquired a fortitude, which does not fear a departure from body. But reason
and intellect having obtained in him a perfect dominion, and the inferior
powers of his soul no longer opposing themselves to purifying justice, his
whole life was adorned with the divine irradiations of genuine virtue.
Our philosopher, therefore, having most happily absolved this form of
virtues, advancing now, as it were, by the highest and most mystical step, he
ascended to the greatest and most consummate or telestic virtues;
employing for this purpose, the felicity of his nature, and a sciential
institution. Hence, being now purified, and the victor of his nativity, and
despising the vain Thyrsus-bearers, and boasters of wisdom, he happily
penetrated into her profound recesses, and enjoyed the contemplation of the
truly blessed spectacles she contains. No longer requiring prolix
dissertations, or demonstrations, for the purpose of collecting the science of
these, but, with a simple vision and energy of intellect, beholding the
exemplar of the divine mind, he obtained a virtue which cannot with
sufficient propriety be called prudence, but is more properly denominated
wisdom, or something, if possible, still more venerable and divine. But the

philosopher energizing, according to this virtue, easily comprehended all
the theology of the Greeks and Barbarians, and whatever is shadowed over
by the figments of fables, and placed it in a clear light, for the use of those
who are willing and able to pursue its latent signification. But having
interpreted divinely every thing of this kind, and shewing the symphony
between them all; at the same time, investigating all the writings of the
ancients, whatever he found in them of genuine wisdom, and approved by
general consent, this he judiciously applied to use; but if he found any thing
of a different and dissonant nature, this he entirely rejected, as vicious and
false. And whatever he met with contrary to wisdom, though endued with a
friendly appearance, this he vigorously subverted by a diligent examination.
Nor did he employ less force and perspicuity in his association with other
men. For he was a man laborious to a miracle; as he often, in one day
absolved five, and sometimes more lectures; and writ besides, many verses,
often to the number of seven hundred. Besides this, he went to other
philosophers, and frequented their company; and again celebrated with
them an evening association, ceasing from the labour of writing. And all
these employments he executed in such a manner, as not to neglect his
nocturnal and vigilant piety to the gods, and assiduously supplicating the
sun when rising, when at his meridian, and when he sets.
But he was the parent of many dogmata, which were never known
before, both in physics, and in intellectual and more divine concerns. For he
first taught, that there was a kind of souls[60], endued with the power of
contemplating many forms at once, which he placed, not without great
probability, between intellect, collectively, and as it were with one intuition
comprehending every thing, and souls, which are alone able to direct their
vision to one particular form. And those who are willing to peruse his works
will meet with a great variety of dogmata, peculiar to him alone; the
relation of which I shall omit, lest I should give a too great extent to my
discourse. But he who evolves his writings, will easily perceive that all we
have above related of him is most true, and much more if he happens to
have known him, to have seen his face, and to have heard him interpreting
in the most excellent dissertations, and delivering the Platonic and Socratic
dogmata in his yearly schools. Nor did he seem destitute of divine
inspiration; for he produced from his wise mouth, words similar to the most
white and thick falling snow[61]; so that his eyes emitted a bright radiance,

and the rest of his countenance was resplendent with a divine light. Hence,
when on a certain time, one Rufinus, a man of a great name in the republic,
who was studious of truth, and in other respects worthy of veneration, came
to him when he was teaching and interpreting, he perceived that the head of
Proclus was surrounded with a light; and when the philosopher had finished
his interpretation, Rufinus rising, adored him; and offered to give a public
testimony, by oath, of the divine vision which he had observed. And much
gold was offered to him, by the same Rufinus, on his return from Asia,
having escaped the danger of the war. But Proclus likewise rejected this
gift, and was by no means willing to receive it.
But that we may return to our first design, having now discoursed
concerning the contemplative wisdom of the philosopher, though in a
manner but little suited to its dignity, it remains that we now speak of the
justice pertaining to this kind of virtues. For this, unlike that of which we
spoke before, is not conversant in distribution, or proportion; but must be
equally removed from the kind of self-energizing justice, by which all
things are alone directed to the rational soul. For to that, concerning which
we now treat, it is alone proper to refer every energy to intellect and the
deity, which our philosopher performed in the most exalted manner. For he
scarcely rested from his diurnal labours, or refreshed his body with sleep,
and perhaps even then was not free from meditation and contemplation.
This is certain, that having very speedily roused himself from sleep, as from
a certain torpor of the soul, he aspired after the morning, the time of prayer;
and lest the greater part of the night, should glide from him without
advantage, as he was lying alone in his bed, he either composed hymns, or
examined and fortified those dogmata which afterwards, in the day time, he
committed to writing.
After a similar manner he pursued that temperance which has an affinity
with these virtues, and which consists in a conversion of the soul to
intellect, so as not to suffer itself to be touched, nor moved with any other
concerns. Lastly, he joined fortitude in alliance with these, by a certain
perfect method, zealously aspiring after that liberty which is ignorant of all
passion, and which he perceived was natural to the divine object of his
contemplation. And thus, through the whole of his conduct, he did not lead
the life of a man merely good, to which, as Plotinus says, the political
virtues may lead, but leaving this far behind him, he endeavoured to change

it for one much more perfect and divine, the life of the gods themselves;
since, to become similar to these, and not to virtuous men, was the great
object of his contention.
And thus he had rendered virtues of this kind familiar to himself, while
he frequented the philosopher Syrianus, and evolved and studied the
commentaries of the ancients. But he received from the mouth of his
preceptor certain small seeds, as it were, of the Orphic and Chaldaic
theology; because he was prevented from hearing the complete
interpretation of his master on the Orphic verses. For Syrianus left to the
choice of Proclus and one Domninus[62], a philosopher of the Syrian nation,
and who afterwards succeeded Syrianus, the exposition of the Orphic
writings, or the oracles. But they were by no means unanimous in their
choice; for Domninus preferred the interpretation of the Orphic verses, and
Proclus that of the oracles. But our philosopher did not perfect his
undertaking, because the death of the great Syrianus happened not long
after. Having therefore, as I have said, received the outlines from the mouth
of his master, he applied himself with the greatest diligence to the written
commentaries of Syrianus upon Orpheus; and being assiduously nourished
with the copious lucubrations of Porphyry and Jamblichus on the oracles,
and similar writings of the Chaldeans, he arrived, as much as is possible to
man, to the top of those highest virtues, which the divine Jamblichus was
accustomed to call after a truly divine manner, theurgic. He laboured
therefore, not without exquisite judgment, in collecting the expositions of
philosophers prior to his time; and contracted into one, other Chaldaic
hypotheses, and the most excellent of the commentaries on the divine
oracles, completing this great work in the space of five years; concerning
which this divine vision appeared to him in his sleep. For he saw the great
Plutarch approach to him, affirming that he should live so many years as he
had composed tetrads or quaternions on the oracles. Afterwards, having
collected the number of these, he found they amounted to seventy. But that
this was a divine dream, was sufficiently evinced by the last part of his life;
for though he lived seventy-five years, as we have mentioned above, yet he
had not the perfect use of his powers, in the five last. But his body, though
naturally of the best constitution, being debilitated by that hard and scarcely
tolerable kind of diet which he used, and by so many frequent labours and
fastings, began to languish exactly at his seventieth year, so that he then
became much more remiss in all his energies than before. Yet, even at this

period, and thus affected, he composed orations and hymns: he also writ
some things, and conversed with his friends; but his ancient vigour was
wanting in each. Hence the memory of the dream excited his wonder, and
he every where said, that he had only lived seventy years. But while he
laboured under this infirmity of body, a certain youth, named Hegias,
rendered him more alert in the business of interpretation. This young man,
who already exhibited from his tender years such egregious signs of all the
virtues of his ancestors, was one of that golden chain of philosophers, who
had formerly appeared to men; and adhered most diligently to Proclus
delivering the Platonic and other theologies. But Proclus, at this advanced
period, was not moderately rejoiced in communicating with the young man,
his own writings, after he understood that he made cubital advances in
every kind of disciplines. And thus we have briefly discoursed concerning
the apparatus of the philosopher in the Chaldean oracles.
But I, on a certain time, having read with him the Orphic verses, and
heard, among his interpretations, not only the recondite theology which is to
be found in Jamblichus and Syrianus, but also, among many other divine
men, I requested the philosopher that he would not leave these divine verses
also without his explanation: but his answer was, That he had often thought
of writing commentaries on Orpheus, but that he had been strongly
prohibited in more than one dream. For Syrianus appearing to him in his
sleep, had deterred him with threats from the design. Having therefore
employed other machines, I intreated that at least he would mark what he
principally approved of in the books of his master; which when this best of
men had performed, in consequence of my persuasions, and had noted some
things in the front of each of Syrianus’s commentaries, we obtained a
collection of all these, and by this means scholia, and commentaries of no
small bulk; though to accomplish this on the whole of that divine poetry,
and on all the Orphic rhapsiodies, was not the intention of Proclus.
But since, as we have said, his incredible study of these concerns,
procured him a greater and more perfect degree of theurgic virtue, hence he
no longer remained in the contemplative order, nor contented with either of
the twofold properties in divine concerns, exercised intellect and thought
alone on the more excellent objects of speculation: but he was careful to
obtain another kind of virtues more divine than the former, and separated
from the politic mode; for he used the Chaldean assemblies and

conferences, and their divine and ineffable concealments. And having
comprehended these, he learned the manner of pronouncing and expressing
them, with their remaining use, from Asclepigenia, the daughter of Plutarch:
for she alone, at that time, preserved the knowledge of the great Orgies, and
of the whole theurgic discipline, delivered to her by her father, who
received it from Nestor. Besides this, our philosopher also being purified in
an orderly manner in the Chaldean lustrations, was an inspector of the lucid
hecatic phasmai (or visions) of which he himself makes mention in one of
his commentaries. But by moving a certain hecatic sphærula[63], he very
opportunely brought down showers of rain, and freed Athens from an
unseasonable heat. Besides this, by certain phylacteria, or charms, he stopt
an earthquake, and had thoroughly tried the energies of a divining tripod,
having learned from certain verses concerning its defect. For when he was
in his fortieth year, he thought in a dream, that he repeated to himself, the
following verses:
High above æther there with radiance bright,
A pure immortal splendor wings its flight;
Whose beams divine with vivid force aspire,
And leap resounding from a fount of fire.

And in the beginning of his forty-second year, he appeared to himself to
pronounce these verses, with a loud voice:
Lo! on my soul a sacred fire descends,
Whose vivid pow’r the intellect extends;
From whence far-beaming through dull bodies night,
It soars to æther deck’d with starry light;
And with soft murmurs through the azure round,
The lucid regions of the gods resound.

Besides, he clearly perceived that he belonged to the Mercurial series;
and was persuaded from a dream that he possessed the soul of[64]
Nicomachus the Pythagorean.
Indeed, if I were willing to be prolix, I could relate many theurgical
operations of this blessed man; from the infinite number of which, I will
only relate one in this place of a very wonderful nature. Asclepigenia, the
daughter of Archiades and Plutarch, but the wife of Theagenes, from whom
I have received many benefits, while she was yet a girl, and was nourished
by her parents, fell into a disease, grievous, and incurable by the physicians.

And Archiades, who had reposed in her alone the hope of his race, was
deeply afflicted with her condition. But the physicians despairing of her
recovery, the father fled to his last anchor, Proclus the philosopher, which
he was accustomed to do in affairs of the greatest moment, considering him
as his best preserver, and earnestly intreated him to intercede by his prayers
with the god for his daughter, whose condition was well known to our
philosopher. Proclus therefore, taking with him Pericles[65] the Lydian, a
man well deserving the name of a philosopher, went to the temple of
Esculapius[66], that he might pray to the god for the health of the sick virgin:
for the city at that time happily possessed this divinity, and rejoiced in the
temple of the saviour god, which was as yet free from the destruction of the
Christians. As soon, therefore, as Proclus had prayed after the ancient
manner, the girl immediately perceived a great change and alleviation of her
disease; for the preserving god had easily restored her to health. The sacred
rites being finished, Proclus went to Asclepigenia, and found her delivered
from the molestations of disease, and in a healthy condition. But this affair,
with many others, was performed privately, so that no traces of it remained
to succeeding investigators; and the house in which he resided greatly
assisted him in this design. For besides his other fortunes, he enjoyed a
most convenient dwelling, which his father Syrianus and his grand-father
Plutarch (for so he did not hesitate to call him) had once inhabited; and this
was situated near the temple of Esculapius, together with that of the
celebrated Sophocles, and of Bacchus, which is next to the theatre, and
looks towards, or is at least seen from, the lofty towers of Minerva.
But how much Proclus was loved by the philosophic goddess is
abundantly evinced by his philosophic life, which he chose through her
persuasions, and that with the great success we have hitherto described. But
she clearly demonstrated her affection to Proclus, by the following
circumstance. When her image, which had been so long dedicated in the
parthenon, or temple, was taken away by those[67] who, without any
hesitation, moved out of their places things the most holy, and which ought
to be immoveable, there appeared to the philosopher in a dream, a woman
of a graceful form, who admonished him to build a temple with great
expedition, for, says she, it pleases Minerva, the presiding deity of
philosophy, to dwell with you. And how familiar he was with Esculapius,
besides what we have mentioned above, may be evinced from his

perceiving the presence of the same god in his last illness. For while he was
between sleeping and waking, he perceived a dragon creeping on his head,
which vision occasioned a remission of his disease, and a mitigation of his
pain; so that it is probable he would have been restored to perfect health, if
his desire and vehement expectation of death had not prevented his
recovery, or his no longer bestowing a diligent attention on his body.
But he also related the following circumstances, (worthy of being
remembered) but not without tears, through the sympathy of his mind. For,
when a young man, he was afraid lest he should be infested with the gout,
which was the disease of his father, and which loves to descend from
parents to their children. Nor was he afraid, as it seems, without reason; for
prior to that which we have already related concerning him, he was
tormented with pains of this kind, when another extraordinary circumstance
happened to this blessed man. In consequence of a certain person’s
prescription, whom he had consulted, he applied a plaster to his tormented
foot, which an unexpected bird flew away with as he was lying on his bed.
And this was certainly a divine and salutary symbol to Proclus, and might
have persuaded him not to be afraid of this calamity in future. But he, as I
have said, was notwithstanding afflicted with the dread of this disease; he
therefore supplicated the healing god concerning this, and intreated him to
afford him a more evident token of his will: and after this he saw in his
sleep (it is bold indeed to conceive such a circumstance in the mind, but we
must dare, nevertheless, nor dread to bring truth to light) Esculapius
approaching to him, and accurately contemplating his leg, nor disdaining,
through his great philanthropy to embrace his knee. Hence, in consequence
of this vision, Proclus was, through the whole of his life, free from the
apprehensions of this disease, and was never more tortured with this kind of
pains.
Again, the god who is worshipped by the Adrotteni, did not less
conspicuously evince his great familiarity with this friend of the gods; for,
more than once, the god benevolently presented himself to the sight of
Proclus, frequenting his temple. And when he was doubtful, and wished to
know what god or gods resided and were worshipped in that place, because
the inhabitants were of different opinions in this affair, some believing that
the temple of Esculapius was there, persuaded of this by various signs, as
voices are certainly said to be heard in that place, and a table is consecrated

to the gods, and oracles are given concerning the recovery of health, and
those who approach are miraculously delivered from the most grievous
dangers of life. Others, on the contrary, think that the Dioscuri reside in that
place, because there have been those who saw in the way tending towards
Adrotta, two young men of the most beautiful form, and riding with great
celerity on horseback; at the same time declaring, that they hastened to the
temple. They add besides, that the countenance of these was indeed human,
but that they immediately gave evidence of a more divine presence; for
when the men were arrived at the temple, the above mentioned youths
appeared to them, making no enquiry concerning the affair, and occupied in
the sacred concerns; but presently after, withdrew themselves from their
sight. As the philosopher, therefore, was in doubt concerning these
divinities, and did not discredit the relations, having requested the gods who
inhabited that place, that they would condescend to manifest who they
were; a god appeared to him in a dream, and clearly spoke to him as
follows: What! Hast thou not heard of Jamblichus teaching concerning, and
celebrating those two, Machaon and Podalirius? And besides this, the god
condescended to afford so great an instance of benevolence to this blessed
man, that he stood after the manner of those who bestow encomiums on
others in the theatres, and with a clear voice, and composed habit, extending
his right hand, did not hesitate to exclaim, (that I may relate the words of
the god,) Proclus the ornament of the city. And what could possibly be a
greater testimony of this blessed man’s friendship with divinity? But he,
indeed, in consequence of a most remarkable sympathy, by which he was
united with the gods, could never refrain from tears when he mentioned this
affair to us, and related the divine encomium of himself.
Indeed, if I were willing to pursue every particular in like manner, and
to relate his familiarity with the Hermetic Pan, together with the great
benevolence and manifold assistance which that divinity condescended to
afford Proclus at Athens, and of that perfectly singular kind of felicity
which he obtained from the mother of the gods, and in which he was
accustomed especially to triumph and rejoice; I should perhaps seem to
many readers, to be rash in my assertions, and to others, the author of things
perfectly incredible. For many and great were the daily instances of this
goddess’s benevolence towards him, in words and actions, which are both
innumerable and unheard of, and concerning which, I do not at present
retain an accurate remembrance. But if any one is desirous of knowing how

great he was in these, he must evolve his book concerning the mother of the
gods, from which he will understand, that it was not without divine
assistance he produced all the theology of that goddess into light; as
likewise, whatever is acted or affirmed concerning the same in the fables of
Athis, which he has explained after a philosophical manner, that vulgar ears
might afterwards cease to be disturbed, on perceiving the lamentations and
other obscure ceremonies with which her mysteries are celebrated.
After having, therefore, run through the theurgic virtue of Proclus, and
its energies, and the happy circumstances which befell him, respecting its
operations, and having shewn, that he did not less excel in every kind of
virtue, and that he was a man such as mortals have not beheld for a long
period of time, it remains that we now bring our discourse concerning him
to a conclusion: for we are not now at the beginning of our narration, nor
does the half of the whole remain to us, according to the proverb, but the
whole itself is now perfect and complete. Since having begun from the
felicity of the philosopher, and proceeded in its exemplification, our
discourse now returns to it again. For we have explained the great goods,
and providential exertions, which were granted to this most excellent man
from the gods, and have shewn their prompt attention to his prayers, the
divine visions which he enjoyed, and the help and solicitude which the gods
testified towards him. We have likewise explained his prosperous fate, and
propitious fortune, his country, parents, the good habit of his body, his
masters, friends, and other external advantages; each of which we have
shewn to have been far greater and more splendid in him than in other men;
and have likewise diligently enumerated such things as cannot be reckoned
amongst outward allurements, but entirely depended on his will, such as are
the upright and illustrious deeds of his soul, according to universal virtue.
And thus we have summarily demonstrated that his soul arrived in reality at
the summit of the most consummate virtue, and was happily established in a
perfect life, by human and divine goods of every kind.
But that the lovers of more elegant studies way be able to conjecture
from the position of the stars under which he was born, that the condition of
his life, was by no means among the last or middling classes, but ranked
among the highest and most happy orders, we have thought fit to expose in
this place, the following scheme of his nativity[68].
☉ 16° 26’ ♒

☽ 17 29 ♊
♄ 24 23 ♉
♃ 24 41 ♉
♂ 29 50 ♐
♀ — 23 ♓
☿ 4 42 ♒
Horos. 8 19 ♈
Mid.
4 42 ♑
Heaven.
☋ or the head of the dragon.
24 33 ♏
The new moon preceding his birth.
8 51 ♒
But Proclus departed from this corporeal life, in the one hundred and
twenty-fourth year after the government of Julian, on the seventeenth day of
the Attic Munichion, or the April of the Romans. Nicagoras, the junior,
being at that time the Athenian archon. The dead body was taken care of
according to the Athenian rites, as he himself while living had directed. And
if that diligent attention which is due to the deceased, was ever paid by any
one, it was certainly bestowed by this most blessed man. For he did not
neglect any particular of the accustomed ceremonies, but every year, on
certain days, visited the sepulchres of the Athenian heroes and philosophers,
and besides this sacrificed himself, and not through the medium of another,
to the manes of his friends and familiars. And having thus exhibited to each
a proper reverence and honour, proceeding into the Academy, he there, in a
certain place, pleased one by one the manes of his ancestors and kindred;
and shortly after, in another part of the same Academy, he supplicated in
common the souls of all philosophers. And these ceremonies being ended,
this most excellent man, having chosen a third place, performed sacred rites
to all the souls of deceased mankind. The dead body of Proclus, therefore,
being disposed of according to his appointment, as we have said, was
carried by his friends and buried in the eastern part of the suburbs[69], near
to Lycabetus, where also the body of his preceptor Syrianus reposes, who,
when he was alive, had requested this of our philosopher, and, in
consequence of this, had taken care to procure a twofold receptacle in his
sepulchre. But when this most holy man was consulted how he would wish

his funeral to be conducted, lest there should be any thing indecent, or
without a proper decorum, he desired flutes, with which he was threatened
in a dream, and nothing besides. An epigram, consisting of four verses, is
inscribed on his tomb, which he composed himself, as follows:
I Proclus, here the debt of nature paid,
(My country Lycia) in the dust am laid;
Great Syrianus form’d my early youth,
And left me his successor in the truth.
One common tomb, our earthly part contains,
One place our kindred souls,—th’ ætherial plains.

Nor were prognostics wanting in the year prior to his decease, such as
an eclipse[70] of the sun, so great that night was produced in the day-time:
for the darkness was so profound, that the stars became visible. And this
happened when the sun was in Capricorn, in the eastern centre. Besides, the
writers of Ephemeride observe that there will be another eclipse of the sun,
at the conclusion of the next year. But such like affections of the heavenly
bodies are said to portend events on the earth: to us indeed, the eclipse
perspicuously signified a privation and, as it were, defection of the
illustrious luminary of philosophy.
And thus much may suffice for the life of the philosopher. It now
remains for those who are willing to undertake the labour, to write
concerning his disciples and friends. For it appears that many resorted to
him from all parts; among which some were only his auditors, but others
firm adherents to his doctrine; and, on account of philosophy, his familiars.
I could likewise wish that some one, who is more laborious than myself,
would give a particular account of his writings. For I have alone delivered
these particulars of his life, that I might satisfy my conscience, and might
testify that I religiously reverence the soul of Proclus, and his good dæmon.
I shall, therefore, add nothing concerning the philosopher’s writings, except
that I have always heard him prefer his commentaries on the Timæus of
Plato beyond all his works. He likewise vehemently approved of his
commentaries on the Theætetus. And he used frequently to say, that if he
was endued with despotic power, he would alone preserve, of all the
writings of the ancients, the Oracles and the Timæus. For, said he, I would
abolish all the rest, and remove them from the present age, because it

happens that many are offended, who undertake to read them rashly, and
without proper institution.

Concerning the Published Writings of PROCLUS.

M arinus having neglected to give us an account of the writings of
Proclus, I thought it would not be unacceptable to the reader, to
present him with a catalogue of his works which are still extant; and the
most perfect relation I am able of such as are unfortunately lost. In the
execution of this design, I shall follow, for the most part, the accurate
Fabricius in the order, and critical account of his works; but shall not
hesitate to dissent from him in deciding on their philosophical merit: for it
is very rare that philology and philosophy are united in the same person,
and coalesce in amicable conjunction.

