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Title: A View of Sir Isaac Newton's Philosophy

Author: Henry Pemberton

Contributor: Richard Glover

Release date: September 28, 2016 [eBook #53161]
Most recently updated: October 23, 2024

Language: English

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*** START OF THE PROJECT GUTENBERG EBOOK A VIEW OF SIR
ISAAC NEWTON'S PHILOSOPHY ***

TRANSCRIBER’S NOTES:
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A
VIEW
OF
Sir I S A A C N E W T O N’s
PHILOSOPHY.

L O N D O N:
Printed by S . P a l m e r, 1728.

To the Noble and Right Honourable
Sir ROBERT WALPOLE.

SIR,
I Take the liberty to send you this view of Sir I s a a c
N e w t o n ’ s philosophy, which, if it were performed suitable to
the dignity of the subject, might not be a present unworthy the
acceptance of the greatest person. For his philosophy operations
of nature, which for so many ages had imployed the curiosity of mankind;
though no one before him was furnished with the strength of mind
necessary to go any depth in this difficult search. However, I am
encouraged to hope, that this attempt, imperfect as it is, to give our
countrymen in general some conception of the labours of a person, who
shall always be the boast of this nation, may be received with indulgence by
one, under whose influence these kingdoms enjoy so much happiness.
Indeed my admiration at the surprizing inventions of this great man, carries
me to conceive of him as a person, who not only must raise the glory of the
country, which gave him birth; but that he has even done honour to human
nature, by having extended the greatest and most noble of our faculties,
reason, to subjects, which, till he attempted them, appeared to be wholly
beyond the reach of our limited capacities. And what can give us a more
pleasing prospect of our own condition, than to see so exalted a proof of the
strength of that faculty, whereon the conduct of our lives, and our happiness

depends; our passions and all our motives to action being in such manner
guided by our opinions, that where these are just, our whole behaviour will
be praise-worthy? But why do I presume to detain you, Sir, with such
reflections as these, who must have the fullest experience within your own
mind, of the effects of right reason? For to what other source can be
ascribed that amiable frankness and unreserved condescension among your
friends, or that masculine perspicuity and strength of argument, whereby
you draw the admiration of the publick, while you are engaged in the most
important of all causes, the liberties of mankind?

I humbly crave leave to make the only acknowledgement within my
power, for the benefits, which I receive in common with the rest of my
countrymen from these high talents, by subscribing my self

S I R,

Your most faithful,

and

Most humble Servant,

H e n r y P e m b e r t o n.

P R E F A C E.

I
Drew up the following papers many years ago at the desire of some
friends, who, upon my taking care of the late edition of Sir I s a a c
N e w t o n ’ s Principia, perswaded me to make them publick. I laid
hold of that opportunity, when my thoughts were afresh employed on
this subject, to revise what I had formerly written. And I now send it abroad
not without some hopes of answering these two ends. My first intention was
to convey to such, as are not used to mathematical reasoning, some idea of
the philosophy of a person, who has acquired an universal reputation, and
rendered our nation famous for these speculations in the learned world. To
which purpose I have avoided using terms of art as much as possible, and
taken care to define such as I was obliged to use. Though this caution was
the less necessary at present, since many of them are become familiar words
to our language, from the great number of books wrote in it upon
philosophical subjects, and the courses of experiments, that have of late
years been given by several ingenious men. The other view I had, was to
encourage such young gentlemen as have a turn for the mathematical
sciences, to pursue those studies the more chearfully, in order to understand
in our author himself the demonstrations of the things I here declare. And to
facilitate their progress herein, I intend to proceed still farther in the
explanation of Sir I s a a c N e w t o n ’ s philosophy. For as I have received
very much pleasure from perusing his writings, I hope it is no illaudable
ambition to endeavour the rendering them more easily understood, that
greater numbers may enjoy the same satisfaction.
It will perhaps be expected, that I should say something particular of a
person, to whom I must always acknowledge my self to be much obliged.
What I have to declare on this head will be but short; for it was in the very

last years of Sir I s a a c’s life, that I had the honour of his acquaintance.
This happened on the following occasion. Mr. Polenus, a Professor in the
University of Padua, from a new experiment of his, thought the common
opinion about the force of moving bodies was overturned, and the truth of
Mr. Libnitz’s notion in that matter fully proved. The contrary of what
Polenus had asserted I demonstrated in a paper, which Dr. M e a d, who
takes all opportunities of obliging his friends, was pleased to shew Sir
I s a a c N e w t o n This was so well approved of by him, that he did me the
honour to become a fellow-writer with me, by annexing to what I had
written, a demonstration of his own drawn from another consideration.
When I printed my discourse in the philosophical transactions, I put what
Sir I s a a c had written in a scholium by it self, that I might not seem to
usurp what did not belong to me. But I concealed his name, not being then
sufficiently acquainted with him to ask whether he was willing I might make
use of it or not. In a little time after he engaged me to take care of the new
edition he was about making if his Principia. This obliged me to be very
frequently with him, and as he lived at some distance from me, a great
number of letters passed between us on this account. When I had the
honour of his conversation, I endeavoured to learn his thoughts upon
mathematical subjects, and something historical concerning his inventions,
that I had not been before acquainted with. I found, he had read fewer of
the modern mathematicians, than one could have expected; but his own
prodigious invention readily supplied him with what he might have an
occasion for in the pursuit of any subject he undertook. I have often heard
him censure the handling geometrical subjects by algebraic calculations;
and his book of Algebra he called by the name of Universal Arithmetic, in
opposition to the injudicious title of Geometry, which Des Cartes had given
to the treatise, wherein he shews, how the geometer may assist his invention
by such kind of computations. He frequently praised Slusius, Barrow and
Huygens for not being influenced by the false taste, which then began to
prevail. He used to commend the laudable attempt of Hugo de Omerique to
restore the ancient analysis, and very much esteemed Apollonius’s book De
sectione rationis for giving us a clearer notion of that analysis than we had
before. Dr. Barrow may be esteemed as having shewn a compass of
invention equal, if not superior to any of the moderns, our author only
excepted; but Sir I s a a c N e w t o n has several times particularly
recommended to me Huygens’s stile and manner. He thought him the most

elegant of any mathematical writer of modern times, and the most just
imitator of the antients. Of their taste, and form of demonstration Sir
I s a a c always professed himself a great admirer: I have heard him even
censure himself for not following them yet more closely than he did; and
speak with regret of his mistake at the beginning of his mathematical
studies, in applying himself to the works of Des Cartes and other algebraic
writers, before he had considered the elements of Euclide with that
attention, which so excellent a writer deserves. As to the history of his
inventions, what relates to his discoveries of the methods of series and
fluxions, and of his theory of light and colours, the world has been
sufficiently informed of already. The first thoughts, which gave rise to his
Principia, he had, when he retired from Cambridge in 1666 on account of
the plague. As he sat alone in a garden, he fell into a speculation on the
power of gravity: that as this power is not found sensibly diminished at the
remotest distance from the center of the earth, to which we can rise, neither
at the tops of the loftiest buildings, nor even on the summits of the highest
mountains; it appeared to him reasonable to conclude, that this power must
extend much farther than was usually thought; why not as high as the
moon, said he to himself? and if so, her motion must be influenced by it;
perhaps she is retained in her orbit thereby. However, though the power of
gravity is not sensibly weakened in the little change of distance, at which
we can place our selves from the center of the earth; yet it is very possible,
that so high as the moon this power may differ much in strength from what
it is here. To make an estimate, what might be the degree of this diminution,
he considered with himself, that if the moon be retained in her orbit by the
force of gravity, no doubt the primary planets are carried round the sun by
the like power. And by comparing the periods of the several planets with
their distances from the sun, he found, that if any power like gravity held
them in their courses, its strength must decrease in the duplicate proportion
of the increase of distance. This be concluded by supposing them to move in
perfect circles concentrical to the sun, from which the orbits of the greatest
part of them do not much differ. Supposing therefore the power of gravity,
when extended to the moon, to decrease in the same manner, he computed
whether that force would be sufficient to keep the moon in her orbit. In this
computation, being absent from books, he took the common estimate in use
among geographers and our seamen, before Norwood had measured the
earth, that 60 English miles were contained in one degree of latitude on the

surface of the earth. But as this is a very faulty supposition, each degree
containing about 69½ of our miles, his computation did not answer
expectation; whence he concluded, that some other cause must at least join
with the action of the power of gravity on the moon. On this account he laid
aside for that time any farther thoughts upon this matter. But some years
after, a letter which he received from Dr. Hook, put him on inquiring what
was the real figure, in which a body let fall from any high place descends,
taking the motion of the earth round its axis into consideration. Such a
body, having the same motion, which by the revolution of the earth the
place has whence it falls, is to be considered as projected forward and at
the same time drawn down to the center of the earth. This gave occasion to
his resuming his former thoughts concerning the moon; and Picart in
France having lately measured the earth, by using his measures the moon
appeared to be kept in her orbit purely by the power of gravity; and
consequently, that this power decreases as you recede from the center of the
earth in the manner our author had formerly conjectured. Upon this
principle he found the line described by a falling body to be an ellipsis, the
center of the earth being one focus. And the primary planets moving in such
orbits round the sun, he had the satisfaction to see, that this inquiry, which
he had undertaken merely out of curiosity, could be applied to the greatest
purposes. Hereupon he composed near a dozen propositions relating to the
motion of the primary planets about the sun. Several years after this, some
discourse he had with Dr. Halley, who at Cambridge made him a visit,
engaged Sir I s a a c N e w t o n to resume again the consideration of this
subject; and gave occasion to his writing the treatise which he published
under the title of mathematical principles of natural philosophy. This
treatise, full of such a variety of profound inventions, was composed by him
from scarce any other materials than the few propositions before
mentioned, in the space of one year and an half.
Though his memory was much decayed, I found he perfectly understood
his own writings, contrary to what I had frequently heard in discourse from
many persons. This opinion of theirs might arise perhaps from his not being
always ready at speaking on these subjects, when it might be expected he
should. But as to this, it may be observed, that great genius’s are frequently
liable to be absent, not only in relation to common life, but with regard to
some of the parts of science they are the best informed of. Inventors seem to
treasure up in their minds, what they have found out, after another manner

than those do the same things, who have not this inventive faculty. The
former, when they have occasion to produce their knowledge, are in some
measure obliged immediately to investigate part of what they want. For this
they are not equally fit at all times: so it has often happened, that such as
retain things chiefly by means of a very strong memory, have appeared off
hand more expert than the discoverers themselves.
As to the moral endowments of his mind, they were as much to be
admired as his other talents. But this is a field I leave others to exspatiate
in. I only touch upon what I experienced myself during the few years I was
happy in his friendship. But this I immediately discovered in him, which at
once both surprized and charmed me: Neither his extreme great age, nor
his universal reputation had rendred him stiff in opinion, or in any degree
elated. Of this I had occasion to have almost daily experience. The Remarks
I continually sent him by letters on his Principia were received with the
utmost goodness. These were so far from being any ways displeasing to
him, that on the contrary it occasioned him to speak many kind things of me
to my friends, and to honour me with a publick testimony of his good
opinion. He also approved of the following treatise, a great part of which
we read together. As many alterations were made in the late edition of his
Principia, so there would have been many more if there had been a
sufficient time. But whatever of this kind may be thought wanting, I shall
endeavour to supply in my comment on that book. I had reason to believe he
expected such a thing from me, and I intended to have published it in his life
time, after I had printed the following discourse, and a mathematical
treatise Sir I s a a c N e w t o n had written a long while ago, containing
the first principles of fluxions, for I had prevailed on him to let that piece go
abroad. I had examined all the calculations, and prepared part of the
figures; but as the latter part of the treatise had never been finished, he was
about letting me have other papers, in order to supply what was wanting.
But his death put a stop to that design. As to my comment on the Principia, I
intend there to demonstrate whatever Sir I s a a c N e w t o n has set down
without express proof, and to explain all such expressions in his book, as I
shall judge necessary. This comment I shall forthwith put to the press,
joined to an english translation of his Principia, which I have had some
time by me. A more particular account of my whole design has already been
published in the new memoirs of literature for the month of march 1727.

I have presented my readers with a copy of verses on Sir I s a a c
N e w t o n, which I have just received from a young Gentleman, whom I am
proud to reckon among the number of my dearest friends. If I had any
apprehension that this piece of poetry stood in need of an apology, I should
be desirous the reader might know, that the author is but sixteen years old,
and was obliged to finish his composition in a very short space of time. But
I shall only take the liberty to observe, that the boldness of the digressions
will be best judged of by those who are acquainted with P i n d a r.

A

POEM
ON
Sir I S A A C N E W T O N.

T
o N e w t o n’s genius, and immortal fame
Th’ advent’rous muse with trembling pinion soars.
Thou, heav’nly truth, from thy seraphick throne
Look favourable down, do thou assist
My lab’ring thought, do thou inspire my song.
Newton, who first th’ almighty’s works display’d,
And smooth’d that mirror, in whose polish’d face
The great creator now conspicuous shines;
Who open’d nature’s adamantine gates,
And to our minds her secret powers expos’d;
Newton demands the muse; his sacred hand
Shall guide her infant steps; his sacred hand
Shall raise her to the Heliconian height,
Where, on its lofty top inthron’d, her head

Shall mingle with the Stars. Hail nature, hail,
O Goddess, handmaid of th’ ethereal power,
Now lift thy head, and to th’ admiring world
Shew thy long hidden beauty. Thee the wise
Of ancient fame, immortal P l a t o’s self,
The Stagyrite, and Syracusian sage,
From black obscurity’s abyss to raise,
(Drooping and mourning o’er thy wondrous works)
With vain inquiry sought. Like meteors these
In their dark age bright sons of wisdom shone:
But at thy N e w t o n all their laurels fade,
They shrink from all the honours of their names.
So glimm’ring stars contract their feeble rays,
When the swift lustre of A u r o r a’s face
Flows o’er the skies, and wraps the heav’ns in light.

The Deity’s omnipotence, the cause,
Th’ original of things long lay unknown.
Alone the beauties prominent to sight
(Of the celestial power the outward form)
Drew praise and wonder from the gazing world.
As when the deluge overspread the earth,
Whilst yet the mountains only rear’d their heads
Above the surface of the wild expanse,
Whelm’d deep below the great foundations lay,
Till some kind angel at heav’n’s high command
Roul’d back the rising tides, and haughty floods,
And to the ocean thunder’d out his voice:
Quick all the swelling and imperious waves,
The foaming billows and obscuring surge,
Back to their channels and their ancient seats
Recoil affrighted: from the darksome main
Earth raises smiling, as new-born, her head,
And with fresh charms her lovely face arrays.
So his extensive thought accomplish’d first
The mighty task to drive th’ obstructing mists
Of ignorance away, beneath whose gloom

Th’ inshrouded majesty of Nature lay.
He drew the veil and swell’d the spreading scene.
How had the moon around th’ ethereal void
Rang’d, and eluded lab’ring mortals care,
Till his invention trac’d her secret steps,
While she inconstant with unsteady rein
Through endless mazes and meanders guides
In its unequal course her changing carr:
Whether behind the sun’s superior light
She hides the beauties of her radiant face,
Or, when conspicuous, smiles upon mankind,
Unveiling all her night-rejoicing charms.
When thus the silver-tressed moon dispels
The frowning horrors from the brow of night,
And with her splendors chears the sullen gloom,
While sable-mantled darkness with his veil
The visage of the fair horizon shades,
And over nature spreads his raven wings;
Let me upon some unfrequented green
While sleep sits heavy on the drowsy world,
Seek out some solitary peaceful cell,
Where darksome woods around their gloomy brows
Bow low, and ev’ry hill’s protended shade
Obscures the dusky vale, there silent dwell,
Where contemplation holds its still abode,
There trace the wide and pathless void of heav’n,
And count the stars that sparkle on its robe.
Or else in fancy’s wild’ring mazes lost
Upon the verdure see the fairy elves
Dance o’er their magick circles, or behold,
In thought enraptur’d with the ancient bards,
Medea’s baleful incantations draw
Down from her orb the paly queen of night.
But chiefly N e w t o n let me soar with thee,
And while surveying all yon starry vault
With admiration I attentive gaze,
Thou shalt descend from thy celestial seat,

And waft aloft my high-aspiring mind,
Shalt shew me there how nature has ordain’d
Her fundamental laws, shalt lead my thought
Through all the wand’rings of th’ uncertain moon,
And teach me all her operating powers.
She and the sun with influence conjoint
Wield the huge axle of the whirling earth,
And from their just direction turn the poles,
Slow urging on the progress of the years.
The constellations seem to leave their seats,
And o’er the skies with solemn pace to move.
You, splendid rulers of the day and night,
The seas obey, at your resistless sway
Now they contract their waters, and expose
The dreary desart of old ocean’s reign.
The craggy rocks their horrid sides disclose;
Trembling the sailor views the dreadful scene,
And cautiously the threat’ning ruin shuns.
But where the shallow waters hide the sands,
There ravenous destruction lurks conceal’d,
There the ill-guided vessel falls a prey,
And all her numbers gorge his greedy jaws.
But quick returning see th’ impetuous tides
Back to th’ abandon’d shores impell the main.
Again the foaming seas extend their waves,
Again the rouling floods embrace the shoars,
And veil the horrours of the empty deep.
Thus the obsequious seas your power confess,
While from the surface healthful vapours rise
Plenteous throughout the atmosphere diffus’d,
Or to supply the mountain’s heads with springs,
Or fill the hanging clouds with needful rains,
That friendly streams, and kind refreshing show’rs
May gently lave the sun-burnt thirsty plains,
Or to replenish all the empty air
With wholsome moisture to increase the fruits
Of earth, and bless the labours of mankind.