1. Four Hymns. In the preceding life we are informed, by Marinus, that
Proclus composed many hymns, in which the divinities, both of Greeks and
Barbarians, were celebrated; but unfortunately there are only four
preserved, the first of which is to the Sun, the second and third to Venus
(which may be found in the Florilegium of Stobus, p. 249. edit. Grotii,) and
the fourth to the Muses. They are collected by Fabricius, in the eighth
volume of his Bibliotheca Græca; and are most happy specimens of
philosophical poetry. Indeed, they bear most evident marks of a mind full of
divine light, and agitated with the fury of the Muses; and possess all that
elegance of composition for which the writings of Proclus are so
remarkable. So that it is very strange Gyraldus should ascribe them to a
Hierophant of Laodicæa, of that name; since, as Fabricius observes, Suidas
mentions other writings of this priest, but does not speak of him as the
author of any hymns. And if he had, these hymns breathe too much of the
spirit and manner of Proclus, to be the production of any other.
2. Two Books concerning the useful Parts of Learning, contained in
Grammar. Three books on this subject are mentioned by Suidas, and four by
Photius; but there are alone extant fragments of two, in the Bibliotheca of
Photius; which were published separate by Andrew Schottus, at the end of
the syntax of Apollonius Alexandrinus, Francof. 1590. Hanov. 1615. quarto.
It appears from these extracts, that Proclus, in this work, not only treats of

every kind of verse, but also of the more celebrated poets. And hence
Fabricius conjectures, that the short life of Homer, which Leo Allatius
published under the name of Proclus, was taken from the first book of this
Chrestomathia.
3. Eighteen Arguments against the Christians. The learned Cave, was
egregiously mistaken, in supposing that this work, as well as the answer to
it by Philoponus, is lost; not considering, probably, that these arguments in
defence of the world’s eternity are (as Fabricius well observes) directly
against one of the Christian dogmata, the creation of the world. These
arguments (except the first, which is lost,) are preserved in the answer of
Philoponus; the Greek edition of which was published at Venice, in 1535,
folio; and in Latin from the version of Joh. Mahatius, at Lyons, 1557. folio.
Simplicius, in his commentaries on the first book of Aristotle de Cœlo, and
elsewhere, opposes this work of Philoponus; at the same time, suppressing
his name. The arguments are, in my opinion, extremely subtle, and
unanswerable; and Philoponus, in his refutation, every where discovers, that
he has by no means fathomed the depth of our philosopher.
4. A Commentary on the Timæus of Plato, in five Books. A most
admirable work, composed in the flower of his age, when he was twenty-
eight years old, according to Marinus. This invaluable treasure, may be said
to contain all the wisdom of the highest antiquity: for Proclus every where
cites the most celebrated interpreters on the Timæus, such as Adrastus the
Peripatetic; Ælianus and Albinus, Platonists; Aristocles, Clearchus, Crantor,
Dercyllides; and especially Jamblichus; likewise Longinus, whom he often
refutes; Numenius the Pythagorean, and Origen, (different from
Adamantius,) together with Plotinus, Porphyry the Platonic, Ptolemy,
Severus, and Theodorus Asinæus; but he always prefers the explications of
his master Syrianus, before all the rest. Add too, that he every where
conciliates the Orphic writings and the Chaldean oracles with the theology
of Plato. And with respect to Orpheus in particular, we are indebted to these
sublime commentaries for the greatest part of the fragments of that divine
poet. He is perpetually on the wing, and rises gradually in elevated
conceptions, until his mind, like Homer’s discord, reaches the heavens. His
style is at the same time nervous and diffuse, accurate and elegant. We are
astonished with the magnificence of his metaphors, delighted with the
copious variety of his matter, and filled with a divine light, by the sacred

truths he unfolds. This great work, however, unfortunately, scarcely
explains a third part of the Timæus; from whence it is probable, as Fabricius
well observes, that several books have been lost through the injury of time.
It was published in Greek, at Basil, in the year 1556; and is full of mistakes,
as is generally the case with the Basil editions of books, so that a more
correct edition is greatly to be desired, though at present not much to be
expected. On concluding my account of this inestimable work, I find my
indignation roused by the following words of Dr. Cudworth, in his
Intellectual System, p. 306. “Proclus (says he) had some peculiar fancies
and whims of his own, and was indeed a confounder of the Platonic
theology, and a mingler of much unintelligible stuff with it.” I must confess,
(and I am neither afraid, nor ashamed of the declaration,) that I never found
any thing in Proclus, but what by patient thought, accompanied with a
sincere and vehement thirst after truth, I have been able to fathom. Had Dr.
Cudworth been endued with these requisites, he would doubtless have had
equal success; but without them, the sublimest truths will certainly appear
to be unintelligible stuff. Besides this consideration is not to be omitted, that
a modern priest makes a bad philosopher.
5. On Plato’s Republic. These commentaries, or rather fragments of
commentaries, are extant in Greek, at the end of Proclus on the Timæus.
Suidas mentions four books of Proclus on Plato’s politics; and some of
Proclus’s dissertations on these books, were found (according to Fabricius)
in the library of Lucas Holstenius. The chief design of this work seems to
be the unveiling the theological mysteries concealed under the fables of
Homer, and other divine poets; which Proclus has accomplished (in my
opinion) in a most wonderful manner. That Homer, indeed, every where
abounds with Egyptian learning, is obvious to every one; but few are
acquainted with the profound wisdom which his fables conceal. The latent
meaning of most of these is unfolded in the present invaluable, though
imperfect work; and he whose mind is sufficiently enlightened by the
ancient philosophy, to comprehend the beauty of these illustrations, will
receive an additional delight from the study of Homer, which it is
impossible to express. An epitome of this work was published in Latin, by
the learned Gesner, 1542. 8vo. under the following title: Apologiæ quædam
pro Homero, et Arte Poetica, Fabularumque aliquot Enarrationes ex
commentariis Procli Lycii Diadochi philosophi Platonici in libros Platonis
de Rep. in quibus plurimæ de Diis Fabulæ non juxta grammaticorum vulgus

historicè, physicè aut ethicè tractantur, sed Theologicis, ut Gentiles
loquntur, ex prima Philosophia rationibus explanantur.
6. On the first Alcibiades of Plato. Ficinus translated parts of this work
into Latin, and published them under the title of Procli de Anima ac
dæmone, de Sacrificio et Magia, Venice 1497. and 1516. fol. by Aldus; and
in a more simple form at Lyons. Fabricius informs us, that the manuscript
commentary of Proclus in Greek, but scarcely explaining the half part of the
Alcibiades, is to be found in various libraries of France, England, and Italy.
Also at Lyons, among the books of Isaac Vossius; and at Hamburgh in the
Johannean library. From the specimen given of this work by Ficinus, it
appears, like all Proclus’s philosophical writings, to be an invaluable
treasury of wisdom; and nothing certainly, reflects greater disgrace on a
nation than suffering such monuments of ancient learning and wisdom to lie
concealed in colleges, covered with dust, and never consulted.
7. Six Books on Plato’s Theology. A most divine work, in which the
philosopher collects into a system the theology dispersed in the writings of
Plato, and establishes it by invincible demonstrations. He deduces, in a
beautiful and connected series, all the divine orders, from the retreats of
ineffable unity; every where connects them by proper mediums, and, after
leading us through the long gradation of principles, brings us back again to
the original from whence they flowed, and to which they constantly tend.
The whole is uncommonly profound and abstruse; and it was not before the
third reading, that I could fathom the depth it contains. Fabricius observes,
“that it is a subtle and learned work, but from which, you will sooner learn
the opinion of Syrianus and Proclus, concerning the deity and divine
concerns, than that of Plato. He adds, that it is usual with the Platonists,
even from Plotinus, to unite to the doctrine of Plato, a thousand dogmata,
foreign from his philosophy, as if Plato, though he did not perceive after
this manner, ought certainly so to perceive.” When men mistake their
abilities, they always act absurdly, and often dangerously. As a laborious
and accurate critic on philological matters, Fabricius merits the highest
commendation such attainments can deserve; but when he leaves the beaten
road in which nature designed him to walk, and attempts the tractless paths
of philosophy, he perpetually stumbles, and often falls on the ground. The
wings of philology, like those of the swallow, were never destined for a
lofty flight:—it must be the eagle wing of genius, which can alone soar to

the sun of philosophy. The Greek and Latin edition of this valuable work,
was published at Hamburgh, by Æmilius Portus, 1618, folio.
8. Theological Institutions; or, as it may be called, the Elements of
Theology. This admirable work contains two hundred and ten propositions,
disposed in a scientific order, and supported by the firmest demonstrations.
They begin from super-essential unity, and proceed gradually through all
the beautiful and wonderful progressions of divine causes, ending in the
self-moving energies of soul. They possess all the accuracy of Euclid, and
all the subtilty and sublimity necessary to a knowledge of the most
profound theology; and may be considered as bearing the same relation to
the Pythagoric and Platonic wisdom, as Euclid’s Elements, to the most
abstruse geometry. Patricius, the first Latin translator of this divine work,
seems to have been very sensible of the truth of this observation: for he
every where carefully distinguishes the propositions from their
demonstrations; and adds the word corollary to such consequencies as merit
that appellation. His edition was published at Ferraria, 1583. quarto, under
the title of Theological Elements. The Greek and Latin edition, is subjoined
to Proclus’s six books on Plato’s Theology, Hamburgh 1618. folio.
9. Two Books concerning Motion. This useful work, collected, as
Fabricius observes, from the third and following books of Aristotle’s
physics, was published in Greek at Basil, 1531, and with the Latin version
of one Justus Velsius, a physician, Basil, 1545. octavo. It was likewise
translated by Patricius, and is annexed to his version of the Theological
Institutions.
10. An Hypotyposis, or Information concerning Astronomical
Hypotheses. This work, which Fabricius observes is a compendium of
Ptolemy’s Almagest, was published in Greek, at Basil, 1540. quarto; and in
Latin by George Valla, folio, 1541. A part of this work, which treats of the
use of the astrolabe, Fabricius informs us, is extant in manuscript, in various
libraries. The same accurate critic likewise observes, that a small treatise,
inscribed Uranodromus, is extant, under the name of Proclus, in some
libraries, as in that of Vindobona, and of Oxford, among the Barrocian
volumes. The comprehensive variety of Proclus’s genius equally demands
our admiration and applause.

11. A small Treatise concerning the Sphere, or Celestial Circles. This
little work is an accurate and elegant introduction to astronomy; and is
almost wholly taken from the Isagoge of Geminus Rhodius, on the
phænomena. The best editions are the Greek and Latin one published at
Paris in 1553, quarto; and that of Bainbridge, professor of astronomy at
Oxford, London 1620. quarto.
12. A Paraphrase in four Books, on the Quadripartite of Ptolemy. This
elegant work must, I should imagine, be an invaluable treasure to the lovers
of astrology. It was first published in Greek by Melancthon; and afterwards
in Greek and Latin by Leo Allatius, at Lyons Bat. 1654. octavo.
13. Four Books, on the first Book of Euclid’s Elements. For an account
of this work, see the introduction, and the following sheets, in which it
speaks for itself, in an English dress.
14. A Commentary on Hesiod’s Works and Days. This work contains a
valuable moral explanation of this great poet’s meaning; and Fabricius
justly observes, that he is often assaulted without occasion, by the petulant
jeers of that vain man Joh. Tzetzes. The best edition of this work is that of
Daniel Heinsius, Lugd. Bat. 1603. quarto.
15. Fabricius informs us, that in some manuscripts, as in the
Vindobonensian and Barrocian, a small treatise is usually ascribed to
Proclus, entitled Epistolic Characters; and is prefixed to the Epistles of
Phalaris and Brutis, and published under the name of Libanius, in Greek,
with the version of Casp. Stibilinus, Commelin. 1597, octavo. But it is
doubtful whether Proclus is the genuine author of this work: from the title, I
should suppose the contrary. And thus much for an account of those
writings of Proclus which have escaped the ravages of time, and have been
fortunately exposed to public inspection: it now remains that we relate such
inestimable works of this philosopher, as are yet preserved in shameful
concealment; or are utterly lost in the ruins of antiquity.

Concerning the Unpublished Writings of PROCLUS.

16. On the Alcibiades of Plato. See num. 6.
17. On Plato’s Politics. See num. 5.

18. On Plato’s Parmenides. A commentary, in seven books; the last of
which was not completed by Proclus, but by Damascius. From occasional
fragments, which have been published of this commentary, it appears to be
a most divine work; and indeed it cannot be otherwise, if we consider it as
the production of one of the greatest philosophers, on the most sublime and
profound of all Plato’s Dialogues. It is dedicated to Asclepiodotus, a
physician and philosopher, and is not only extant in Greek MS. in the
library of the German emperor, according to Lambecius, lib. vii. p. 41. but
also in Latin, from the unpublished version of one Antonius Hermannus
Gogava, as the same Lambecius informs us, p. 41. Four books of this work
are extant in Greek, in the Bodleian library at Oxford; and it is much to be
lamented that Thomson did not publish these, instead of his trifling edition
of the Parmenides. Fabricius likewise informs us, that Livius Galantes
mentions his having found six of these books in some of the Italian
libraries. They are also extant in the Medicæan library of the great Etruscan
commander.
19. On the Cratylus of Plato. We have already observed, in the
dissertation on the Orphic theology, p. 105. what a great treasure of ancient
mythology, must be contained in this work; but there is little hope of its
ever emerging from the obscurity of public libraries. It is extant in Greek,
not only in the Italian libraries, but also among the manuscript books of
Isaac Vossius.
20. Uranodromus. See above, num. 10.
21. Ten Doubts concerning Providence, in one Book. Philoponus
mentions this work, in his second book against Proclus on the eternity of
the world; and a Latin version of it is extant by one William de Morbeka, in
the Johannean library of Hamburgh. Extracts from this translation are
preserved by Fabricius, in his Greek Library; and they are in every respect
worthy of the genius of Proclus.
22. Concerning Providence and Fate, and that which is in our Power,
one Book. This work is dedicated to one Theodorus, a mechanist; and is
extant in the Latin translation of the same Morbeka, in the Greek Library of
Fabricius. The translation is for the most part barbarous, but is, however,
sufficiently legible to discover that it is a most valuable treatise, replete
with the usual elegance, subtilty, and sublimity of our philosopher.

23. Concerning the Hypostasis, or Subsistence of Evil. This book is
extant in Latin, in the Johannean library; and fragments of it are preserved
by Fabricius, in his Greek library. It is to be regretted, that Fabricius did not
preserve the whole in that excellent philological work.

Concerning the Lost Writings of PROCLUS.

24. On the Speech of Diotima, in Plato’s Banquet, concerning the
Subsistence of the Beautiful. Fabricius informs us, that this work is
distributed into many books; and Holstenius observes, that it is mentioned
in a certain scholium of the Medicean copy of Proclus’s commentaries on
Plato’s politics; but it is unfortunately no where extant.
25. On the Philebus of Plato; as may be inferred from the narration of
Damascius in Photius, p. 550; and Suidas in Marinus. For Damascius
relates, that Marinus having composed a commentary on this dialogue, on
shewing it to Isidorus for his approbation, that philosopher observed, that
those of his master were sufficient; which words Fabricius, with great
propriety, applies to the commentaries of Proclus on the Philebus.
26. On the Theætetus of Plato. This work is praised by Marinus, in the
last chapter of the preceding life; and no doubt with great propriety: for this
abstruse and sublime dialogue would naturally call forth all the divine fire
and elegance of our philosopher.
27. Commentaries on the Enneads of Plotinus. This work is mentioned
by Gyraldus, in his second dialogue on ancient poets; by Ficinus on
Plotinus; by Philip Labbeus, in his account of MS. books, p. 286; and in the
notes of Bullialdus to Theo of Smyrna, p. 224. But also in a certain note
prefixed to an ancient manuscript of Jamblichus, on the Egyptian mysteries,
to this effect: “The philosopher Proclus, commenting on the Enneads of the
great Plotinus, says, that it is the divine Jamblichus who answers the epistle
of Porphyry.” This note is in Greek, in the original, and is (in my opinion)
of itself sufficient to prove that such a work was once extant, though now
unfortunately lost. How much the want of these commentaries is to be
regretted, must be deeply felt by every lover of the Platonic philosophy. For
the unequalled profundity, and divine mysteries, contained in the writings
of Plotinus, could never be more happily illustrated than by the irradiations
of such a genius as Proclus.

28. Lectures on Aristotle’s Book Περὶ Ερμηνείας, or concerning
Interpretation. This work, it seems, was never published; but Ammonius
Hermeas, the disciple of Proclus, has inserted in his valuable commentary
on this book all that he could retain in his memory of Proclus’s lectures.
29. Hymns, not a few, see num. 1.
30. Chrestomathea. See num. 3.
31. On the Mother of the Gods, one book, mentioned by Marinus, in the
preceding Life.
32. On the Theology of Orpheus. This work is mentioned by Marinus, in
the preceding Life, and by Suidas; and its loss must be particularly regretted
by all the lovers of recondite theology.
33. Ten Books, on the Chaldean Oracles. This most valuable work is
mentioned by Marinus, in the preceding Life, and by Proclus himself on
Plato’s Politics, p. 359. It was doubtless not extant at the time when Psellus
and Pletho undertook the illustration of a few of these oracles: at least the
inconsiderable merit of their commentaries, strongly favours this
supposition.
34. A Commentary on the whole of Homer. Suidas. A specimen of the
great value of this work may be seen in our philosopher’s commentaries on
Plato’s republic. The works of Homer are not only the great fountain of
poetry, but likewise of philosophy; and are no less admirable for inspiring
the fury of the Muses than for containing the mysteries of the most
recondite theology.
35. Concerning the Gods, according to Homer. Had this work been
preserved, we should doubtless have been furnished with a defence of the
heathen religion, which would have silenced the ignorant clamours of its
opponents.
36. The Symphony or Concord of Orpheus, Pythagoras, and Plato.
Suidas. Proclus, in his published writings, is every where studious of
reconciling the doctrines of these great men, and is always successful in this
undertaking. Indeed, the same divine genius seems to have irradiated and
inspired these wonderful heroes, but in different ways: in Orpheus it was
accompanied with the fire of the Muses; in Pythagoras it shone through the
mysterious veil of numbers; and in Plato, combining the preceding modes,

it was seen enshrined in awful majesty of thought, clothed with the graces
of poetical diction, and resplendent with ineffable light.
37. Two Books on the Theurgic Discipline. Suidas. How much Proclus
excelled in this art, may be seen in the preceding Life.
38. Concerning the Oppositions of Aristotle to Plato’s Timæus. This
work is mentioned by Proclus in the 3d book of his commentary on the
Timæus, p. 226. and seems to have escaped the notice of the accurate
Fabricius. Aristotle may, no doubt, in many particulars be reconciled with
Plato; but it is also certain, that in some he is perfectly dissonant. And thus
much for the Life and Writings of Proclus.

C O M M E N TA R I E S
OF

P R O C L U S .

BOOK I.

C H A P. I.
On the Middle Nature of the Mathematical Essence.

I t is necessary that the mathematical essence should neither be separated
from the first nor last genera of things, nor from that which obtains a
simplicity of essence; but that it should obtain a middle situation between
substances destitute of parts, simple, incomposite and indivisible, and such
as are subject to partition, and are terminated in manifold compositions and
various divisions. For since that which subsists in its inherent reasons
remains perpetually the same, is firm and durable, and cannot be confuted,
it evidently declares it is superior to the forms existing in matter. But that
power of progression which apprehends, and which besides uses the
dimensions of subjects, and prepares different conclusions from different
principles, gives it an order inferior to that nature which is allotted an
indivisible essence, perfectly constituted in itself. Hence (as it appears to
me)[71] Plato also divides the knowledge of things which are, into first,
middle, and last substances. And to indivisible natures, indeed, he attributes
an intelligence, which, in a collective manner, and by a certain simple
power, divides the objects of intellectual perception; so that being divested
of matter, and endued with the greatest purity, it apprehends things
themselves, by a certain unifying perception, and excels the other kinds of
knowledge. But to divisible essences, and such as are allotted the lowest
nature, and to all sensible beings, he attributes opinion, which obtains an
obscure and imperfect truth. But to middle essences (and such are
mathematical forms), and to things inferior to an indivisible and superior to
a divisible nature, he attributes cogitation. For this, indeed, is inferior to
intellect, and the supreme science dialectic; but is more perfect than
opinion, and more certain and pure. For it advances by a discursive
procession, expands the indivisibility of intellect, and unfolds that which
was involved in the unity of intellectual apprehension: but it collects things
which are divided, and brings them back to mind. Hence, as knowledges
differ among themselves, so the objects of knowledge are distinguished by
nature. So that intelligible essences having an uniform subsistence,
evidently excel all others. But sensibles are entirely excelled by primary
essences: and mathematical natures, and whatever falls under cogitation, are

allotted a middle order: for they are excelled by the division of intelligibles;
but because destitute of matter, they are superior to sensible natures; and by
a certain simple power, they are excelled by the first; but by a certain reason
are more exalted than the last. Hence they possess notions of an intellectual
essence, which are more manifest than sensibles, but which are, at the same
time, only the images of an intellectual nature; and they imitate divisibly the
indivisible, and, in a multiform manner, the uniform exemplars of things.
And, that I may sum up the whole in a few words, they are placed in the
vestibules or entrances of primary forms, and disclose their indivisible and
prolific subsistence collected into one, but they do not yet excel the division
and composition of reasons, and an essence accommodated to the obscurity
of images; nor are they capable of passing beyond the various notions of the
soul, endued with a discursive power, and of adhering to intellections
perfectly simple, and purified from all material imperfection. After this
manner then, is the middle nature of mathematical genera and forms to be
understood; as filling up the medium between essences entirely indivisible,
and such as are divisible about matter.

C H A P. II.
Concerning the common Principles of Beings, and of the Mathematical
Essence,[72] bound and infinite.

B ut it is necessary that, considering the principles of the whole
mathematical essence, we should return to those general principles,
which pervade through and produce all things from themselves, I mean
bound and infinite. For from these two after that cause of one, which can
neither be explained, nor entirely comprehended, every other thing, as well
as the nature of the mathematical disciplines, is constituted. In the former,
indeed, producing all things collectively and separately; but in these
proceeding in a convenient measure, and receiving a progression in a
becoming order; and in some, subsisting among primary, but in others
among middle, and in others again among posterior natures. For intelligible
genera, by their simplicity of power, are the first participants of bound and
infinite: because, on account of their union and identity, and their firm and
stable existence, they are perfected by bound: but on account of their
division into multitude, their copious power of generation, and their divine
diversity and progression, they obtain the nature of infinite. But
mathematical genera originate, indeed, from bound and infinite, yet not
from primary, intelligible, and occult principles only; but also from those
principles which proceed from the first to a secondary order, and which are
sufficient to produce the middle ornaments of beings, and the variety which
is alternately found in their natures. Hence, in these also, the reasons and
proportions advance to infinity, but are restrained and confined by that
which is the cause of bound. For number rising from the retreats of unity,
receives an incessant increase, but that which is received as it stops in its
progression, is always finite. Magnitude also suffers an infinite division, yet
all the parts which are divided are bounded, and the particles of the whole
exist finite in energy. So that without the being of infinity, all magnitudes
would be commensurable, and no one would be found but what might either
be explained by words, or comprehended by reason (in which indeed
geometrical subjects appear to differ from such as are arithmetical;) and
numbers would be very little able to evince the prolific power of unity, and
all the multiplex and super-particular proportions which they contain. For

every number changes its proportion, looking back upon, and diligently
enquiring after unity, and a reason prior to itself. But bound being taken
away, the commensurability and communication of reasons, and one and
the same perpetual essence of forms, together with equality, and whatever
regards a better co-ordination, would never appear in mathematical
anticipations: nor would there be any science of these; nor any firm and
certain comprehensions. Hence then, as all other genera of beings require
these two principles, so likewise the mathematical essences. But such things
as are last in the order of beings, which subsist in matter, and are formed by
the plastic hand of nature, are manifestly seen to enjoy these two principles
essentially. Infinite as the subject seat of their forms; but bound as that
which invests them with reasons, figures, and forms. And hence it is
manifest that mathematical essences have the same pre-existent principles
with all the other genera of beings.

C H A P. III.
What the common Theorems are of the Mathematical Essences.

B ut as we have contemplated the common principles of things, which are
diffused through all the mathematical genera, after the same manner we
must consider those common and simple theorems, originating from one
science, which contains all mathematical knowledge in one. And we must
investigate how they are capable of according with all numbers, magnitudes
and motions. But of this kind are all considerations respecting proportions,
compositions, divisions, conversions, and alternate changes: also the
speculation of every kind of reasons, multiplex, super-particular, super-
partient, and the opposite to these: together with the common and universal
considerations respecting equal and unequal, not as conversant in figures, or
numbers, or motions, but so far as each of these possesses a common nature
essentially, and affords a more simple knowledge of itself. But beauty and
order are also common to all the mathematical disciplines, together with a
passage from things more known, to such as are sought for, and a transition
from these to those which are called resolutions and compositions. Besides,
a similitude and dissimilitude of reasons are by no means absent from the
mathematical genera: for we call some figures similar, and others
dissimilar; and the same with respect to numbers. And again, all the
considerations which regard powers, agree in like manner to all the
mathematical disciplines, as well the powers themselves, as things subject
to their dominion: which, indeed, Socrates, in the Republic, dedicates to the
Muses, speaking things arduous and sublime, because he had embraced
things common to all mathematical reasons, in terminated limits, and had
determined them in given numbers, in which the measures both of
abundance and sterility appear.

C H A P. IV.
How these Common Properties subsist, and by what Science they are
considered.

B ut it is requisite to believe, that these common properties do not
primarily subsist in many and divided forms, nor originate from things
many and the last: but we ought to place them as things preceding in a
certain simplicity and excellence. For the knowledge of these antecedes
many knowledges, and supplies them with principles; and the multitude of
sciences subsist about this, and are referred to it as their source. Thus the
geometrician affirms, that when four magnitudes are proportional, they
shall be alternately proportional; and he demonstrates this from principles
peculiar to his science, and which the arithmetician never uses. In like
manner, the arithmetician affirms, that when four numbers are proportional,
they shall be so alternately: and this he evinces from the proper principles
of his science. For who is he that knows alternate ratio considered by itself,
whether it subsists in magnitudes or in numbers? And the division of
composite magnitudes or numbers, and in like manner, the composition of
such as are divided? For surely it cannot be said that there are sciences and
cognitions of things divisible: but that we have no science of things
destitute of matter, and which are assigned a more intellectual
contemplation; for the knowledge of these is by a much greater priority
science, and from these the common reasons of many sciences are derived.
And there is a gradual ascent in cognitions from things more particular to
more universal, till we revert to the science of that which is, considered as it
is, abstracted from all secondary properties. For this sublime science does
not think it suitable to its dignity, to contemplate the common properties
which are essentially inherent in numbers, and are common to all quantities;
but it contemplates the one, and firm essence of all the things which are.
Hence, it is the most capacious of all sciences, and from this all the rest
assume their own peculiar principles. For the superior sciences always
afford the first suppositions of demonstrations to such as are subordinate.
But that which is the most perfect of all the sciences, distributes from itself
principles to all the rest, to some indeed, such as are more universal, but to
others, such as are more particular. Hence, Socrates, in the Theætetus,

mingling the jocose with the serious, compares the sciences which reside in
us to doves: but he says they fly away, some in flocks, but others separate
from one another. For such, indeed, as are more common and more
capacious, comprehend in themselves many such as are more particular: but
such as being distributed into forms, touch things subject to knowledge, are
distant from one another, and can by no means be copulated together, since
they are excited by different primary principles. One science, therefore,
precedes all sciences and disciplines, since it knows the common properties
which pervade through all the genera of beings, and supplies principles to
all the mathematical sciences. And thus far our doctrine concerning
dialectic[73] is terminated.

C H A P. V.
What the Instrument is, which judges of the Mathematical Genera and
Species.

L et us now consider what that instrument is[74], adapted to the judgment
of mathematical concerns; and let us appoint Plato as our guide in this
affair, who, in his Republic, divides cognitions separately from such things
as are the objects of knowledge; and distributes cognitions in conjunction
with things subject to knowledge. For of the things which are, some he
ranks among intelligibles, and others among sensibles. And of intelligibles,
some are again pure intelligibles, and others subject to cogitation. And of
sensibles, some are purely sensibles, but others conjectural. To intelligibles,
indeed, which are the first of the four genera, he assigns an intelligible
knowledge; but to those which are subject to cogitation, he attributes
thought: to sensibles, faith; but to conjecturals, a conjectural or assimilatory
power. And he shews, that the assimilatory power has the same proportion
to sense as thought to intelligence. For the conjectural power knows the
spectres of sensible forms, while they are beheld in water and other bodies,
which perspicuously represent their image: since, by their situation in water,
they are after a manner, allotted the last seat in the gradations of forms, and
truly become the resemblances of resemblances. In like manner, thought
beholds the images of intelligibles in a degraded state, fallen from primary
simple and indivisible forms, into multitude and division. Hence, a
knowledge of this kind, depends on other more ancient hypotheses; but
intelligence arrives at that principle which is no longer supposed. If then,
mathematical concerns are neither allotted an essence separate from all
division and variety, nor that nature which is apprehended by sense, which
is obnoxious to many mutations, and is in every proportion divisible, it
must be manifest to every one, that they are essentially subject to
cogitation: but cogitation presides over these as an instrument adapted to
judgment, in the same manner as sense to sensibles, and the assimilatory
power to conjecturals. From whence, indeed, Socrates determines that the
knowledge of these is more obscure than the first science, but is more
evident than the impulsive apprehension of opinion. For in this the
mathematical sciences are inferior to intelligence, because they contemplate

that which is evolved, and is endued with a power of progression; but they
are superior to opinion, by that stability of reasons which they contain, and
which cannot be confuted. And they originate from supposition, through a
diminution of the first science; but they contain forms independent of
matter, from their possessing a knowledge more perfect than that of
sensibles. We have therefore determined an instrument adapted to the
judgment of all mathematical concerns, i. e. cogitation, according to the
mind of Plato; which places itself indeed above opinion, but is excelled by
intelligence.

C H A P. VI.
Concerning the Essence of Mathematical Genera and Species[75].

I t now remains, that we consider what subsistence or essence ought to be
assigned to mathematical genera and species? Whether we must deduce
their origin and subsistence from sensible objects, or from abstraction, or
from a collection of such things as are dispersed by parts into one common
definition; or must allow them an existence prior to that of sensibles, as
Plato affirms, and as the progression of universal being demonstrates? First
then, if we affirm that mathematical species are composed from sensibles;
whilst the soul from material triangles or circles, forms in herself the
trigonic, or circular species, by a kind of secondary generation; I would ask
from whence is derived the great certainty and accuracy of definitions? For
it must either proceed from sensibles, or from the soul herself. But from
sensibles is impossible, for these, in a continual flow of generation and
decay, do not for a moment retain an exact sameness of being; and
consequently fall far short of the exactness contained in the definitions
themselves. It must therefore proceed from the soul, which, by her
immaterial nature, procures perfection from the imperfect, accurate subtilty
from that which is neither accurate nor subtle, and rekindles the light of
ideas from the obscure and unreal objects of sense.
For where shall we find, amongst sensible objects, an indivisible nature,
such as that of a point, or a line without the dimension of breadth, or a
superficies without depth, or the ever constant proportion of sides, and
exact rectitude of angles? For my part, I cannot see where, since all
divisible natures are thus mixed and confused together, nothing sincere,
nothing free from its contrary, but things every where yielding to
separation, as well such as are removed by distance of place, as those which
are united together. How then shall we obtain this durable essence for these
immoveable natures from the ever fluctuating forms of sense? For whatever
derives its existence from moveable beings, must of necessity be mutable
and frail. And how shall we gain this perfect accuracy for the stable species,
from the inaccurate and imperfect? For whatever is the cause of a
conception, always immutable, is itself much more stable than its effect. We
must therefore admit the soul to be the generator of these mathematical

species and reasons. But if she contains them in herself, as first exemplars,
she gives them an essential being, so that the generations are nothing else
than propagations of species, which had a prior subsistence in herself: and
thus we shall speak agreeably to the sentiments of Plato, and discover the
true essence of mathematical entities. But if the soul, though she neither
possesses nor received the mathematical reasons prior to the energies of
sense, yet fabricates this admirable immaterial building, and generates this
fair series of speculations; how can she discern whether her productions are
stable and constant, or things which the winds may dissipate, and phantoms
rather than realities? What standard can she apply as the measure of their
truth? Or how, since she is destitute of their essence, can she generate such
a variety of reasons? For from such an hypothesis, we make their
subsistence fortuitous, not tending to any scientific bound. Mathematical
species are therefore the genuine offspring of the soul: nor does she derive
from sensible objects the definitions she frames, but rather the first are
propagated from the second; they are the energies of soul, which, as it were,
pregnant with forms, delivers her immaterial progeny into the dark and
fluctuating regions of matter, as evidences of the permanent duration of her
species.
Again, if we collect mathematical reasons from externals, why are not
demonstrations composed from sensibles, better than the demonstrations of
universal and simple species? For we say, in order to the investigation of
any thing sought, that the principles and propositions, should be allied to
the conclusions. If then, particulars are the causes of universals, and
sensibles the sources of reasoning, why does the boundary of demonstration
always refer to that which is more universal, and not to that which is partial
and particular? And how can we prove that the essence of intelligibles is
more allied to demonstration than the essence of sensibles? For thus they
speak[76]: his knowledge is not legitimate, who demonstrates that the
isosceles, the equilateral, or the scalene triangle, have angles equal to two
right; but he possesses science, properly so called, who demonstrates this of
every triangle simply, or of triangle itself. And again, that universals, for the
purpose of demonstration, are superior to particulars; that demonstrations
concern things more universal; but that the principles from which
demonstrations are composed, have a priority of existence, and a
precedency in nature to singulars, and are the causes of the propositions
they prove. It is very remote, therefore, from the nature of Apodictical

sciences, that from converse with things of posterior origin, and from the
dark perceptions of sense, they should falsely collect their indubitable
propositions. I add farther, that they who affirm this, make the soul of a
baser nature than the material species themselves. For if matter derives
from nature beings essential, and participating a high degree of entity and
evidence; but the soul, by a posterior energy, receives these from sensible
objects, and fashions in herself resemblances and images of posterior
origin, contemplating vile essences, and abstracting from matter, the forms
inseparable from its nature; do they not make the soul more obscure and
indigent than matter itself? For matter is the receptacle of forms
materialized, as the soul is of species immaterialized. But in this case,
matter would be the place of primary beings, and the soul of such as are
secondary and subordinate: matter and its forms obtaining the lead in being,
and existing as the sources of the subsistence of immaterial forms. Lastly,
the material forms would have an essential existence, the others only an
intentional denomination. How then can the soul, which is the first
participant of intellect, and an intellective essence, and which derives from
thence consummate knowledge, and a plenitude of life, become the
receptacle of the most obscure species, the lowest in the order of things, and
participating the most imperfect existence. But this opinion, which has been
sufficiently exploded by others, needs no farther confutation.
If then, mathematical species do not subsist by material abstraction, nor
by a collection of those common properties inherent in individuals; nor are
at all, in their origin, posterior to sensibles, nor derived in any manner from
them: it is necessary that the soul should either deduce them from herself,
or from intellect; or lastly, from herself and intellect united. But if from
herself alone, Whence do the images of intellectual species arise; whence
do they derive their middle nature, linking, as it were, the divisible and
indivisible essence together; if they do not participate the fullness of entity
from primary essences? Lastly, how, upon this hypothesis, are the first
exemplars, paradigms, or ideas, which subsist in intellect, the principles of
universals? But if they are derived from intellect alone into the soul, how
can the soul remain self-operative, and self-motive, if her inherent reasons
flow from an external source, and are regulated by its operations? And in
what respect does the soul differ from matter, which is all things in mere
dormant capacity, but generates nothing appertaining to material species? It
remains, therefore, that the soul deduces these species from herself, and

intellect; and that she is the absolute consummation of the forms which
originate from intellectual exemplars, but which are allotted from
themselves a transition to permanent being. The soul, therefore, is by no
means to be compared to a smooth tablet, void of all reasons; but she is an
ever-written tablet, herself inscribing the characters in herself, of which she
derives an eternal plenitude from intellect. For soul is a certain subordinate
intellect, revolving round an intellect prior to herself, formed to its image,
and participating its divine irradiations. If then, this superior intellect is all
things intellectually, soul will all things animally; if the first exists as the
exemplar, soul will be as its image; if as contracted and united in itself, soul
as divisible and expanded. And this is what Plato understood, when in his
Timæus, he composes the soul of the world from all things, dividing her
according to harmonical reasons, and analogies; assigning to her the first
principles effective of figures, I mean the right and circular line, and giving
an intellectual motion to her inherent circles. All mathematical species,
therefore, have a primary subsistence in the soul: so that, before sensible
numbers, there are to be found in her inmost recesses, self-moving
numbers; vital figures, prior to the apparent, ideal proportions of harmony
previous to concordant sounds; and invisible orbs, prior to the bodies which
revolve in a circle. So that soul is the prolific abundance of all these, and is
another ornament producing herself, and produced from a proper principle,
filling herself with life, and at the same time filled from the demiurgus of
the universe, is an incorporeal and indistant manner. When, therefore, she
produces and unfolds her latent reasons, she then detects every science and
virtue. The essence of soul then consists in these species, nor must we
suppose her inherent numbers to be a multitude of units, nor her archytipal
ideas of divisible forms to be corporeal: but we must conceive all these as
subsisting ever vitally, and intellectually, as the exemplars of apparent
numbers, figures, reasons and motions. And here we must follow the
doctrine of Timæus, who derives the origin, and consummates the fabric of
the soul, from mathematical forms, and reposes in her nature the causes of
every thing which exists. For the seven bounding terms[77], comprehending
the principles of all numbers, lines, planes and solids, pre-exist in soul
according to cause. And again, the principles of figures are placed in her
essence, according to a demiurgical power. And lastly, the first of all
motions, which embraces every other motion in its comprehensive ambit, is
co-existent with soul. For the principle of every thing which is moved is a

circle, and the circular motion. The mathematical reasons, therefore, which
fully consummate the soul, are essential, and self-moving: and the soul, by
her cogitative power, diffusing, propagating, and evolving these, from her
profound recesses, constitutes all the fair variety of mathematical sciences.
Nor will she ever cease to generate, and waken into energy, succeeding
species, while she divests her indivisible reasons of their intellectual
simplicity. For she previously received all things, after a primary manner;
and according to her infinite power, from pre-existent principles, deduces a
beautiful series of various speculations.

C H A P. VII.
What the Employments and Powers are of the Mathematical Science, and
how far they extend themselves in their Energies.

B ut, after contemplating the essence of mathematical forms, it is
necessary we should recur to that one master-science of these, which
we have shewn is prior to a multitude of others, and that we should
contemplate what its employment is, what are its powers, and how far it
advances in its energies. The employment, therefore, of the whole
mathematical science, possessing, as we have before said, the power of
cogitation, must not be placed so high as that of intelligence; which is
firmly seated in its own stable essence, is perfect, is contained by itself, and
in itself continually verges. Nor must it be situated so low as that of opinion
and sense, since these cognitions dwell upon external concerns, energize
upon them, and do not possess the causes of the objects of their knowledge.
But the mathematical science, receives its commencement, indeed,
extrinsically from[78] recollection, but ends in the most intimate reasons,
residing in the depths of the soul; and is excited, indeed, from things
posterior, but arrives by gradual advances at the principal essence of forms.
Nor is its energy immoveable, like that of intelligence, nor is it affected
with local motion and alteration, like sense, but it revolves with a vital
energy, and runs through the ornament of incorporeal reasons, sometimes
advancing from principles to such things as are perfected by principles, but
at other times yielding in a retrograde progression from conclusions to their
forming principles: and sometimes proceeding from things previously
known, to such as are the subject of investigation: but at other times, from
things placed in the question, to such as precede in cognition. Besides, it
does not excel all inquisition, as if it were perfect from itself, like intellect,
nor is it perfected from others, like sense, but it proceeds by enquiry to
invention, and ascends from the imperfect to perfection. But it likewise
possesses twofold powers, one kind of these deducing principles into
multitude, and generating the different paths of contemplation: but the other
endued with a power of collecting many transitions into proper
suppositions. For since it proposes to itself as principles, as well unity, and
multitude, as bound and infinite, and such things as are subject to its

comprehension, are allotted a middle order, between forms indivisible and
every way divisible; with great propriety (I think) the gnostic powers of the
whole science of these are essentially twofold. One species indeed, hastens
to union, and contracts the expansion of multitude: but the other possesses a
power of distinguishing things simple into such as are various, more
universals into more particulars, and reasons digested in their principle, into
things secondary and multifariously multiplied from these principles. For
rising higher from its commencement it penetrates even to such things as
are the perfections of sensible concerns, is joined with nature, and
demonstrates many things together with natural science. Since ascending
from inferiors, it accedes in a certain respect proximate to intellectual
knowledge, and touches the contemplation of things primary and divine.
And hence, in the limits which flow from its essence, it produces the whole
mechanic, optic, and catoptric speculation, together with many other
sciences which are inwoven and entangled with sensible concerns, and
which operate through their assistance. Besides, in its ascensions from
corporeal natures, it derives intelligences indivisible and destitute of matter:
and with these it perfects its divisible apprehensions, those cognitions
which subsist in progressions, and its own genera and forms: it likewise
indicates the truth respecting the gods themselves, and in its peculiar
treatises exhibits a contemplation of the things which are. And thus much
concerning the employment and powers of the Mathematical Science.

C H A P. VIII.
Concerning the Utility of the Mathematical Science.

B ut let us now consider the utility of this Science, which extends itself
from the most principal to the last cognitions. Timæus, therefore, calls
the knowledge of the mathematical disciplines the path of erudition,
because, indeed, it has the same proportion to universal science, and the
first philosophy, which learning has to virtue. For this last frames our soul
to a perfect life, by the possession of worthy manners; but the former
prepares our cogitation, and the divine eye of our soul to an elevation from
the obscurity of sensible information. Hence, Socrates in the Republic, says,
“That the eye of the soul, which is darkened and buried by other studies,
can by the mathematical disciplines alone be invigorated, and again excited
to the contemplation of that which is, and transferred from resemblances to
real beings, from an obscure light to that light which has the power of
intelligence, and from a cave, and those bonds which exist in it as the
authors of generation, and from material impediments be able to rise to an
incorporeal and indivisible essence. For the beauty and order of
mathematical reasons, and the firmness and stability of the contemplations
they afford, conjoins us with intelligible objects, and perfectly determines
us in their essences; which perpetually remain the same, ever shining with
divine beauty, and preserving a mutual order without end. But Socrates, in
the Phædrus, delivers to us three characters who are elevated from sense,
because they fill up and accomplish the primary life of the soul, i. e. the
philosopher, the lover, and the musician. But the beginning and path of
elevation to the lover, is a progression from apparent beauty, using as
excitations the middle forms of beautiful objects. But to the musician, who
is allotted the third seat, the way consists in a transition from sensible to
invisible harmonies, and to the reasons existing in these. So that to the one,
sight is the instrument of reminiscence, and to the other, hearing. But to him
who is by nature a philosopher, from whence and by what means is
reminiscence the prelude of intellectual knowledge, and an excitation to that
which truly is, and to truth itself? For this character also, on account of its
imperfection, requires a proper principle: for it is allotted a natural virtue,
an imperfect eye, and a degraded manner. It must therefore be excited from

itself; and he who is of such a nature, rejoices in that which is. But to the
philosopher, says Plotinus, the mathematical disciplines must be exhibited,
that they may accustom him to an incorporeal nature, and that afterwards
using these as figures, he may be led to dialectic reasons, and to the
contemplation of all the things which are. And thus it is manifest, from
hence, that the mathematics are of the greatest utility to philosophy. But it is
requisite that we should be more explicit, and mention the several
particulars to which they conduce, and evince that they prepare the
intellectual apprehensions of theology. For whatever to imperfect natures
appears difficult and arduous in obtaining the true knowledge of the gods,
the mathematical reasons render, by their images, credible, manifest, and
certain. Thus, in numbers, they indicate the significations of super-essential
properties, but they evince the powers of intellectual figures, in those
figures which fall under cogitation. Hence it is, that Plato, by mathematical
forms teaches us many and admirable sentences concerning the gods, and
the philosophy of the Pythagoreans, using these as veils, conceals from
vulgar inspection the discipline of divine sentences. For such is the whole
of the Sacred and Divine Discourse[79], that of Philolaus in his Bacchics,
and the universal method of the Pythagoric narration concerning the Gods.
But it especially refers to the contemplation of nature, since it discloses the
order of those reasons by which the universe is fabricated, and that
proportion which binds, as Timæus says, whatever the world contains, in
union and consent; besides, it conciliates in amity things mutually opposing
each other, and gives convenience and consent to things mutually
disagreeing, and exhibits to our view simple and primary elements, from
which the universe is composed, on every side comprehended by
commensurability and equality, because it receives convenient figures in its
proportions, and numbers proper to every production, and finds out their
revolutions and renovations, by which we are enabled to reason concerning
the best origin, and the contrary dissolution of particulars. In consequence
of this, as it appears to me, Timæus discloses the contemplation concerning
the nature of the universe, by mathematical names, adorns the origin of the
elements with numbers and figures, referring to these their powers,
passions, and energies; and esteeming as well the acuteness as the
obtuseness of angles, the levity of sides, or contrary powers, and their
multitude and paucity to be the cause of the all-various mutation of the
elements. But why may we not say, that it profits much, and in an admirable

manner, to that philosophy which is called Politic, as well by measuring the
times of actions as affording the various revolutions of the universe, and
numbers convenient to things rising into being; I mean the assimilating, and
authors of dissimilitude, the prolific too and the perfect, and the contraries
to these; together with orderly and elegant ministers of life, and inelegance;
and finally, such numbers as procure fertility and sterility. Which, indeed,
the speech of the Muses in the Republic[80] evinces, placing the universal
Geometric Number as the author of better and more debased generations,
and as the cause of the indissoluble perseverance of good manners, and of
the mutation of the best Republics into such as are remote from reason, and
are given to affections. For it is sufficiently evident, that it belongs to the
whole mathematical discipline to deliver the science of this number which
is called geometrical, and not to one particular science, such as arithmetic,
or geometry: since the reasons or proportions of abundance and sterility,
permeate through all the mathematical disciplines. Again, it is the means of
our institution in moral philosophy which it brings to its ultimate perfection,
and gives order and an elegant life to our manners. Besides this, it delivers
to us figures, and modulations and motions convenient to virtue, by which
the Athenian guest wishes those to be instituted and perfected, who are
destined to pursue moral virtue from their early youth. Add too, that it
places before our view the reasons of virtues, in one manner, indeed, in
numbers, in another in figures, but differently in musical symphonies; and
lastly, it indicates the excess and defect of vices, by which we are enabled to
moderate and adorn our manners. Hence it is, that Socrates, in the Gorgias,
accusing Calicles of an inordinate and intemperate life, says to him, “You
neglect geometry and geometric equality:” but, in the Republic, he finds out
the proportion of tyrannic pleasure to a royal interval, according to a plane
and solid generation. But we shall learn what great utility is derived to other
sciences and arts from the mathematical science, when we consider that it
adds order and perfection to contemplative arts; I mean rhetoric, and all
such as consist in discourse. But it proposes to the poetic arts, the reasons of
poems in the place of an example, because it presides over the measures
existing in these. But to the active arts it determines action and motion, by
its own abiding and immoveable forms. For all arts, as Socrates says, in the
Philebus, require arithmetic, mensuration, and statics, either in all, or in
some of their operations. But all these are contained in the discourses of the
mathematical science, and are terminated according to their diversity. For

from this science the divisions of numbers, and the variety of dimensions,
and the difference of weights are known. The utility, therefore, of the whole
mathematical science to philosophy itself, and to other sciences and arts,
may be from hence known to intelligent hearers.”

C H A P. IX.
A Solution of an Objection raised by some against the Utility of the
Mathematical Sciences.