O N e w t o n, whether flies thy mighty soul,
How shall the feeble muse pursue through all
The vast extent of thy unbounded thought,
That even seeks th’ unseen recesses dark
To penetrate of providence immense.
And thou the great dispenser of the world
Propitious, who with inspiration taught’st
Our greatest bard to send thy praises forth;
Thou, who gav’st N e w t o n thought; who smil’dst serene,
When to its bounds he stretch’d his swelling soul;
Who still benignant ever blest his toil,
And deign’d to his enlight’ned mind t’ appear
Confess’d around th’ interminated world:
To me O thy divine infusion grant
(O thou in all so infinitely good)
That I may sing thy everlasting works,
Thy inexhausted store of providence,
In thought effulgent and resounding verse.
O could I spread the wond’rous theme around,
Where the wind cools the oriental world,
To the calm breezes of the Zephir’s breath,
To where the frozen hyperborean blasts.
To where the boist’rous tempest-leading south
From their deep hollow caves send forth their storms.
Thou still indulgent parent of mankind,
Left humid emanations should no more
Flow from the ocean, but dissolve away
Through the long series of revolving time;
And left the vital principle decay,
By which the air supplies the springs of life;
Thou hast the fiery visag’d comets form’d
With vivifying spirits all replete,
Which they abundant breathe about the void,
Renewing the prolifick soul of things.
No longer now on thee amaz’d we call,
No longer tremble at imagin’d ills,
When comets blaze tremendous from on high,

Or when extending wide their flaming trains
With hideous grasp the skies engirdle round,
And spread the terrors of their burning locks.
For these through orbits in the length’ning space
Of many tedious rouling years compleat
Around the sun move regularly on;
And with the planets in harmonious orbs,
And mystick periods their obeysance pay
To him majestick ruler of the skies
Upon his throne of circled glory fixt.
He or some god conspicuous to the view,
Or else the substitute of nature seems,
Guiding the courses of revolving worlds.
He taught great N e w t o n the all-potent laws
Of gravitation, by whose simple power
The universe exists. Nor here the sage
Big with invention still renewing staid.
But O bright angel of the lamp of day,
How shall the muse display his greatest toil?
Let her plunge deep in Aganippe’s waves,
Or in Castalia’s ever-flowing stream,
That re-inspired she may sing to thee,
How N e w t o n dar’d advent’rous to unbraid
The yellow tresses of thy shining hair.
Or didst thou gracious leave thy radiant sphere,
And to his hand thy lucid splendours give,
T’ unweave the light-diffusing wreath, and part
The blended glories of thy golden plumes?
He with laborious, and unerring care,
How different and imbodied colours form
Thy piercing light, with just distinction found.
He with quick sight pursu’d thy darting rays,
When penetrating to th’ obscure recess
Of solid matter, there perspicuous saw,
How in the texture of each body lay
The power that separates the different beams.
Hence over nature’s unadorned face

Thy bright diversifying rays dilate
Their various hues: and hence when vernal rains
Descending swift have burst the low’ring clouds,
Thy splendors through the dissipating mists
In its fair vesture of unnumber’d hues
Array the show’ry bow. At thy approach
The morning risen from her pearly couch
With rosy blushes decks her virgin cheek;
The ev’ning on the frontispiece of heav’n
His mantle spreads with many colours gay;
The mid-day skies in radiant azure clad,
The shining clouds, and silver vapours rob’d
In white transparent intermixt with gold,
With bright variety of splendor cloath
All the illuminated face above.
When hoary-headed winter back retires
To the chill’d pole, there solitary sits
Encompass’d round with winds and tempests bleak
In caverns of impenetrable ice,
And from behind the dissipated gloom
Like a new Venus from the parting surge
The gay-apparell’d spring advances on;
When thou in thy meridian brightness sitt’st,
And from thy throne pure emanations flow
Of glory bursting o’er the radiant skies:
Then let the muse Olympus’ top ascend,
And o’er Thessalia’s plain extend her view,
And count, O Tempe, all thy beauties o’er.
Mountains, whose summits grasp the pendant clouds,
Between their wood-invelop’d slopes embrace
The green-attired vallies. Every flow’r
Here in the pride of bounteous nature clad
Smiles on the bosom of th’ enamell’d meads.
Over the smiling lawn the silver floods
Of fair Peneus gently roul along,
While the reflected colours from the flow’rs,
And verdant borders pierce the lympid waves,

And paint with all their variegated hue
The yellow sands beneath. Smooth gliding on
The waters hasten to the neighbouring sea.
Still the pleas’d eye the floating plain pursues;
At length, in Neptune’s wide dominion lost,
Surveys the shining billows, that arise
Apparell’d each in Phœbus’ bright attire:
Or from a far some tall majestick ship,
Or the long hostile lines of threat’ning fleets,
Which o’er the bright uneven mirror sweep,
In dazling gold and waving purple deckt;
Such as of old, when haughty Athens power
Their hideous front, and terrible array
Against Pallene’s coast extended wide,
And with tremendous war and battel stern
The trembling walls of Potidæa shook.
Crested with pendants curling with the breeze
The upright masts high bristle in the air,
Aloft exalting proud their gilded heads.
The silver waves against the painted prows
Raise their resplendent bosoms, and impearl
The fair vermillion with their glist’ring drops:
And from on board the iron-cloathed host
Around the main a gleaming horrour casts;
Each flaming buckler like the mid-day sun,
Each plumed helmet like the silver moon,
Each moving gauntlet like the light’ning’s blaze,
And like a star each brazen pointed spear.
But lo the sacred high-erected fanes,
Fair citadels, and marble-crowned towers,
And sumptuous palaces of stately towns
Magnificent arise, upon their heads
Bearing on high a wreath of silver light.
But see my muse the high Pierian hill,
Behold its shaggy locks and airy top,
Up to the skies th’ imperious mountain heaves
The shining verdure of the nodding woods.

See where the silver Hippocrene flows,
Behold each glitt’ring rivulet, and rill
Through mazes wander down the green descent,
And sparkle through the interwoven trees.
Here rest a while and humble homage pay,
Here, where the sacred genius, that inspir’d
Sublime M æ o n i d e s and P i n d a r ’ s breast,
His habitation once was fam’d to hold.
Here thou, O H o m e r, offer’dst up thy vows,
Thee, the kind muse C a l l i o p æ a heard,
And led thee to the empyrean feats,
There manifested to thy hallow’d eyes
The deeds of gods; thee wise M i n e r v a taught
The wondrous art of knowing human kind;
Harmonious P h œ b u s tun’d thy heav’nly mind,
And swell’d to rapture each exalted sense;
Even M a r s the dreadful battle-ruling god,
M a r s taught thee war, and with his bloody hand
Instructed thine, when in thy sounding lines
We hear the rattling of Bellona’s carr,
The yell of discord, and the din of arms.
P i n d a r, when mounted on his fiery steed,
Soars to the sun, opposing eagle like
His eyes undazled to the fiercest rays.
He firmly seated, not like G l a u c u s ’ son,
Strides his swift-winged and fire-breathing horse,
And born aloft strikes with his ringing hoofs
The brazen vault of heav’n, superior there
Looks down upon the stars, whose radiant light
Illuminates innumerable worlds,
That through eternal orbits roul beneath.
But thou all hail immortalized son
Of harmony, all hail thou Thracian bard,
To whom A p o l l o gave his tuneful lyre.
O might’st thou, O r p h e u s, now again revive,
And N e w t o n should inform thy list’ning ear
How the soft notes, and soul-inchanting strains

Of thy own lyre were on the wind convey’d.
He taught the muse, how sound progressive floats
Upon the waving particles of air,
When harmony in ever-pleasing strains,
Melodious melting at each lulling fall,
With soft alluring penetration steals
Through the enraptur’d ear to inmost thought,
And folds the senses in its silken bands.
So the sweet musick, which from O r p h e u s’ touch
And fam’d A m p h i o n ’ s, on the sounding string
Arose harmonious, gliding on the air,
Pierc’d the tough-bark’d and knotty-ribbed woods,
Into their saps soft inspiration breath’d
And taught attention to the stubborn oak.
Thus when great H e n r y, and brave M a r l b ’ r o u g h led
Th’ imbattled numbers of B r i t a n n i a ’ s sons,
The trump, that swells th’ expanded cheek of fame,
That adds new vigour to the gen’rous youth,
And rouzes sluggish cowardize it self,
The trumpet with its Mars-inciting voice,
The winds broad breast impetuous sweeping o’er
Fill’d the big note of war. Th’ inspired host
With new-born ardor press the trembling G a u l;
Nor greater throngs had reach’d eternal night,
Not if the fields of Agencourt had yawn’d
Exposing horrible the gulf of fate;
Or roaring Danube spread his arms abroad,
And overwhelm’d their legions with his floods.
But let the wand’ring muse at length return;
Nor yet, angelick genius of the sun,
In worthy lays her high-attempting song
Has blazon’d forth thy venerated name.
Then let her sweep the loud-resounding lyre
Again, again o’er each melodious string
Teach harmony to tremble with thy praise.
And still thine ear O favourable grant,
And she shall tell thee, that whatever charms,

Whatever beauties bloom on nature’s face,
Proceed from thy all-influencing light.
That when arising with tempestuous rage,
The North impetuous rides upon the clouds
Dispersing round the heav’ns obstructive gloom,
And with his dreaded prohibition stays
The kind effusion of thy genial beams;
Pale are the rubies on A u r o r a ’ s lips,
No more the roses blush upon her cheeks,
Black are Peneus’ streams and golden sands
In Tempe’s vale dull melancholy sits,
And every flower reclines its languid head.
By what high name shall I invoke thee, say,
Thou life-infusing deity, on thee
I call, and look propitious from on high,
While now to thee I offer up my prayer.
O had great N e w t o n, as he found the cause,
By which sound rouls thro’ th’ undulating air,
O had he, baffling times resistless power,
Discover’d what that subtle spirit is,
Or whatsoe’er diffusive else is spread
Over the wide-extended universe,
Which causes bodies to reflect the light,
And from their straight direction to divert
The rapid beams, that through their surface pierce.
But since embrac’d by th’ icy arms of age,
And his quick thought by times cold hand congeal’d,
Ev’n N e w t o n left unknown this hidden power;
Thou from the race of human kind select
Some other worthy of an angel’s care,
With inspiration animate his breast,
And him instruct in these thy secret laws.
O let not N e w t o n, to whose spacious view,
Now unobstructed, all th’ extensive scenes
Of the ethereal ruler’s works arise;
When he beholds this earth he late adorn’d,
Let him not see philosophy in tears,

Like a fond mother solitary sit,
Lamenting him her dear, and only child.
But as the wise P y t h a g o r a s, and he,
Whose birth with pride the fam’d Abdera boasts,
With expectation having long survey’d
This spot their ancient seat, with joy beheld
Divine philosophy at length appear
In all her charms majestically fair,
Conducted by immortal N e w t o n ’ s hand.
So may he see another sage arise,
That shall maintain her empire: then no more
Imperious ignorance with haughty sway
Shall stalk rapacious o’er the ravag’d globe:
Then thou, O N e w t o n, shalt protect these lines.
The humble tribute of the grateful muse;
Ne’er shall the sacrilegious hand despoil
Her laurel’d temples, whom his name preserves:
And were she equal to the mighty theme,
Futurity should wonder at her song;
Time should receive her with extended arms,
Seat her conspicuous in his rouling carr,
And bear her down to his extreamest bound.

F a b l e s with wonder tell how Terra’s sons
With iron force unloos’d the stubborn nerves
Of hills, and on the cloud-inshrouded top
Of Pelion Ossa pil’d. But if the vast
Gigantick deeds of savage strength demand
Astonishment from men, what then shalt thou,
O what expressive rapture of the soul,
When thou before us, N e w t o n, dost display
The labours of thy great excelling mind;
When thou unveilest all the wondrous scene,
The vast idea of th’ eternal king,
Not dreadful bearing in his angry arm
The thunder hanging o’er our trembling heads;
But with th’ effulgency of love replete,

And clad with power, which form’d th’ extensive heavens.
O happy he, whose enterprizing hand
Unbars the golden and relucid gates
Of th’ empyrean dome, where thou enthron’d
Philosophy art seated. Thou sustain’d
By the firm hand of everlasting truth
Despisest all the injuries of time;
Thou never know’st decay when all around,
Antiquity obscures her head. Behold
Th’ Egyptian towers, the Babylonian walls,
And Thebes with all her hundred gates of brass,
Behold them scatter’d like the dust abroad.
Whatever now is flourishing and proud,
Whatever shall, must know devouring age.
Euphrates’ stream, and seven-mouthed Nile,
And Danube, thou that from Germania’s soil
To the black Euxine’s far remoted shore,
O’er the wide bounds of mighty nations sweep’st
In thunder loud thy rapid floods along.
Ev’n you shall feel inexorable time;
To you the fatal day shall come; no more
Your torrents then shall shake the trembling ground,
No longer then to inundations swol’n
Th’ imperious waves the fertile pastures drench,
But shrunk within a narrow channel glide;
Or through the year’s reiterated course
When time himself grows old, your wond’rous streams
Lost ev’n to memory shall lie unknown
Beneath obscurity, and Chaos whelm’d,
But still thou sun illuminatest all
The azure regions round, thou guidest still
The orbits of the planetary spheres;
The moon still wanders o’er her changing course,
And still, O N e w t o n, shall thy name survive:
As long as nature’s hand directs the world,
When ev’ry dark obstruction shall retire,
And ev’ry secret yield its hidden store,

Which thee dim-sighted age forbad to see
Age that alone could stay thy rising soul.
And could mankind among the fixed stars,
E’en to th’ extremest bounds of knowledge reach,
To those unknown innumerable suns,
Whose light but glimmers from those distant worlds,
Ev’n to those utmost boundaries, those bars
That shut the entrance of th’ illumin’d space
Where angels only tread the vast unknown,
Thou ever should’st be seen immortal there:
In each new sphere, each new-appearing sun,
In farthest regions at the very verge
Of the wide universe should’st thou be seen.
And lo, th’ all-potent goddess N a t u r e takes
With her own hand thy great, thy just reward
Of immortality; aloft in air
See she displays, and with eternal grasp
Uprears the trophies of great N e w t o n’s fame.

R. Glover.

THE
C O N T E N T S.

I
NTRODUCTION concerning Sir I s a a c N e w t o n’s
method of reasoning in philosophy pag. 1

Book I.
C h a p . 1 . Of the laws of motion
The first law of motion proved p. 29
The second law of motion proved p. 29
The third law of motion proved p. 31
C h a p . 2 . Further proofs of the laws of motion
The effects of percussion p. 49
The perpendicular descent of bodies p. 55
The oblique descent of bodies in a straight line p. 57
The curvilinear descent of bodies p. 58
The perpendicular ascent of bodies ibid.
The oblique ascent of bodies p. 59
The power of gravity proportional to the quantity of matter in each
body p. 60
The centre of gravity of bodies p. 62
The mechanical powers p. 69
The lever p. 71
The wheel and axis p. 77
The pulley p. 80
The wedge p. 83
The screw ibid.

The inclined plain p. 84
The pendulum p. 86
Vibrating in a circle ibid.
Vibrating in a cycloid p. 91
The line of swiftest descent p. 93
The centre of oscillation p. 94
Experiments upon the percussion of bodies made by pendulums p. 98
The centre of percussion p. 100
The motion of projectiles p. 102
The description of the conic sections p. 106
The difference between absolute and relative motion, as also
between absolute and relative time p. 112
C h a p . 3 . Of centripetal forces p. 117
C h a p . 4 . Of the resistance of fluids p. 143
Bodies are resisted in the duplicate proportion of their velocities p. 147
Of elastic fluids and their resistance p. 149
How fluids may be rendered elastic p. 150
The degree of resistance in regard to the proportion between the
density of the body and of the fluid
In rare and uncompressed fluids p. 153
In compressed fluids p. 155
The degree of resistance as it depends upon the figure of bodies
In rare and uncompressed fluids p. 155
In compressed fluids p. 158

Book II.
C h a p . 1 . That the planets move in a space empty of sensible
matter p. 161
The system of the world described p. 162
The planets suffer no sensible resistance in their motion p. 166
They are not kept in motion by a fluid p. 168
That all space is not full of matter without vacancies p. 169
C h a p . 2 . Concerning the cause that keeps in motion the primary
planets p. 171
They are influenced by a centripetal power directed to the sun p. 171
The strength of this power is reciprocally in the duplicate
proportion of the distance ibid.
The cause of the irregularities in the motions of the planets p. 175

A correction of their motions p. 178
That the frame of the world is not eternal p. 180
C h a p . 3 . Of the motion of the moon and the other secondary
planets
That they are influenced by a centripetal force directed toward
their primary, as the primary are influenced by the sun p. 182
That the power usually called gravity extends to the moon p. 189
That the sun acts on the secondary planets p. 190
The variation of the moon p. 193
That the circuit of the moons orbit is increased by the sun in the
quarters, and diminished in the conjunction and opposition p. 198
The distance of the moon from the earth in the quarters and in
the conjunction and opposition is altered by the sun p. 200
These irregularities in the moon’s motion varied by the change
of distance between the earth and sun p. 201
The period of the moon round the earth and her distance varied
by the same means ibid.
The motion of the nodes and the inclination of the moons orbit p. 202
The motion of the apogeon and change of the eccentricity p. 218
The inequalities of the other secondary planets deducible from
these of the moon p. 229
C h a p . 4 . Of comets
They are not meteors, nor placed totally without the planetary
system p. 230
The sun acts on them in the same manner as on the planets p. 231
Their orbits are near to parabola’s p. 233
The comet that appeared at the end of the year 1680, probably
performs its period in 575 years, and another comet in 75
years p. 234
Why the comets move in planes more different from one another
than the planets p. 235
The tails of comets p. 238
The use of them p. 243
244
The possible use of the comet it self p. 245
246
C h a p . 5 . Of the bodies of the sun and planets
That each of the heavenly bodies is endued with an attractive
power, and that the force of the same body on others is
proportional to the quantity of matter in the body attracted p. 247
This proved in the earth p. 248
In the sun p. 250

In the rest of the planets p. 251
That the attractive power is of the same nature in the sun and in
all the planets, and therefore is the same with gravity p. 252
That the attractive power in each of these bodies is proportional
to the quantity of matter in the body attracting ibid.
That each particle of which the sun and planets are composed is
endued with an attracting power, the strength of which is
reciprocally in the duplicate proportion of the distance p. 257
The power of gravity universally belongs to all matter p. 259
The different weight of the same body upon the surface of the
sun, the earth, Jupiter and Saturn; the respective densities of
these bodies, and the proportion between their diameters p. 261
C h a p . 6 . Of the fluid parts of the planets
The manner in which fluids press p. 264
The motion of waves on the surface of water p. 269
The motion of sound through the air p. 270
The velocity of sound p. 282
Concerning the tides p. 283
The figure of the earth p. 296
The effect of this figure upon the power of gravity p. 300
The effect it has upon pendulums p. 302
Bodies descend perpendicularly to the surface of the earth p. 304
The axis of the earth changes its direction twice a year, and
twice a month p. 313
The figure of the secondary planets ibid.

Book III.
C h a p . 1 . Concerning the cause of colours inherent in the light
The sun’s light is composed of rays of different colours p. 318
The refraction of light p. 319
320
Bodies appear of different colour by day-light, because some
reflect one kind of light more copiously than the rest, and
other bodies other kinds of light p. 329
The effect of mixing rays of different colours p. 334
C h a p . 2 . Of the properties of bodies whereon their colours
depend.
Light is not reflected by impinging against the solid parts of
bodies p. 339

The particles which compose bodies are transparent p. 341
Cause of opacity p. 342
Why bodies in the open day-light have different colours p. 344
The great porosity of bodies considered p. 355
C h a p . 3 . Of the refraction, reflection, and inflection of light.
Rays of different colours are differently refracted p. 357
The sine of the angle of incidence in each kind of rays bears a
given proportion to the sine of refraction p. 361
The proportion between the refractive powers in different
bodies p. 366
Unctuous bodies refract most in proportion to their density p. 368
The action between light and bodies is mutual p. 369
Light has alternate fits of easy transmission and reflection p. 371
The fits found to return alternately many thousand times p. 375
Why bodies reflect part of the light incident upon them and
transmit another part ibid.
Sir I s a a c N e w t o n’s conjecture concerning the cause of
this alternate reflection and transmission of light p. 376
The inflection of light p. 377
C h a p . 4 . Of optic glasses.
How the rays of light are refracted by a spherical surface of
glass p. 378
How they are refracted by two such surfaces p. 380
How the image of objects is formed by a convex glass p. 381
Why convex glasses help the sight in old age, and concave
glasses assist short-sighted people p. 383
The manner in which vision is performed by the eye p. 385
Of telescopes with two convex glasses p. 386
Of telescopes with four convex glasses p. 388
Of telescopes with one convex and one concave glass ibid.
Of microscopes p. 389
Of the imperfection of telescopes arising from the different
refrangibility of the light p. 390
Of the reflecting telescope p. 393
C h a p . 5 . Of the rainbow
Of the inner rainbow p. 394
395
398
399
Of the outter bow p. 396
397

400
Of a particular appearance in the inner rainbow p. 401
Conclusion p. 405

E R R ATA .