B ut some, who are prone to contradiction through those who wish to
subvert geometry, endeavour to destroy the dignity of this science. One
part, indeed, depriving it of ornament and good, because it does not
discourse on these. But another part[81] affirming that sensible experiments
are more useful than the universal objects of its speculation; I mean, that
Geodesia (for instance,) or the mensuration of the earth, is preferable to
geometry, and vulgar arithmetic to that arithmetic which is conversant with
theorems alone: and that nautical astrology is more useful than that which
teaches universally, abstracted from any application to sensible concerns.
For we are not, say they, made rich by our knowledge of riches, but by
using them; nor are we happy by the merely understanding felicity, but by
living happily. Hence we must confess that those mathematical sciences,
which are conversant with cognition, do not profit human life, and confer to
action, but those only which are engaged in exercise. For those who are
ignorant of the reasons of things, but are exercised in particular and sensible
experiments, are in every respect more excellent, for the purposes of human
life, than those who are employed in contemplation alone. Against
objections then, of this kind, we shall reply, by shewing the beauty of the
mathematical disciplines from those arguments by which Aristotle
endeavours to persuade us. We must therefore confess that there are three
things which especially cause beauty, both in bodies and souls; I mean,
order, convenience, and determination. Since corporeal baseness, indeed,
arises from material inordination, deformity, and inconvenience, and from
the dominion of the indefinite in the composite body. But the baseness of
the soul originates from its irrationality, and inordinate motion, and from its
being in a state of discord with reason, and not receiving from thence its
proper limitation. Hence, beauty exists even in contraries, by means of
order, convenience and determination. But we may behold these in a more
eminent degree in the mathematical science; order, indeed, in the perpetual
exhibition of things posterior and more various, from such as are primary
and more simple; for things subsequent are always annexed to their

precedents, the latter ranking as principles, and the former as the first
suppositions of things consequent: but convenience is evinced in the mutual
consonance of things demonstrated, and in the relation of all of them to
intellect, since intellect is the common measure of all science, from which it
receives its principles, and to which it converts the learner: but
determination is perceived in its perpetually abiding and immoveable
reasons, for the objects of its knowledge are not, at times, subject to
variation, like those of opinion and sense, but present themselves for ever
the same, and are bounded by intellectual forms. If such then, are the
principal requisites of beauty, it is evident, that in these sciences that
illustrious ornament and gracefulness is found. For how is it possible this
should not be the case with a science receiving a supernal illumination from
intellect, to which it continually advances, hastening to transfer us from the
obscure light of sensible information? With respect to the second objection,
we think it proper to judge of its utility, without regarding the conveniencies
and necessities of human life. For otherwise, we must confess that
contemplative virtue is also useless, which separates itself from human
concerns, which it is very little desirous to look down upon and understand.
Indeed Socrates, in the Theætetus, affirming this concerning noblemen
endued with the prophetic power, says, “that it withdraws them from all
regard to human life, and raises their thoughts, properly liberated, from all
necessity and use, to the very summit of all true being.” The mathematical
science, therefore, must be considered as desirable for its own sake, and for
the contemplation it affords, and not on account of the utility it administers
to human concerns. But if it is necessary to refer the utility it produces to
something different from itself, it must be referred to intellectual
knowledge. For it leads us to this, and prepares the eye of the soul for the
knowledge of universals, removing and obliterating the impediments arising
from the senses, and from corporeal involution. As therefore we call the
whole of purgative virtue useful, or the contrary, not regarding the use of
the sensible life, but of that which is contemplative, so indeed it is requisite
to refer the end of mathematics to intellect, and universal wisdom. Hence its
energy is worthy our study, both on its own account, and on account of an
intellectual life. But it appears, as Aristotle[82] says, that this science is
desirable of itself to its votaries, because though no reward is proposed to
its enquirers, yet the mathematical contemplation receives, in a small time,
an abundant increase. Besides, this is farther evident from hence, that all

men are willingly employed in its pursuit, and wish to dwell on its
speculations, omitting every other concern; even those who have, with their
lips, as it were, but just touched its utility. And hence it follows, that they
who despise the knowledge of the mathematical disciplines, have very little
tasted of the pleasures they contain. The mathematics, therefore, are not to
be despised because their speculative parts do not immediately confer to
human utility, (for the ultimate limits of its progressions, and whatever
operates with matter, consider a use of this kind;) but on the contrary we
should admire its immateriality, and the good which it contains, considered
by itself alone. For when mankind were entirely disengaged from the care
of necessary concerns, they converted themselves to the investigation of the
mathematical disciplines; and this, indeed, with the greatest propriety. Since
affairs familiar to human life in its most imperfect state, and which are
immediately connected with its origin, first of all employed the studies of
mankind: but, in the second place, those concerns succeeded which separate
the soul from generation, and restore its memory of that which IS. After this
manner, then, we are engaged in necessaries, before things honourable for
their own sakes, on account of their intrinsic dignity and worth; and in
things related to sense, before such as are apprehended by the nobler
energies of mind. For every origin and life of the soul which is converted
into herself, is naturally adapted to proceed from the imperfect to the
perfect. And thus much against those who despise the mathematical
science.”

C H A P. X.
A Solution of another Objection of certain Platonists, against the Utility of
the Mathematical Sciences.

B ut, perhaps, some of our own family will here rise up against us, and,
proposing Plato as a witness, will endeavour to provoke ruder
understandings into a contemptuous disregard of the mathematical
disciplines. For they will say, that this philosopher entirely excludes (in his
Republic) the mathematical knowledge from the choir of the sciences, and
that he accuses it as being ignorant of its own principles, that its very
principle is to itself unknown, and its ends and mediums composed from
things of which it is ignorant. To these objections they may likewise add
whatever other reproaches are there urged by Socrates against this
contemplation. In answer then, to the objections of our friends, we shall
recall into their memory, that Plato himself perspicuously asserts the
mathematical science to be the purgation of the soul, and that it is endued
with a power of leading it on high; because, like the Homeric Minerva, it
removes the darkness of a sensible nature from the intellectual light of
thought, which is better worth saving than ten thousand corporeal eyes, and
which not only participates of a mercurial gift, (preserving us from the
incantations and delusions of this material abode, which is similar to the
fascinating realms of Circe,) but also of the more divine arts of Minerva. He
likewise every where calls it by the name of science, and asserts that it is
the cause of the greatest felicity to those who are exercised in its
contemplation. But I will briefly explain why, in the Republic he takes from
it the surname of science: for my present discourse is addressed to the
learned. Plato, indeed, in most places, calls all the knowledge (as I may say)
of universals by the name of science, opposing it in a division to sense
which apprehends only particulars, whether such a mode of cognition is
accomplished by art or experience. And in this sense, as it appears to me in
the Civil Dialogue, and in the Sophista, he seems to use the name of
science; placing likewise the illustrious Sophistic science, which Socrates in
the Gorgias, says, is a certain experience: also, the adulatory, and many
others, which are experiences, but not true sciences. But again, dividing this
knowledge of universals into that which knows causes, and into that which

understands without a cause, he thinks that the one should be called science,
but the other experience. And hence, to arts he sometimes attributes the
name of science, but to experience never. For how (says he in the Banquet)
can a thing which possesses no reason be science? All knowledge,
therefore, which contains the reason and cause of the things known, is a
certain science. Again, therefore, he divides this science which is endued
with a power from the cause of knowing, by the peculiarity of its subjects,
and he places one, conjectural of things divisible; but the other of such as
subsist by themselves, and are ever knowable after the same manner. And
according to this division he separates from science, medicine, and every
faculty which is conversant with material concerns. But he calls
mathematical knowledge, and whatever possesses a power of contemplating
eternal objects, by the name of science. Lastly, dividing this science, which
we distinguished from arts, he considers one part as void of supposition; but
the other as flowing from supposition. And that the one which is void of
supposition, has a power of knowing universals: that it rises to good, and
the supreme cause of all; and that it considers good as the end of its
elevation: but that the other, which previously fabricates for itself definite
and determinate principles, from which it evinces things consequent to such
principles, does not tend to the principle, but to the conclusion. And hence
he asserts, that mathematical knowledge, because it makes use of
supposition, falls short of that science which is without supposition, and is
perfect. For there is one true science, by means of which we are disposed to
know all the things which are, and from which also principles emerge to all
sciences; to some, indeed, constituted more proximately, but to others more
remotely. We must not say, therefore, that Plato expels mathematical
knowledge from the number of the sciences, but that he asserts it to be the
second from that one science, which possesses the supreme seat of all: nor
must we affirm, that he accuses it as ignorant of its own principles, but that
receiving these from the master science dialectic, and possessing them
without any demonstration, it demonstrates from these its consequent
propositions. For, indeed, he sometimes allows the soul, which is
constituted from mathematical reasons, to be the principle of motion: and
sometimes he affirms, that it receives its motion from genera which are
subject to intelligence. And these variations accord among themselves. For
to such things as are moved by another, the soul is a certain cause of
motion, but it is not the cause of every motion. After the same manner, the

mathematical science is indeed the second from the first of all sciences,
and, with reference to it, imperfect: but it is, nevertheless, a science, not as
being free from supposition, but as knowing the peculiar reasons resident in
the soul, and as bringing the causes of conclusions, and containing the
reason of such things as are subject to its knowledge. And thus much for the
opinion of Plato respecting mathematics.

C H A P. XI.

B ut let us now consider what are the things which may be required of a
mathematician, and how any one may rightly judge concerning his
distinguishing peculiarities. For[83] Aristotle indeed, says, that he who is
simply learned in all disciplines, is adapted to judge of all: but that he who
is alone skilled in the mathematical sciences, can alone determine
concerning the magnitude of reasons inherent in these. It is requisite,
therefore, that we should previously assume the terms of judging, and that
we should know, in the first place, in what things it is proper to demonstrate
generally, and in what to regard the peculiarities of singulars. For many of
the same properties reside in things differing in species, as two right angles
in all triangles: but many have indeed the same predicament, yet differ in
their individuals in a common species, as similitude in figures and numbers.
But one demonstration is not to be sought for by the mathematician in
these, for the principles of figures and numbers are not the same, but differ
in their subject genus. And if the essential accident is one, the
demonstration will also be one[84]: for the possession of two right angles is
the same in all triangles, and that general something to which this pertains
is the same in all, I mean triangle, and a triangular reason. In the same
manner, likewise, the possession of external angles to four right ones, not
only pertains to triangles, but also to all right-lined figures; and the
demonstration, so far as they are right-lined, agrees in all. For every reason
brings with it, at the same time, a certain property and passion, of which all
participate through that reason, whether triangular, or rectilinear, or
universally figure. But the second limit by which a mathematician is to be
judged, is, if he demonstrates according to his subject-matter, and renders
necessary reasons, and such as cannot be confuted, but are at the same time
neither probable, nor replenished with a similitude of truth. For, says
Aristotle, it is just the same to require demonstrations from a rhetorician,
and to assent to a mathematician disputing probably; since every one,
endued with science and art, ought to render reasons adapted to the subjects
of his investigation. In like manner also, Plato in the Timæus, requires
credible reasons of the natural philosopher, as one who is employed in the
resemblances of truth: but of him who discourses concerning intelligibles,

and a stable essence, he demands reasons which can neither be confuted nor
moved. For subjects every where cause a difference in sciences and arts,
since, if some of them are immoveable, others are conversant with motion;
and some are more simple, but others more composite; and some are
intelligibles, but others sensibles. Hence we must not require the same
certainty from every part of the mathematical science. For if one part, after
a manner, borders upon sensibles, but another part is the knowledge of
intelligible subjects, they cannot both be equally certain, but one must
inherit a higher degree of evidence than the other. And hence it is, that we
call arithmetic more certain than the science of harmony. Nor must we think
it just that mathematics and other sciences should use the same
demonstrations; for their subjects afford them no small variety. In the third
place, we must affirm, that he who rightly judges mathematical reasons,
must consider sameness and difference, what subsists by itself, and what is
accidental, what proportion is, and every consideration of a similar kind.
For almost all errors of this sort happen to those who think they
demonstrate mathematically, when at the same time they by no means
demonstrate, since they either demonstrate the same thing as if different in
each species, or that which is different as if it were the same: or when they
regard that which is accidental, as if it were an essential property; or that
which subsists by itself, as if it were accidental. For instance, when they
endeavour to demonstrate that the circumference of a circle is more
beautiful than a right line, or an equilateral than an isosceles triangle. For
the determination of these does not belong to the mathematician, but to the
first philosopher alone. Lastly, in the fourth place, we must affirm, that
since the mathematical science obtains a middle situation between
intelligibles and sensibles, and exhibits in itself many images of divine
concerns, and many exemplars of natural reasons, we may behold in it three
kinds of demonstration[85], one approaching nearer to intellect, the second
more accommodated to cogitation, and the third bordering on opinion. For
it is requisite that demonstrations should differ according to the varieties of
problems, and receive a division correspondent to the genera of beings,
since the mathematical science is connected with all these, and adapts its
reasons to the universality of things. And thus much for a discussion of the
subject proposed.

C H A P. XII.
What and how many the Species of the whole Mathematical Science are,
according to the Opinion of the Pythagoreans.

B ut after these considerations, it is requisite to determine concerning the
parts of the mathematical science, what, and how many they are. For it
is just, after speculating its whole and entire genus, to consider the
differences of its more particular sciences, according to their species. The
Pythagoreans[86], therefore, thought that the whole mathematical science
should receive a fourfold distribution, attributing one of its parts to the how-
many, but the other to the how-much; and they assigned to each of these
parts a twofold division. For they said, that discrete quantity, or the how-
many, either subsists by itself, or must be considered with relation to some
other; but that continued quantity, or the how-much, is either stable or in
motion. Hence they affirmed, that arithmetic contemplates that discrete
quantity which subsists by itself, but music that which is related to another;
and that geometry considers continued quantity so far as it is immoveable;
but spherics contemplates continued quantity as moving from itself, in
consequence of its union with a self-motive nature. They affirmed besides,
that these two sciences, discrete and continued quantity, did not consider
either magnitude or multitude absolutely, but that alone which in each of
these is definite from the participation of bound. For sciences alone
speculate the definite, rejecting as vain the comprehension of infinite
quantity. But when these wise men assigned this distribution, we must not
suppose they understood that discrete quantity which is found in sensible
natures, nor that continued quantity which subsists about the fluctuating
order of bodies. For, I think, the contemplation of these pertains to the
natural and not to the mathematical science. But because the demiurgus of
the universe, employed the union, division, and identity of general natures,
together with difference, station, and motion, for the purpose of completing
the essence of the soul, and composed it from these genera, as Timæus
informs us, we must affirm, that cogitation, abiding according to its
diversity, its division of reasons, and its multitude, and understanding itself
to be both one and many, proposes indeed to itself, and produces numbers,
together with an arithmetical knowledge of these: but it provides for itself

music according to an union of its multitude, and a communication and
junction with itself; and hence it is that arithmetic excels music in antiquity;
since, according to the narration of Plato, the demiurgus first divided the
soul, and afterwards collected it in harmonical proportions. Again, thought
establishing its energy according to the stability which it contains, draws
from its inmost retreats geometry, together with one essential figure, and the
demiurgical principles of all figures[87]: but, according to its inherent
motion, it produces the spherical science. For it is moved also by circles,
but abides perpetually the same from the causes of circles. Hence, likewise,
geometry precedes spherics, in the same manner as station is prior to
motion. But because cogitation itself produces these sciences, not by
looking back upon its convolution of forms, endued with an infinite power,
but upon the inclosure of bound according to its definite genera; hence they
say, that the mathematical sciences take away infinite from multitude and
magnitude, and are only conversant about finite quantity. Indeed, intellect
has placed in cogitation all the principles both of multitude and magnitude.
For since it wholly consists, with reference to itself, of similar parts, and is
one and indivisible, and again divisible, educing the ornament of forms, it
participates of bound and infinite, from intelligible essences themselves.
But it understands, indeed, from its participation of bound, and generates
vital energies, and various reasons from the nature of infinite. The
intellections, therefore, of thought, constitute these sciences according to
the bound which they contain, and not according to an infinity of life; since
they bring with them an image of intellect, but not of life. Such then is the
opinion of the Pythagoreans, and the division of the four mathematical
sciences.

C H A P. XIII.
Another Division of the Mathematical Science, according to Geminus.

A gain, some think (among whom is Geminus) that the mathematical
science is to be divided in a different manner from the preceding. And
they consider that one of its parts is conversant with intelligibles only, but
the other with sensibles, upon which it borders; denominating as
intelligibles whatever inspections the soul rouses into energy by herself,
when separating herself from material forms. And of that which is
conversant with intelligibles they establish two, by far the first and most
principal parts, arithmetic and geometry: but of that which unfolds its office
and employment in sensibles, they appoint six parts, mechanics, astrology,
optics, geodæsia, canonics, and logistics, or the art of reckoning. But they
do not think that the military art, or tactics, should be called any one part of
mathematics, according to the opinion of some[88]; but they consider it as
using at one time the art of reckoning, as in the numbering of legions; but at
another time geodæsia, as in dividing and measuring the spaces filled by a
field of camps. As, say they, neither the art of writing, nor the art of healing,
are any part of mathematics, though frequently both the historian and
physician use mathematical theorems. This is the case with historians
indeed, when relating the situation of climates, or collecting the magnitudes
and dimensions of cities, or their compass and circuit: but with physicians,
when elucidating by ways of this kind, many things in their art. For
Hippocrates himself shews the utility derived to medicine from astrology,
and almost all who speak of opportune times and places. By the same
reason he also, who accommodates his work to tactics, uses indeed
mathematical theorems, yet is not on this account a mathematician,
although he is sometimes willing that a numerous camp should exhibit a
very small multitude, and forms his army according to a circular figure; but
sometimes in a quadrangular, quinquangular, or some other multangular
figure, when he desires it to appear numerous. But since these are the
species of the whole mathematical science, geometry is again divided into
the contemplation of planes, and the dimension of solids, which is called
stereometry. For there is not any peculiar treatise about points and lines,
because no figure can be produced from these without planes or solids. For

geometry treats of nothing else in every one of its parts, than that it may
constitute either planes or solids: or that when constituted, it may compare
and divide them among themselves. In like manner, arithmetic is distributed
into the contemplation of linear, plane, and solid numbers. For it considers
the species of numbers separate from sensible connections, proceeding from
unity, and the origin of plane numbers; I mean of the similar, dissimilar, and
solid, even to the third increase. But geodæsia, and the art of reckoning, are
divided similarly to arithmetic and geometry, as they do not discourse
concerning intelligible numbers or figures, but of such as are sensible alone.
For neither is it the office of geodesia to measure the cylinder or the cone,
but material masses as if they were cones, and wells as if they were
cylinders. Neither does it accomplish this purpose by intelligible right lines,
but by such as are sensible, sometimes indeed by a more certain means, as
by the solar rays: but at other times by grosser ones, as by a line and
perpendicular. In like manner, the reckoner does not survey the passions of
numbers by themselves, but as they are resident in sensible objects. From
whence he also imposes a name upon these derived from the things which
he reckons, calling them μηλίαι, & φιαλίται. Besides this, he does not,
admit of any least, like the arithmetician, who receives that minimum, as a
genus of relation. For some one man is considered by him as the measure of
the whole multitude of men, as unity also is the common measure of all
numbers. Again, optics and canonics are produced from geometry and
arithmetic. And optics uses the visual rays which are constituted by the rays
of the eyes, as lines and angles. But it is divided into that which is properly
called optics (because it renders the cause of these appearances, which are
accustomed to present themselves to us different from their reality, on
account of the different situations and distances of visible objects, as the
coincidence of parallel lines, or the appearance of quadrangles as if they
were circles); and into universal catoptrics, which is conversant about
various and manifold refractions, and is connected with imaginative or
conjectural knowledge: as also into that which is called sciography[89], or
the delineation of shadows, which shews how appearances in images may
seem neither inelegant nor deformed, on account of the distances and
altitudes of the things designed. But canonics (music) or the regular art,
considers the apparent reasons of harmonies, finding out the sections of
rules, every where using the assistance of sense, and, as Plato says, seeming
to prefer the testimony of the ears to intellect itself. But to the parts we have

hitherto enumerated, mechanics must he added, as it is a certain part of the
whole science, and of the knowledge of sensible objects, and of things
united with matter. But under this exists the art effective of instruments,
which is called (ὀργανoποιητικὴ) I mean of those instruments proper for the
purposes of war: such, indeed, as Archimedes is reported to have
constructed, resisting the besiegers of sea and land; and that which is
effective of miracles, and which is called (θαυματοποιητικὴ.) One part of
this constructs with the greatest artifice pneumatic engines, such as
Ctesibius and Heron fabricated: but another operates with weights, the
motion of which is reckoned to be the cause of inequilibrity; but their
station of equilibrity, as Timæus also has determined: and again, another
part imitates animate foldings and motions by strings and ropes. Again,
under mechanics is placed the knowledge of equilibriums, and of such
instruments as are called centroponderants: also (σφαιροποιία) or the art
effective of spheres, imitating the celestial revolutions, such as Archimedes
fabricated; and lastly, every thing endued with a power of moving matter.
But the last of all is astrology, which treats of the mundane motions, of the
magnitudes of the celestial bodies, their figures and illuminations, their
distances from the earth, and every thing of this kind; assuming many
things indeed to itself from sense, but communicating much with the natural
speculation. One part of this is gnomonics, which is exercised in settling the
dimension of hororary gnomons: but the other is metheoroscopics, which
finds out the differences of elevations, and the distances of the stars, and
also teaches many other and various astrological theorems. The third part is
dioptrics, which ascertains by dioptric instruments of this kind the distances
of the sun and moon, and of the five other stars. And such is the account of
the parts of the mathematical science, delivered by the ancients, and
transmitted to our memory by the informing hand of time.

C H A P. XIV.
How Dialectic is the Top of the Mathematical Sciences, and what their
Conjunction is, according to Plato.

L et us again consider after what manner Plato, in his Republic, calls
dialectic the top of the mathematical disciplines; and what their
conjunction is, according to the tradition of the author of the Epinomis[90].
And in order to this we must assert, that as intellect is superior to cogitation,
supplying it with supernal principles, and from itself giving perfection to
cogitation; in the same manner dialectic also, being the purest part of
philosophy, excels in simplicity the mathematical disciplines, to which it is
proximate, and with which it is conjoined. Indeed it embraces the complete
circle of these sciences, to which it elevates from itself various energies,
endued with a power of causing perfection, judgment, and intelligence. And
these energies consist in resolving, dividing, defining, and demonstrating;
by which mathematics itself, receiving assistance and perfection, invents
some things by resolution, but others by composition: and some things it
explains by division, others by definition: but collects other subjects of its
investigation by demonstration; accommodating, indeed, these ways to its
subjects, but using each of them for the purpose of beholding its middle
enquiries. From whence indeed, both the resolutions, definitions, divisions,
and demonstrations which it contains, are peculiar, and adapted to its
nature, and revolve according to the mode of mathematical cognition. Not
undeservedly, therefore, is dialectic the vertex as it were, and summit of
mathematics. Since it perfects all which mathematics contains of
intelligence; renders its certainty free from reprehension, preserves the
stability of its immovable essence, and refers what it contains destitute of
matter and pure to the simplicity of intellect, and a nature separated from
material connections. Besides, it distinguishes the first principles of these
sciences, by definitions: exhibits the separations of genera and forms
contained under the genera themselves: and besides this, teaches the
compositions, which, from principles, produce things consequent to
principles: and the resolutions which rise and mount up to things first, and
to principles themselves. But with respect to what remains, proportion itself
is not to be considered (as Eratosthenes thought it was) as the conjunction

of the mathematical disciplines. Since proportion is said to be, and indeed is
one of those things common to the mathematics. But in short, many other
things besides proportion regard all the mathematical disciplines, which are
essentially inherent in the common nature of the mathematics. But as it
appears to me, we should say, that there is one proximate conjunction of
these, and of the whole mathematical science, which especially embraces in
itself, in a more simple manner, the principles of all sciences; which
considers their community and difference; teaches whatever is found in
these the same; together with what things are inherent in a many, and what
in a few. So that to those who aptly learn there is a reversion from many
other sciences to this alone[91]. But, dialectic is a conjunction of the
mathematical disciplines superior to the preceding; which Plato, as I have
already observed, calls in his Republic their vertex: for, indeed, it perfects
the whole of mathematics, brings it back to intellect by its powers, shews it
to be a true science, and causes it to be certain and obnoxious to no reproof.
But, intellect obtains the third order between these conjunctions, which
comprehends in itself uniformly all the dialectic powers, contracts their
variety by its simplicity, their partition by its indivisible knowledge, and
their multitude by its occult union. Hence, intellect itself congregates
indeed the involutions and deviations of the dialectic paths, into an
intelligible essence, but it collects supernally all the progression of
mathematical discourses: and it is the best end both of the elevating power
of the soul, and of the energy consisting in cognition. And such are the
sentiments declared by me on the present enquiry.

C H A P. XV.
From whence the Name Mathematics originated.