P
AGE 25. line 4. read In these Precepts. p. 40. l. 24. for I read K. p. 53. l. penult. f. Æ.
r. F. p. 82. l. ult. f. 40. r. 41. p. 83 l. ult. f. 43. r. 45. p. 91. l. 3. f. 48. r. 50. ibid. l. 25.
for 49. r. 51. p. 92. l. 18. f. A G F E. r. H G F C. p. 96. l. 23. dele the comma after
{⅓}. p. 140. l. 12. dele and. p. 144. l. 15. f. threefold. r. two-fold. p. 162. l. 25. f.
{⅓}. r. {⅞}. p. 193. 1. 2. r. always. p. 199. l. penult. and p. 200. l. 3. 5. f. F. r. C. p. 201. l.
8. f. ascends. r. must ascend. ibid. l. 10. f. it descends. r. descend. p. 208. l. 14. f. W T O. r.
N T O. In fig. 110. draw a line from I through T, till it meets the circle A D C B, where
place W. p. 216. l. penult. f. action. r. motion. p. 221. l. 23. f. A F. r. A H. p. 232. l. 23. after
invention put a full point. p. 253. l. penult. delete the comma after remarkable. p. 255. l.
ult. f. D E. r. B E. p. 278. l. 17. f. ξ τ. r. ξ π. p. 299. l. 19 r. the. p. 361. l. 12. f. I. r. t. p. 369.
l. 2, 3. r. Pseudo-topaz. p. 378. l. 12. f. that. r. than. p. 379. l. 15. f. converge. r. diverge. p.
384. l. 7. f. optic-glass. r. optic-nerve. p. 391. l. 18. r. as 50 to 78. p. 392. l. 18. after
telescope add be about 100 feet long and the. in fig. 161. f. δ put ε. p. 399. l. 8. r. A n, A x.
&c. p. 400. 1. 19. r. A π, A ρ. A σ, A τ. A φ. p. 401. l. 14. r. fig. 163. The pages 374, 375,
376 are erroneously numbered 375, 376, 377; and the pages 382, 383 are numbered 381,
382.

A LIST of such of the
SUBSCRIBERS NAMES
As are come to the H a n d of the
AUTHOR.

A

M
Onseigneur d’Aguesseau, Chancelier de France
Reverend Mr Abbot, of Emanuel Coll. Camb.
Capt. George Abell
The Hon. Sir John Anstruther, Bar.
Thomas Abney, Esq;
Mr. Nathan Abraham
Sir Arthur Acheson, Bart.
Mr William Adair
Rev. Mr John Adams, Fellow of Sidney Coll. Cambridge
Mr William Adams
Mr George Adams
Mr William Adamson, Scholar of Caius Coll. Camb.
Mr Samuel Adee, Fell. of Corp. Chr. Coll. Oxon
Mr Andrew Adlam
Mr John Adlam
Mr Stephen Ainsworth
Mrs Aiscot
Mr Robert Akenhead, Bookseller at Newcastle upon Tyne

S. B. Albinus, M. D. Anatom. and Chirurg in Acad. L. B. Prof.
George Aldridge, M. D.
Mr George Algood
Mr Aliffe
Robert Allen, Esq;
Mr Zach. Allen
Rev. Mr Allerton, Fellow of Sidney Coll. Cambridge
Mr St. Amand
Mr John Anns
Thomas Anson, Esq;
Rev. Dr. Christopher Anstey
Mr Isaac Antrabus
Mr Joshua Appleby
John Arbuthnot, M. D.
William Archer, Esq;
Mr John Archer, Merchant of Amsterdam
Thomas Archer, Esq;
Coll. John Armstrong, Surveyor-General of His Majesty’s Ordnance
Mr Armytage
Mr Street Arnold, Surgeon
Mr Richard Arnold
Mr Ascough
Mr Charles Asgill
Richard Ash, Esq; of Antigua
Mr Ash, Fellow-Commoner of Jesus Coll. Cambridge
William Ashurst, Esq; of Castle Henningham, Essex
Mr Thomas Ashurst
Mr Samuel Ashurst
Mr John Askew, Merchant
Mr Edward Athawes, Merchant
Mr Abraham Atkins
Mr Edward Kensey Atkins
Mr Ayerst
Mr Jonathan Ayleworth, Jun.
Rowland Aynsworth, Esq;

B

His Grace the Duke of Bedford
Right Honourable the Marquis of Bowmont
Right Hon. the Earl of Burlington
Right Honourable Lord Viscount Bateman
Rt. Rev. Ld. Bp. of Bath and Wells
Rt. Rev. Lord Bishop of Bristol
Right Hon. Lord Bathurst
Richard Backwell, Esq;
Mr William Backshell, Merch.
Edmund Backwell, Gent.
Sir Edmund Bacon
Richard Bagshaw, of Oakes, Esq;
Tho. Bagshaw, of Bakewell, Esq;
Rev. Mr. Bagshaw
Sir Robert Baylis
Honourable George Baillie, Esq;
Giles Bailly, M. D. of Bristol
Mr Serjeant Baines
Rev. Mr. Samuel Baker, Residen. of St. Paul’s.
Mr George Baker
Mr Francis Baker
Mr Robert Baker
Mr John Bakewell
Anthony Balam, Esq;
Charles Bale, M. D.
Mr Atwell, Fellow of Exeter Coll. Oxon
Mr Savage Atwood
Mr John Atwood
Mr James Audley
Sir Robert Austen, Bart.
Sir John Austen
Benjamin Avery, L. L. D.
Mr Balgay
Rev. Mr Tho. Ball, Prebendary of Chichester

Mr Pappillon Ball, Merchant
Mr Levy Ball
Rev. Mr Jacob Ball, of Andover
Rev. Mr Edward Ballad, of Trin. Coll. Cambridge
Mr Baller
John Bamber, M. D.
Rev. Mr Banyer, Fellow of Emanuel Coll. Cambridge
Mr Henry Banyer, of Wisbech, Surgeon
Mr John Barber, Apothecary in Coventry
Henry Steuart Barclay, of Colairny, Esq;
Rev. Mr Barclay, Canon of Windsor
Mr David Barclay
Mr Benjamin Barker, Bookseller in London
—— Barker, Esq;
Mr Francis Barkstead
Rev. Mr Barnard
Thomas Barrett, Esq;
Mr Barrett
Richard Barret, M. D.
Mr Barrow, Apothecary
William Barrowby, M. D.
Edward Barry, M. D. of Corke
Mr Humphrey Bartholomew, of University College, Oxon
Mr Benjamin Bartlett
Mr Henry Bartlett
Mr James Bartlett
Mr Newton Barton, of Trinity College, Cambridge
Rev. Mr. Barton
William Barnsley, Esq;
Mr Samuel Bateman
Mr Thomas Bates
Peter Barhurst, Esq;
Mark Barr, Esq;
Thomas Bast, Esq;
Mr Batley, Bookseller in London
Mr Christopher Batt, jun.
Mr William Batt, Apothecary

Rev. Mr Battely, M. A. Student of Christ Church, Oxon
Mr Edmund Baugh
Rev. Mr. Thomas Bayes
Edward Bayley, M. D. of Havant
John Bayley, M. D. of Chichester
Mr. Alexander Baynes, Professor of Law in the University of Edinburgh
Mr Benjamin Beach
Thomas Beacon, Esq;
Rev. Mr Philip Bearcroft
Mr Thomas Bearcroft
Mr William Beachcroft
Richard Beard, M. D. of Worcester
Mr Joseph Beasley
Rev. Mr Beats, M. A. Fellow of Magdalen College, Cambridge
Sir George Beaumont
John Beaumont, Esq; of Clapham
William Beecher, of Howberry, Esq;
Mr Michael Beecher
Mr Finney Beifield, of the Inner-Temple
Mr Benjamin Bell
Mr Humphrey Bell
Mr Phineas Bell
Leonard Belt, Gent.
William Benbow, Esq;
Mr Martin Bendall
Mr George Bennet, of Cork, Bookseller
Rev. Mr Martin Benson, Archdeacon of Berks
Samuel Benson, Esq;
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Rev. Richard Bently, D. D. Master of Trinity Coll. Cambridge
Thomas Bere, Esq;
The Hon. John Berkley, Esq;
Mr Maurice Berkley, sen. Surgeon
John Bernard, Esq;
Mr Charles Bernard
Hugh Bethell, of Rise in Yorkshire, Esq;
Hugh Bethell, of Swindon in Yorkshire, Esq;

Mr Silvanus Bevan, Apothecary
Mr Calverly Bewick, jun.
Henry Bigg, B. D. Warden of New College, Oxon
Sir William Billers
—— Billers, Esq;
Mr John Billingsley
Mr George Binckes
Rev. Mr Birchinsha, of Exeter College, Oxon
Rev. Mr Richard Biscoe
Mr Hawley Bishop, Fellow of St. John’s College, Oxon
Dr Bird, of Reading
Henry Blaake, Esq;
Mr Henry Blaake
Rev. Mr George Black
Steward Blacker, Esq;
William Blacker, Esq;
Rowland Blackman, Esq;
Rev. Mr Charles Blackmore, of Worcester
Rev Mr Blackwall, of Emanuel College, Cambridge
Jonathan Blackwel, Esq;
James Blackwood, Esq;
Mr Thomas Blandford
Arthur Blaney, Esq;
Mr James Blew
Mr William Blizard
Dr Blomer
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Mr Charles Boehm, Merchant
Mr William Bogdani
Mr John Du Bois, Merchant
Mr Samuel Du Bois
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Mr John Bond
John Bonithon, M. A.
Mr James Bonwick, Bookseller in London

Thomas Boone, Esq;
Rev. Mr Pennystone, M. A.
Mrs Judith Booth
Thomas Bootle, Esq;
Thomas Borret, Esq;
Mr Benjamin Boss
Dr Bostock
Henry Bosville, Esq;
Mr John Bosworth
Dr George Boulton
Hon. Bourn M. D. of Chesterfield
Mrs Catherine Bovey
Mr Humphrey Bowen
Mr Bower
John Bowes, Esq;
William Bowles, Esq;
Mr John Bowles
Mr Thomas Bowles
Mr Duvereux Bowly
Duddington Bradeel, Esq;
Rev. Mr James Bradley, Professor of Astronomy, in Oxford
Mr Job Bradley, Bookseller in Chesterfield
Rev. Mr John Bradley
Rev. Mr Bradshaw, Fellow of Jesus College, Cambridge
Mr Joseph Bradshaw
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Champion Bramfield, Esq;
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Mr Thomas Brand
Mr Braxton
Capt. David Braymer
Rev Mr Charles Brent, of Bristol
Mr William Brent
Mr Edmund Bret
John Brickdale, Esq;

Rev. Mr John Bridgen A. M.
Abraham Bridges, Esq;
George Briggs, Esq;
John Bridges, Esq;
Brook Bridges, Esq;
Orlando Bridgman, Esq;
Mr Charles Bridgman
Mr William Bridgman, of Trinity College, Cambridge
Sir Humphrey Briggs, Bart.
Robert Bristol, Esq;
Mr Joseph Broad
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Mr Jacob Brook
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Mr Thomas Brookes
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Noel Broxholme, M. D.
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Rev. Mr Brydam
Christopher Buckle, Esq;
Samuel Buckley, Esq;
Mr Budgen
Sir John Bull
Josiah Bullock, of Faulkbourn-Hall, Essex, Esq;
Rev. Mr Richard Bullock
Rev. Mr Richard Bundy

Mr Alexander Bunyan
Rev. Mr D. Burges
Ebenezer Burgess, Esq;
Robert Burleston, M. B.
Gilbert Burnet, Esq;
Thomas Burnet, Esq;
Rev. Mr Gilbert Burnet
His Excellency Will. Burnet, Esq; Governour of New-York
Mr Trafford Burnston, of Trin. College, Cambridge
Peter Burrel Esq;
John Burridge, Esq;
James Burrough, Esq; Beadle and Fellow of Caius Coll. Cambr.
Mr Benjamin Burroughs
Jeremiah Burroughs, Esq;
Rev. Mr Joseph Burroughs
Christopher Burrow, Esq;
James Burrow, Esq;
William Burrow, A. M.
Francis Burton, Esq;
John Burton, Esq;
Samuel Burton, of Dublin, Esq;
William Burton, Esq;
Mr Burton.
Richard Burton, Esq;
Dr Simon Burton
Rev. Mr Thomas Burton, M.A. Fellow of Caius College, Cambridge
John Bury, jun. Esq;
Rev. Mr Samuel Bury
Mr William Bush
Rev. Mr Samuel Butler
Mr Joseph Button, of Newcastle upon Tyne
Hon. Edward Byam, Governour of Antigua
Mr Edward Byam, Merchant
Mr John Byrom
Mr Duncumb Bristow, Merch.
Mr William Bradgate

C

His Grace the Archbishop of Canterbury
Right Hon. the Lord Chancellor
His Grace the Duke of Chandois
The Right Hon. the Earl of Carlisle
Right Hon. Earl Cowper
Rt. Rev. Lord Bishop of Carlisle
Rt. Rev. Lord Bishop of Chichester
Rt. Rev. Lord Bish. of Clousert in Ireland
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I

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K

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Cambridge.
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L

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Henry Liddell, Esq;

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M.

His Grace the Duke of Montague
His Grace the Duke of Montrosse
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The Rt. Hon. Lord Viscount Molesworth
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The Rt. Rev. Ld. Bishop of Meath
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Ireland
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Library of Merton College

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Mr Mustapha
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N

His Grace the Duke of Newcastle
Rt. Rev. Ld. Bishop of Norwich
Stephen Napleton, M. D.
Mr Robert Nash, M. A. Fellow of Wadham College, Oxon
Mr Theophilus Firmin Nash
Dr. David Natto
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Books
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Sir Isaac Newton, 12 Books
Sir Michael Newton
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O

Right Hon. Lord Orrery
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P

Right Hon. Countess of Pembroke, 10 Books
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Thomas Parratt, M. D.
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—— Powis, Esq;
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George Pye, M. D.
Samuel Pye, M. D.
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Q.

His Grace the Duke of Queenborough
Rev. Mr. Question, M. A. of Exeter College, Oxon
Jeremiah Quare, Merchant

R.

His Grace the Duke of Richmond
The Rt. Rev. Ld. Bishop of Raphoe
The Rt. Hon. Lord John Russel
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S

His Grace the Duke of St. Albans
Rt. Hon. Earl of Sunderland
Rt. Hon. Earl of Scarborough
Rt. Rev. Ld. Bp. of Salisbury
Rt. Rev. Lord Bishop of St. Asaph
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Rt. Hon. Lord Shaftsbury
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Denmark
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Prof.
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T.

His Grace the Archbishop of Tuam
Right Hon. Earl of Tankerville
Rt. Hon. Ld. Viscount Townshend, One of His Majesty’s Principal
Secretaries of State
Right Honourable Lady Viscountess Townshend
Right Hon Ld Viscount Tyrconnel
The Honourable Lord Trevor Charles Talbot, Esq; Solicitor-General.
Francis Talbot, Esq;
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Mr Talbot
Thomas Tanner, D. D. Chancellor of Norwich
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Mr Henry Tatham
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Mr John Tayleur
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Col. Tyrrell
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V

Rt. Hon. Lord Viscount Vane
Rev. Mr Thomas Valentine

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W

Rt. Hon. the Earl of Winchelsea
Rt. Rev. Lord Bishop of Winchester
Rev. Mr Wade
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Rev. Mr Wagstaffe
Rev. Dr. Edward Wake
Mr Jasper Wakefield
Mr Samuel Walbank
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Rev. Mr Isaac Watts
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INTRODUCTION.
T HE manner, in which Sir I s a a c N e w t o n has published his
philosophical discoveries, occasions them to lie very much
concealed from all, who have not made the mathematics
particularly their study. He once, indeed, intended to deliver, in
a more familiar way, that part of his inventions, which relates to the system
of the world; but upon farther consideration he altered his design. For as the
nature of those discoveries made it impossible to prove them upon any
other than geometrical principles; he apprehended, that those, who should
not fully perceive the force of his arguments, would hardly be prevailed on
to exchange their former sentiments for new opinions, so very different
from what were commonly received[1]. He therefore chose rather to explain
himself only to mathematical readers; and declined the attempting to
instruct such in any of his principles, who, by not comprehending his
method of reasoning, could not, at the first appearance of his discoveries,
have been persuaded of their truth. But now, since Sir I s a a c N e w t o n’s
doctrine has been fully established by the unanimous approbation of all,
who are qualified to understand the same; it is without doubt to be wished,
that the whole of his improvements in philosophy might be universally
known. For this purpose therefore I drew up the following papers, to give a
general notion of our great philosopher’s inventions to such, as are not
prepared to read his own works, and yet might desire to be informed of the
progress, he has made in natural knowledge; not doubting but there were
many, besides those, whose turn of mind had led them into a course of
mathematical studies, that would take great pleasure in tasting of this
delightful fountain of science.
2. I t is a just remark, which has been made upon the human mind, that
nothing is more suitable to it, than the contemplation of truth; and that all
men are moved with a strong desire after knowledge; esteeming it

honourable to excel therein; and holding it, on the contrary, disgraceful to
mistake, err, or be in any way deceived. And this sentiment is by nothing
more fully illustrated, than by the inclination of men to gain an
acquaintance with the operations of nature; which disposition to enquire
after the causes of things is so general, that all men of letters, I believe, find
themselves influenced by it. Nor is it difficult to assign a reason for this, if
we consider only, that our desire after knowledge is an effect of that taste
for the sublime and the beautiful in things, which chiefly constitutes the
difference between the human life, and the life of brutes. These inferior
animals partake with us of the pleasures, that immediately flow from the
bodily senses and appetites; but our minds are furnished with a superior
sense, by which we are capable of receiving various degrees of delight,
where the creatures below us perceive no difference. Hence arises that
pursuit of grace and elegance in our thoughts and actions, and in all things
belonging to us, which principally creates imployment for the active mind
of man. The thoughts of the human mind are too extensive to be confined
only to the providing and enjoying of what is necessary for the support of
our being. It is this taste, which has given rise to poetry, oratory, and every
branch of literature and science. From hence we feel great pleasure in
conceiving strongly, and in apprehending clearly, even where the passions
are not concerned. Perspicuous reasoning appears not only beautiful; but,
when set forth in its full strength and dignity, it partakes of the sublime, and
not only pleases, but warms and elevates the soul. This is the source of our
strong desire of knowledge; and the same taste for the sublime and the
beautiful directs us to chuse particularly the productions of nature for the
subject of our contemplation: our creator having so adapted our minds to
the condition, wherein he has placed us, that all his visible works, before we
inquire into their make, strike us with the most lively ideas of beauty and
magnificence.
3. B u t if there be so strong a passion in contemplative minds for
natural philosophy; all such must certainly receive a particular pleasure in
being informed of Sir I s a a c N e w t o n’s discoveries, who alone has
been able to make any great advancements in the true course leading to
natural knowledge: whereas this important subject had before been usually
attempted with that negligence, as cannot be reflected on without surprize.
Excepting a very few, who, by pursuing a more rational method, had gained
a little true knowledge in some particular parts of nature; the writers in this

science had generally treated of it after such a manner, as if they thought,
that no degree of certainty was ever to be hoped for. The custom was to
frame conjectures; and if upon comparing them with things, there appeared
some kind of agreement, though very imperfect, it was held sufficient. Yet
at the same time nothing less was undertaken than intire systems, and
fathoming at once the greatest depths of nature; as if the secret causes of
natural effects, contrived and framed by infinite wisdom, could be searched
out by the slightest endeavours of our weak understandings. Whereas the
only method, that can afford us any prospect of success in this difficult
work, is to make our enquiries with the utmost caution, and by very slow
degrees. And after our most diligent labour, the greatest part of nature will,
no doubt, for ever remain beyond our reach.
4. T h i s neglect of the proper means to enlarge our knowledge, joined
with the presumption to attempt, what was quite out of the power of our
limited faculties, the Lord Bacon judiciously observes to be the great
obstruction to the progress of science[2]. Indeed that excellent person was
the first, who expresly writ against this way of philosophizing; and he has
laid open at large the absurdity of it in his admirable treatise, intitled Novum
organon scientiarum; and has there likewise described the true method,
which ought to be followed.
5. T h e r e are, saith he, but two methods, that can be taken in the
pursuit of natural knowledge. One is to make a hasty transition from our
first and slight observations on things to general axioms, and then to
proceed upon those axioms, as certain and uncontestable principles, without
farther examination. The other method; (which he observes to be the only
true one, but to his time unattempted;) is to proceed cautiously, to advance
step by step, reserving the most general principles for the last result of our
inquiries[3]. Concerning the first of these two methods; where objections,
which happen to appear against any such axioms taken up in haste, are
evaded by some frivolous distinction, when the axiom it self ought rather to
be corrected[4]; he affirms, that the united endeavours of all ages cannot
make it successful; because this original error in the first digestion of the
mind (as he expresses himself) cannot afterwards be remedied[5]: whereby
he would signify to us, that if we set out in a wrong way; no diligence or
art, we can use, while we follow so erroneous a course, will ever bring us to
our designed end. And doubtless it cannot prove otherwise; for in this

spacious field of nature, if once we forsake the true path, we shall
immediately lose our selves, and must for ever wander with uncertainty.
6. T h e impossibility of succeeding in so faulty a method of
philosophizing his Lordship endeavours to prove from the many false
notions and prejudices, to which the mind of man is exposed[6]. And since
this judicious writer apprehends, that men are so exceeding liable to fall
into these wrong tracts of thinking, as to incur great danger of being misled
by them, even while they enter on the true course in pursuit of nature[7]; I
trust, I shall be excused, if, by insisting a little particularly upon this
argument, I endeavour to remove whatever prejudice of this kind, might
possibly entangle the mind of any of my readers.
7. H i s Lordship has reduced these prejudices and false modes of
conception under four distinct heads[8].
8. T h e first head contains such, as we are subject to from the very
condition of humanity, through the weakness both of our senses, and of the
faculties of the mind[9]; seeing, as this author well observes, the subtilty of
nature far exceeds the greatest subtilty of our senses or acutest
reasonings[10]. One of the false modes of conception, which he mentions
under this head, is the forming to our selves a fanciful simplicity and
regularity in natural things. This he illustrates by the following instances;
the conceiving the planets to move in perfect circles; the adding an orb of
fire to the other three elements, and the supposing each of these to exceed
the other in rarity, just in a decuple proportion[11]. And of the same nature is
the assertion of D e s C a r t e s, without any proof, that all things are made
up of three kinds of matter only[12]. As also this opinion of another
philosopher; that light, in passing through different mediums, was refracted,
so as to proceed by that way, through which it would move more speedily,
than through any other[13]. The second erroneous turn of mind, taken notice
of by his Lordship under this head, is, that all men are in some degree prone
to a fondness for any notions, which they have once imbibed; whereby they
often wrest things to reconcile them to those notions, and neglect the
consideration of whatever will not be brought to an agreement with them;
just as those do, who are addicted to judicial astrology, to the observation of
dreams, and to such-like superstitions; who carefully preserve the memory
of every incident, which serves to confirm their prejudices, and let slip out
of their minds all instances, that make against them[14]. There is also a

farther impediment to true knowledge, mentioned under the same head by
this noble writer, which is; that whereas, through the weakness and
imperfection of our senses, many things are concealed. from us, which have
the greatest effect in producing natural appearances; our minds are
ordinarily most affected by that, which makes the strongest impression on
our organs of sense; whereby we are apt to judge of the real importance of
things in nature by a wrong measure[15]. So, because the figuration and the
motion of bodies strike our senses more immediately than most of their
other properties, Des Cartes and his followers will not allow any other
explication of natural appearances, than from the figure and motion of the
parts of matter. By which example we see how justly his Lordship observes
this cause of error to be the greatest of any[16]; since it has given rise to a
fundamental principle in a system of philosophy, that not long ago obtained
almost an universal reputation.
9. T h e s e are the chief branches of those obstructions to knowledge,
which this author has reduced under his first head of false conceptions. The
second head contains the errors, to which particular persons are more
especially obnoxious[17]. One of these is the consequence of a preceding
observation: that as we are exposed to be captivated by any opinions, which
have once taken possession of our minds; so in particular, natural
knowledge has been much corrupted by the strong attachment of men to
some one part of science, of which they reputed themselves the inventers,
or about which they have spent much of their time; and hence have been apt
to conceive it to be of greater use in the study of natural philosophy than it
was: like Aristotle, who reduced his physics to logical disputations; and
the chymists, who thought, that nature could be laid open only by the force
of their fires[18]. Some again are wholly carried away by an excessive
veneration for antiquity; others, by too great fondness for the moderns; few
having their minds so well balanced, as neither to depreciate the merit of
the ancients, nor yet to despise the real improvements of later times[19]. To
this is added by his Lordship a difference in the genius of men, that some
are most fitted to observe the similitude, there is in things, while others are
more qualified to discern the particulars, wherein they disagree; both which
dispositions of mind are useful: but to the prejudice of philosophy men are
apt to run into excess in each; while one sort of genius dwells too much

upon the gross and sum of things, and the other upon trifling minutenesses
and shadowy distinctions[20].
10. U n d e r the third head of prejudices and false notions this writer
considers such, as follow from the lax and indefinite use of words in
ordinary discourse; which occasions great ambiguities and uncertainties in
philosophical debates (as another eminent philosopher has since shewn
more at large[21];) insomuch that this our author thinks a strict defining of
terms to be scarce an infallible remedy against this inconvenience[22]. And
perhaps he has no small reason on his side: for the common inaccurate
sense of words, notwithstanding the limitations given them by definitions,
will offer it self so constantly to the mind, as to require great caution and
circumspection for us not to be deceived thereby. Of this we have a very
eminent instance in the great disputes, that have been raised about the use
of the word attraction in philosophy; of which we shall be obliged hereafter
to make particular mention[23]. Words thus to be guarded against are of two
kinds. Some are names of things, that are only imaginary[24]; such words
are wholly to be rejected. But there are other terms, that allude to what is
real, though their signification is confused[25]. And these latter must of
necessity be continued in use; but their sense cleared up, and freed, as much
as possible, from obscurity.
11. T h e last general head of these errors comprehends such, as follow
from the various sects of false philosophies; which this author divides into
three sorts, the sophistical, empirical, and superstitious[26]. By the first of
these he means a philosophy built upon speculations only without
experiments[27]; by the second, where experiments are blindly adhered to,
without proper reasoning upon them[28]; and by the third, wrong opinions of
nature fixed in mens minds either through false religions, or from
misunderstanding the declarations of the true[29].
12. T h e s e are the four principal canals, by which this judicious author
thinks, that philosophical errors have flowed in upon us. And he rightly
observes, that the faulty method of proceeding in philosophy, against which
he writes[30], is so far from assisting us towards overcoming these
prejudices; that he apprehends it rather suited to rivet them more firmly to
the mind[31]. How great reason then has his Lordship to call this way of
philosophizing the parent of error, and the bane of all knowledge[32]? For,

indeed, what else but mistakes can so bold and presumptuous a treatment of
nature produce? have we the wisdom necessary to frame a world, that we
should think so easily, and with so slight a search to enter into the most
secret springs of nature, and discover the original causes of things? what
chimeras, what monsters has not this preposterous method brought forth?
what schemes, or what hypothesis’s of the subtilest wits has not a stricter
enquiry into nature not only overthrown, but manifested to be ridiculous
and absurd? Every new improvement, which we make in this science, lets
us see more and more the weakness of our guesses. Dr. H a r v e y, by that
one discovery of the circulation of the blood, has dissipated all the
speculations and reasonings of many ages upon the animal oeconomy.
A s e l l i u s, by detecting the lacteal veins, shewed how little ground all
physicians and philosophers had in conjecturing, that the nutritive part of
the aliment was absorbed by the mouths of the veins spread upon the
bowels: and then Pecquet, by finding out the thoracic duct, as evidently
proved the vanity of the opinion, which was persisted in after the lacteal
vessels were known, that the alimental juice was conveyed immediately to
the liver, and there converted into blood.
13. A s these things set forth the great absurdity of proceeding in
philosophy on conjectures, by informing us how far the operations of nature
are above our low conceptions; so on the other hand, such instances of
success from a more judicious method shew us, that our bountiful maker
has not left us wholly without means of delighting our selves in the
contemplation of his wisdom. That by a just way of inquiry into nature, we
could not fail of arriving at discoveries very remote from our
apprehensions; the Lord B a c o n himself argues from the experience of
mankind. If, says he, the force of guns should be described to any one
ignorant of them, by their effects only, he might reasonably suppose, that
those engines of destruction were only a more artificial composition, than
he knew, of wheels and other mechanical powers: but it could never enter
his thoughts, that their immense force should be owing to a peculiar
substance, which would enkindle into so violent an explosion, as we
experience in gunpowder: since he would no where see the least example of
any such operation; except perhaps in earthquakes and thunder, which he
would doubtless look upon as exalted powers of nature, greatly surpassing
any art of man to imitate. In the same manner, if a stranger to the original of
silk were shewn a garment made of it, he would be very far from imagining

so strong a substance to be spun out of the bowels of a small worm; but
must certainly believe it either a vegetable substance, like flax or cotton; or
the natural covering of some animal, as wool is of sheep. Or had we been
told, before the invention of the magnetic needle among us, that another
people was in possession of a certain contrivance, by which they were
inabled to discover the position of the heavens, with vastly more ease, than
we could do; what could have been imagined more, than that they were
provided with some fitter astronomical instrument for this purpose than we?
That any stone should have so amazing a property, as we find in the
magnet, must have been the remotest from our thoughts[33].
14. B u t what surprizing advancements in the knowledge of nature may
be made by pursuing the true course in philosophical inquiries; when those
searches are conducted by a genius equal to so divine a work, will be best
understood by considering Sir I s a a c N e w t o n discoveries. That my’s
reader may apprehend as just a notion of these, as can be conveyed to him,
by the brief account, which I intend to lay before him; I have set apart this
introduction for explaining, in the fullest manner I am able, the principles,
whereon Sir I s a a c N e w t o n proceeds. For without a clear conception
of these, it is impossible to form any true idea of the singular excellence of
the inventions of this great philosopher.
15. T h e principles then of this philosophy are; upon no consideration to
indulge conjectures concerning the powers and laws of nature, but to make
it our endeavour with all diligence to search out the real and true laws, by
which the constitution of things is regulated. The philosopher’s first care
must be to distinguish, what he sees to be within his power, from what is
beyond his reach; to assume no greater degree of knowledge, than what he
finds himself possessed of; but to advance by slow and cautious steps; to
search gradually into natural causes; to secure to himself the knowledge of
the most immediate cause of each appearance, before he extends his views
farther to causes more remote. This is the method, in which philosophy
ought to be cultivated; which does not pretend to so great things, as the
more airy speculations; but will perform abundantly more: we shall not
perhaps seem to the unskilful to know so much, but our real knowledge will
be greater. And certainly it is no objection against this method, that some
others promise, what is nearer to the extent of our wishes: since this, if it
will not teach us all we could desire to be informed of, will however give us
some true light into nature; which no other can do. Nor has the philosopher

any reason to think his labour lost, when he finds himself stopt at the cause
first discovered by him, or at any other more remote cause, short of the
original: for if he has but sufficiently proved any one cause, he has entered
so far into the real constitution of things, has laid a safe foundation for
others to work upon, and has facilitated their endeavours in the search after
yet more distant causes; and besides, in the mean time he may apply the
knowledge of these intermediate causes to many useful purposes. Indeed
the being able to make practical deductions from natural causes, constitutes
the great distinction between the true philosophy and the false. Causes
assumed upon conjecture, must be so loose and undefined, that nothing
particular can be collected from them. But those causes, which are brought
to light by a strict examination of things, will be more distinct. Hence it
appears to have been no unuseful discovery, that the ascent of water in
pumps is owing to the pressure of the air by its weight or spring; though the
causes, which make the air gravitate, and render it elastic, be unknown: for
notwithstanding we are ignorant of the original, whence these powers of the
air are derived; yet we may receive much advantage from the bare
knowledge of these powers. If we are but certain of the degree of force,
wherewith they act, we shall know the extent of what is to be expected from
them; we shall know the greatest height, to which it is possible by pumps to
raise water; and shall thereby be prevented from making any useless efforts
towards improving these instruments beyond the limits prescribed to them
by nature; whereas without so much knowledge as this, we might probably
have wasted in attempts of this kind much time and labour. How long did
philosophers busy themselves to no purpose in endeavouring to perfect
telescopes, by forming the glasses into some new figure; till Sir I s a a c
N e w t o n demonstrated, that the effects of telescopes were limited from
another cause, than was supposed; which no alteration in the figure of the
glasses could remedy? What method Sir I s a a c N e w t o n himself has
found for the improvement of telescopes shall be explained hereafter[34].
But at present I shall proceed to illustrate, by some farther instances, this
distinguishing character of the true philosophy, which we have now under
consideration. It was no trifling discovery, that the contraction of the
muscles of animals puts their limbs in motion, though the original cause of
that contraction remains a secret, and perhaps may always do so; for the
knowledge of thus much only has given rise to many speculations upon the
force and artificial disposition of the muscles, and has opened no narrow

prospect into the animal fabrick. The finding out, that the nerves are great
agents in this action, leads us yet nearer to the original cause, and yields us
a wider view of the subject. And each of these steps affords us assistance
towards restoring this animal motion, when impaired in our selves, by
pointing out the seats of the injuries, to which it is obnoxious. To neglect all
this, because we can hitherto advance no farther, is plainly ridiculous. It is
confessed by all, that G a l i l e o greatly improved philosophy, by shewing,
as we shall relate hereafter, that the power in bodies, which we call gravity,
occasions them to move downwards with a velocity equably accelerated[35];
and that when any body is thrown forwards, the same power obliges it to
describe in its motion that line, which is called by geometers a parabola[36]:
yet we are ignorant of the cause, which makes bodies gravitate. But
although we are unacquainted with the spring, whence this power in nature
is derived, nevertheless we can estimate its effects. When a body falls
perpendicularly, it is known, how long time it takes in descending from any
height whatever: and if it be thrown forwards, we know the real path, which
it describes; we can determine in what direction, and with what degree of
swiftness it must be projected, in order to its striking against any object
desired; and we can also ascertain the very force, wherewith it will strike.
Sir I s a a c N e w t o n has farther taught, that this power of gravitation
extends up to the moon, and causes that planet to gravitate as much towards
the earth, as any of the bodies, which are familiar to us, would, if placed at
the same distance[37]: he has proved likewise, that all the planets gravitate
towards the sun, and towards one another; and that their respective motions
follow from this gravitation. All this he has demonstrated upon indisputable
geometrical principles, which cannot be rendered precarious for want of
knowing what it is, which causes these bodies thus mutually to gravitate:
any more than we can doubt of the propensity in all the bodies about us, to
descend towards the earth; or can call in question the forementioned
propositions of G a l i l e o, which are built upon that principle. And as
G a l i l e o has shewn more fully, than was known before, what effects
were produced in the motion of bodies by their gravitation towards the
earth; so Sir I s a a c N e w t o n, by this his invention, has much advanced
our knowledge in the celestial motions. By discovering that the moon
gravitates towards the sun, as well as towards the earth; he has laid open
those intricacies in the moon’s motion, which no astronomer, from
observations only, could ever find out[38]: and one kind of heavenly bodies,

the comets, have their motion now clearly ascertained; whereof we had
before no true knowledge at all[39].
16. D o u b t l e s s it might be expected, that such surprizing success
should have silenced, at once, every cavil. But we have seen the contrary.
For because this philosophy professes modestly to keep within the extent of
our faculties, and is ready to confess its imperfections, rather than to make
any fruitless attempts to conceal them, by seeking to cover the defects in
our knowledge with the vain ostentation of rash and groundless conjectures;
hence has been taken an occasion to insinuate that we are led to miraculous
causes, and the occult qualities of the schools.
17. B u t the first of these accusations is very extraordinary. If by calling
these causes miraculous nothing more is meant than only, that they often
appear to us wonderful and surprizing, it is not easy to see what difficulty
can be raised from thence; for the works of nature discover every where
such proofs of the unbounded power, and the consummate wisdom of their
author, that the more they are known, the more they will excite our
admiration: and it is too manifest to be insisted on, that the common sense
of the word miraculous can have no place here, when it implies what is
above the ordinary course of things. The other imputation, that these causes
are occult upon the account of our not perceiving what produces them,
contains in it great ambiguity. That something relating to them lies hid, the
followers of this philosophy are ready to acknowledge, nay desire it should
be carefully remarked, as pointing out proper subjects for future inquiry.
But this is very different from the proceeding of the schoolmen in the
causes called by them occult. For as their occult qualities were understood
to operate in a manner occult, and not apprehended by us; so they were
obtruded upon us for such original and essential properties in bodies, as
made it vain to seek any farther cause; and a greater power was attributed to
them, than any natural appearances authorized. For instance, the rise of
water in pumps was ascribed to a certain abhorrence of a vacuum, which
they thought fit to assign to nature. And this was so far a true observation,
that the water does move, contrary to its usual course, into the space, which
otherwise would be left void of any sensible matter; and, that the procuring
such a vacuity was the apparent cause of the water’s ascent. But while we
were not in the least informed how this power, called an abhorrence of a
vacuum, produced the visible effects; instead of making any advancement
in the knowledge of nature, we only gave an artificial name to one of her

operations: and when the speculation was pushed so beyond what any
appearances required, as to have it concluded, that this abhorrence of a
vacuum was a power inherent in all matter, and so unlimited as to render it
impossible for a vacuum to exist at all; it then became a much greater
absurdity, in being made the foundation of a most ridiculous manner of
reasoning; as at length evidently appeared, when it came to be discovered,
that this rise of the water followed only from the pressure of the air, and
extended it self no farther, than the power of that cause. The scholastic stile
in discoursing of these occult qualities, as if they were essential differences
in the very substances, of which bodies consisted, was certainly very
absurd; by reason it tended to discourage all farther inquiry. But no such ill
consequences can follow from the considering of any natural causes, which
confessedly are not traced up to their first original. How shall we ever come
to the knowledge of the several original causes of things, otherwise than by
storing up all intermediate causes which we can discover? Are all the
original and essential properties of matter so very obvious, that none of
them can escape our first view? This is not probable. It is much more likely,
that, if some of the essential properties are discovered by our first
observations, a stricter examination should bring more to light.
18. B u t in order to clear up this point concerning the essential
properties of matter, let us consider the subject a little distinctly. We are to
conceive, that the matter, out of which the universe of things is formed, is
furnished with certain qualities and powers, whereby it is rendered fit to
answer the purposes, for which it was created. But every property, of which
any particle of this matter is in it self possessed, and which is not barely the
consequence of the union of this particle with other portions of matter, we
may call an essential property: whereas all other qualities or attributes
belonging to bodies, which depend on their particular frame and
composition, are not essential to the matter, whereof such bodies are made;
because the matter of these bodies will be deprived of those qualities, only
by the dissolution of the body, without working any change in the original
constitution of one single particle of this mass of matter. Extension we
apprehend to be one of these essential properties, and impenetrability
another. These two belong universally to all matter; and are the principal
ingredients in the idea, which this word matter usually excites in the mind.
Yet as the idea, marked by this name, is not purely the creature of our own
understandings, but is taken for the representation of a certain substance

without us; if we should discover, that every part of the substance, in which
we find these two properties, should likewise be endowed universally with
any other essential qualities; all these, from the time they come to our
notice, must be united under our general idea of matter. How many such
properties there are actually in all matter we know not; those, of which we
are at present apprized, have been found out only by our observations on
things; how many more a farther search may bring to light, no one can say;
nor are we certain, that we are provided with sufficient methods of
perception to discern them all. Therefore, since we have no other way of
making discoveries in nature, but by gradual inquiries into the properties of
bodies; our first step must be to admit without distinction all the properties,
which we observe; and afterwards we must endeavour, as far as we are able,
to distinguish between the qualities, wherewith the very substances
themselves are indued, and those appearances, which result from the
structure only of compound bodies. Some of the properties, which we
observe in things, are the attributes of particular bodies only; others
universally belong to all, that fall under our notice. Whether some of the
qualities and powers of particular bodies, be derived from different kinds of
matter entring their composition, cannot, in the present imperfect state of
our knowledge, absolutely be decided; though we have not yet any reason
to conclude, but that all the bodies, with which we converse, are framed out
of the very same kind of matter, and that their distinct qualities are
occasioned only by their structure; through the variety whereof the general
powers of matter are caused to produce different effects. On the other hand,
we should not hastily conclude, that whatever is found to appertain to all
matter, which falls under our examination, must for that reason only be an
essential property thereof, and not be derived from some unseen disposition
in the frame of nature. Sir I s a a c N e w t o n has found reason to
conclude, that gravity is a property universally belonging to all the
perceptible bodies in the universe, and to every particle of matter, whereof
they are composed. But yet he no where asserts this property to be essential
to matter. And he was so far from having any design of establishing it as
such, that, on the contrary, he has given some hints worthy of himself at a
cause for it[40]; and expresly says, that he proposed those hints to shew, that
he had no such intention[41].
19. I t appears from hence, that it is not easy to determine, what
properties of bodies are essentially inherent in the matter, out of which they