A gain, from whence shall we say this name of mathematics, and
mathematical disciplines, was assigned by the ancients, and what apt
reason can we render of its position? Indeed, it appears to me, that such an
appellation of a science which respects cogitative reasons, was not, like
most names, invented by indifferent persons, but (as the truth of the case is,
and according to report) by the Pythagoreans alone. And this, when they
perceived, that whatever is called mathesis or discipline, is nothing more
than reminiscence; which does not approach the soul extrinsically, like the
images which rising from sensible objects are formed in the phantasy: nor is
it adventitious and foreign, like the knowledge consisting in opinion, but it
is excited, indeed, from apparent objects, and is perfected within, by
thought intimately converted to itself. And when they likewise perceived
that though reminiscence might be shewn from many particulars, yet it was
evinced in a more eminent manner (as Plato also says[92]) from the
mathematical disciplines. For if any one, says he, is led into the
descriptions, he will there easily prove that discipline is reminiscence. From
whence Socrates also, in the Meno, shews by this method of arguing, that
learning is nothing else than the soul’s recollection of her inherent reasons.
And this, because that which recollects, is alone the cogitative part of the
soul; but this perfects her essence in the reasons of the mathematical
disciplines, the sciences of which she previously received into herself,
though she does not always energize on their fair variety. Indeed, she
contains them all essentially and occultly; but she produces each of them
when she is freed from the impediments originating from sense. For sense
unites her with divisible objects: the phantasy fills her with forming
motions, and appetite bends her to an indulgent and luxurious life. But
every thing divisible is an obstacle to our self-conversion. And whatever
invests with form, disturbs and offends that knowledge which is destitute of
form. And whatever is obnoxious to perturbations is an impediment to that
energy which is unimpaired by affections. When, therefore, we have moved
all these from the cogitative power, then shall we be able to understand by
thought itself, the reasons which thought contains: then shall we become

scientific in energy; and unfold our essential knowledge. But whilst we are
captive and bound, and winking with the eye of the soul, we cannot by any
means attain to a perfection convenient to our nature. Such then is mathesis
or discipline: a reminiscence of the eternal reasons contained in the soul.
And the mathematical or disciplinative science is on this account
particularly denominated that knowledge which especially confers to our
reminiscence of these essential reasons. Hence, the business and office of
this science[93], is apparent from its name. For its duty is to move the
inherent knowledge of the soul; to awaken its intelligence; to purify its
cogitation; to call forth its essential forms from their dormant retreats; to
remove that oblivion and ignorance, which are congenial with our birth; and
to dissolve the bonds arising from our union with an irrational nature. It
plainly leads us to a similitude of that divinity who presides over this
science, who manifests intellectual gifts, and fills the universe with divine
reasons; who elevates souls to intellect, wakens them as from a profound
sleep, converts them by enquiry to themselves; and by a certain obstetric
art, and invention of pure intellect, brings them to a blessed life. To whom
indeed, dedicating the present work, we here conclude our contemplation of
the mathematical science.

B O O K II.

C H A P. I.
What Part Geometry is of Mathematics, and what the Matter is of which it
consists.

I n the preceding discourses we have considered those common properties
which respect the whole of the mathematical science; and this we have
done agreeable to the doctrine of Plato; at the same time collecting such
particulars as pertain to our present design. But consequent to this it is
requisite that we should discourse on geometry itself, and on the proposed
institution of the elements, for the sake of which we have undertaken the
whole of the present work. That geometry then, is a part of the whole of
mathematics, and that it obtains the second place after arithmetic, since it is
perfected and bounded by this, (for whatever in geometry may be expressed
and known, is determined by arithmetical reasons) has been asserted by the
ancients, and requires no long discussion in the present enquiry. But we also
may be able to relate our opinion on this particular, if we consider what
place, and what essence its subject matter[94] is allotted among the
universality of things. For from a proper survey of this, the power of the
science which knows this subject matter, the utility arising from it, and the
good acquired by its learners, will immediately appear. Indeed, some one
may doubt in what genus of things he ought to place geometrical matter, so
as not to deviate from the truth it contains. For if the figures concerning
which geometry discourses, exist in sensible natures, and cannot be
separated from the dark receptacle of matter; how can we assert that
geometry frees us from sensible objects, that it brings us to an incorporeal
essence, that it accustoms us to an inspection of intelligibles, and prepares
us for intellectual energy? Where shall we ever survey among sensible
objects a point without parts, or a line destitute of breadth, or a superficies
without profundity, or the equality of lines from the centre to the
circumference; or the multangles, and all the figures of many bases,
concerning which geometry informs us? Lastly, after what manner can the
reasons of such a science remain free from all possible confutation; since,
indeed, sensible forms and figures are susceptive of the more and the less,
are all moveable and mutable, and are full of material variety; among which
equality subsists mixt and confused with its contrary inequality, and into

which things without parts have proceeded into partition, and interval,
darkened with the shades of matter, and lost in its infinite folds? But if the
subjects of geometry are removed from matter, are pure forms, and are
separated from sensible objects: they will be all of them, without doubt,
void of parts, incorporeal, and destitute of magnitude. For extension, tumor,
and interval, approach to forms, on account of the material receptacle in
which they are involved, and which receives things destitute of parts,
distributed into parts; things void of dimension, extended into dimension;
and immoveable natures accompanied with motion. How then, if this is the
case, shall we cut a right line, triangle, and circle? How can we speak of the
diversities of angles, and the increments and decrements of triangular and
quadrangular figures? Or how exhibit the contacts of circles or right lines?
For all these evince that the geometric matter consists of parts, and does not
reside among indivisible reasons. Such then are the doubts concerning the
matter of geometry, to which we may add, that Plato considers the forms of
geometry as placed in cogitation; and grants, that we advance from
sensibles to forms of this kind, and that we rise from sensibles to intellect,
though (as we have previously observed) the reasons subsisting in
cogitation are indivisible, are separated by no interval, and subsist
according to the peculiarity of the soul. But if reasons are to be rendered
agreeable to things themselves, and to the doctrine of Plato, the following
division must be adopted. [95]Every universal, and one thing containing
many, is either naturally disposed to be thought of in particulars, or to
appear such, because it possesses its existence in these; is inseparable from
them; is disposed and distributed in them; and together with these is either
moved, or firmly and immoveably abides. Or it is adapted to subsist prior to
many, and to possess a power of generating multitude, affording to many
things images from itself, being furnished with a nature destitute of parts,
from the essences which it participates, and raising various participations to
secondary natures: or it is disposed to be formed by thought, from the many,
to possess a generating existence, and to reside in the last place in the many.
For, according to these three modes of subsistence, we shall find, I think,
that some subsist before the many, others in the many, and others from the
relation and predication which they possess to these. But, that I may
absolve all in one word, universal forms being threefold, we shall consider
the differences of that form which many participate, which exists in many,
and fills particular natures according to its subject matter. Besides this,

establishing a twofold order of participants, one subsisting in sensible
objects, but the other in the phantasy, (since matter is twofold; one indeed,
of things united with sense, but the other of such as fall under the inspection
of phantasy, as Aristotle asserts, in a certain place[96]) we must allow that
the universal, which is distributed in the many, is likewise twofold. The one,
indeed, sensible, as being that which sensible objects participate; but the
other imaginative, as that which subsists in the many of the phantasy. For
the phantasy, on account of its forming motion, and because it subsists with,
and in body, always receives impressions which are both divided and
figured. So that whatever is known by it, is allotted a correspondent
existence: on which account, Aristotle[97] does not hesitate to call it passive
intellect. But if it is intellect, why is it not impassive, and destitute of
matter? And if it operates with passion, how can it with propriety be called
intellect? For impassivity, indeed, properly belongs to intellect and an
intelligent nature: but passivity is very remote from such an essence. But
(unless I am deceived) Aristotle being willing to explain its middle nature
between cognitions the most primary, and such as are the last, calls it at the
same time intellect, because similar to primary cognitions, and passive from
that alliance which it possesses with such as are posterior. For first
cognitions are indeed destitute of figures and forms; comprehending in
themselves, intelligible natures, energizing about themselves, united with
the objects of knowledge, and free from all extrinsical impression and
passion. But last cognitions exercise themselves through the medium of
instruments, are rather passions than energies, admit extrinsical knowledge,
and move themselves together with their various subjects. For such (says
Plato) are the sensations which arise from violent passions. But the
phantasy, obtaining a middle centre in the order of cognitions, is excited,
indeed, by itself, and produces that which falls under cogitation: but
because it is not separate from body, it deduces into partition, interval, and
figure, the objects of its knowledge, from the indivisibility of an intellectual
life. Hence, whatever it knows, is a certain impression and form of
intelligence. For it understands the circle, together with its interval, void,
indeed, of external matter, but possessing intelligible matter. On this
account, like sensible matter, it does not contain one circle only: for we
behold in its receptacle, distance, together with the more and the less, and a
multitude of circles and triangles. If then an universal nature is distributed
in sensible circles, since each of these completes a circular figure, and they

are all mutually similar, subsisting in one reason, but differing in
magnitudes or subjects: in like manner, there is a common something in the
circles, which subsist in the receptacle of the phantasy, of which all its
circles participate, and according to which they all possess the same form;
but in the phantasy they possess but one difference only, that of magnitude.
For when you imagine many circles about the same centre, they all of them
exist in one immaterial subject and life, which is inseparable from a simple
body, which, by the possession of interval, exceeds an essence destitute of
parts; but they differ in magnitude and parvitude, and because they are
contained and contain. Hence, that universal is twofold, which is
understood as subsisting in the many: one, indeed, in sensible forms; but the
other in such as are imaginative. And the reason of a circular and triangular
figure, and of figure universal, is twofold. The one subsisting in intelligible,
but the other in sensible matter. But prior to these is the reason which
resides in cogitation, and that which is seated in nature herself. The former
being the author of imaginative circles, and of the one form which they
contain; but the other, of such as are sensible. For there are circles existing
in the heavens, and universally those produced by nature, the reason of
which does not fall under a cogitative distribution. For in incorporeal
causes, things possessing interval, are distinguished by no intervals: such as
are endued with parts, subsist without parts: and magnitudes without the
diffusion of magnitude, as on the contrary in corporeal causes, things
without parts subsist divisibly, and such as are void of magnitude with the
extension of magnitude. Hence, the circle resident in cogitation, is one,
simple and free from interval: and magnitude itself is there destitute of
magnitude; and figure expressed by no figure: for such are reasons separate
from matter. But the circle subsisting in the phantasy, is divisible, figured,
endued with interval, not one only, but one and many, nor form alone, but
distributed with form. And the circle, in sensible objects, is composite,
distant with magnitude, diminished by a certain reason, full of ineptitude,
and very remote from the purity of immaterial natures. We must therefore
say, that geometry, when it asserts any thing of circle and diameter, and of
the passions and affections which regard the circle; as of contacts, divisions,
and the like: neither teaches nor discourses concerning sensible forms,
(since it endeavours to separate us from these), nor yet concerning the form
resident in cogitation, (for here the circle is one, but geometry discourses of
many, proposing something of each, and contemplating the same of all: and

here it is indivisible, but the geometric circle is divisible); but we must
confess, that it considers universal itself; yet as distributed in imaginative
circles. And that it beholds, indeed, one circle[98]: and by the medium of
another, contemplates the circle resident in the depths of cogitation: but by
another, different from the preceding, fabricates the fair variety of its
demonstrations. For since cogitation is endued with reasons, but cannot
behold them contractedly, separated from material figure; it distributes and
removes them, and draws them forth seated in the shadowy bosom of the
phantasy, and placed in the vestibules of primary forms; revolving in it, or
together with it, the knowledge of these: loving, indeed, a separation from
sensibles, but finding imaginative matter proper for the reception of its
universal forms. Hence, its intellection does not subsist without the
phantasy. And the compositions and divisions of figures are imaginative;
and their knowledge is the way which leads us to that essence pursued by
cogitation: but cogitation itself, does not yet arrive at this stable essence,
while it looks abroad to externals, contemplates its internal forms according
to these, uses the impressions of reasons, and is moved from itself to
external and material forms. But if it should ever be able to return to itself,
when it has contracted intervals and impressions, and beholds multitude
without impression, and subsisting uniformly; then it will excellently
perceive geometrical reasons, void of division and interval, essential and
vital, of which there is a copious variety. And this energy will be the best
end of the geometric study; and truly the employment of a Mercurial gift,
bringing it back as from a certain Calypso, and her detaining charms, to a
more intellectual knowledge; and freeing it from those forming
apprehensions with which the mirror of the phantasy is replete. Indeed, it is
requisite that a true geometrician should be employed in this meditation,
and should establish, as his proper end, the excitation and transition from
the phantasy to cogitation alone; and that he should accomplish this by
separating himself from intervals, and the passive intellect to that energy
which cogitation contains. For by this means he will perceive all things
without an interval, the circle and diameter without a part, the polygons in
the circle, all in all, and yet every one separate and apart. Since, on this
account, we exhibit also in the phantasy, both circles inscribed in polygons,
and polygons in circles; imitating the alternate exhibition of reasons
destitute of parts. Hence, therefore, we describe the constitutions, the origin,
divisions, positions, and applications of figures: because we use the

phantasy, and distances of this kind proceeding from its material nature;
since form itself is immoveable, without generation, indivisible, and free
from every subject. But whatever form contains occultly, and in an indistant
manner, is produced into the phantasy subsisting with intervals, divisibly
and expanded. And that which, indeed, produces the forms of geometric
speculation, is cogitation: but that from which they are produced, is the
form resident in cogitation: and that in which the produced figure resides is
what is called the passive intellect. Which folds itself about the impartibility
of true intellect, separates from itself the power of pure intelligence free
from interval; conforms itself according to all formless species, and
becomes perfectly every thing from which cogitation itself, and our
indivisible reason consists. And thus much concerning the geometric matter,
as we are not ignorant of whatever Porphyry the Philosopher has observed
in his miscellanies, and whatever many of the Platonists describe. But we
think that the present discussions are more agreeable to geometric
dissertations, and to Plato himself, who subjects to geometry the objects of
cogitation. For these mutually agree among themselves; because the causes,
indeed, of geometrical forms, by which cogitation produces demonstrations,
pre-exist in demonstration itself: but the particular figures which are
divided and compounded, are situated in the receptacle of the phantasy.

C H A P. II.
What kind of Science Geometry is.

B ut let us now speak of that science which possesses a power of
contemplating the universal forms participated by imaginative matter.
Geometry, therefore, is endued with the knowledge of magnitudes and
figures, and of the terms and reasons subsisting in these; together with the
passions, various positions and motions which are contingent about these.
For it proceeds, indeed, from an impartible point, but descends even to
solids, and finds out their multiform diversities. And again, runs back from
things more composite, to things more simple, and to the principles of
these: since it uses compositions and resolutions, always beginning from
suppositions, and assuming its principles from a previous science; but
employing all the dialectic ways. In principles, by the divisions of forms
from their genera, and by defining its orations. But in things posterior to
principles, by demonstrations and resolutions. As likewise, it exhibits things
more various, proceeding from such as are more simple, and returning to
them again. Besides this, it separately discourses of its subjects; separately
of its axioms; from which it rises to demonstrations; and separately of
essential accidents, which it shews likewise are resident in its subjects. For
every science has, indeed, a genus, about which it is conversant, and whose
passions it proposes to consider: and besides this, principles, which it uses
in demonstrations; and essential accidents. Axioms, indeed, are common to
all sciences (though each employs them in its peculiar subject matter), but
genus and essential accident vary according to the sciential variety. The
subjects of geometry are therefore, indeed, triangles, quadrangles, circles,
and universally figures and magnitudes, and the boundaries of these. But its
essential accidents are divisions, ratios, contacts, equalities, applications,
excesses, defects, and the like. But its petitions and axioms, by which it
demonstrates every particular are, this, to draw a right line from any point
to any point; and that, if from equals you take away equals, the remainders
will be equal; together with the petitions and axioms consequent to these.
Hence, not every problem nor thing sought is geometrical, but such only as
flow from geometric principles. And he who is reproved and convicted
from these, is convinced as a geometrician. But whoever is convinced from

principles different from these, is not a geometrician, but is foreign from the
geometric contemplation. But the objects of the non-geometric
investigation, are of two kinds. For the thing sought for, is either from
entirely different principles, as we say that a musical enquiry is foreign
from geometry, because it emanates from other suppositions, and not from
the principles of geometry: or it is such as uses, indeed, geometrical
principles, but at the same time perversely, as if any one should say, that
parallels coincide. And on this account, geometry also exhibits to us
instruments of judging, by which we may know what things are consequent
to its principles, and what those are which fall from the truth of its
principles: for some things attend geometrical, but others arithmetical
principles. And why should we speak of others, since they are far distant
from these? For one science is more certain than another (as Aristotle
says[99]) that, indeed, which emanates from more simple suppositions, than
that which uses more various principles; and that which tells the why, than
that which knows only the simple existence of a thing; and that which is
conversant about intelligibles, than that which touches and is employed
about sensibles. And according to these definitions of certainty, arithmetic
is, indeed, more certain than geometry, since its principles excel by their
simplicity. For unity is void of position, with which a point is endued. And a
point, indeed, when it receives position, is the principle of geometry: but
unity, of arithmetic. But geometry is more certain than spherics; and
arithmetic, than music. For these render universally the causes of those
theorems, which are contained under them. Again, geometry is more certain
than mechanics, optics, and catoptrics. Because these discourse only on
sensible objects. The principles, therefore, of geometry and arithmetic,
differ, indeed, from the principles of other sciences; but the hypotheses of
these two, alternately differ and agree according to the difference we have
already described. Hence, also, with respect to the theorems which are
demonstrated in these sciences, some are, indeed, common to them, but
others peculiar. For the theorem which says, every proportion may be
expressed, alone belongs to arithmetic; but by no means to geometry: since
this last science contains things which cannot be expressed[100]. That
theorem also, which affirms, that the gnomons of quadrangles are
terminated according to the least[101], is the property of arithmetic: for in
geometry, a minimum cannot be given. But those things are peculiar to
geometry, which are conversant about positions; for numbers have no

position: which respect contacts; for contact is found in continued
quantities: and which are conversant about ineffable proportions; for where
division proceeds to infinity, there also that which is ineffable is found[102].
But things common to both these sciences, are such as respect divisions,
which Euclid treats of in the second book; except that proposition which
divides a right line into extreme and mean proportion[103]. Again, of these
common theorems, some, indeed, are transferred from geometry into
arithmetic; but others, on the contrary, from arithmetic into geometry: and
others similarly accord with both, which are derived into them from the
whole mathematical science. For the permutation, indeed, conversions,
compositions, and divisions of ratios are, after this manner, common to
both. But such things as are commensurable, arithmetic first beholds; but
afterwards geometry, imitating arithmetic. From whence, also, it determines
such things to be commensurables of this kind, which have the same mutual
ratio to one another, as number to number; because commensurability
principally subsists in numbers. For where number is, there also that which
is commensurable is found; and where commensurable is, there also
number. Lastly, geometry first inspects triangles and quadrangles: but,
arithmetic, receiving these from geometry, considers them according to
proportion. For in numbers, figures reside in a causal manner. Being
excited, therefore, from effects, we pass to their causes, which are contained
in numbers. And at one time, we indifferently behold the same accidents, as
when every polygon is resolved by us into triangles[104]: but, at another
time, we are content with what is nearest to the truth, as when we find in
geometry one quadrangle the double of another, but not finding this in
numbers, we say that one square is double of another, except by a
deficience of unity. As for instance, the square from 7, is double the square
from 5, wanting one. But we have produced our discussion to this length,
for the purpose of evincing the communion and difference in the principles
of these two sciences. Since it belongs to a geometrician to survey from
what common principles common theorems are divided; and from what
principles such as are peculiar proceed; and thus to distinguish between the
geometrical, and non-geometrical, referring each of them to different
sciences.

C H A P. III.
From whence the whole of Geometry originated, how far it proceeds, and in
what its Utility consists.

B ut, beginning still higher, let us contemplate the whole of geometry,
from whence it originated, and how far it proceeds in its energies: for
thus we shall properly perceive the ornament which it contains. Indeed, it is
necessary to understand that it is extended through the universality of
things: that it accommodates its animadversions[105] to all beings; and
contains in itself the forms of all things: that, according to its supreme part,
and which is endued with the highest power of intelligence, it surveys true
beings; and teaches by images the properties of divine ornaments, and the
powers of intellectual forms: for it contains the reasons of these also in its
peculiar contemplations. And it exhibits what figures are convenient to the
god, to primary essences, and to the natures of souls. But, according to its
middle cognitions, it evolves cogitative reasons; explains and beholds the
variety which they contain; exhibits their existence, and inherent passions;
as also, their communities and diversities. From which, indeed, it
comprehends, in terminated bounds, the imaginative formations of figures,
and reduces them to the essential substance of reasons. But, according to
the third propagations of cogitative intelligence, it considers nature, and
delivers the manner in which the forms of sensible elements, and the
powers which they contain, are previously received according to cause, in
the reasons themselves. For it possesses, indeed, the images of universal
intelligible genera; but the exemplars of such as are sensible: and completes
its own essence, according to such things as are subject to cogitation. And
through these, as through proper mediums, it ascends and descends to those
universals which truly are, and to sensible forms which are in a state of
perpetual formation. But always geometrically philosophising concerning
the things which are, it comprehends in all the proportions of virtues, the
images of intellectual, animal, and natural concerns. And it delivers, in an
orderly manner, all the ornaments of republics: and exhibits in itself their
various mutations. Such then are its energies arising from a certain
immaterial power of cognition: but when it touches upon matter, it produces
from itself a multitude of sciences; such as geodæsia, mechanics, and

perspective: by which it procures the greatest benefit to the life of mortals.
For it constructs by these sciences, war-instruments, and the bulwarks of
cities; and makes known the circuits of mountains, and the situations of
places. Lastly, it instructs us in measures: at one time of the diversified
ways of the earth; and at another, of the restless paths of the deep. Add too,
that it constructs balances and scales, by which it renders to cities a sure
equality according to the invariable standard of number. Likewise, it clearly
expresses, by images, the order of the whole orb of the earth; and by these,
manifests many things incredible to mankind, and renders them credible to
all. Such, indeed, as Hiero of Syracuse is reported to have said of
Archimedes[106], when he had fabricated a ship furnished with three sails,
which he had prepared to send to Ptolemy king of Egypt. For when all the
Syracusians together, were unable to draw this ship, Archimedes enabled
Hiero to draw it himself, without any assistance from others. But he, being
astonished, said, From this day, Archimedes shall be believed in whatever
he shall affirm. They also report, that Gelo said the same, when Archimedes
discovered the weight of the several materials from which his crown was
composed, without dissolving their union. And such are the narrations
which many of the ancients have delivered to our memory, who were
willing to speak in praise of the mathematics: and, on this account, we have
placed before the reader, for the present, a few out of the many, as not
foreign from our design of exhibiting the knowledge and utility of
geometry.

C H A P. IV.
On the Origin of Geometry, and its Inventors.

B ut let us now explain the origin of geometry, as existing in the present
age of the world. For the demoniacal Aristotle[107] observes, that the
same opinions often subsist among men, according to certain orderly
revolutions of the world: and that sciences did not receive their first
constitution in our times, nor in those periods which are known to us from
historical tradition, but have appeared and vanished again in other
revolutions of the universe; nor is it possible to say how often this has
happened in past ages, and will again take place in the future circulations of
time. But, because the origin of arts and sciences is to be considered
according to the present revolution of the universe, we must affirm, in
conformity with the most general tradition, that geometry was first invented
by the Egyptians, deriving its origin from the mensuration of their fields:
since this, indeed, was necessary to them, on account of the inundation of
the Nile washing away the boundaries of land belonging to each. Nor ought
it to seem wonderful that the invention of this as well as of other sciences,
should receive its commencement from convenience and opportunity. Since
whatever is carried in the circle of generation, proceeds from the imperfect
to the perfect. A transition, therefore, is not undeservedly made from sense
to consideration, and from this to the nobler energies of intellect[108].
Hence, as the certain knowledge of numbers received its origin among the
Phœnicians, on account of merchandise and commerce, so geometry was
found out among the Egyptians from the distribution of land. When Thales,
therefore, first went into Egypt, he transferred this knowledge from thence
into Greece: and he invented many things himself, and communicated to his
successors the principles of many. Some of which were, indeed, more
universal, but others extended to sensibles. After him Ameristus, the brother
of Stesichorus the poet, is celebrated as one who touched upon, and tasted
the study of geometry, and who is mentioned by Hippias the Elean, as
restoring the glory of geometry. But after these, Pythagoras changed that
philosophy, which is conversant about geometry itself, into the form of a
liberal doctrine, considering its principles in a more exalted manner; and
investigating its theorems immaterially and intellectually; who likewise

invented a treatise of such things as cannot be explained[109] in geometry,
and discovered the constitution of the mundane figures. After him,
Anaxagoras the Clazomenian succeeded, who undertook many things
pertaining to geometry. And Oenopides the Chian, was somewhat junior to
Anaxagoras, and whom Plato mentions in his Rivals, as one who obtained
mathematical glory. To these, succeeded Hippocrates, the Chian, who
invented the quadrature of the lunula[110], and Theodorus the Cyrenean,
both of them eminent in geometrical knowledge. For the first of these,
Hippocrates composed geometrical elements: but Plato, who was posterior
to these, caused as well geometry itself, as the other mathematical
disciplines, to receive a remarkable addition, on account of the great study
he bestowed in their investigation. This he himself manifests, and his books,
replete with mathematical discourses, evince: to which we may add, that he
every where excites whatever in them is wonderful, and extends to
philosophy. But in his time also lived Leodamas the Thasian, Archytas the
Tarantine, and Theætetus the Athenian; by whom theorems were increased,
and advanced to a more skilful constitution. But Neoclides was junior to
Leodamas, and his disciple was Leon; who added many things to those
thought of by former geometricians. So that Leon also constructed elements
more accurate, both on account of their multitude, and on account of the use
which they exhibit: and besides this, he discovered a method of determining
when a problem, whose investigation is sought for, is possible, and when it
is impossible. But Eudoxus the Cnidian, who was somewhat junior to Leon,
and the companion of Plato, first of all rendered the multitude of those
theorems which are called universals more abundant; and to three
proportions added three others; and things relative to a section, which
received their commencement from Plato, he diffused into a richer
multitude, employing also resolutions in the prosecution of these. Again,
Amyclas the Heracleotean, one of Plato’s familiars, and Menæchmus, the
disciple, indeed, of Eudoxus, but conversant with Plato, and his brother
Dinostratus, rendered the whole of geometry as yet more perfect. But
Theudius, the Magnian, appears to have excelled, as well in mathematical
disciplines, as in the rest of philosophy. For he constructed elements
egregiously, and rendered many particulars more universal. Besides,
Cyzicinus the Athenian, flourished at the same period, and became
illustrious in other mathematical disciplines, but especially in geometry.
These, therefore, resorted by turns to the Academy, and employed

themselves in proposing common questions. But Hermotimus, the
Colophonian, rendered more abundant what was formerly published by
Eudoxus and Theætetus, and invented a multitude of elements, and wrote
concerning some geometrical places. But Philippus the Mendæan[111], a
disciple of Plato, and by him inflamed in the mathematical disciplines, both
composed questions, according to the institutions of Plato, and proposed as
the object of his enquiry whatever he thought conduced to the Platonic
philosophy. And thus far historians produce the perfection of this science.
But Euclid was not much junior to these, who collected elements, and
constructed many of those things which were invented by Eudoxus; and
perfected many which were discovered by Theætetus. Besides, he reduced
to invincible demonstrations, such things as were exhibited by others with a
weaker arm. But he lived in the times of the first Ptolemy: for Archimedes
mentions Euclid, in his first book, and also in others. Besides, they relate
that Euclid was asked by Ptolemy, whether there was any shorter way to the
attainment of geometry than by his elementary institution, and that he
answered, there was no other royal path which led to geometry. Euclid,
therefore, was junior to the familiars of Plato, but more ancient than
Eratosthenes and Archimedes (for these lived at one and the same time,
according to the tradition of Eratosthenes) but he was of the Platonic sect,
and familiar with its philosophy: and from hence he appointed the
constitution of those figures which are called Platonic[112], as the end of his
elementary institutions.

C H A P. V.
What Mathematical Volumes Euclid composed.

T here are, therefore, many other mathematical volumes of this man, full
of admirable diligence, and skilful consideration: for such are his
Optics[113], and Catoptrics: and such also, are his elementary institutions,
which conduce to the attainment of music[114]; and his book concerning
divisions[115]. But his geometrical institution of the Elements is especially
admirable, on account of the order and election of those theorems and
problems, which are distributed through the Elements. For he does not
assume all which might be said, but that only which could be delivered in
an elementary order. Besides this, he exhibits modes of syllogisms of every
kind; some, indeed, receiving credibility from causes, but others proceeding
from certain signs; but all of them invincible and sure, and accommodated
to science. But, besides these, he employs all the dialectic ways, dividing,
indeed, in the inventions of forms; but defining in essential reasons: and
again, demonstrating in the progressions from principles to things sought,
but resolving in the reversions from things sought to principles. Besides
this, we may view in his geometrical elements, the various species of
conversions, as well of such as are simple as of such as are more composite.
And again, what wholes may be converted with wholes: what wholes with
parts; and on the other hand, what as parts with parts[116]. Besides this, we
must say, that in the continuation of inventions, the dispositions and order
of things preceding and following, and in the power with which he treats
every particular, he is not deceived, as if falling from science, and carried to
its contrary, falsehood and ignorance. But because we may imagine many
things as adhering to truth, and which are consequent to principles
producing science, which nevertheless tend to that error which flows from
the principles, and which deceives ruder minds, he has also delivered
methods of the perspicacious prudence belonging to these. From the
possession of which, we may exercise those in the invention of fallacies,
who undertake this inspection, and may preserve ourselves from all
deception. And this book, by which he procures us this preparation, is
inscribed ψευδαρίος, or, concerning fallacies[117]. Because he enumerates in
order their various modes, and in each exercises our cogitation with various

theorems. And he compares truth with falsehood, and adapts the confutation
of deception to experience itself. This book, therefore, contains a purgative
and exercising power. But the institution of his elementary, skilful
contemplation of geometrical concerns, possesses an invincible and perfect
narration.

C H A P. VI.
Concerning the Purport of Geometry.

B ut, perhaps, some one may enquire in what the design of this treatise
consists? To this I answer, that its design is to be distinguished as well
according to the objects of enquiry, as according to the learner. And, indeed,
regarding the subject, we must affirm, that all the discourse of geometry is
concerning the mundane figures. Because it begins from such things as are
simple, but ends in the variety of their constitution. And, indeed, it
constitutes each of them separately, but at the same time delivers their
inscriptions in a sphere, and the proportions which they contain. On which
account some have thought, that the design of each of the books is to be
referred to the world; and they have delivered to our memory, the utility
which they afford us in the contemplation of the universe. But
distinguishing the design with respect to the learner, we must affirm, that its
purpose is the institution of elements; and the perfection of the learners
cogitative powers in universal geometry. For beginning from these, we are
enabled to understand the other parts of this science, and to comprehend the
variety which they contain. And, indeed, without these, the discipline of the
rest, is to us impossible and incomprehensible. For such theorems as are
most principal and simple, and are most allied to first suppositions, are here
collected in a becoming order. And the demonstrations of other
mathematicians, use these as most known, and advance from these in their
most complicated progressions. For thus Archimedes, in what he has writ
concerning the sphere and cylinder, and Apollonius, and the rest of
mathematicians, use, as evident principles, the things exhibited in this
treatise. Its purpose, therefore, is the institution of learners in the whole
geometric science, and to deliver the determinate constitutions of the
mundane figures.

C H A P. VII.
From whence the Name of Elementary Institution originated, and why
Euclid is called the Institutor of Elements.

B ut what gave rise to the name of elementary institution, and of element
itself, from which elementary institution was derived? To this we shall
reply, by observing, that of theorems some are usually called elements, but
others elementary, and others again are determined beyond the power of
these. Hence, an element is that whose consideration passes to the science
of other things, and from which we derive a solution of the doubts incident
to the particular science we investigate. For as there are certain first
principles of speech, most simple and indivisible, which we denominate
elements, and from which all discourse is composed; so there are certain
principal theorems of the whole of geometry, denominated elements, which
have the respect of principles to the following theorems; which regard all
the subsequent propositions, and afford the demonstrations of many
accidents essential to the subjects of geometric speculation. But things
elementary are such as extend themselves to a multitude of propositions,
and possess a certain simplicity and sweetness, yet are not of the same
dignity with elements; because their contemplation is not common to all the
science to which they belong, as is the case in the following theorem, that in
triangles, perpendiculars, drawn from their angles to their sides, coincide in
one point[118]. Lastly, whatever neither possesses a knowledge extended
into multitude, nor exhibits any thing skilful and elegant, falls beyond the
elementary power. Again, an element, as Menæchmus says, may have a
twofold definition. For that which confirms, is an element of that which is
confirmed; as the first proposition of Euclid with respect to the second, and
the fourth with regard to the fifth. And thus, indeed, many things may be
mutually called elements one of another; for they are mutually confirmed.
Thus, because the external angles of right-lined figures, are equal to four
right angles, the multitude of internal ones equal to right angles; and, on the
contrary, that from this is exhibited[119]. Besides, an element is otherwise
called that into which, because it is more simple, a composite is dissolved.
But it must be observed, that every element cannot be called the element of
every thing: but such as are more principal are the elements of such as are

constituted in the reason of the thing effected; as petitions are the elements
of theorems. And, according to this signification of an element, Euclid’s
elements are constructed. Some, indeed, of that geometry which is
conversant about planes; but others of stereometry. In the same manner,
likewise, in arithmetic and astronomy, many have composed elementary
institutions. But it is difficult, in each science, to chuse and conveniently
ordain elements, from which all the peculiarities of that science originate,
and into which they may be resolved. And among those who have
undertaken this employment, some have been able to collect more, but
others fewer elements. And some, indeed, have used shorter
demonstrations; but others have extended their treatise to an infinite length.
And some have omitted the method by an impossibility; but others that by
proportion; and others, again, have attempted preparations against
arguments destroying principles. So that many methods of elementary
institution have been invented by particular writers on this subject. But it is
requisite that this treatise should entirely remove every thing superfluous,
because it is an impediment to science. But every thing should be chosen,
which contains and concludes the thing proposed; for this is most
convenient and useful in science. The greatest care, likewise, should be paid
to clearness and brevity; for the contraries to these, disturb our cogitation.
Lastly, it should vindicate to itself, the universal comprehension of
theorems, in their proper bounds: for such things as divide learning into
particular fragments, produce an incomprehensible knowledge. But in all
these modes, any one may easily find, that the elementary institution of
Euclid excels the institutions of others. For its utility, indeed, especially
confers to the contemplation of primary figures: but the transition from
things more simple to such as are more various, and also that perception,
which from axioms possesses the beginning of knowledge, produces
clearness, and an orderly tradition: and the migration from first and
principal theorems to the objects of enquiry, effects the universality of
demonstration. For whatever he seems to omit, may either be known by the
same ways, as the construction of a scalene and isosceles triangle[120]: or
because they are difficult, and capable of infinite variety, they are far
remote from the election of elements, such as the doctrine of perturbate
proportions, which Apollonius has copiously handled: or, lastly, because
they may be easily constructed from the things delivered, as from causes,
such as many species of angles and lines. For these, indeed, were omitted

by Euclid, and are largely discoursed of by others, and are known from
simple propositions. And thus much concerning the universal elementary
institution of geometry.

C H A P. VIII.
Concerning the Order of Geometrical Discourses.

B ut let us now explain the universal order of the discourses contained in
geometry. Because then, we assert that this science consists from
hypothesis[121], and demonstrates its consequent propositions from definite
principles (for one science only, I mean the first philosophy, is without
supposition, but all the rest assume their principles from this) it is necessary
that he who constructs the geometrical institution of elements, should
separately deliver the principles of the science, and separately the
conclusions which flow from those principles; and that he should render no
reason concerning the nature or truth of the principles, but should confirm
by reasons, the things consequent to these geometric principles. For no
science demonstrates its own principles, nor discourses concerning them;
but procures to itself a belief of their reality, and they become more evident
to the particular science to which they belong than the things derived from
them as their source. And these, indeed, science knows by themselves; but
their consequents, through the medium of these. For thus, also, the natural
philosopher propagates his reasons from a definite principle, supposing the
existence of motion. Thus too, the physician, and he who is skilled in any of
the other sciences and arts. For if any one mingles principles, and things
flowing from principles into one and the same, he disturbs the whole order
of knowledge, and conglutinates things which can never mutually agree;
since a principle, and its emanating consequent, are naturally distinct from
each other. In the first place, therefore (as I have said), principles in the
geometric institution are to be distinguished from their consequents, which
is performed by Euclid in each of his books; who, before every treatise,
exhibits the common principles of this science; and afterwards divides these
common principles into hypotheses, petitions, and axioms. For all these
mutually differ; nor is an axiom, petition, and hypothesis the same,
according to the demoniacal Aristotle; but when that which is assumed in
the order of a principle, is indeed known to the learner, and credible by
itself, it is an axiom: such as, that things equal to the same, are mutually
equal to each other. But when any one, hearing another speak concerning
that of which he has no self-evident knowledge, gives this assent to its

assumption, this is hypothesis. For that a circle is a figure of such a
particular kind, we presume (not according to any common conception)
without any preceding doctrine. But when, again, that which is asserted was
neither known, nor admitted by the learner, yet is assumed, then (says he)
we call it petition; as the assumption that all right angles are equal. But the
truth of this is evinced by those who study to treat of some petition, as of
that which cannot by itself be admitted by any one. And thus, according to
the doctrine of Aristotle[122], are axiom, petition, and supposition
distinguished. But oftentimes, some denominate all these hypotheses, in the
same manner as the Stoics call every simple enunciation an axiom. So that,
according to their opinion, hypotheses also will be axioms; but, according
to the opinion of others, axioms will be called suppositions. Again, such
things as flow from principles are divided into problems and theorems. The
first, indeed, containing the origin, sections, ablations, or additions of
figures, and all the affections with which they are conversant; but the other
exhibiting the accidents essential to each figure. For, as things effective of
science, participate of contemplation, in the same manner things
contemplative previously assume problems in the place of operations. But
formerly some of the ancient mathematicians thought that all geometrical
propositions should be called theorems, as the followers of Speusippus and
Amphinomus, believing, that to contemplative sciences, the appellation of
theorems is more proper than that of problems; especially since they
discourse concerning eternal and immutable objects. For origin does not
subsist among things eternal: on which account, problems cannot have any
place in these sciences; since they enunciate origin, and the production of
that which formerly had no existence, as the construction of an equilateral
triangle, or the description of a square on a given right line, or the position
of a right line at a given point. It is better, therefore (say they), to assert that
all propositions are of the speculative kind; but that we perceive their
origin, not by production, but by knowledge, receiving things eternal as if
they were generated; and on this account we ought to conceive all those
theorematically, but not problematically. But others, on the contrary, think
that all should be called problems; as those mathematicians who have
followed Menæchmus. But that the office of problems is twofold,
sometimes, indeed, to procure the thing sought; but at other times when
they have received the determinate object of enquiry, to see, either what it
is, or of what kind it is, or what affection it possesses, or what its relation is

to another. And, indeed, the assertions of each are right; for the followers of
Speusippus well perceive. Since the problems of geometry are not of the
same kind, with such as are mechanical. For these are sensibles, and are
endued with origin, and mutation of every kind. And, on the other hand,
those who follow Menæchmus do not dissent from truth: since the
inventions of theorems cannot by any means take place without an approach
into matter; I mean intelligible matter. Reasons, therefore, proceeding into
this, and giving form to its formless nature, are not undeservedly said to be
assimilated to generations. For we say that the motion of our cogitation, and
the production of its inherent reasons, is the origin of the figures situated in
the phantasy, and of the affections with which they are conversant: for there
constructions and sections, positions and applications, additions and
ablations, exist: but every thing resident in cogitation, subsists without
origin and mutation. There are, therefore, both geometrical problems and
theorems. But, because contemplation abounds in geometry, as production
in mechanics, all problems participate of contemplation; but every thing
contemplative is not problematical. For demonstrations are entirely the
work of contemplation; but every thing in geometry posterior to the
principles, is assumed by demonstration. Hence, a theorem is more
common: but all theorems do not require problems; for there are some
which possess from themselves the demonstration of the thing sought. But
others, distinguishing a theorem from a problem, say, that indeed every
problem receives whatever is predicated of its matter, together with its own
opposite: but that every theorem receives, indeed, its symptom predicate,
but not its opposite. But I call the matter of these, that genus which is the
subject of enquiry; as for instance, a triangle, quadrangle, or a circle: but the
symptom predicate, that which is denominated an essential accident, as
equality, or section, or position, or some other affection of this kind. When,
therefore, any one proposes to inscribe an equilateral triangle in a circle, he
proposes a problem: for it is possible to inscribe one that is not equilateral.
But when any one asserts that the angles at the base of an isosceles triangle
are equal, we must affirm that he proposes a theorem; for it is not possible
that the angles at the base of an isosceles triangle should be unequal to each
other. On which account, if any one forming problematically, should say
that he wishes to inscribe a right angle in a semi-circle, he must be
considered as ignorant of geometry; since every angle in a semi-circle is
necessarily a right one. Hence, propositions which have an universal

symptom, attending the whole matter, must be called theorems; but those in
which the symptom is not universal, and does not attend its subject, must be
considered as problems. As to bisect a given terminated right line, or to cut
it into equal parts: for it is possible to cut it into unequal parts. To bisect
every rectilinear angle, or divide it into equal parts; for a division may be
given into unequal parts. On a given right line to describe a quadrangle; for
a figure that is not quadrangular may be described. And, in short, all of this
kind belong to the problematical order. But the followers of Zenodotus, who
was familiar with the doctrine of Oenopides, but the disciple of Andron,
distinguish a theorem from a problem, so far as a theorem enquires what the
symptom is which is predicated of the matter it contains; but a problem
enquires what that is, the existence of which is granted. From whence the
followers of Possidonius define a theorem a proposition, by which it is
enquired whether a thing exists or not; but a problem, a proposition, in
which it is enquired what a thing is, or the manner of its existence. And they
say that we ought to form the contemplating proposition by enunciating, as
that every triangle has two sides greater than the remaining one, and that the
angles at the base of every isosceles triangle are equal: but we must form
the problematical proposition, as if enquiring whether a triangle is to be
constructed upon this right line. For there is a difference, say they,
absolutely and indefinitely, to enquire whether the thing proposed is from a
given point to erect a right line at right angles to a given line, and to behold
what the perpendicular is. And thus, from what has been said, it is manifest
there is some difference between a problem and a theorem. But that the
elementary institution of Euclid, also, consists partly of problems, and
partly of theorems, will be manifest from considering the several
propositions. Since, in the conclusion of his demonstrations, he sometimes
adds (which was to be shewn) sometimes (which was to be done) the latter
sentence being the mark or symbol of problems, and the former of
theorems. For although, as we have said, demonstration takes place in
problems, yet it is often for the sake of generation; for we assume
demonstration in order to shew, that what was commanded is accomplished:
but sometimes it is worthy by itself, since the nature of the thing sought
after may be brought into the midst. But you will find Euclid sometimes
combining theorems with problems, and using them alternately, as in the
first book; but sometimes abounding with the one and not the other. For the

fourth book is wholly problematical; but the fifth is entirely composed from
theorems. And thus much concerning the order of geometrical propositions.

C H A P. IX.
Concerning the Design of the first Book,—its Division,—and a previous
Admonition to the Reader.

B ut, after these considerations, when we have determined the design of
the first book, and have exhibited its division, we shall enter upon the
treatise of the definitions. The design, then, of this book, is to deliver the
principles of the contemplation of right lines. For though a circle, and its
consideration, is more excellent than the essence and knowledge of right
lines, yet the doctrine concerning these is more adapted to us, who are
hastening to transfer our cogitation from more imperfect and sensible
natures, to such as are intelligible. For, indeed, right lined figures are proper
to sensibles, but a circle to intelligibles. Because that which is simple,
uniform, and definite, is proper to the nature of the things which are: but
that which is various, and which increases indefinitely from the number of
its containing sides, regards the fluctuating essence of sensible particulars.
Hence, in this book, the first and most principal of right lined figures are
delivered; I mean the triangle and parallelogram. For in these, as under their
proper genus, the causes of the elements are contained: viz. the isosceles
and scalene, and those which are formed from these, the equilateral triangle,
and the quadrangle, from which the four figures of the elements are
composed. We shall find, therefore, as well the origin of the equilateral
triangle as of the quadrangle; of the last, indeed, upon, but of the first from
a given right line. [[123]An equilateral triangle, therefore, is the proximate
cause of the three elements, fire, air, and water: but a quadrangle is annexed
to earth.] And lastly, the design of the first book is adapted to the whole
treatise, and confers to the universal knowledge of the mundane elements.
Besides, it instructs learners in the science concerning right-lined figures;
since it rightly invents, and accurately collects, the first principles of these.
But this book is divided into three greatest parts, of which the first
declares the origin and properties of triangles, as well according to angles,
as also according to sides. Besides, it makes mutual comparisons of these,
and beholds every one by itself. For receiving one triangle, sometimes it
considers the angles from the sides; but sometimes the sides from the

angles: and this according to equality and inequality. And supposing two
triangles, it discovers the same property again, by various methods. But the
second part combines the contemplation of parallelograms, describing their
properties and generations. And the third part shews the communication of
triangles and parallelograms, both in symptoms and mutual comparisons.
For it shews that triangles and parallelograms constituted on the same and
on equal bases, are affected with the same passions; and by complication,
when both stand upon one base: and again, after what manner a
parallelogram may be made equal to a triangle; and lastly, concerning the
proportion which in right angled triangles, the square made from the side
subtending, has to the squares containing the right angle. And such is the
division of the first Book.
But, previous to our enquiry into each of these parts, we think it
requisite to admonish the reader, that he must not require of us, those small
assumptions, and cases, and whatever else there may be of that kind, which
has been divulged by our predecessors. For we are satiated with these, and
shall, therefore, but rarely adopt them in our discourse. But whatever has a
more difficult contemplation, and regards universal philosophy, of this we
shall make a particular relation: imitating the Pythagoreans, with whom this
ænigma was common, “a[124] figure and a step: but not a figure and three
oboli,” shewing by this, that it is requisite to pursue that philosophy which
ascends every theorem by a step, and raises the soul on high; but does not
suffer it to remain among sensibles, to fill up the use attendant on mortals,
and, consulting for this, to neglect the elevation which rises from hence to
an intelligible essence.

D E F I N I T I O N S .
DEFINITION I.

A Point is that which has no Parts.