are made, and what depend upon their frame and composition. But certainly
whatever properties are found to belong either to any particular systems of
matter, or universally to all, must be considered in philosophy; because
philosophy will be otherwise imperfect. Whether those properties can be
deduced from some other appertaining to matter, either among those, which
are already known, or among such as can be discovered by us, is afterwards
to be sought for the farther improvement of our knowledge. But this inquiry
cannot properly have place in the deliberation about admitting any property
of matter or bodies into philosophy; for that purpose it is only to be
considered, whether the existence of such a property has been justly proved
or not. Therefore to decide what causes of things are rightly received into
natural philosophy, requires only a distinct and clear conception of what
kind of reasoning is to be allowed of as convincing, when we argue upon
the works of nature.
20. T h e proofs in natural philosophy cannot be so absolutely
conclusive, as in the mathematics. For the subjects of that science are
purely the ideas of our own minds. They may be represented to our senses
by material objects, but they are themselves the arbitrary productions of our
own thoughts; so that as the mind can have a full and adequate knowledge
of its own ideas, the reasoning in geometry can be rendered perfect. But in
natural knowledge the subject of our contemplation is without us, and not
so compleatly to be known: therefore our method of arguing must fall a
little short of absolute perfection. It is only here required to steer a just
course between the conjectural method of proceeding, against which I have
so largely spoke; and demanding so rigorous a proof, as will reduce all
philosophy to mere scepticism, and exclude all prospect of making any
progress in the knowledge of nature.
21. T h e concessions, which are to be allowed in this science, are by Sir
I s a a c N e w t o n included under a very few simple precepts.
22. T h e first is, that more causes are not to be received into philosophy,
than are sufficient to explain the appearances of nature. That this rule is
approved of unanimously, is evident from those expressions so frequent
among all philosophers, that nature does nothing in vain; and that a variety
of means, where fewer would suffice, is needless. And certainly there is the
highest reason for complying with this rule. For should we indulge the
liberty of multiplying, without necessity, the causes of things, it would
reduce all philosophy to mere uncertainty; since the only proof, which we

can have, of the existence of a cause, is the necessity of it for producing
known effects. Therefore where one cause is sufficient, if there really
should in nature be two, which is in the last degree improbable, we can
have no possible means of knowing it, and consequently ought not to take
the liberty of imagining, that there are more than one.
23. T h e second precept is the direct consequence of the first, that to
like effects are to be ascribed the same causes. For instance, that respiration
in men and in brutes is brought about by the same means; that bodies
descend to the earth here in Europe, and in America from the same
principle; that the light of a culinary fire, and of the sun have the same
manner of production; that the reflection of light is effected in the earth, and
in the planets by the same power; and the like.
24. T h e third of these precepts has equally evident reason for it. It is
only, that those qualities, which in the same body can neither be lessened
nor increased, and which belong to all bodies that are in our power to make
trial upon, ought to be accounted the universal properties of all bodies
whatever.
25. I n this precept is founded that method of arguing by induction,
without which no progress could be made in natural philosophy. For as the
qualities of bodies become known to us by experiments only; we have no
other way of finding the properties of such bodies, as are out of our reach to
experiment upon, but by drawing conclusions from those which fall under
our examination. The only caution here required is, that the observations
and experiments, we argue upon, be numerous enough, and that due regard
be paid to all objections, that occur, as the Lord Bacon very judiciously
directs[42]. And this admonition is sufficiently complied with, when by
virtue of this rule we ascribe impenetrability and extension to all bodies,
though we have no sensible experiment, that affords a direct proof of any of
the celestial bodies being impenetrable; nor that the fixed stars are so much
as extended. For the more perfect our instruments are, whereby we attempt
to find their visible magnitude, the less they appear; insomuch that all the
sensible magnitude, which we observe in them, seems only to be an optical
deception by the scattering of their light. However, I suppose no one will
imagine they are without any magnitude, though their immense distance
makes it undiscernable by us. After the same manner, if it can be proved,
that all bodies here gravitate towards the earth, in proportion to the quantity

of solid matter in each; and that the moon gravitates to the earth likewise, in
proportion to the quantity of matter in it; and that the sea gravitates towards
the moon, and all the planets towards each other; and that the very comets
have the same gravitating faculty; we shall have as great reason to conclude
by this rule, that all bodies gravitate towards each other. For indeed this rule
will more strongly hold in this case, than in that of the impenetrability of
bodies; because there will more instances be had of bodies gravitating, than
of their being impenetrable.
25. T h i s is that method of induction, whereon all philosophy is
founded; which our author farther inforces by this additional precept, that
whatever is collected from this induction, ought to be received,
notwithstanding any conjectural hypothesis to the contrary, till such times
as it shall be contradicted or limited by farther observations on nature.

B O O K I.
Concerning the

MOTION of BODIES
IN GENERAL.

Chap. I.
Of the LAWS of MOTION.
H AVING thus explained Sir I s a a c N e w t o n ’ s method of
reasoning in philosophy, I shall now proceed to my intended
account of his discoveries. These are contained in two treatises.
In one of them, the Mathematical principles of natural
philosophy, his chief design is to shew by what laws the heavenly motions
are regulated; in the other, his Optics, he discourses of the nature of light
and colours, and of the action between light and bodies. This second treatise
is wholly confined to the subject of light: except some conjectures proposed
at the end concerning other parts of nature, which lie hitherto more
concealed. In the other treatise our author was obliged to smooth the way to
his principal intention, by explaining many things of a more general nature:
for even some of the most simple properties of matter were scarce well
established at that time. We may therefore reduce Sir I s a a c N e w t o n’s
doctrine under three general heads; and I shall accordingly divide my
account into three books. In the first I shall speak of what he has delivered
concerning the motion of bodies, without regard to any particular system of
matter; in the second I shall treat of the heavenly motions; and the third
shall be employed upon light.
2. I n the first part of my design, we must begin with an account of the
general laws of motion.
3. T h e s e laws are some universal affections and properties of matter
drawn from experience, which are made use of as axioms and evident
principles in all our arguings upon the motion of bodies. For as it is the
custom of geometers to assume in their demonstrations some propositions,
without exhibiting the proof of them; so in philosophy, all our reasoning
must be built upon some properties of matter, first admitted as principles
whereon to argue. In geometry these axioms are thus assumed, on account
of their being so evident as to make any proof in form needless. But in
philosophy no properties of bodies can be in this manner received for self-
evident; since it has been observed above, that we can conclude nothing

concerning matter by any reasonings upon its nature and essence, but that
we owe all the knowledge, we have thereof, to experience. Yet when our
observations on matter have inform’d us of some of its properties, we may
securely reason upon them in our farther inquiries into nature. And these
laws of motion, of which I am here to speak, are found so universally to
belong to bodies, that there is no motion known, which is not regulated by
them. These are by Sir I s a a c N e w t o n reduced to three[43].
4. T h e first law is, that all bodies have such an indifference to rest, or
motion, that if once at rest they remain so, till disturbed by some power
acting upon them: but if once put in motion, they persist in it; continuing to
move right forwards perpetually, after the power, which gave the motion, is
removed; and also preserving the same degree of velocity or quickness, as
was first communicated, not stopping or remitting their course, till
interrupted or otherwise disturbed by some new power impressed.
5. T h e second law of motion is, that the alteration of the state of any
body, whether from rest to motion, or from motion to rest, or from one
degree of motion to another, is always proportional to the force impressed.
A body at rest, when acted upon by any power, yields to that power, moving
in the same line, in which the power applied is directed; and moves with a
less or greater degree of velocity, according to the degree of the power; so
that twice the power shall communicate a double velocity, and three times
the power a threefold velocity. If the body be moving, and the power
impressed act upon the body in the direction of its motion, the body shall
receive an addition to its motion, as great as the motion, into which that
power would have put it from a state of rest; but if the power impressed
upon a moving body act directly opposite to its former motion, that power
shall then take away from the body’s motion, as much as in the other case it
would have added to it. Lastly, if the power be impressed obliquely, there
will arise an oblique motion differing more or less from the former
direction, according as the new impression is greater or less. For example, if
the body A (in fig. 1.) be moving in the direction A B, and when it is at the
point A, a power be impressed upon it in the direction A C, the body shall
from henceforth neither move in its first direction A B, nor in the direction
of the adventitious power, but shall take a course as A D between them: and
if the power last impressed be just equal to that, which first gave to the
body its motion; the line A D shall pass in the middle between A B and A C,
dividing the angle under B A C into two equal parts; but if the power last

impressed be greater than the first, the line A D shall incline most to A C;
whereas if the last impression be less than the first, the line A D shall
incline most to A B. To be more particular, the situation of the line A D is
always to be determined after this manner. Let A E be the space, which the
body would have moved through in the line A B during any certain portion
of time; provided that body, when at A, had received no second impulse.
Suppose likewise, that A F is the part of the line A C, through which the
body would have moved during an equal portion of time, if it had been at
rest in A, when it received the impulse in the direction A C: then if from E
be drawn a line parallel to, or equidistant from A C, and from F another line
parallel to A B, those two lines will meet in the line A D.
6. T h e third and last of these laws of motion is, that when any body
acts upon another, the action of that body upon the other is equalled by the
contrary reaction of that other body upon the first.
7. T h e s e laws of motion are abundantly confirmed by this, that all the
deductions made from them, in relation to the motion of bodies, how
complicated soever, are found to agree perfectly with observation. This
shall be shewn more at large in the next chapter. But before we proceed to
so diffusive a proof; I chuse here to point out those appearances of bodies,
whereby the laws of motion are first suggested to us.
8. D a i l y observation makes it appear to us, that any body, which we
once see at rest, never puts it self into fresh motion; but continues always in
the same place, till removed by some power applied to it.
9. A g a i n, whenever a body is once in motion, it continues in that
motion some time after the moving power has quitted it, and it is left to it
self. Now if the body continue to move but a single moment, after the
moving power has left it, there can no reason be assigned, why it should
ever stop without some external force. For it is plain, that this continuance
of the motion is caused only by the body’s having already moved, the sole
operation of the power upon the body being the putting it in motion;
therefore that motion continued will equally be the cause of its farther
motion, and so on without end. The only doubt that can remain, is, whether
this motion communicated continues intire, after the power, that caused it,
ceases to act; or whether it does not gradually languish and decrease. And
this suspicion cannot be removed by a transient and slight observation on

bodies, but will be fully cleared up by those more accurate proofs of the
laws of motion, which are to be considered in the next chapter.
10. L a s t l y, bodies in motion appear to affect a straight course
without any deviation, unless when disturbed by some adventitious power
acting upon them. If a body be thrown perpendicularly upwards or
downwards, it appears to continue in the same straight line during the whole
time of its motion. If a body be thrown in any other direction, it is found to
deviate from the line, in which it began to move, more and more continually
towards the earth, whither it is directed by its weight: but since, when the
weight of a body does not alter the direction of its motion, it always moves
in a straight line, without doubt in this other case the body’s, declining from
its first course is no more, than what is caused by its weight alone. As this
appears at first sight to be unquestionable, so we shall have a very distinct
proof thereof in the next chapter, where the oblique motion of bodies will
be particularly considered.
11. T h u s we see how the first of the laws of motion agrees with what
appears to us in moving bodies. But here occurs this farther consideration,
that the real and absolute motion of any body is not visible to us: for we are
our selves also in constant motion along with the earth whereon we dwell;
insomuch that we perceive bodies to move so far only, as their motion is
different from our own. When a body appears to us to lie at rest, in reality it
only continues the motion, it has received, without putting forth any power
to change that motion. If we throw a body in the course or direction,
wherein we are carried our selves; so much motion as we seem to have
given to the body, so much we have truly added to the motion, it had, while
it appeared to us to be at rest. But if we impel a body the contrary way,
although the body appears to us to have received by such an impulse as
much motion, as when impelled the other way; yet in this case we have
taken from the body so much real motion, as we seem to have given it. Thus
the motion, which we see in bodies, is not their real motion, but only
relative with respect to us; and the forementioned observations only shew
us, that this first law of motion has place in this relative or apparent motion.
However, though we cannot make any observation immediately on the
absolute motion of bodies, yet by reasoning upon what we observe in
visible motion, we can discover the properties and effects of real motion.
12. W i t h regard to this first law of motion, which is now under
consideration, we may from the foregoing observations most truly collect,

that bodies are disposed to continue in the absolute motion, which they have
once received, without increasing or diminishing their velocity. When a
body appears to us to lie at rest, it really preserves without change the
motion, which it has in common with our selves: and when we put it into
visible motion, and we see it continue that motion; this proves, that the
body retains that degree of its absolute motion, into which it is put by our
acting upon it: if we give it such an apparent motion, which adds to its real
motion, it preserves that addition; and if our acting on the body takes off
from its real motion, it continues afterwards to move with no more real
motion, than we have left it.
13. A g a i n, we do not observe in bodies any disposition or power
within themselves to change the direction of their motion; and if they had
any such power, it would easily be discovered. For suppose a body by the
structure or disposition of its parts, or by any other circumstance in its
make, was indued with a power of moving it self; this self-moving
principle, which should be thus inherent in the body, and not depend on any
thing external, must change the direction wherein it would act, as often as
the position of the body was changed: so that for instance, if a body was
lying before me in such a position, that the direction, wherein this principle
disposes the body to move, was pointed directly from me; if I then
gradually turned the body about, the direction of this self-moving principle
would no longer be pointed directly from me, but would turn about along
with the body. Now if any body, which appears to us at rest, were furnished
with any such self-moving principle; from the body’s appearing without
motion we must conclude, that this self-moving principle lies directed the
same way as the earth is carrying the body; and such a body might
immediately be put into visible motion only by turning it about in any
degree, that this self-moving principle might receive a different direction.
14. F r o m these considerations it very plainly follows, that if a body
were once absolutely at rest; not being furnished with any principle,
whereby it could put it self into motion, it must for ever continue in the
same place, till acted upon by something external: and also that when a
body is put into motion, it has no power within it self to make any change in
the direction of that motion; and consequently that the body must move on
straight forward without declining any way whatever. But it has before been
shewn, that bodies do not appear to have in themselves any power to
change the velocity of their motion: therefore this first law of motion has

been illustrated and confirmed, as much as can be from the transient
observations, which have here been discoursed upon; and in the next
chapter all this will be farther established by more correct observations.
15. B u t I shall now pass to the second law of motion; wherein, when it
is asserted, that the velocity, with which any body is moved by the action of
a power upon it, is proportional to that power; the degree of power is
supposed to be measured by the greatness of the body, which it can move
with a given celerity. So that the sense of this law is, that if any body were
put into motion with that degree of swiftness, as to pass in one hour the
length of a thousand yards; the power, which would give the same degree of
velocity to a body twice as great, would give this lesser body twice the
velocity, causing it to describe in the same space of an hour two thousand
yards. But by a body twice as great as another, I do not here mean simply of
twice the bulk, but one that contains a double quantity of solid matter.
16. W h y the power, which can move a body twice as great as another
with the same degree of velocity, should be called twice as great as the
power, which can give the lesser body the same velocity, is evident. For if
we should suppose the greater body to be divided into two equal parts, each
equal to the lesser body, each of these halves will require the same degree
of power to move them with the velocity of the lesser body, as the lesser
body it self requires; and therefore both those halves, or the whole greater
body, will require the moving power to be doubled.
17. T h a t the moving power being in this sense doubled, should just
double likewise the velocity of the same body, seems near as evident, if we
consider, that the effect of the power applied must needs be the same,
whether that power be applied to the body at once, or in parts. Suppose then
the double power not applied to the body at once, but half of it first, and
afterwards the other half; it is not conceivable for what reason the half last
applied should come to have a different effect upon the body, from that
which is applied first; as it must have, if the velocity of the body was not
just doubled by the application of it. So far as experience can determine, we
see nothing to favour such a supposition. We cannot indeed (by reason of
the constant motion of the earth) make trial upon any body perfectly at rest,
whereby to see whether a power applied in that case would have a different
effect, from what it has, when the body is already moving; but we find no
alteration in the effect of the same power on account of any difference there
may be in the motion of the body, when the power is applied. The earth

does not always carry bodies with the same degree of velocity; yet we find
the visible effects of any power applied to the same body to be, at all times
the very same: and a bale of goods, or other moveable body lying in a ship
is as easily removed from place to place, while the ship is under sail, if its
motion be steady, as when it is fixed at anchor.
18. N o w this experience is alone sufficient to shew to us the whole of
this law of motion.
19. S i n c e we find, that the same power will always produce the same
change in the motion of any body, whether that body were before moving
with a swifter or slower motion; the change wrought in the motion of a
body depends only on the power applied to it, without any regard to the
body’s former motion: and therefore the degree of motion, which the body
already possesses, having no influence on the power applied to disturb its
operation, the effects of the same power will not only be the same in all
degrees of motion of the body; but we have likewise no reason to doubt, but
that a body perfectly at rest would receive from any power as much motion,
as would be equivalent to the effect of the same power applied to that body
already in motion.
20. A g a i n, suppose a body being at rest, any number of equal powers
should be successively applied to it; pushing it forward from time to time in
the same course or direction. Upon the application of the first power the
body would begin to move; when the second power was applied, it appears
from what has been said, that the motion of the body would become double;
the third power would treble the motion of the body; and so on, till after the
operation of the last power the motion of the body would be as many times
the motion, which the first power gave it, as there are powers in number.
and the effect of this number of powers will be always the same, without
any regard to the space of time taken up in applying them: so that greater or
lesser intervals between the application of each of these powers will
produce no difference at all in their effects. Since therefore the distance of
time between the action of each power is of no consequence; without doubt
the effect will still be the same, though the powers should all be applied at
the very same instant; or although a single power should be applied equal in
strength to the collective force of all these powers. Hence it plainly follows,
that the degree of motion, into which any body will be put out of a state of
rest by any power, will be proportional to that power. A double power will
give twice the velocity, a treble power three times the velocity, and so on.

The foregoing reasoning will equally take place, though the body were not
supposed to be at rest, when the powers began to be applied to it; provided
the direction, in which the powers were applied, either conspired with the
action of the body, or was directly opposite to it. Therefore if any power be
applied to a moving body, and act upon the body either in the direction
wherewith the body moves, so as to accelerate the body; or if it act directly
opposite to the motion of the body, so as to retard it: in both these cases the
change of motion will be proportional to the power applied; nay, the
augmentation of the motion in one case, and the diminution thereof in the
other, will be equal to that degree of motion, into which the same power
would put the body, had it been at rest, when the power was applied.
21. F a r t h e r, a power may be so applied to a moving body, as to act
obliquely to the motion of the body. And the effects of such an oblique
motion may be deduced from this observation; that as all bodies are
continually moving along with the earth, we see that the visible effects of
the same power are always the same, in whatever direction the power acts:
and therefore the visible effects of any power upon a body, which seems
only to be at rest, is always to appearance the same as the real effect would
be upon a body truly at rest. Now suppose a body were moving along the
line A B (in fig. 2.) and the eye accompanied it with an equal motion in the
line C D equidistant from A B; so that when the body is at A, the eye shall
be at C, and when the body is advanced to E in the line A B, the eye shall be
advanced to F in the line C D, the distances A E and C F being equal. It is
evident, that here the body will appear to the eye to be at rest; and the line
F E G drawn from the eye through the body shall seem to the eye to be
immoveable; though as the body and eye move forward together, this line
shall really also move; so that when the body shall be advanced to H and
the eye to K, the line F E G shall be transferred into the situation K H L,
this line K H L being equidistant from F E G. Now if the body when at E
were to receive an impulse in the direction of the line F E G; while the eye
is moving on from F to K and carrying along with it the line F E G, the
body will appear to the eye to move along this line F E G: for this is what
has just now been said; that while bodies are moving along with the earth,
and the spectator’s eye partakes of the same motion, the effect of any power
upon the body will appear to be what it would really have been, had the
body been truly at rest, when the power was applied. From hence it follows,
that when the eye is advanced to K, the body will appear somewhere in the

line K H L. Suppose it appear in M; then it is manifest, from what has been
premised at the beginning of this paragraph, that the distance H M is equal
to what the body would have run upon the line E G, during the time,
wherein the eye has passed from F to K, provided that the body had been at
rest, when acted upon in E. If it be farther asked, after what manner the
body has moved from E to M? I answer, through a straight line; for it has
been shewn above in the explication of the first law of motion, that a
moving body, from the time it is left to it self, will proceed on in one
continued straight line.
22. I f E N be taken equal to H M and N M be drawn; since H M is
equidistant from E N, N M will be equidistant from E H. Therefore the
effect of any power upon a moving body, when that power acts obliquely to
the motion of the body, is to be determined in this manner. Suppose the
body is moving along the straight line A E B, if when the body is come to
E, a power gives it an impulse in the direction of the line E G, to find what
course the body will afterwards take we must proceed thus. Take in E B any
length E H, and in E G take such a length E N, that if the body had been at
rest in E, the power applied to it would have caused it to move over E N in
the same space of time, as it would have employed in passing over E H, if
the power had not acted at all upon it. Then draw H L equidistant from E G,
and N M equidistant from E B. After this, if a line be drawn from E to the
point M, where these two lines meet, the line E M will be the course into
which the body will be put by the action of the power upon it at E.
23. A m a t h e m a t i c a l reader would here expect in some particulars
more regular demonstrations; but as I do not at present address my self to
such, so I hope, what I have now written will render my meaning evident
enough to those, who are unacquainted with that kind of reasoning.
24. N o w as we have been shewing, that some actual force is necessary
either to put bodies out of a state of rest into motion, or to change the
motion, which they have once received; it is proper here to observe, that
this quality in bodies, whereby they preserve their present state, with regard
to motion or rest, till some active force disturb them, is called the v i s
i n e r t i a e of matter: and by this property, matter, sluggish and unactive of
it self, retains all the power impressed upon it, and cannot be made to cease
from action, but by the opposition of as great a power, as that which first
moved it. By the degree of this v i s i n e r t i a e, or power of inactivity, as
we shall henceforth call it, we primarily judge of the quantity of solid

matter in each body; for as this quality is inherent in all the bodies, upon
which we can make any trial, we conclude it to be a property essential to all
matter; and as we yet know no reason to suppose, that bodies are composed
of different kinds of matter, we rather presume, that the matter of all bodies
is the same; and that the degree of this power of inactivity is in every body
proportional to the quantity of the solid matter in it. But although we have
no absolute proof, that all the matter in the universe is uniform, and
possesses this power of inactivity in the same degree; yet we can with
certainty compare together the different degrees of this power of inactivity
in different bodies. Particularly this power is proportional to the weight of
bodies, as Sir I s a a c N e w t o n has demonstrated[44]. However,
notwithstanding that this power of inactivity in any body can be more
certainly known, than the quantity of solid matter in it; yet since there is no
reason to suspect that one is not proportional to the other, we shall hereafter
speak without hesitation of the quantity of matter in bodies, as the measure
of the degree of their power of inactivity.
25. T h i s being established, we may now compare the effects of the
same power upon different bodies, as hitherto we have shewn the effects of
different powers upon the same body. And here if we limit the word motion
to the peculiar sense given to it in philosophy, we may comprehend all that
is to be said upon this head under one short precept; that the same power, to
whatever body it is applied, will always produce the same degree of motion.
But here motion does not signify the degree of celerity or velocity with
which a body moves, in which sense only we have hitherto used it; but it is
made use of particularly in philosophy to signify the force with which a
body moves: as if two bodies A and B being in motion, twice the force
would be required to stop A as to stop B, the motion of A would be
esteemed double the motion of B. In moving bodies, these two things are
carefully to be distinguished; their velocity or celerity, which is measured
by the space they pass through during any determinate portion of time; and
the quantity of their motion, or the force, with which they will press against
any resistance. Which force, when different bodies move with the same
velocity, is proportional to the quantity of solid matter in the bodies; but if
the bodies are equal, this force is proportional to their respective velocities,
and in other cases it is proportional both to the quantity of solid matter in
the body, and also to its velocity. To instance in two bodies A and B: if A be
twice as great as B, and they have both the same velocity, the motion of A

shall be double the motion of B; and if the bodies be equal, and the velocity
of A be twice that of B, the motion of A shall likewise be double that of B;
but if A be twice as large as B, and move twice as swift, the motion of A
will be four times the motion of B; and lastly, if A be twice as large as B,
and move but half as fast, the degree of their motion shall be the same.
26. T h i s is the particular sense given to the word motion by
philosophers, and in this sense of the word the same power always produces
the same quantity or degree of motion. If the same power act upon two
bodies A and B, the velocities, it shall give to each of them, shall be so
adjusted to the respective bodies, that the same degree of motion shall be
produced in each. If A be twice as great as B, its velocity shall be half that
of B; if A has three times as much solid matter as B, the velocity of A shall
be one third of the velocity of B; and generally the velocity given to A shall
bear the same proportion to the velocity given to B, as the quantity of solid
matter contained in the body B bears to the quantity of solid matter
contained in A.
27. T h e reason of all this is evident from what has gone before. If a
power were applied to B, which should bear the same proportion to the
power applied to A, as the body B bears to A, the bodies B and A would
both receive the same velocity; and the velocity, which B will receive from
this power, will bear the same proportion to the velocity, which it would
receive from the action of the power applied to A, as the former of these
powers bears to the latter: that is, the velocity, which A receives from the
power applied to it, will bear to the velocity, which B would receive from
the same power, the same proportion as the body B bears to A.
28. F r o m hence we may now pass to the third law of motion, where
this distinction between the velocity of a body and its whole motion is
farther necessary to be regarded, as shall immediately be shewn; after
having first illustrated the meaning of this law by a familiar instance. If a
stone or other load be drawn by a horse; the load re-acts upon the horse, as
much as the horse acts upon the load; for the harness, which is strained
between them, presses against the horse as much as against the load; and
the progressive motion of the horse forward is hindred as much by the load,
as the motion of the load is promoted by the endeavour of the horse: that is,
if the horse put forth the same strength, when loosened from the load, he
would move himself forwards with greater swiftness in proportion to the

difference between the weight of his own body and the weight of himself
and load together.
29. T h i s instance will afford some general notion of the meaning of
this law. But to proceed to a more philosophical explication: if a body in
motion strike against another at rest, let the body striking be ever so small,
yet shall it communicate some degree of motion to the body it strikes
against, though the less that body be in comparison of that it impinges upon,
and the less the velocity is, with which it moves, the smaller will be the
motion communicated. But whatever degree of motion it gives to the
resting body, the same it shall lose it self. This is the necessary consequence
of the forementioned power of inactivity in matter. For suppose the two
bodies equal, it is evident from the time they meet, both the bodies are to be
moved by the single motion of the first; therefore the body in motion by
means of its power of inactivity retaining the motion first given it, strikes
upon the other with the same force, wherewith it was acted upon it self: but
now both the bodies being to be moved by that force, which before moved
one only, the ensuing velocity will be the same, as if the power, which was
applied to one of the bodies, and put it into motion, had been applied to
both; whence it appears, that they will proceed forwards, with half the
velocity, which the body first in motion had: that is, the body first moved
will have lost half its motion, and the other will have gained exactly as
much. This rule is just, provided the bodies keep contiguous after meeting;
as they would always do, if it were not for a certain cause that often
intervenes, and which must now be explained. Bodies upon striking against
each other, suffer an alteration in their figure, having their parts pressed
inwards by the stroke, which for the most part recoil again afterwards, the
bodies endeavouring to recover their former shape. This power, whereby
bodies are inabled to regain their first figure, is usually called their
elasticity, and when it acts, it forces the bodies from each other, and causes
them to separate. Now the effect of this elasticity in the present case is such,
that if the bodies are perfectly elastic, so as to recoil with as great a force as
they are bent with, that they recover their figure in the same space of time,
as has been taken up in the alteration made in it by their compression
together; then this power will separate the bodies as swiftly, as they before
approached, and acting upon both equally, upon the body first in motion
contrary to the direction in which it moves, and upon the other as much in
the direction of its motion, it will take from the first, and add to the other

equal degrees of velocity: so that the power being strong enough to separate
them with as great a velocity, as they approached with, the first will be quite
stopt, and that which was at rest, will receive all the motion of the other. If
the bodies are elastic in a less degree, the first will not lose all its motion,
nor will the other acquire the motion of the first, but fall as much short of it,
as the other retains. For this rule is never deviated from, that though the
degree of elasticity determines how much more than half its velocity the
body first in motion shall lose; yet in every case the loss in the motion of
this body shall be transferred to the other, that other body always receiving
by the stroke as much motion, as is taken from the first.
30. This is the case of a body striking directly against an equal body at
rest, and the reasoning here used is fully confirmed by experience. There
are many other cases of bodies impinging against one another: but the
mention of these shall be reserved to the next chapter, where we intend to
be more particular and diffusive in the proof of these laws of motion, than
we have been here.

Chap. II.
Farther proofs of the Laws of Motion.

H
AV I N G in the preceding chapter deduced the three laws of motion,
delivered by our great philosopher, from the most obvious
observations, that suggest them to us; I now intend to give more
particular proofs of them, by recounting some of the discoveries
which have been made in philosophy before Sir Isaac Newton. For as they
were all collected by reasoning upon those laws; so the conformity of these
discoveries to experience makes them so many proofs of the truth of the
principles, from which they were derived.
2. L e t us begin with the subject, which concluded the last chapter.
Although the body in motion be not equal to the body at rest, on which it
strikes; yet the motion after the stroke is to be estimated in the same manner
as above. Let A (in fig. 3.) be a body in motion towards another body B
lying at rest. When A is arrived at B, it cannot proceed farther without
putting B into motion; and what motion it gives to B, it must lose it self,
that the whole degree of motion of A and B together, if neither of the bodies
be elastic, shall be equal, after the meeting of the bodies, to the single
motion of A before the stroke. Therefore, from what has been said above, it
is manifest, that as soon as the two bodies are met, they will move on
together with a velocity, which will bear the same proportion to the original
velocity of A, as the body A bears to the sum of both the bodies.
3. I f the bodies are elastic, so that they shall separate after the stroke, A
must lose a greater part of its motion, and the subsequent motion of B will
be augmented by this elasticity, as much as the motion of A is diminished
by it. The elasticity acting equally between both the bodies, it will
communicate to each the same degree of motion; that is, it will separate the

bodies by taking from the body A and adding to the body B different
degrees of velocity, so proportioned to their respective quantities of matter,
that the degree of motion, wherewith A separates from B, shall be equal to
the degree of motion, wherewith B separates from A. It follows therefore,
that the velocity taken from A by the elasticity bears to the velocity, which
the same elasticity adds to B, the same proportion, as B bears to A:
consequently the velocity, which the elasticity takes from A, will bear the
same proportion to the whole velocity, wherewith this elasticity causes the
two bodies to separate from each other, as the body B bears to the sum of
the two bodies A and B; and the velocity, which is added to B by the
elasticity, bears to the velocity, wherewith the bodies separate, the same
proportion, as the body A bears to the sum of the two bodies A and B. Thus
is found, how much the elasticity takes from the velocity of A, and adds to
the velocity of B; provided the degree of elasticity be known, whereby to
determine the whole velocity wherewith the bodies separate from each
other after the stroke[45].
4. A f t e r this manner is determined in every case the result of a body
in motion striking against another at rest. The same principles will also
determine the effects, when both bodies are in motion.
5. L e t two equal bodies move against each other with equal swiftness.
Then the force, with which each of them presses forwards, being equal
when they strike; each pressing in its own direction with the same energy,
neither shall surmount the other, but both be stopt, if they be not elastic: for
if they be elastic, they shall from thence recover new motion, and recede
from each other, as swiftly as they met, if they be perfectly elastic; but more
slowly, if less so. In the same manner, if two bodies of unequal bigness
strike against each other, and their velocities be so related, that the velocity
of the lesser body shall exceed the velocity of the greater in the same
proportion, as the greater body exceeds the lesser (for instance, if one body
contains twice the solid matter as the other, and moves but half as fast) two
such bodies will entirely suppress each other’s motion, and remain from the
time of their meeting fixed; if, as before, they are not elastic: but, if they are
so in the highest degree, they shall recede again, each with the same
velocity, wherewith they met. For this elastic power, as in the preceding
case, shall renew their motion, and pressing equally upon both, shall give
the same motion to both; that is, shall cause the velocity, which the lesser
body receives, to bear the same proportion to the velocity, which the greater

receives, as the greater body bears to the lesser: so that the velocities shall
bear the same proportion to each other after the stroke, as before. Therefore
if the bodies, by being perfectly elastic, have the sum of their velocities
after the stroke equal to the sum of their velocities before the stroke, each
body after the stroke will receive its first velocity. And the same proportion
will hold likewise between the velocities, wherewith they go off, though
they are elastic but in a less degree; only then the velocity of each will be
less in proportion to the defect of elasticity.
6. I f the velocities, wherewith the bodies meet, are not in the proportion
here supposed; but if one of the bodies, as A, has a swifter velocity in
comparison to the velocity of the other; then the effect of this excess of
velocity in the body A must be joined to the effect now mentioned, after the
manner of this following example. Let A be twice as great as B, and move
with the same swiftness as B. Here A moves with twice that degree of
swiftness, which would answer to the forementioned proportion. For A
being double to B, if it moved but with half the swiftness, wherewith B
advances, it has been just now shewn, that the two bodies upon meeting
would stop, if they were not elastic; and if they were elastic, that they
would each recoil, so as to cause A to return with half the velocity,
wherewith B would return. But it is evident from hence, that B by
encountring A will annul half its velocity, if the bodies be not elastic; and
the future motion of the bodies will be the same, as if A had advanced
against B at rest with half the velocity here assigned to it. If the bodies be
elastic, the velocity of A and B after the stroke may be thus discovered. As
the two bodies advance against each other, the velocity, with which they
meet, is made up of the velocities of both bodies added together. After the
stroke their elasticity will separate them again. The degree of elasticity will
determine what proportion the velocity, wherewith they separate, must bear
to that, wherewith they meet. Divide this velocity, with which the bodies
separate into two parts, that one of the parts bear to the other the same
proportion, as the body A bears to B; and ascribe the lesser part to the
greater body A, and the greater part of the velocity to the lesser body B.
Then take the part ascribed to A from the common velocity, which A and B
would have had after the stroke, if they had not been elastic; and add the
part ascribed to B to the same common velocity. By this means the true
velocities of A and B after the stroke will be made known.

7. I f the bodies are perfectly elastic, the great H u y g e n s has laid
down this rule for finding their motion after concourse[46]. Any straight line
C D (in fig. 4, 5.) being drawn, let it be divided in E, that C E bear the same
proportion to E D, as the swiftness of A bore to the swiftness of B before
the stroke. Let the same line C D be also divided in F, that C F bear the
same proportion to F D, as the body B bears to the body A. Then F G being
taken equal to F E, if the point G falls within the line C D, both the bodies
shall recoil after the stroke, and the velocity, wherewith the body A shall
return, will bear the same proportion to the velocity, wherewith B shall
return, as G C bears to G D; but if the point G falls without the line C D,
then the bodies after their concourse shall both proceed to move the same
way, and the velocity of A shall bear to the velocity of B the same
proportion, that G C bears to G D, as before.
8. I f the body B had stood still, and received the impulse of the other
body A upon it; the effect has been already explained in the case, when the
bodies are not elastic. And when they are elastic, the result of their collision
is found by combining the effect of the elasticity with the other effect, in the
same manner as in the last case.
9. W h e n the bodies are perfectly elastic, the rule of H u y g e n s[47]
here is to divide the line C D (fig. 6.) in E as before, and to take E G equal
to E D. And by these points thus found, the motion of each body after the
stroke is determined, as before.
10. I n the next place, suppose the bodies A and B were both moving the
same way, but A with a swifter motion, so as to overtake B, and strike
against it. The effect of the percussion or stroke, when the bodies are not
elastic, is discovered by finding the common motion, which the two bodies
would have after the stroke, if B were at rest, and A were to advance against
it with a velocity equal to the excess of the present velocity of A above the
velocity of B; and by adding to this common velocity thus found the
velocity of B.
11. I f the bodies are elastic, the effect of the elasticity is to be united
with this other, as in the former cases.
12. W h e n the bodies are perfectly elastic, the rule of Huygens[48] in
this case is to prolong C D (fig. 7.) and to take in it thus prolonged C E in
the same proportion to E D, as the greater velocity of A bears to the lesser
velocity of B; after which F G being taken equal to F E, the velocities of the

two bodies after the stroke will be determined, as in the two preceding
cases.
13. T h u s I have given the sum of what has been written concerning the
effects of percussion, when two bodies freely in motion strike directly
against each other; and the results here set down, as the consequence of our
reasoning from the laws of motion, answer most exactly to experience. A
particular set of experiments has been invented to make trial of these effects
of percussion with the greatest exactness. But I must defer these
experiments, till I have explained the nature of pendulums[49]. I shall
therefore now proceed to describe some of the appearances, which are
caused in bodies from the influence of the power of gravity united with the
general laws of motion; among which the motion of the pendulum will be
included.
14. T h e most simple of these appearances is, when bodies fall down
merely by their weight. In this case the body increases continually its
velocity, during the whole time of its fall, and that in the very same
proportion as the time increases. For the power of gravity acts constantly on
the body with the same degree of strength: and it has been observed above
in the first law of motion, that a body being once in motion will perpetually
preserve that motion without the continuance of any external influence
upon it: therefore, after a body has been once put in motion by the force of
gravity, the body would continue that motion, though the power of gravity
should cease to act any farther upon it; but, if the power of gravity
continues still to draw the body down, fresh degrees of motion must
continually be added to the body; and the power of gravity acting at all
times with the same strength, equal degrees of motion will constantly be
added in equal portions of time.
15. T h i s conclusion is not indeed absolutely true: for we shall find
hereafter[50], that the power of gravity is not of the same strength at all
distances from the center of the earth. But nothing of this is in the least
sensible in any distance, to which we can convey bodies. The weight of
bodies is the very same to sense upon the highest towers or mountains, as
upon the level ground; so that in all the observations we can make, the
forementioned proportion between the velocity of a falling body and the
time, in which it has been descending, obtains without any the least
perceptible difference.