T hat geometry, according to the transition which takes place from things
more composite to such as are more simple, runs from body, which is
diffused into distance by three dimensions, to a superficies by which it is
bounded; but from superficies to a line, the boundary of superficies; and
from a line to a point destitute of all dimension, has been often said, and is
perfectly manifest. But because these terms, in many places, on account of
their simplicity, appear to be more excellent than the nature of composites;
but in many, as when they subsist in things which they terminate, they are
similar to accidents, it is necessary to determine in what genera of beings
each of these may be beheld[125]. I say then, that such things as are destitute
of matter, which subsist in separate reasons, and in those forms which are
placed under themselves, are always allotted a subsistence of more simple
essences, superior to the subsistence of such as are more composite. On this
account, both in intellect, and in the ornaments, as well of the middle kind
as among those peculiar to the soul, and in natures themselves, the terms
which proximately vivify bodies, excel according to essence the things
which are terminated; and are more impartible, more uniform, and more
primary than these. For in immaterial forms, unity is more perfect than
multitude; that which is impartible, than that which is endued with
unbounded progression; and that which terminates, than that which receives
bound from another. But such things as are indigent of matter, and abide in
others, and degenerate from the perfection of their essence, which are
scattered about subjects, and have an unnatural union, are allotted more
composite reasons, prior to such as are more simple. Hence, things which
appear in the phantasy invested with form, and the matter of the figures
which the phantasy contains, and whatever in sensibles is generated by
nature, have, in a preceding order, the reasons of the things terminated; but
the reasons which terminate, in a following and adventitious rank[126].] For
lest that which is distributed into three dimensions, should be extended into
infinite magnitude, either according to intelligence or sense, it was every
way terminated by superficies. And lest a plane superficies should conceal

itself in an infinite progression, a line approaching opposed its diffusion,
and gave bound to its indefinite extension. And, in like manner, a point
limited the progressions of a line; composite natures deriving their
subsistence from such as are simple. For this also is again manifest, that in
separate forms the reasons of terms subsist in themselves, but not in those
which are terminated; and abiding such as they are in reality, possess a
power of constituting secondary natures. But, in inseparable forms they
give themselves up to things which are terminated, reside in them, become,
as it were, their parts, and are replenished with baser natures. On which
account, that which is impartible is there endued with a partible essence,
and that which is void of latitude is diffused into breadth. And terms are no
longer able to preserve their simplicity and purity. For since they abide in
another, they necessarily change their own nature into the matter of their
containing subject. Matter, indeed, disturbs the perfection of these, and
causes the reason of a plane to become a profound plane; but obscuring the
one dimension of a line, causes it to be every way partible; and gives
corporeity to the indivisibility of a point, and separates it together with the
natures which it terminates. For all these reasons falling into matter, the one
kind from cogitation into intelligible matter, but the other from nature into
that which is sensible, are replenished with their containing subjects; and
depart from their own simplicity, into foreign compositions and intervals.
But here a doubt arises how all these, existing in intellect and soul in an
impartible manner, and without any dimension, are distributed into matter,
some indeed, principally, but others on account of its nature? Shall we say
that there is a certain order in immaterial forms, so that some are allotted
the first, some the middle, and others the last place; and that of forms some
are more uniform, but that others are more multiplied; and that some have
their powers collected together, but others tending into interval; and that
some, again, border upon bound, but that others are proximate to infinity?
For though all participate of these two principles, yet some originate from
bound, but others from infinity, of which they more largely participate.
Hence, a point is entirely impartible, since it subsists according to bound,
yet it occultly contains an infinite power, by which it produces every
interval, and the progression of all intervals, unfolds its infinite power. But
body, and the reason of body, participates more of an infinite nature; on
which account it is among the number of things terminated by another, and
divisible in infinitum, according to all dimensions. But the mediums

between these, according to the distance of the extremes, are either among
the number of things which have an abundance of bound; or among such as
have an affluence of infinity: on which account they both terminate and are
terminated. For, indeed, so far as they consist from bound, they are able to
terminate others; but so far as they participate of infinity, they are indigent
of termination from others, Hence, since a point is also a bound, it preserves
its proper power in participation: but since it likewise contains infinity
occultly, and is compelled to be every where present with the natures which
it terminates, it resides with them infinitely. And, because among
immaterial forms there was a certain infinite power capable of producing
things distant from each other by intervals, a point is present with its
participants in capacity. For infinity in intelligibles is the primary cause and
prolific power of the universe; but in material natures it is imperfect, and is
alone all things in dormant capacity. And in short, those forms which, on
account of their simplicity and impartibility, hold a superior rank among
principles, preserve, indeed, (in conformity to their nature,) their own
property in their participations, but become worse than more composite
reasons. For matter is able to participate these more clearly, and to be
prepared for their reception, rather than that of the most simple causes of
beings. On which account, the vestigies of separate principles descend into
matter; but the participations of those in a second and third order, become
more conspicuous. Hence, matter participates more of the cause of body,
than of a plane; and of this more than the form of a line; and of this still
more than that of a point, which contains all these, and is the boundary of
them all. For the reason of a point presides over this whole series, unites
and contains all partible natures, terminates their progressions, produces
them all by its infinite power, and comprehends them in its indivisible
bound. On which account also, in the images of immaterial forms, some are
the boundaries of others; but a point is the limit of them all. But that we
must not think with the Stoics, that these boundaries of bodies alone subsist
from cogitation; but that there are certain natures of this kind among beings,
which previously contain the demiurgical reasons of things, we shall be
enabled to remember, if we regard the whole world, the convolutions of its
parts, the centres of those convolutions, and the axes which penetrate
through the whole of these revolving circles. For the centres subsist in
energy, since they contain the spheres, preserve them in their proper state,
unite their intervals, and bind and establish to themselves the powers which

they possess. But the axes themselves being in an immoveable position,
evolve the spheres, give them a circular motion, and a revolution round
their own abiding nature. And the poles of the spheres, which both
terminate the axes, and bind in themselves the other convolutions, do they
not perspicuously evince, that points are endued with demiurgical and
capacious powers, that they are perfective of every thing distant by
intervals, and are the sources of union, and an unceasing motion? From
whence, indeed, Plato[127] also says, that they have an adamantine
subsistence; shewing by this, the immutable, eternal, and stable power of
their essence, ever preserving itself in the same uniform mode of existence.
He adds too, that the whole spindle of the Fates, is turned about these, and
leaps round their coercive union. But other more recondite and abstruse
discourses affirm, that the demiurgus presides over the world, seated in the
poles, and, by his divine love, converting the universe to himself. But the
Pythagoreans thought that the pole should be called the Seal of Rhea[128];
because the zoogonic, or vivific goddess, pours through these into the
universe, an inexplicable and efficacious power. And the centre they called
the prison of Jupiter; because, since Jupiter has placed a demiurgical guard
in the bosom of the world, he has firmly established it in the midst. For,
indeed, the centre abiding, the universe possesses its immoveable ornament,
and unceasing convolution: and the gods who preside over the poles, obtain
a power collective of divisible natures, and unific of such as are multiplied:
and those who are allotted the government of the axes, restrain and eternally
evolve their perpetual convolutions. And, if it is lawful to offer our own
opinion on this subject, the centres and poles of all the spheres are the
symbols of the conciliating gods, shadowing forth their imperceptible and
unifying composition. But the axes express the coherencies of the universal
ornaments; and are endued with a power of comprehending the mundane
integrities and periods, in the same manner as their presiding deities, of
such as are intellectual. But the spheres themselves are images of the gods,
called perfectors of works, copulating the principle with the end, and
excelling all figures in simplicity, similitude, and perfection. But we have
been thus prolix, that we might evince the power of impartibles, and of the
terms which the world contains, and that so far as they bear an image of
primary and most principal causes, they are allotted the most excellent
order in the universe. For centres and poles are not of the same kind with
things which are terminated; but they subsist in energy, and possess an

essence, and perfect power, which pervades through all partible natures. But
many beholding those terms which imperfectly subsist in terminated
essences, consider them as endued with a slender subsistence; and some
indeed say, that they are alone separated from sensibles by thought; but
others, that they have an essence no where but in our thoughts. However,
since the forms of all these are found both in the nature of intellect, in the
ornaments of soul, in the nature of things, and in inferior bodies, let us
consider how, according to the order they contain, they subsist in the genera
of beings. And indeed, all of them pre-exist in intellect, but in an impartible
and uniform manner: so that they all subsist according to one form, the
reason of a point, which exists occultly and impartibly. But they all subsist
in soul according to the form of a line: on which account Timæus also
composes the soul from right and circular lines: for every circle is a line
alone[129]. But they all subsist in natures, according to the reason of a plane;
and on this account, Plato commands us to manifest those natural reasons,
which are endued with a power of constituting bodies by a plane. And the
resolution of bodies into planes leads us to the proximate cause of
appearances. Lastly, they all subsist in bodies, but in a corporeal manner;
since all forms have their being in these, according to the partible nature of
bodies. Hence, all of them appear every where, and each according to its
proper order; and diversity arises from pre-dominating power. The point,
indeed, is every where impartible, and when that which is divisible into
parts, excels according to the diminution of beings, it vindicates to itself, an
illustrious subsistence of partible natures. And sometimes the point is
entirely superior, according to the excellence of cause; but sometimes it is
connected with divisibles, and sometimes it is allotted in them an
adventitious existence; and, as if swallowed up by the partition of the
lowest natures, loses its own proper impartibility. As, therefore, with respect
to the monad, one[130] is the mother of number, but the other is as matter
spread under, and the receptacle of numbers; and each of them a principle,
(yet neither of them is number), but in a different respect: in the same
manner a point also, is partly the parent and author of magnitudes; but is
partly a principle in another respect, and not according to a generative
cause. But is a point, then, the only impartible? Or may we affirm this of the
now in time, and of unity in numbers? Shall we not say, that to the
philosopher, indeed, discoursing concerning the universality of things, it is
proper to behold every thing, however falling under distribution; but that to

him who is endued with the science of particulars, who produces his
contemplation from certain definite principles, and runs back even to these,
but very little scrutinizes the progressions of beings, it is requisite to
attempt, consider, and treat concerning that impartible nature alone, which
regards his first principles; and to behold that simplicity which presides
over all the particular subjects of his knowledge? In consequence of this
reasoning, therefore, a point alone, according to the geometric matter, is
destitute of partition; but unity according to that which is arithmetical. And
the reason of a point, however in some other respects it may be imperfect,
yet is perfect in the present science. For, indeed, the physician also says,
that the elements of bodies are fire and water, and things similar to these;
and as far as to these the resolution of bodies proceeds. But the natural
philosopher passes on to more simple elements; and the one defines an
element simple as to sense, but the other simple as to reason; and both of
them properly as to their peculiar science. We must not, therefore, think that
the definition of a point is faulty, nor determine it as imperfect; for so far as
pertains to the geometric matter, and its principles, it is sufficiently
delivered. This alone, indeed, is wanting to its completion, that the
definition does not clearly say, that which is impartible with me is a point;
and my principle, and that which I contain as most simple, is nothing else
than this. And after this manner it is proper to hear the geometrician
addressing us. Euclid, therefore, from a negation of parts, declares to us a
principle, leading to the theory of its whole subject nature. For negative
discourses are proper to principles, as Parmenides teaches us, who delivers
the doctrine concerning the first and last cause, by negations alone. Since
every principle consists of an essence different from its flowing
consequents; and the negations of these exhibit to us the property of their
source. For that it is, indeed, the cause of these, yet at the same time has
nothing in common with these, becomes perspicuous from a doctrine of this
kind. But here a doubt may arise, how, since the phantasy receives all things
invested with forms, and in a partible manner, the geometrician beholds in it
the point destitute of parts? For it is not because they are reasons existing in
cogitation, but the phantasy receives the resemblances of intellectual and
divine forms according to its own proper nature, exhibiting in its shadowy
bosom the forms of formless natures, and clothing with figure things
entirely free from the affections of figure. To this ambiguity we must say,
that the species of imaginative motion is neither alone partible, nor

impartible; but that it proceeds from the impartible to the partible, and from
the formless nature to that which is expressed by form. For if it was partible
alone, it could not preserve in itself many impressions of forms, since the
subsequent would obscure the pre-existent figures: for no body can contain
at once, and according to the same situation, a multitude of figures; but the
former will be blotted out by the succession of the latter. But if it was alone
impartible, it would not be inferior to cogitation, and to soul, which surveys
all things in an impartible manner. Hence, it is necessary that it should
indeed begin from an impartible according to its motion, and from thence
draw forth the folded and scattered form of every thing falling under
cogitation, and penetrating to its shadowy receptacle: but, that it should at
length end in form, figure, and interval. And if it be allotted a nature of this
kind, it will, after a certain manner, contain an impartible essence: and a
point, according to this, must be said to have its principal subsistence: for
the form of a line is contracted in the phantasy according to this. Hence,
because it possesses a twofold power, impartible and partible, it will indeed
contain a point in an impartible, and intervals in a partible manner. But as
the Pythagoreans define a point to be unity having position, let us consider
what they mean. That numbers, indeed, are more immaterial and more pure
than magnitudes, and that the principle of numbers is more simple than the
principle of magnitudes, is manifest to every one: but when they say that a
point is unity endued with position, they appear to me to evince that unity
and number subsist in opinion: I mean monadic number[131]. On which
account, every number, as the pentad and the heptad, is one in every soul,
and not many; and they are destitute of figure and adventitious form. But a
point openly presents itself in the phantasy, subsists, as it were, in place,
and is material according to intelligible matter. Unity, therefore, has no
position, so far as it is immaterial, and free from all interval and place: but a
point has position, so far as it appears seated in the bosom of the phantasy,
and has a material subsistence. But unity is still more simple than a point,
on account of the community of principles. Since a point exceeds unity
according to position; but appositions in incorporeals produce diminutions
of those natures, by which the appositions are received.

D E F I N I T I O N II.

A Line is a Length without Breadth.

A Line obtains the second place in the Definitions, as it is by far the first
and most simple interval, which the geometrician calls a length, adding
also without breadth; since a line, in respect of a superficies, ranks as a
principle. For he defines a point, as it is the principle of all magnitudes, by
negation alone; but a line, as well by affirmation as by negation. Hence it is
a length, and by this exceeds the impartibility of a point; but it is without
breadth, because it is separated from other dimensions. For, indeed, every
thing which is void of breadth, is also destitute of bulk, but the contrary is
not true, that every thing void of bulk is also destitute of breadth. Since,
therefore, he has removed breadth from a line, he has also removed at the
same time bulk. On which account he does not add, that a line also has no
thickness, because this property is consequent to the notion of being
without breadth. But it is defined by others in various ways: for some call it
the flux of a point, but others a magnitude contained by one interval. And
this definition, indeed; is perfect, and sufficiently explains the essence of a
line; but that which calls it the flux of a point, appears to manifest its nature
from its producing cause; and does not express every line, but alone that
which is immaterial. For this is produced by a point, which though
impartible itself, is the cause of being to partible natures. But the flux of a
point, shews its progression and prolific power, approaching to every
interval, receiving no detriment, perpetually abiding the same, and affording
essence to all partible magnitudes. However, these observations are known,
and manifest to every one. But we shall recall into our memory, discourses
more Pythagorical, which determine a point as analogous to unity, a line to
the duad, a superficies to the triad, and body to the tetrad. [[132]Yet when we
compare those which receive interval together, we shall find a line monadic;
but a superficies dyadic, and a solid body triadic.] From whence also,
Aristotle[133] says; that body is perfected by the ternary number. And,
indeed, this is not wonderful, that a point, on account of its impartibility,
should be assimilated to unity; but that things subsequent to a point, should
subsist according to numbers proceeding from unity, and should preserve
the same proportion to a point, as numbers to unity; and that every one
should participate of its proximate superior, and have the same proportion

to its kindred, and following degree, as the superior to this, which is the
immediate consequent. [[134]For example, that a line has the order of the
duad with respect to the point, but of unity to a superficies; and that this last
has the relation of a triad to the point, but of the duad to a solid.] And on
this account, body is tetradic, with respect to a point, but triadic as to a line.
Each order, therefore, has its proportion; but the order of the Pythagoreans
is the more principal, which receives its commencement from an exalted
source, and follows the nature of beings. For a point is indeed twofold;
since it either subsists by itself, or in a line; in which last respect also, since
as a boundary it is alone and one, neither having a whole nor parts, it
imitates the supreme nature of beings. On which account too, it was placed
in a correspondent proportion to unity. [135]For as the oracle says, Unity is
there first, where the paternal unity abides. But a line is the first endued
with parts and a whole, and it is monadic because it is distant by one
interval only; and dyadic on account of its progression: for if it be infinite, it
participates of the indefinite duad; but if finite, it requires two terms, from
whence and to what place; since, on account of these it imitates totality, and
is allotted an order among totals. For unity, according to the oracle, is
extended[136], and generates two; and this produces a progression into
longitude, together with that which is distant extendedly, and with one
interval, and the matter of the duad. But superficies, since it is both a triad
and duad, as also the receptacle of the primary figures, and that which
receives the first form and species, is in a certain respect similar to the
triadic nature, which first terminates beings; and to the duad, by which they
are divided and dispersed. But a solid, since it has a triple distance, and is
distinguished by the tetrad, which is endued with a power of
comprehending all reasons, is reduced to that order in which the distinction
of corporeal ornaments appears; as also the division of the universe into
three parts, together with the tetradic property, which is generative and
female. And these observations, indeed, might be more largely discussed,
but for the present, must be omitted. Again, the discourse of the
Pythagoreans, not undeservedly, calls a line, which is the second in order,
and is constituted according to the first motion from an impartible nature,
dyadic. And that a point is posterior to unity, a line to the duad, and a
superficies to the triad, Parmenides himself shews, by first of all taking
away multitude from one by negation, and afterwards the whole. Because,
if multitude is before that which is a whole, number also will be prior to

that which is continuous, and the duad to the line, and unity to the point:
since the epithet not many, belongs to unity which generates multitude, but
to the point, the term not a whole, is proper, because it produces a whole;
for this is said to have no part. And these things are affirmed of a line, while
we more accurately contemplate its nature. But we should also admit the
followers of Apollonius, who say, that we obtain a notion of a line, when we
are ordered to measure the lengths alone, either of ways or walls; for we do
not then subjoin either breadth or bulk, but only make one distance the
object of our consideration. In the same manner we perceive superficies,
when we measure fields; and a solid, when we take the dimensions of wells.
For then, collecting all the distances together, we say, that the space of the
well is so much, according to length, breadth, and depth. But a line may
become the object of our sensation, if we behold the divisions of lucid
places from those which are dark, and survey the moon when dichotomized:
for this medium has no distance with respect to latitude; but is endued with
longitude, which is extended together with the light and shadow.

D E F I N I T I O N III.

But the Extremities of a Line are Points.

E very composite receives its bound from that which is simple, and every
thing partible from that which is impartible; and the images of these
openly present themselves in mathematical principles. For when it is said
that a line is terminated by points, it seems manifestly to make it of itself
infinite, because, on account of its proper progression, it has no extremity.
As, therefore, the duad is terminated by unity, and reduces its own
intolerable boldness under bound, when it is restrained in its comprehensive
embrace: so a line also is limited by the points which it contains. For, since
it is similar to the duad, it participates of a point having the relation of unity,
according to the nature of the duad. Indeed, in imaginative, as well as in
sensible forms, the points themselves terminate the lines in which they
reside. But in immaterial forms, the reason of the impartible point pre-exists
separate and apart; but when proceeding from thence by far the first of all,
by determining itself with interval, moving itself, and flowing in infinite
progression, and imitating the indefinite duad, it is restrained indeed, by its
proper principle, is united by its power, and on every side seized by its
coercive bound. Hence it is, at the same time, both infinite and finite:
infinite, indeed, according to its progression; but finite according to its
participation of a terminating cause. So that, when it approaches to this
cause, it is detained in its comprehension, and is terminated according to its
union. Hence too, in the images of incorporeal forms, a point is said to
terminate a line, by occupying its beginning and end. Bound, therefore, in
immaterials, is separated from that which is bounded: but here it is twofold;
for it subsists in that which is terminated. And this affords a wonderful
symptom, that forms; indeed, abiding in themselves, precede their
participants according to cause; but when giving themselves up to their
subordinate natures, subsist according to their diversified properties: since
they are multiplied and distributed together with these, and receive the
division of their subjects. Besides, this also must be previously received
concerning a line, that our geometrician uses it in a threefold acceptation.
As terminated on both sides, and finite; as in the problem[137] which says,
Upon a given terminated right line to construct an equilateral triangle. And
as partly infinite and partly finite; as in the problem which commands us

from three right lines, which are equal to three given right lines, to construct
a triangle; for in the construction of the problem, he says, Let there be
placed a certain right line, on one part finite, but on the other part infinite.
And again, a line is received by Euclid as on both sides infinite; as in the
problem which says, Upon a given infinite right line, from a given point,
which is not in that line, to let fall a perpendicular. But, besides this, the
following doubts, since they are worthy of solution, must not be omitted.
How are points called the extremities of a line? and of what line, since they
can neither be the bounds of one that is infinite, nor of every finite? For
there is a certain line, which is both finite, and has not points for its
extremities. And such is a circular line, which returns into itself, and is not
bounded by points, like a right line. And such also is the ellipsis, or line like
a shield. Is it therefore requisite to behold a line, considered as a line? for
we must receive a certain circumference, which is terminated by points, and
a part of the elliptic line; having, in like manner, its extremities bounded by
points. But every circular and elliptic line, assumes to itself another certain
property, by which it is not line alone, but is also endued with a power of
perfecting figure[138]. Lines, themselves, therefore, have their extremities
terminated by points; but those which are effective of such like figures,
return into themselves. And, indeed, if you conceive them to be described,
you will also find how they are bounded by points; but if you receive them
already described, and connect the end with the beginning, you can no
longer behold their extremes.

D E F I N I T I O N IV.

A Right Line, is that which is equally situated between its bounding Points.

P lato establishing two most simple and principal species of lines, the right
and the circular, composes all the rest from the mixture of these; I mean
such as are called curve lines, some of which are formed from planes; but
others subsist about solids; and whatever species of curve lines are
produced by the sections of solids. And it seems, indeed, that a point (if it
be lawful so to speak) bears an image of the one itself, according to Plato:
for unity has no part, as he also shews in the Parmenides. But, because after
unity itself there are three hypostases, or substances, bound, infinite, and
that which is mixed from these, the species of lines, angles, and figures,
which subsist in the nature of things originate from thence. And, indeed, a
circumference and a circular angle, and a circle among plane figures, and a
sphere among solids, are analogous to bound. But a right line corresponds
to infinity, according to all these; for it properly belongs to all, if it is beheld
as existing in each. But that which is mixed in all these, is analogous to the
mixt which subsists among intelligibles. For lines are mixed, as those which
are called spirals. And angles, as the semi-circular and cornicular[139]. And
plane figures, as segments and apsides; but solids, as cones and cylinders,
and others of that kind. Bound, therefore, infinite, and that which is mixed,
are participated by all these. But Aristotle[140] likewise assents to Plato; for
every species of lines, says he, is either right or circular, or mixed from
these two. From whence also there are three motions, one according to a
right line; the other circular; and the third mixed. But some oppose this
division, and say that there are not two simple lines alone, but that there is a
certain third line given, i. e. a helix or spiral, which is described about a
cylinder[141], when, whilst a right line is moved round the superficies of the
cylinder, a point in the line is carried along with an equal celerity. For by
this means, a helix, or circumvolute line, is produced, which adapts all the
parts of itself to all, according to a similitude of parts, as Apollonius shews
in his book concerning the Cochlea; which passion, among all spirals,
agrees to this alone. For the parts of a plane helix are dissimilar among
themselves; as also of those which are described about a cone and sphere.
But the cylindric spiral alone, consists of similar parts in the same manner
as a right and circular line. Are there, then, three simple lines, and not two

only? To which doubt we reply, that a helix of this kind is, indeed, of
similar parts, as Apollonius teaches, but is by no means simple; since
among natural productions, gold and silver are composed of similar parts,
but are not simple bodies. But the generation of the cylindric helix evinces
that its mixture is from things simple; for it originates while a right line is
circularly moved round the axis of the cylinder, a point at the same time
flowing along in the right line. Two simple motions, therefore, compose its
nature; and, on this account, it is among the number of mixt lines, and not
among such as are simple: for that which is composed from dissimilars is
not simple, but mixt. Hence, Geminus, with great propriety, when he admits
that some simple lines may be produced from many motions, does not grant
that every such line is mixt; but that alone, which arises from dissimilar
motions. For if you conceive a square, and two motions which are
performed with an equal celerity, one according to the length, but the other
according to the breadth, a right line or the diameter will be produced; but
the right line will not, on this account, be mixed: for no other line precedes
it, formed by a simple motion, as we asserted of the cylindric helix. Nor yet,
if you suppose a right line, moving in a right angle, and by a bisection to
describe a circle[142], is the circular line, on this account, produced with
mixture: for the extremities of that which is moved after this manner, since
they are equally moved, will describe a right line; and the bisection, since it
is unequally devolved, will delineate a circle; but the other points will
describe an ellipsis. On which account, the generation of a circular line is
the consequence of that inequality of lation arising from the bisection;
because a right line was supposed to be moved in a right angle, but not in a
natural manner. And thus much concerning the generation of lines. But it
seems, that of the two simple lines, the right and the circular, the right line
is the more simple; for in this, dissimilitude cannot be conceived, even in
opinion. But in the circular line, the concave and the convex, indicate
dissimilitude. And a right line, indeed, does not infer a circumference
according to thought; but a circumference brings with it a right line, though
not according to its generation, yet with respect to its centre. But what if it
should be said that a circumference requires a right line to its construction!
For if either extreme of a right line remains fixt, but the other is moved, it
will doubtless describe a circle, whose centre will be the abiding extreme of
the right line. Shall we say that the generator of the circle is the point which
is carried about the abiding point, but not the right line itself? For the line

only determines the distance, but the point composes the circular line, while
it is moved in a circular manner: but of this enough. Again, a circumference
appears to be proximate to bound, and to have the same proportion to other
lines, as bound to the universality of things. For it is finite, and is alone
among simple lines perfective of figure. But a right line is proximate to
infinity; for its capacity of infinite extension never fails: and as all the rest
are produced from bound and infinite, in the same manner from the circular
and right line, every mixt genus of lines is composed, as well of planes as of
those which consist in solid bodies. And on this account, the soul also[143]
previously assumed into herself the right and circular according to her
essence, that she might moderate all the co-ordination of infinite, and all the
nature of bound, which the world contains. By a right line, indeed,
constituting the progression of these principles into the universe; but by a
circular line, their return to their original source: and by the one, producing
all things into multitude; but by the other, collecting them into one. And not
only the soul, but he also who produced the soul, and endued her with these
powers, contains in himself both these primary causes. For when he
previously assumed the beginning, middle, and end of all things, he
terminated right lines (says Plato[144]), by a circular progression according
to nature. And proceeding to all things by provident energies, and returning
to himself, he established himself, says Timæus, after his own peculiar
manner. But a right line is the mark or symbol of a providence,
indeclinable, incapable of perversion, immaculate, never-failing,
omnipotent, and present to all beings, and to every part of the universe. But
a circumference, and that which environs, is the symbol of an energy
retiring into union with itself, and which rules over all things according to
one intellectual bound. When, therefore, the demiurgus of the universe had
established in himself these two principles, the right and the circular line,
and had given them dominion, he produced from himself two unities; the
one, indeed, energizing according to the circular line, and being effective of
intellectual essences; but the other according to the right line, and affording
an origin to sensible natures. But because the soul is allotted a middle
situation between intellectuals and sensibles, so far, indeed, as she adheres
to an intellectual nature, she energizes according to the circle; but so far as
she presides over sensibles, she provides for their welfare according to the
right line: and thus much concerning the similitude of these forms to the
universality of things. But Euclid, indeed, has properly delivered the present

definition of a line; by which he shews that a right line alone occupies a
space equal to that which is situated between its points: for as much as is
the distance of one point from another, so great is the magnitude of the lines
terminated by the points. And this is the meaning of being equally situated
between its extremes. For if you take two points in a circumference, or in
any other certain line, the space of line which is included between these,
exceeds their distance from each other; and every line, besides a right one,
appears to suffer this property. Hence, according to a common conception,
the vulgar also say, that he who walks by a right line, performs only a
necessary journey: but that they necessarily wander much, who do not
proceed in a right line. But Plato thus defines it; a right line is that whose
middle parts darken its extremes. For this passion necessarily attends things
which have a direct position; but it is not necessary that things situated in
the circumference of a circle, or in another interval, should be endued with
this property. Hence, the astrologers also say, that the sun then suffers an
eclipse when that luminary, the moon, and our eye are in one right line; for
it is then darkened through the middle position of the moon between us and
its orb. And perhaps, the passion of a right line will evince, that in the
things which are, according to processions emanating from causes, the
mediums are endued with a power of dividing the distance of the extremes,
and their mutual communication with each other. As also, according to
regressions, such things as are distant from the extremes, are converted by
mediums to their primary causes. But Archimedes defines a right line the
least of things having the same bounds. For since, according to Euclid, a
right line is equally situated between its points, it is on this account, the
least of things having the same bounds: for if a less line could be given, it
would not lie equally between its extremes: but all the other definitions of a
right line, fall into the same conclusions; as for instance, that it is
constituted in its extremities, and that one part of it is not in its subject
plane, but another, in one more sublime: and that all its parts similarly agree
to all: and that its extremes abiding, it also abides. Lastly, that it does not
perfect figure, with one line similar in species to itself: for all these
definitions express the property of a right line, which it possesses from the
simplicity of its essence, and from its having one progression the shortest of
all from one extremity to another. And thus much concerning the definitions
of a right line. But again, Geminus divides a line first into an incomposite
and composite; calling a composite, that which is refracted, and forms an

angle; but all the rest of them, he denominates incomposites. Afterwards, he
divides a composite line into that which produces figure, and that which
may be infinitely extended. And he calls that which produces figure, a
circular line, and the line of a shield[145], and that which is similar to an ivy
leaf[146]; but that which is not effective of figure, the section of a
rectangular and obtuse angular cone, the line similar to a shell[147], the right
line, and all of that kind. And again, after another manner, of the
incomposite line, one sort is simple, but the other mixt. And of the simple,
one produces figure, as the circular; but the other is indefinite, as the right
line. But of the mixt, one subsists in planes, but the other in solids. And of
that which is in planes, one coincides in itself, as the figure of the ivy leaf,
which is called the cissoid; but the other may be produced in infinitum, as
the helix. But of that which is in solids, one may be considered in the
sections of solids; but the other as consisting about the solids themselves.
For the helix, indeed, which is described about a sphere or a cone, consists
about solids; but conic, or spirical sections are generated from a particular
section of solids. But, with respect to these sections, the conic were
invented by Mænechmus, which also Eratosthenes relating, says,
“Nor in a cone Mænechmian ternaries divide.”