16. F r o m hence it follows, that the space, through which a body falls,
is not proportional to the time of the fall; for since the body increases its
velocity, a greater space will be passed over in the same portion of time at
the latter part of the fall, than at the beginning. Suppose a body let fall from
the point A (in fig. 8.) were to descend from A to B in any portion of time;
then if in an equal portion of time it were to proceed from B to C; I say, the
space B C is greater than A B; so that the time of the fall from A to C being
double the time of the fall from A to B, A C shall be more than double of
A B.
17. T h e geometers have proved, that the spaces, through which bodies
fall thus by their weight, are just in a duplicate or two-fold proportion of the
times, in which the body has been falling. That is, if we were to take the
line D E in the same proportion to A B, as the time, which the body has
imployed in falling from A to C, bears to the time of the fall from A to B;
then A C will be to D E in the same proportion. In particular, if the time of
the fall through A C be twice the time of the fall through A B; then D E will
be twice A B, and A C twice D E; or A C four times A B. But if the time of
the fall through A C had been thrice the time of the fall through A B; D E
would have been treble of A B, and A C treble of D E; that is, A C would
have been equal to nine times A B.
18. I f a body fall obliquely, it will approach the ground by slower
degrees, than when it falls perpendicularly. Suppose two lines A B, A C (in
fig. 9.) were drawn, one perpendicular, and the other oblique to the ground
D E: then if a body were to descend in the slanting line A C; because the
power of gravity draws the body directly downwards, if the line A C
supports the body from falling in that manner, it must take off part of the
effect of the power of gravity; so that in the time, which would have been
sufficient for the body to have fallen through the whole perpendicular line
A B, the body shall not have passed in the line A C a length equal to A B;
consequently the line A C being longer than A B, the body shall most
certainly take up more time in passing through A C, than it would have
done in falling perpendicularly down through A B.
19. T h e geometers demonstrate, that the time, in which the body will
descend through the oblique straight line A C, bears the same proportion to
the time of its descent through the perpendicular A B, as the line it self A C
bears to A B. And in respect to the velocity, which the body will have
acquired in the point C, they likewise prove, that the length of the time

imployed in the descent through A C so compensates the diminution of the
influence of gravity from the obliquity of this line, that though the force of
the power of gravity on the body is opposed by the obliquity of the line
A C, yet the time of the body’s descent shall be so much prolonged, that the
body shall acquire the very same velocity in the point C, as it would have
got at the point B by falling perpendicularly down.
20. I f a body were to descend in a crooked line, the time of its descent
cannot be determined in so simple a manner; but the same property, in
relation to the velocity, is demonstrated to take place in all cases: that is, in
whatever line the body descends, the velocity will always be answerable to
the perpendicular height, from which the body has fell. For instance,
suppose the body A (in fig. 10.) were hung by a string to the pin B. If this
body were let fall, till it came to the point C perpendicularly under B, it will
have moved from A to C in the arch of a circle. Then the horizontal line A D
being drawn, the velocity of the body in C will be the same, as if it had
fallen from the point D directly down to C.
21. I f a body be thrown perpendicularly upward with any force, the
velocity, wherewith the body ascends, shall continually diminish, till at
length it be wholly taken away; and from that time the body will begin to
fall down again, and pass over a second time in its descent the line, wherein
it ascended; falling through this line with an increasing velocity in such a
manner, that in every point thereof, through which it falls, it shall have the
very same velocity, as it had in the same place, when it ascended; and
consequently shall come down into the place, whence it first ascended, with
the velocity which was at first given to it. Thus if a body were thrown
perpendicularly up in the line A B (in fig. II.) with such a force, as that it
should stop at the point B, and there begin to fall again; when it shall have
arrived in its descent to any point as C in this line, it shall there have the
same velocity, as that wherewith it passed by this point C in its ascent; and
at the point A it shall have gained as great a velocity, as that wherewith it
was first thrown upwards. As this is demonstrated by the geometrical
writers; so, I think, it will appear evident, by considering only, that while
the body descends, the power of gravity must act over again, in an inverted
order, all the influence it had on the body in its ascent; so as to give again to
the body the same degrees of velocity, which it had taken away before.
22. A f t e r the same manner, if the body were thrown upwards in the
oblique straight line C A (in fig. 9.) from the point C, with such a degree of

velocity as just to reach the point A; it shall by its own weight return again
through the line A C by the same degrees, as it ascended.
23. A n d lastly, if a body were thrown with any velocity in a line
continually incurvated upwards, the like effect will be produced upon its
return to the point, whence it was thrown. Suppose for instance, the body A
(in fig. 12.) were hung by a string A B. Then if this body be impelled any
way, it must move in the arch of a circle. Let it receive such an impulse, as
shall cause it to move in the arch A C; and let this impulse be of such
strength, that the body may be carried from A as far as D, before its motion
is overcome by its weight: I say here, that the body forthwith returning from
D, shall come again into the point A with the same velocity, as that
wherewith it began to move.
24. I t will be proper in this place to observe concerning the power of
gravity, that its force upon any body does not at all depend upon the shape
of the body; but that it continues constantly the same without any variation
in the same body, whatever change be made in the figure of the body: and if
the body be divided into any number of pieces, all those pieces shall weigh
just the same, as they did, when united together in one body: and if the body
be of a uniform contexture, the weight of each piece will be proportional to
its bulk. This has given reason to conclude, that the power of gravity acts
upon bodies in proportion to the quantity of matter in them. Whence it
should follow, that all bodies must fall from equal heights in the same space
of time. And as we evidently see the contrary in feathers and such like
substances, which fall very slowly in comparison of more solid bodies; it is
reasonable to suppose, that some other cause concurs to make so manifest a
difference. This cause has been found by particular experiments to be the
air. The experiments for this purpose are made thus. They set up a very tall
hollow glass; within which near the top they lodge a feather and some very
ponderous body, usually a piece of gold, this metal being the most weighty
of any body known to us. This glass they empty of the air contained within
it, and by moving a wire, which passes through the top of the glass, they let
the feather and the heavy body fall together; and it is always found, that as
the two bodies begin to descend at the same time, so they accompany each
other in the fall, and come to the bottom at the very same instant, as near as
the eye can judge. Thus, as far as this experiment can be depended on, it is
certain, that the effect of the power of gravity upon each body is
proportional to the quantity of solid matter, or to the power of inactivity in

each body. For in the limited sense, which we have given above to the word
motion, it has been shown, that the same force gives to all bodies the same
degree of motion, and different forces communicate different degrees of
motion proportional to the respective powers[51]. In this case, if the power
of gravity were to act equally upon the feather, and upon the more solid
body, the solid body would descend so much slower than the feather, as to
have no greater degree of motion than the feather: but as both bodies
descend with equal swiftness, the degree of motion in the solid body is
greater than in the feather, bearing the same proportion to it, as the quantity
of matter in the solid body to the quantity of matter in the feather. Therefore
the effect of gravity on the solid body is greater than on the feather, in
proportion to the greater degree of motion communicated; that is, the effect
of the power of gravity on the solid body bears the same proportion to its
effect on the feather, as the quantity of matter in the solid body bears to the
quantity of matter in the feather. Thus it is the proper deduction from this
experiment, that the power of gravity acts not on the surface of bodies only,
but penetrates the bodies themselves most intimately, and operates alike on
every particle of matter in them. But as the great quickness, with which the
bodies fall, leaves it something uncertain, whether they do descend
absolutely in the same time, or only so nearly together, that the difference in
their swift motion is not discernable to the eye; this property of the power
of gravity, which has here been deduced from this experiment, is farther
confirmed by pendulums, whose motion is such, that a very minute
difference would become sufficiently sensible. This will be farther
discoursed on in another place[52]; but here I shall make use of the principle
now laid down to explain the nature of what is called the center of gravity
in bodies.
25. T h e center of gravity is that point, by which if a body be
suspended, it shall hang at rest in any situation. In a globe of a uniform
texture the center of gravity is the same with the center of the globe; for as
the parts of the globe on every side of its center are similarly disposed, and
the power of gravity acts alike on every part; it is evident, that the parts of
the globe on each side of the center are drawn with equal force, and
therefore neither side can yield to the other; but the globe, if supported at its
center, must of necessity hang at rest. In like manner, if two equal bodies A
and B (in fig. 13.) be hung at the extremities of an inflexible rod C D, which
should have no weight; these bodies, if the rod be supported at its middle E,

shall equiponderate; and the rod remain without motion. For the bodies
being equal and at the same distance from the point of support E, the power
of gravity will act upon each with equal strength, and in all respects under
the same circumstances; therefore the weight of one cannot overcome the
weight of the other. The weight of A can no more surmount the weight of B,
than the weight of B can surmount the weight of A. Again, suppose a body
as A B (in fig. 14.) of a uniform texture in the form of a roller, or as it is
more usually called a cylinder, lying horizontally. If a straight line be drawn
between C and D, the centers of the extreme circles of this cylinder; and if
this straight line, commonly called the axis of the cylinder, be divided into
two equal parts in E: this point E will be the center of gravity of the
cylinder. The cylinder being a uniform figure, the parts on each side of the
point E are equal, and situated in a perfectly similar manner; therefore this
cylinder, if supported at the point E, must hang at rest, for the same reason
as the inflexible rod above-mentioned will remain without motion, when
suspended at its middle point. And it is evident, that the force applied to the
point E, which would uphold the cylinder, must be equal to the cylinder’s
weight. Now suppose two cylinders of equal thickness A B and C D to be
joined together at C B, so that the two axis’s E F, and F G lie in one straight
line. Let the axis E F be divided into two equal parts at H, and the axis F G
into two equal parts at I. Then because the cylinder A B would be upheld at
rest by a power applied in H equal to the weight of this cylinder, and the
cylinder C D would likewise be upheld by a power applied in I equal to the
weight of this cylinder; the whole cylinder A D will be supported by these
two powers: but the whole cylinder may likewise be supported by a power
applied to K, the middle point of the whole axis E G, provided that power
be equal to the weight of the whole cylinder. It is evident therefore, that this
power applied in K will produce the same effect, as the two other powers
applied in H and I. It is farther to be observed, that H K is equal to half F G,
and K I equal to half E F; for E K being equal to half E G, and E H equal to
half E F, the remainder H K must be equal to half the remainder F G; so
likewise G K being equal to half G E, and G I equal to half G F, the
remainder I K must be equal to half the remainder E F. It follows therefore,
that H K bears the same proportion to K I, as F G bears to E F. Besides, I
believe, my readers will perceive, and it is demonstrated in form by the
geometers, that the whole body of the cylinder C D bears the same
proportion to the whole body of the cylinder A B, as the axis F G bears to

the axis E F[53]. But hence it follows, that in the two powers applied at H
and I, the power applied at H bears the same proportion to the power
applied at I, as K I bears to K H. Now suppose two strings H L and I M
extended upwards, one from the point H and the other from I, and to be laid
hold on by two powers, one strong enough to hold up the cylinder A B, and
the other of strength sufficient to support the cylinder C D. Here as these
two powers uphold the whole cylinder, and therefore produce an effect,
equal to what would have been produced by a power applied to the point K
of sufficient force to sustain the whole cylinder: it is manifest, that if the
cylinder be taken away, the axis only being left, and from the point K a
string, as K N, be extended, which shall be drawn down by a power
equivalent to the weight of the cylinder, this power shall act against the
other two powers, as much as the cylinder acted against them; and
consequently these three powers shall be upon a balance, and hold the axis
H I fixed between them. But if these three powers preserve a mutual
balance, the two powers applied to the strings H L and I M are a balance to
each other; the power applied to the string H L bearing the same proportion
to the power applied to the string I M, as the distance I K bears to the
distance K H. Hence it farther appears, that if an inflexible rod A B (in fig.
15.) be suspended by any point C not in the middle thereof; and if at A the
end of the shorter arm be hung a weight, and at B the end of the longer arm
be also hung a weight less than the other, and that the greater of these
weights bears to the lesser the same proportion, as the longer arm of the rod
bears to the shorter; then these two weights will equiponderate: for a power
applied at C equal to both these weights will support without motion the rod
thus charged; since here nothing is changed from the preceding case but the
situation of the powers, which are now placed on the contrary sides of the
line, to which they are fixed. Also for the same reason, if two weights A and
B (in fig. 16.) were connected together by an inflexible rod C D, drawn
from C the center of gravity of A to D the center of gravity of B; and if the
rod C D were to be so divided in E, that the part D E bear the same
proportion to the other part C E, as the weight A bears to the weight B: then
this rod being supported at E will uphold the weights, and keep them at rest
without motion. This point E, by which the two bodies A and B will be
supported, is called their common center of gravity. And if a greater number
of bodies were joined together, the point, by which they could all be
supported, is called the common center of gravity of them all. Suppose (in

fig. 17.) there were three bodies A, B, C, whose respective centers of
gravity were joined by the three lines D E, D F, E F: the line D E being so
divided in G, that D G bear the same proportion to G E, as B bears to A; G
is the center of gravity common to the two bodies A and B; that is, a power
equal to the weight of both the bodies applied to G would support them, and
the point G is pressed as much by the two weights A and B, as it would be,
if they were both hung together at that point. Therefore, if a line be drawn
from G to F, and divided in H, so that G H bear the same proportion to H F,
as the weight C bears to both the weights A and B, the point H will be the
common center of gravity of all the three weights; for H would be their
common center of gravity, if both the weights A and B were hung together
at G, and the point G is pressed as much by them in their present situation,
as it would be in that case. In the same manner from the common center of
these three weights, you might proceed to find the common center, if a
fourth weight were added, and by a gradual progress might find the
common center of gravity belonging to any number of weights whatever.
26. A s all this is the obvious consequence of the proposition laid down
for assigning the common center of gravity of any two weights, by the same
proposition the center of gravity of all figures is found. In a triangle, as
A B C (in fig. 18.) the center of gravity lies in the line drawn from the
middle point of any one of the sides to the opposite angle, as the line B D is
drawn from D the middle of the line A C to the opposite angle B[54]; so that
if from the middle of either of the other sides, as from the point E in the
side A B, a line be drawn, as E C, to the opposite angle; the point F, where
this line crosses the other line B D, will be the center of gravity of the
triangle[55]. Likewise D F is equal to half F B, and E F equal to half F C[56].
In a hemisphere, as A B C (fig. 19.) if from D the center of the base the line
D B be erected perpendicular to that base, and this line be so divided in E,
that D E be equal to three fifths of B E, the point E is the center of gravity
of the hemisphere[57].
27. I t will be of use to observe concerning the center of gravity of
bodies; that since a power applied to this center alone can support a body
against the power of gravity, and hold it fixed at rest; the effect of the power
of gravity on a body is the same, as if that whole power were to exert itself
on the center of gravity only. Whence it follows, that, when the power of
gravity acts on a body suspended by any point, if the body is so suspended,

that the center of gravity of the body can descend; the power of gravity will
give motion to that body, otherwise not: or if a number of bodies are so
connected together, that, when any one is put into motion, the rest shall, by
the manner of their being joined, receive such motion, as shall keep their
common center of gravity at rest; then the power of gravity shall not be able
to produce any motion in these bodies, but in all other cases it will. Thus, if
the body A B (in fig. 20, 21.) whose center of gravity is C, be hung on the
point A, and the center C be perpendicularly under A (as in fig. 20.) the
weight of the body will hold it still without motion, because the center C
cannot descend any lower. But if the body be removed into any other
situation, where the center C is not perpendicularly under A (as in fig. 21.)
the body by its weight will be put into motion towards the perpendicular
situation of its center of gravity. Also if two bodies A, B (in fig. 22.) be
joined together by the rod C D lying in an horizontal situation, and be
supported at the point E; if this point be the center of gravity common to the
two bodies, their weight will not put them into motion; but if this point E is
not their common center of gravity, the bodies will move; that part of the
rod C D descending, in which the common center of gravity is found. So in
like manner, if these two bodies were connected together by any more
complex contrivance; yet if one of the bodies cannot move without so
moving the other, that their common center of gravity shall rest, the weight
of the bodies will not put them in motion, otherwise it will.
28. I s h a l l proceed in the next place to speak of the mechanical
powers. These are certain instruments or machines, contrived for the
moving great weights with small force; and their effects are all deducible
from the observation we have just been making. They are usually reckoned
in number five; the lever, the wheel and axis, the pulley, the wedge, and the
screw; to which some add the inclined plane. As these instruments have
been of very ancient use, so the celebrated A r c h i m e d e s seems to have
been the first, who discovered the true reason of their effects. This, I think,
may be collected from what is related of him, that some expressions, which
he used to denote the unlimited force of these instruments, were received as
very extraordinary paradoxes: whereas to those, who had understood the
cause of their great force, no expressions of that kind could have appeared
surprizing.
29. A l l the effects of these powers may be judged of by this one rule,
that, when two weights are applied to any of these instruments, the weights

will equiponderate, if, when put into motion, their velocities will be
reciprocally proportional to their respective weights. And what is said of
weights, must of necessity be equally understood of any other forces
equivalent to weights, such as the force of a man’s arm, a stream of water,
or the like.
30. B u t to comprehend the meaning of this rule, the reader must know,
what is to be understood by reciprocal proportion; which I shall now
endeavour to explain, as distinctly as I can; for I shall be obliged very
frequently to make use of this term. When any two things are so related,
that one increases in the same proportion as the other, they are directly
proportional. So if any number of men can perform in a determined space
of time a certain quantity of any work, suppose drain a fish-pond, or the
like; and twice the number of men can perform twice the quantity of the
same work, in the same time; and three times the number of men can
perform as soon thrice the work; here the number of men and the quantity
of the work are directly proportional. On the other hand, when two things
are so related, that one decreases in the same proportion, as the other
increases, they are said to be reciprocally proportional. Thus if twice the
number of men can perform the same work in half the time, and three times
the number of men can finish the same in a third part of the time; then the
number of men and the time are reciprocally proportional. We shewed
above[58] how to find the common center of gravity of two bodies, there the
distances of that common center from the centers of gravity of the two
bodies are reciprocally proportional to the respective bodies. For C E in fig.
16. being in the same proportion to E D, as B bears to A; C E is so much
greater in proportion than E D, as A is less in proportion than B.
31. N o w this being understood, the reason of the rule here stated will
easily appear. For if these two bodies were put in motion, while the point E
rested, the velocity, wherewith A would move, would bear the same
proportion to the velocity, wherewith B would move, as E C bears to E D.
The velocity therefore of each body, when the common center of gravity
rests, is reciprocally proportional to the body. But we have shewn above[59],
that if two bodies are so connected together, that the putting them in motion
will not move their common center of gravity; the weight of those bodies
will not produce in them any motion. Therefore in any of these mechanical
engines, if, when the bodies are put into motion, their velocities are

reciprocally proportional to their respective weights, whereby the common
center of gravity would remain at rest; the bodies will not receive any
motion from their weight, that is, they will equiponderate. But this perhaps
will be yet more clearly conceived by the particular description of each
mechanical power.
32. T h e lever was first named above. This is a bar made use of to
sustain and move great weights. The bar is applied in one part to some
strong support; as the bar A B (in fig. 23, 24.) is applied at the point C to the
support D. In some other part of the bar, as E, is applied the weight to be
sustained or moved; and in a third place, as F, is applied another weight or
equivalent force, which is to sustain or move the weight at E. Now here, if,
when the level should be put in motion, and turned upon the point C, the
velocity, wherewith the point F would move, bears the same proportion to
the velocity, wherewith the point E would move, as the weight at E bears to
the weight or force at F; then the lever thus charged will have no propensity
to move either way. If the weight or other force at F be not so great as to
bear this proportion, the weight at E will not be sustained; but if the force at
F be greater than this, the weight at E will be surmounted. This is evident
from what has been said above[60], when the forces at E and F are placed (as
in fig. 23.) on different sides of the support D. It will appear also equally
manifest in the other case, by continuing the bar B C in fig. 24. on the other
side of the support D, till C G be equal to C F, and by hanging at G a weight
equivalent to the power at F; for then, if the power at F were removed, the
two weights at G and E would counterpoize each other, as in the former
case: and it is evident, that the point F will be lifted up by the weight at G
with the same degree of force, as by the other power applied to F; since, if
the weight at E were removed, a weight hung at F equal to that at G would
balance the lever, the distances C G and C F being equal.
33. I f the two weights, or other powers, applied to the lever do not
counterbalance each other; a third power may be applied in any place
proposed of the lever, which shall hold the whole in a just counterpoize.
Suppose (in fig. 25.) the two powers at E and F did not equiponderate, and
it were required to apply a third power to the point G, that might be
sufficient to balance the lever. Find what power in F would just
counterbalance the power in E; then if the difference between this power
and that, which is actually applied at F, bear the same proportion to the third
power to be applied at G, as the distance C G bears to C F; the lever will be

counterpoized by the help of this third power, if it be so applied as to act the
same way with the power in F, when that power is too small to
counterbalance the power in E; but otherwise the power in G must be so
applied, as to act against the power in F. In like manner, if a lever were
charged with three, or any greater number of weights or other powers,
which did not counterpoize each other, another power might be applied in
any place proposed, which should bring the whole to a just balance. And
what is here said concerning a plurality of powers, may be equally applied
to all the following cases.
34. I f the lever should consist of two arms making an angle at the point
C (as in fig. 26.) yet if the forces are applied perpendicularly to each arm,
the same proportion will hold between the forces applied, and the distances
of the center, whereon the lever rests, from the points to which they are
applied. That is, the weight at E will be to the force in F in the same
proportion, as C F bears to C E.
35. B u t whenever the forces applied to the lever act obliquely to the
arm, to which they are applied (as in fig. 27.) then the strength of the forces
is to be estimated by lines let fall from the center of the lever to the
directions, wherein the forces act. To balance the levers in fig. 27, the
weight or other force at F will bear the same proportion to the weight at E,
as the distance C E bears to C G the perpendicular let fall from the point C
upon the line, which denotes the direction wherein the force applied to F
acts: for here, if the lever be put into motion, the power applied to F will
begin to move in the direction of the line F G; and therefore its first motion
will be the same, as the motion of the point G.
36. W h e n two weights hang upon a lever, and the point, by which the
lever is supported, is placed in the middle between the two weights, that the
arms of the lever are both of equal length; then this lever is particularly
called a balance; and equal weights equiponderate as in common scales.
When the point of support is not equally distant from both weights, it
constitutes that instrument for weighing, which is called a steelyard.
Though both in common scales, and the steelyard, the point, on which the
beam is hung, is not usually placed just in the same straight line with the
points, that hold the weights, but rather a little above (as in fig. 28.) where
the lines drawn from the point C, whereon the beam is suspended, to the
points E and F, on which the weights are hung, do not make absolutely one
continued line. If the three points E, C, and F were in one straight line,

those weights, which equiponderated, when the beam hung horizontally,
would also equiponderate in any other situation.