But the spiric by Perseus, who composed an epigram on their invention, to
this purpose, “When Perseus had invented three spiral lines in five sections,
he sacrificed to the gods on the occasion.” And the three sections of a cone,
are the parabola, hyperbola, and ellipsis: but of spiral sections, one kind is
twisted and involved, like the fetlock of a horse; but another is dilated in the
middle, and deficient in each extremity: and another which is oblong, has
less space in the middle, but is dilated on each side. But the multitude of the
other mixt lines is infinite. For there is an innumerable multitude of solid
figures, from which there are constituted multiform sections. For a right
line, while it is circularly moved[148], does not make a certain determinate
superficies, nor yet conical, nor conchoidal lines, nor circumferences
themselves. Hence, if these solids are multifariously cut, they will exhibit
various species of lines. Lastly, of those lines which consist about solids,
some are of similar parts, as the helixes about a cylinder; but others of
dissimilar parts, as all the rest. From these divisions, therefore, we may
collect, that there are only three lines of similar parts, the right, the circular,

and the cylindric helix. The two simple ones, indeed, existing in a plane, but
the one mixt, about a solid. And this Geminus evidently demonstrates, when
he shews, that if two right lines are extended from one point, to a line of
similar parts, so as to make equal angles upon that line, they shall be equal
to each other. And the demonstrations of this may be received by the
studious, from his volumes; since in these he delivers the origin of spiral,
conchoidal, and cissoidal lines. But we have barely related the names and
divisions of these lines, for the purpose of exciting the ingenious to their
investigation; as we think, that an accurate enquiry after the method of
detecting the properties of each, would be superfluous in the present
undertaking: since the geometrician only unfolds to us in this work, simple
and primary lines, i.e. the right line, in the present definition; but a circular
line, in the tradition of a circle. For he then says, that the line terminating
the circle, is the circumference. But he makes no mention of mixt lines,
though he was well acquainted with mixt angles, I mean, the semi-circular
and cornicular: as also with plane mixt figures, i.e. segments and sectors;
and with solids, viz. cones and cylinders. Of each of the rest, therefore, he
delivers three species; but of lines only two, i. e. the right and circular: for
he thought it requisite in discourses concerning things simple, to assume
simple species; and all the rest are more composite than lines. Hence, in
imitation of the geometrician, we also shall terminate their explanation with
simple lines.

D E F I N I T I O N V.

A Superficies is that which has only Length and Breadth.

A fter a point and a line, a superficies is placed, which is distant by a
twofold interval, length and breadth. But this also remaining destitute
of thickness or bulk, possesses a nature more simple than body, which is
distant by a triple dimension. On which account the geometrician adds to
the two intervals the particle only, because the third interval does not exist
in superficies. And this is equivalent to a negation of bulk, as here also he
shews the excellency of superficies compared to a solid with respect to
simplicity, by negation, or by an addition equivalent to negation: but the
diminution which it possesses, if compared with the preceding terms, by the
affirmations themselves. But others define a superficies to be the boundary
of body, which is almost affirming the same as the definition of Euclid;
since that which terminates is exceeded in one dimension, by that which is
terminated. And others, a magnitude different by two intervals. Lastly,
others declaring the same affection, form its assignation in a somewhat
different manner. But they say we have a knowledge of superficies when we
measure fields, and distinguish their extremities according to length and
breadth; but that we receive a certain sensation of it, when we behold
shadows. For as they are without bulk, because they cannot penetrate into
the interior part of the earth, they have only length and breadth. But the
Pythagoreans say, that it is assimilated to the triad; because the ternary is by
far the first cause to all the figures; which a superficies contains. For a
circle, which is the principle of orbicular figures, occultly possesses the
ternary, by its centre, interval, and circumference. But a triangle, which
ranks as the first among all right-lined figures, on every side evinces that it
is enclosed by the triad, and receives its form from its perfect nature.

D E F I N I T I O N VI.

The Extremities of a Superficies are Lines.

F rom these also, as images, we may understand, that things more simple
procure bound and an end to every one of their proximate natures: for
soul perfects and determines the operations of nature; and nature the motion
of bodies. And prior to these, intellect measures the convolutions of soul;
and unity the life of intellect; for that is the measure of all. Just as in these
also, a solid is terminated by a superficies; but a superficies by a line; and a
line by a point; for that is the boundary of them all. Hence, the line existing
uniformly in immaterial forms and impartible reasons, terminates and
restrains the various motion of a superficies in its progression, and
proximately unites its infinity. But in the images of these, when that which
bounds supervenes that which is bounded, it causes, by this means, its
limitation and bound. But if it should be enquired how lines are the
extremities of every superficies, since they are not the extremes of every
finite figure; for the superficies of a sphere is terminated indeed, yet not by
lines, but by itself? In answer to this, we must say, that by receiving a
superficies so far as it is distant by a twofold interval, we shall find it
terminated by lines according to length and breadth. But if we behold a
spherical superficies, we must receive it as that which is endued with figure;
which possesses another quality, and conjoins the end with the beginning;
and loses its two extremities in the comprehensive embraces of one: and
this one extremity subsists in capacity only, and not in energy.

D E F I N I T I O N VII.

A Plane Superficies is that which is equally situated between its bounding
Lines.

I t was not agreeable to the ancient philosophers to establish a plane
species of superficies; but they considered superficies in general, as the
representative of magnitude, which is distant by a twofold interval. For thus
the divine Plato[149] says, that geometry is contemplative of planes,
opposing it in division to stereometry, as if a plane and a superficies were
the same. And this was likewise the opinion of the demoniacal
Aristotle[150]. But Euclid and his followers consider superficies as a genus,
but a plane as its species, in the same manner as rectitude of a line. And on
this account he defines a plane separate from a superficies, after the
similitude of a right line. For he defines this last as equal to the space,
placed between its points. And in like manner, he says, that two right lines
being given, a plane superficies occupies a place equal to the space situated
between those two lines. For this is equally situated between its lines; and
others also explaining the same boundary, assert that it is constituted in its
extremities. But others define it as that to all the parts of which a right line
may be adapted[151]. But perhaps others will say, that it is the shortest of
superficies, having the same boundaries; and that its middle parts darken its
extremities; and that all the definitions of a right line may be transferred
into a plane superficies, by only changing the genus: since a right, circular,
and mixt line, commencing from lines, arrive even at solids, as we have
asserted above; for they are proportionally, both in superficies and solids.
Hence also, Parmenides says, that every figure is either right, or circular, or
mixt. But if you wish to consider the right in superficies, take a plane, to
which a right line agrees in various ways; but if a circular receive a
spherical superficies; and if a mixt, a conic or cylindric, or some one of that
genus. But it is requisite (says Geminus) since a line, and also a superficies
is called mixt, to know the measure of mixture, because it is various. For
mixture in lines, is neither by composition, nor by temperament only: since,
indeed, a helix is mixed, yet one part of it is not straight, and another part
circular, like those things which are mixed by composition: nor if a helix is
cut after any manner, does it exhibit an image of things simple, such as
those which are mixed through temperament; but in these the extremes are,

at the same time, corrupted and confused. Hence, Theodorus the
mathematician, does not rightly perceive, in thinking that this mixture is in
lines. But mixture in superficies, is neither by composition, nor by
confusion; but subsists rather by a certain temperament. For conceiving a
circle in a subject plane, and a point on high, and producing a right line
from the point to the circumference of the circle, the revolution of this line
will produce a conical superficies which is mixt. And we again resolve it
into its simple elements, by a parallel section: for by drawing a section
between the vertex and the base, which shall cut the plane of the generative
right line, we effect a circular line. But the idea of lines, shews that the
mode of mixture is not by temperament; for neither does it send us back to
the simple nature of elements: on the contrary, when superficies are cut,
they immediately exhibit to us their producing lines. The mode of mixture,
therefore, is not the same in lines and superficies. But as among lines there
were some simple, that is, the right and circular, of which the vulgar also
possess an anticipated knowledge without any previous instruction; but the
species of mixt lines require a more artificial apprehension: so among
superficies, we possess an innate notion of those which are especially
elementary, the plane and spherical; but science and its reason investigates
the variety of those which are composed through mixture. But this is an
admirable property of superficies, that their mixture in generation is
oftentimes produced from a circular line; and this also happens to a spiral
superficies. For this is understood by the revolution of a circle remaining
erect, and turning itself about the same point which is not its centre. And on
this account, a spiral also is threefold; for its centre is either in a
circumference, or within, or external to a circumference. If the centre is in
the circumference, a continued spiral is produced: if within the
circumference, an intangled one; if without, a divided one. And there are
three spiral sections corresponding to these three differences. But every
spiral line is mixt, although the motion from which it is produced is one and
circular. And mixt superficies are produced as well from simple lines, (as
we have said,) while they are moved with a motion of this kind, as from
mixt lines. Since, therefore, there are three conic lines, they produce four
mixt superficies, which they call conoids. For a rectangular conoid, is
produced from the revolution of the parabola about its axis: but that which
is formed by the ellipsis, is called a spheroid; and is the revolution is made
about the greater axis, it is an oblong; but if about the lesser a broad

spheroid. Lastly, an obtuse-angled conoid is generated from the revolution
of the hyperbola. But it is requisite to know, that sometimes we arrive at the
knowledge of superficies from lines, and sometimes the contrary; for from
conical and spiral superficies, we apprehend conical and spiral lines.
Besides, this also must be previously received concerning the difference of
lines and superficies, that there are three lines of similar parts (as we have
already observed), but only two superficies, the plane and the spherical. For
this is not true of the cylindric, since all parts of the cylindric superficies
cannot agree to all. And thus much concerning the differences of
superficies, one of which the geometrician having chosen (I mean the
plane), this also he has defined; and in this, as a subject, he contemplates
figures, and their attendant passions: for his discourse is more copious in
this than in other superficies: since, indeed, we may understand right lines,
and circles, and helixes in a plane; also the sections of circles and right
lines, contacts, and applications, and the constructions of angles of every
kind. But in other superficies, all these cannot be beheld. For how in one
that is spherical, can we apprehend a right line, or a right-lined angle? How,
lastly, in a conic or cylindric superficies, can we behold sections of circles
or right lines? Not undeservedly, therefore, does he both define this
superficies, and discuss his geometrical concerns, by exhibiting every thing
in this as in a subject; for from hence he calls the present treatise plane.
And, after this manner, it is requisite to understand that which is plane, as
projected and constituted before the eyes: but cogitation as describing all
things in this, the phantasy corresponding to a plane mirror, and the reasons
resident in cogitation as dropping their images[152] into its shadowy
receptacle.

D E F I N I T I O N VIII.
[153]A Plane Angle, is the inclination of two Lines to each other in a Plane, which
meet together, but are not in the same direction.

S ome of the ancient philosophers, placing an angle in the predicament of
relation, have said, that it is the mutual inclination of lines or planes to
each other. But others, including this in quality, as well as rectitude and
obliquity, say, that it is a certain passion of a superficies or a solid. And
others, referring it to quantity, confess that it is superficies or a solid. For
the angle which subsists in superficies is divided by a line; but that which is
in solids, by a superficies. But (say they) that which is divided by these, is
no other than magnitude, and this is not linear, since a line is divided by a
point; and therefore it follows that it must be either a superficies or a solid.
But if it is magnitude, and all finite magnitudes of the same kind have a
mutual proportion; all angles of the same kind, i. e. which subsist in
superficies, will have a mutual proportion. And hence, the cornicular will be
proportionable to a right-lined angle. But things which have a mutual
proportion, may, by multiplication, exceed each other; and therefore it may
be possible for the cornicular to exceed a right-lined angle, which, it is well
known, is impossible, since it is shewn to be less than every right-lined
angle. But if it is quality alone, like heat and cold, how is it divisible into
equal parts? For equality, inequality, and divisibility, are not less resident in
angles than in magnitudes; but they are, in like manner, essential. But if the
things in which these are essentially inherent, are quantities, and not
qualities, it is manifest that angles also are not qualities. Since the more and
the less are the proper passions of quality[154], but not equal and unequal.
On this hypothesis, therefore, angles ought not to be called unequal, and
this greater, but the other less; but they ought to be denominated dissimilars,
and one more an angle, but the other less. But that these appellations are
foreign from the essence of mathematical concerns, is obvious to every one:
for every angle receives the same definition, nor is this more an angle, but
that less. Thirdly, if an angle is inclination, and belongs to the category of
relation, it must follow, that from the existence of one inclination, there will
also be one angle, and not more than one. For if it is nothing else than the
relation of lines or planes, how is it possible there can be one relation of
lines or planes, but many angles? If, therefore, we conceive a cone cut by a

triangle from the vertex to the base, we shall behold one inclination of the
triangular lines in the semicone to the vertex; but two distinct angles: one of
which is plane, I mean that of the triangle; but the other subsists in the mixt
superficies of the cone, and both are comprehended by the two triangular
lines. The relation, therefore, of these, do not make the angle. Again, if is
necessary to call an angle either quality or quantity, or relation; for figures,
indeed, are qualities, but their mutual proportions belong to relation. It is
necessary, therefore, that an angle should be reduced under one of these
three genera. Such doubts, then, arising concerning an angle, and Euclid
calling it inclination, but Apollonius the collection of a superficies, or a
solid in one point, under a refracted line or superficies (for he seems to
define every angle universally), we shall affirm, agreeable to the sentiments
of our preceptor Syrianus, that an angle is of itself none of the aforesaid; but
is constituted from the concurrence of them all. And that, on this account, a
doubt arises among those who regard one category alone. But this is not
peculiar to an angle, but is likewise the property of a triangle. For this, too,
participates of quantity, and is called equal and unequal; because it has to
quantity the proportion of matter. But quality also, is present with this, in
consequence of its figure (since triangles are called as well similar as
equal); but it possesses this from one category, and that from another.
Hence, an angle is perfectly indigent of quantity, the subject of magnitude.
But it is also indigent of quality, by which it possesses, as it were, its proper
form and figure, Lastly, it is indigent of the relation of lines terminating, or
of superficies comprehending its form. So that an angle consists from all
these, yet is not any one of them in particular. And it is indeed divisible, and
capable of receiving equality and inequality, according to the quantity
which it contains. But it is not compelled to admit the proportion of
magnitudes of the same kind, since it has also a peculiar quantity, by which
angles are also incapable of a comparison with each other. Nor can one
inclination perfect one angle: since the quantity also, which is placed
between the inclined lines, completes its essence. If then we regard these
distinctions, we shall dissolve all absurdities, and discover that the property
of an angle is not the collection of a superficies or solid, according to
Apollonius (since these also complete its essence,) but that it is nothing else
than a superficies itself, collected into one point, and comprehended by
inclined lines, or by one line inclined to itself: and that a solid angle is the
collection of superficies mutually inclined to each other. Hence, we shall

find that a formed quantum, constituted in a certain relation, supplies its
perfect definition. And thus much we have thought requisite to assert
concerning the substance of angles, previously contemplating the common
essence of every triangle, before we divide it into species. But since there
are three opinions of an angle, Eudemus the Peripatetic, who composed a
book concerning an angle, affirms that it is quality. For, considering the
origin of an angle, he says that it is nothing else than the fraction of lines:
because, if rectitude is quality, fraction also will be quality. And hence,
since its generation is in quality, an angle will be entirely quality. But
Euclid, and those who call it inclination, place it in the category of relation.
But they call it quantity, who say that it is the first interval under a point,
that is immediately subsisting after a point. In the number of which is
Plutarch, who constrains Apollonius also into the same opinion. For it is
requisite (says he) there should be some first interval, under the inclination
of containing lines or superficies. But since the interval, which is under a
point, is continuous, it is not possible that a first interval can be assumed;
since every interval is divisible in infinitum. Besides, if we any how
distinguish a first interval, and through it draw a right line, a triangle is
produced, and not one angle. But Carpus Antiochenus says, that an angle is
quantity, and is the distance of its comprehending lines, or superficies; and
that this is distant by one interval, and yet an angle is not on that account a
line: since it is not true that every thing which is distant by only one
interval, is a line. But this surely is the most absurd of all, that there should
be any magnitude except a line, which is distant only by one interval. And
thus much concerning the nature of an angle. But with respect to the
division of angles, some consist in superficies, but others in solids. And of
those which are in superficies, some are in simple ones, but others in such
as are mixt. For an angle may be produced in a cylindric, conic, spherical,
and plane superficies. But of those which consist in simple superficies,
some are constituted in the spherical; but others in the plane. For the zodiac
itself forms angles, dividing the equinoctial in two parts, at the vertex of the
cutting superficies. And angles of this kind subsist in a spherical superficies.
But of those which are in planes, some are comprehended by simple lines,
others by mixt ones; and others, again, by both. For in the shield-like
figure[155], an angle is comprehended by the axis, and the line of the shield:
but one of these lines is mixt, and the other simple. But if a circle cuts the
shield, the angle will be comprehended by the circumference, and the

ellipsis. And when cissoids, or lines similar to an ivy leaf, closing in one
point like the leaves of ivy (from whence they derive their appellation)
make an angle, such an angle is comprehended by mixt lines. Also, when
the hippopede, or line familiar to the foot of a mare, which is one of the
spirals, inclining to another line, forms an angle, it is comprehended by
mixt lines. Lastly, the angles contained by a circumference and a right line,
are comprehended by simple lines. But of these again, some are contained
by such as are similar in species, but others by such as are dissimilar. For
two circumferences, mutually cutting, or touching each other, produce
angles: and these triple, for they are either on both sides convex, when the
convexities of the circumferences are external: or on both sides concave,
when both the concavities are external; which they call sistroides; or mixt
from convex and concave lines, as the lines called lunulas. But besides this,
angles are contained in a twofold manner, by a right line and a
circumference: for they are either contained by a right line, and a concave
circumference, as the semi-circular angle; or by a right line and a convex
circumference, as the cornicular angle. But all those which are
comprehended by two right lines, are called rectilinear angles, which have
likewise a triple difference[156]. The geometrician, therefore, in the present
hypothesis, defines all those angles which are constituted in plane
superficies, and gives them the common name of a plane angle. And the
genus of these he denominates inclination: but the place, the plane itself, for
angles have position: but their origin such, that it is requisite there should
be two lines at least, and not three as in a solid. And that these should touch
each other, and by touching, must not lie in a right line, as an angle is the
inclination and comprehension of lines: but is not distance only, according
to one interval. But if we examine this definition, in the first place it
appears that it does not admit, an angle can be perfected by one line; though
a cissoid, which is but one, perfects an angle. And, in like manner, the
hippopede. For we call the whole a cissoid, and not its portions (lest any
one should say, that the conjunction of these forms an angle) and the whole
a spiral, but not its parts. Each, therefore, since it is one, forms an angle to
itself, and not to another. But after this, he is faulty, in defining an angle to
be inclination. For how, on this hypothesis, will there be two angles, from
one inclination? How can we call angles equal and unequal? And whatever
else is usually objected against this opinion. Thirdly, and lastly, that part of
the definition, which says, and not placed in a right line, is superfluous in

certain angles, as in those which are formed from orbicular lines. For
without the assistance of this part, the definition is perfect; since the
inclination of one of the lines to the other, forms the angle. And it is not
possible that orbicular angles should be placed in a right line. And thus
much we have thought proper to say concerning the definition of Euclid;
partly, indeed, interpreting, and partly doubting its truth.

D E F I N I T I O N IX.

But when the Lines containing the Angle, are right, the Angle is called
Rectilinear.

A n angle is the symbol and image of the connection and compression,
which subsists in the divine genera, and of that order which collects
divisibles into one, particles into an impartible nature, and the many into
conciliating community. For it is the bond of a multitude of lines and
superficies, the collector of magnitude into the impartibility of points, and
the comprehender of every figure which is composed by its confining
nature. On which account, the oracles[157] call the angular junctions of
figures, knots, so far as they bring with them an image of connecting union,
and divine conjunctions, by which discrete natures mutually cohere with
each other. The angles, therefore, subsisting in superficies, express the more
immaterial, simple, and perfect unions which superficies contain: but those
which are in solids, represent the unions, which proceed even to inferiors,
and supply a community to things disjunct, and a construction of the same
nature, to things which on every side receive a perfect partition. But of the
angles in superficies, some shadow forth primary and unmixt unions; but
others, such as comprehend in themselves, an infinity of progressions. And
some, indeed, are the sources of union to intellectual forms; but others, to
sensible reasons; and others, again, are copulative of those forms which
obtain between these, a middle situation. Hence, the angles which are made
from circumferences, imitate those causes which envelop intellectual
variety in coercive union; for circumferences, hastening to coalesce with
each other, are images of intellect, and intellectual forms. On the contrary,
rectilineal angles, are the symbols of those unions which preside over
sensibles, and afford a conjunction of the reasons subsisting in these: but
mixt angles represent the preservers of the communion, as well of sensible,
as of intellectual forms, according to one immoveable union. It is requisite,
therefore, by regarding these paradigms, or exemplars, to render the causes
of each. For among the Pythagoreans we shall find various angles dedicated
to various gods. Thus, Philolaus, consecrates to some a triangular, but to
others a quadrangular angle; and to others, again, different angles.
Likewise, he permits the same to many gods, and many to the same god,
according to the different powers which they contain. And with a view to