But we see in these instruments, when they are charged with weights,
which equiponderate with the beam hanging horizontally; that, if the beam
be inclined either way, the weight most elevated surmounts the other, and
descends, causing the beam to swing, till by degrees it recovers its
horizontal position. This effect arises from the forementioned structure: for
by this structure these instruments are levers composed of two arms, which
make an angle at the point of support (as in fig. 29, 30.) the first of which
represents the case of the common balance, the second the case of the
steelyard. In the first, where C E and C F are equal, equal weights hung at E

and F will equiponderate, when the points E and F are in an horizontal
situation. Suppose the lines E G and F H to be perpendicular to the horizon,
then they will denote the directions, wherein the forces applied to E and F
act. Therefore the proportion between the weights at E and F, which shall
equiponderate, are to be judged of by perpendiculars, as C I, C K, let fall
from C upon E G and F H: so that the weights being equal, the lines C I,
C K, must be equal also, when the weights equiponderate. But I believe my
readers will easily see, that since C E and C F are equal, the lines C I and
C K will be equal, when the points E and F are horizontally situated.
37. I f this lever be set into any other position (as in fig. 31.) then the
weight, which is raised highest, will outweigh the other. Here, if the point F
be raised higher than E, the perpendicular C K will be longer than C I: and
therefore the weights would equiponderate, if the weight at F were less than
the weight at E. But the weight at F is equal to that at E; therefore is greater,
than is necessary to counterbalance the weight at E, and consequently will
outweigh it, and draw the beam of the lever down.
38. I n like manner in the case of the steelyard (fig. 32.) if the weights at
E and F are so proportioned, as to equiponderate, when the points E and F
are horizontally situated; then in any other situation of this lever the weight,
which is raised highest, will preponderate. That is, if in the horizontal
situation of the points E and F the weight at F bears the same proportion to
the weight at E, as C I bears to C K; then, if the point F be raised higher
than E (as in fig. 32.) the weight at F shall bear a greater proportion to the
weight at E, than C I bears to C K.
39. F a r t h e r a lever may be hung upon an axis, and then the two arms
of the lever need not be continuous, but fixed to different parts of this axis;
as in fig. 33, where the axis A B is supported by its two extremities A and
B. To this axis one arm of the lever is fixed at the point C, the other at the
point D. Now here, if a weight be hung at E, the extremity of that arm,
which is fixed to the axis at the point C; and another weight be hung at F,
the extremity of the arm, which is fixed on the axis at D; then these weights
will equiponderate, when the weight at E bears the same proportion to the
weight at F, as the arm D F bears to C E.
40. T h i s is the case, if both the arms are perpendicular to the axis, and
lie (as the geometers express themselves) in the same plane; or, in other
words, if the arms are so fixed perpendicularly upon the axis, that, when

one of them lies horizontally, the other shall also be horizontal. If either arm
stand not perpendicular to the axis; then, in determining the proportion
between the weights, instead of the length of that arm, you must use the
perpendicular let fall upon the axis from the extremity of that arm. If the
arms are not so fixed as to become horizontal, at the same time; the method
of assigning the proportion between the weights is analogous to that made
use of above in levers, which make an angle at the point, whereon they are
supported.
41. F r o m this case of the lever hung on an axis, it is easy to make a
transition to another mechanical power, the wheel and axis.
42. T h i s instrument is a wheel fixed on a roller, the roller being
supported at each extremity so as to turn round freely with the wheel, in the
manner represented in fig. 34, where A B is the wheel, C D the roller, and
E F its two supports. Now suppose a weight G hung by a cord wound round
the roller, and another weight H hung by a cord wound about the wheel the
contrary way: that these weights may support each other, the weight H must
bear the same proportion to the weight G, as the thickness of the roller bears
to the diameter of the wheel.
43. S u p p o s e the line k l to be drawn through the middle of the roller;
and from the place of the roller, where the cord, on which the weight G
hangs, begins to leave the roller, as at m, let the line m n be drawn
perpendicularly to k l; and from the point, where the cord holding the
weight H begins to leave the wheel, as at o, let the line o p be drawn
perpendicular to k l. This being done, the two lines o p and m n represent
two arms of a lever fixed on the axis k l; consequently the weight H will
bear to the weight G the same proportion, as m n bears to o p. But m n bears
the same proportion to o p, as the thickness of the roller bears to the
diameter of the wheel; for m n is half the thickness of the roller, and o p half
the diameter of the wheel.
44. I f the wheel be put into motion, and turned once round, that the
cord, on which the weight G hangs, be wound once more round the axis;
then at the same time the cord, whereon the weight H hangs, will be wound
off from the wheel one circuit. Therefore the velocity of the weight G will
bear the same proportion to the velocity of the weight H, as the
circumference of the roller to the circumference of the wheel. But the
circumference of the roller bears the same proportion to the circumference

of the wheel, as the thickness of the roller bears to the diameter of the
wheel, consequently the velocity of the weight G bears to the velocity of the
weight H the same proportion, as the thickness of the roller bears to the
diameter of the wheel, which is the proportion that the weight H bears to the
weight G. Therefore as before in the lever, so here also the general rule laid
down above is verified, that the weights equiponderate, when their
velocities would be reciprocally proportional to their respective weights.
45. I n like manner, if on the same axis two wheels of different sizes are
fixed (as in fig. 35.) and a weight hung on each; the weights will
equiponderate, if the weight hung on the greater wheel bear the same
proportion to the weight hung on the lesser, as the diameter of the lesser
wheel bears to the diameter of the greater.
46. I t is usual to join many wheels together in the same frame, which
by the means of certain teeth, formed in the circumference of each wheel,
shall communicate motion to each other. A machine of this nature is
represented in fig. 36. Here A B C is a winch, upon which is fixed a small
wheel D indented with teeth, which move in the like teeth of a larger wheel
E F fixed on the axis G H. Let this axis carry another wheel I, which shall
move in like manner a greater wheel K L fixed on the axis M N. Let this
axis carry another small wheel O, which after the same manner shall turn
about a larger wheel P Q fixed on the roller R S, on which a cord shall be
wound, that holds a weight, as T. Now the proportion required between the
weight T and a power applied to the winch at A sufficient to support the
weight, will most easily be estimated, by computing the proportion, which
the velocity of the point A would bear to the velocity of the weight. If the
winch be turned round, the point A will describe a circle as A V. Suppose
the wheel E F to have ten times the number of teeth, as the wheel D; then
the winch must turn round ten times to carry the wheel E F once round. If
wheel K L has also ten times the number of teeth, as I, the wheel I must turn
round ten times to carry the wheel K L once round; and consequently the
winch A B C must turn round an hundred times to turn the wheel K L once
round. Lastly, if the wheel P Q has ten times the number of teeth, as the
wheel O, the winch must turn about one thousand times in order to turn the
wheel P Q, or the roller R S once round. Therefore here the point A must
have gone over the circle A V a thousand times, in order to lift the weight T
through a space equal to the circumference of the roller R S: whence it
follows, that the power applied at A will balance the weight T, if it bear the

same proportion to it, as the circumference of the roller to one thousand
times the circle A V; or the same proportion as half the thickness of the
roller bears to one thousand times A B.
47. I s h a l l now explain the effect of the pulley. Let a weight hang by
a pulley, as in fig. 37. Here it is evident, that the power A, by which the
weight B is supported, must be equal to the weight; for the cord C D is
equally strained between them; and if the weight B move, the power A must
move with equal velocity. The pulley E has no other effect, than to permit
the power A to act in another direction, than it must have done, if it had
been directly applied to support the weight without the intervention of any
such instrument.
48. A g a i n, let a weight be supported, as in fig. 38; where the weight A
is fixed to the pulley B, and the cord, by which the weight is upheld, is
annexed by one extremity to a hook C, and at the other end is held by the
power D. Here the weight is supported by a cord doubled; insomuch that
although the cord were not strong enough to hold the weight single, yet
being thus doubled it might support it. If the end of the cord held by the
power D were hung on the hook C, as well as the other end; then, when
both ends of the cord were tied to the hook, it is evident, that the hook
would bear the whole weight; and each end of the string would bear against
the hook with the force of half the weight only, seeing both ends together
bear with the force of the whole. Hence it is evident, that, when the power
D holds one end of the weight, the force, which it must exert to support the
weight, must be equal to just half the weight. And the same proportion
between the weight and power might be collected from comparing the
respective velocities, with which they would move; for it is evident, that the
power must move through a space equal to twice the distance of the pulley
from the hook, in order to lift the pulley up to the hook.
49. I t is equally easy to estimate the effect, when many pulleys are
combined together, as in fig. 39, 40; in the first of which the under set of
pulleys, and consequently the weight is held by six strings; and in the latter
figure by five: therefore in the first of these figures the power to support the
weight, must be one sixth part only of the weight, and in the latter figure the
power must be one fifth part.
50. T h e r e are two other ways of supporting a weight by pulleys,
which I shall particularly consider.

51. O n e of these ways is represented in fig. 41. Here the weight being
connected to the pulley B, a power equal to half the weight A would support
the pulley C, if applied immediately to it. Therefore the pulley C is drawn
down with a force equal to half the weight A. But if the pulley D were to be
immediately supported by half the force, with which the pulley C is drawn
down, this pulley D will uphold the pulley C; so that if the pulley D be
upheld with a force equal to one fourth part of the weight A, that force will
support the weight. But, for the same reason as before, if the power in E be
equal to half the force necessary to uphold the pulley D; this pulley, and
consequently the weight A, will be upheld: therefore, if the power in E be
one eighth part of the weight A, it will support the weight.
52. A n o t h e r way of applying pulleys to a weight is represented in
fig. 42. To explain the effect of pulleys thus applied, it will be proper to
consider different weights hanging, as in fig. 43. Here, if the power and
weights balance each other, the power A is equal to the weight B; the weight
C is equal to twice the power A, or the weight B; and for the same reason
the weight D is equal to twice the weight C, or equal to four times the
power A. It is evident therefore, that all the three weights B, C, D together
are equal to seven times the power A. But if these three weights were joined
in one, they would produce the case of fig. 40: so that in that figure the
weight A, where there are three pulleys, is seven times the power B. If there
had been but two pulleys, the weight would have been three times the
power; and if there had been four pulleys, the weight would have been
fifteen times the power.
53. T h e wedge is next to be considered. The form of this instrument is
sufficiently known. When it is put under any weight (as in fig. 44.) the
force, with which the wedge will lift the weight, when drove under it by a
blow upon the end A B, will bear the same proportion to the force,
wherewith the blow would act on the weight, if directly applied to it; as the
velocity, which the wedge receives from the blow, bears to the velocity,
wherewith the weight is lifted by the wedge.
54. T h e screw is the fifth mechanical power. There are two ways of
applying this instrument. Sometimes it is screwed into a hole, as in fig. 45,
where the screw A B is screwed through the plank C D. Sometimes the
screw is applied to the teeth of a wheel, as in fig. 46, where the thread of the
screw A B turns in the teeth of a wheel C D. In both these cases, if a bar, as
A E, be fixed to the end A of the screw; the force, wherewith the end B of

the screw in fig. 45 is forced down, and the force, wherewith the teeth of
the wheel C D in fig. 44 are held, bears the same proportion to the power
applied to the end E of the bar; as the velocity, wherewith the end E will
move, when the screw is turned, bears to the velocity, wherewith the end B
of the screw in fig. 43, or the teeth of the wheel C D in fig. 46, will be
moved.
55. T h e inclined plane affords also a means of raising a weight with
less force, than what is equal to the weight it self. Suppose it were required
to raise the globe A (in fig. 47.) from the ground B C up to the point, whose
perpendicular height from the ground is E D. If this globe be drawn along
the slant D F, less force will be required to raise it, than if it were lifted
directly up. Here if the force applied to the globe bear the same proportion
only to its weight, as E D bears to F D, it will be sufficient to hold up the
globe; and therefore any addition to that force will put it in motion, and
draw it up; unless the globe, by pressing against the plane, whereon it lies,
adhere in some degree to the plane. This indeed it must always do more or
less, since no plane can be made so absolutely smooth as to have no
inequalities at all; nor yet so infinitely hard, as not to yield in the least to the
pressure of the weight. Therefore the globe cannot be laid on such a plane,
whereon it will slide with perfect freedom, but they must in some measure
rub against each other; and this friction will make it necessary to imploy a
certain degree of force more, than what is necessary to support the globe, in
order to give it any motion. But as all the mechanical powers are subject in
some degree or other to the like impediment from friction; I shall here only
shew what force would be necessary to sustain the globe, if it could lie upon
the plane without causing any friction at all. And I say, that if the globe
were drawn by the cord G H, lying parallel to the plane D F; and the force,
wherewith the cord is pulled, bear the same proportion to the weight of the
globe, as E D bears to D F; this force will sustain the globe. In order to the
making proof of this, let the cord G H be continued on, and turned over the
pulley I, and let the weight K be hung to it. Now I say, if this weight bears
the same proportion to the globe A, as D E bears to D F, the weight will
support the globe. I think it is very manifest, that the center of the globe A
will lie in one continued line with the cord H G. Let L be the center of the
globe, and M the center of gravity of the weight K. In the first place let the
weight hang so, that a line drawn from L to M shall lie horizontally; and I
say, if the globe be moved either up or down the plane D F, the weight will

so move along with it, that the center of gravity common to both the
weights shall continue in this line L M, and therefore shall in no case
descend. To prove this more fully, I shall depart a little from the method of
this treatise, and make use of a mathematical proportion or two: but they are
such, as any person, who has read E u c l i d ’ s E l e m e n t s, will fully
comprehend; and are in themselves so evident, that, I believe, my readers,
who are wholly strangers to geometrical writings, will make no difficulty of
admitting them. This being premised, let the globe be moved up, till its
center be at G, then will M the center of gravity of the weight K be sunk to
N; so that M N shall be equal to G L. Draw N G crossing the line M L in O;
then I say, that O is the common center of gravity of the two weights in this
their new situation. Let G P be drawn perpendicular to M L; then G L will
bear the same proportion to G P, as D F bears to D E; and M N being equal
to G L, M N will bear the same proportion to G P, as D F bears to D E. But
N O bears the same proportion to O G, as M N bears to G P; consequently
N O will bear the same proportion to O G, as D F bears to D E. In the last
place, the weight of the globe A bears the same proportion to the other
weight K, as D F bears to D E; therefore N O bears the same proportion to
O G, as the weight of the globe A bears to the weight K. Whence it follows,
that, when the center of the globe A is in G, and the center of gravity of the
weight K is in N, O will be the center of gravity common to both the
weights. After the same manner, if the globe had been caused to descend,
the common center of gravity would have been found in this line M L.
Since therefore no motion of the globe either way will make the common
center of gravity descend, it is manifest, from what has been said above,
that the weights A and K counterpoize each other.
56. I s h a l l now consider the case of pendulums. A pendulum is made
by hanging a weight to a line, so that it may swing backwards and forwards.
This motion the geometers have very carefully considered, because it is the
most commodious instrument of any for the exact measurement of time.
57. I h a v e observed already[61], that if a body hanging
perpendicularly by a string, as the body A (in fig. 48.) hangs by the string
A B, be put so into motion, as to be made to ascend up the circular arch
A C; then as soon as it has arrived at the highest point, to which the motion,
that the body has received, will carry it; it will immediately begin to
descend, and at A will receive again as great a degree of motion, as it had at
first. This motion therefore will carry the body up the arch A D, as high as it

ascended before in the arch A C. Consequently in its return through the arch
D A it will acquire again at A its original velocity, and advance a second
time up the arch A C as high as at first; by this means continuing without
end its reciprocal motion. It is true indeed, that in fact every pendulum,
which we can put in motion, will gradually lessen its swing, and at length
stop, unless there be some power constantly applied to it, whereby its
motion shall be renewed; but this arises from the resistance, which the body
meets with both from the air, and the string by which it is hung: for as the
air will give some obstruction to the progress of the body moving through
it; so also the string, whereon the body hangs, will be a farther impediment;
for this string must either slide on the pin, whereon it hangs, or it must bend
to the motion of the weight; in the first there must be some degree of
friction, and in the latter the string will make some resistance to its
inflection. However, if all resistance could be removed, the motion of a
pendulum would be perpetual.
58. B u t to proceed, the first property, I shall take notice of in this
motion, is, that the greater arch the pendulous body moves through, the
greater time it takes up: though the length of time does not increase in so
great a proportion as the arch. Thus if C D be a greater arch, and E F a
lesser, where C A is equal to A D, and E A equal to A F; the body, when it
swings through the greater arch C D, shall take up in its swing from C to D
a longer time than in swinging from E to F, when it moves only in that
lesser arch; or the time in which the body let fall from C will descend
through the arch C A is greater than the time, in which it will descend
through the arch E A, when let fall from E. But the first of these times will
not hold the same proportion to the latter, as the first arch C A bears to the
other arch E A; which will appear thus. Let C G and E H be two horizontal
lines. It has been remarked above[62], that the body in falling through the
arch C A will acquire as great a velocity at the point A, as it would have
gained by falling directly down through G A; and in falling through the arch
E A it will acquire in the point A only that velocity, which it would have got
in falling through H A. Therefore, when the body descends through the
greater arch C A, it shall gain a greater velocity, than when it passes only
through the lesser; so that this greater velocity will in some degree
compensate the greater length of the arch.
59. T h e increase of velocity, which the body acquires in falling from a
greater height, has such an effect, that, if straight lines be drawn from A to

C and E, the body would fall through the longer straight line C A just in the
same time, as through the shorter straight line E A. This is demonstrated by
the geometers, who prove, that if any circle, as A B C D (fig. 49.) be placed
in a perpendicular situation; a body shall fall obliquely through every line,
as A B drawn from the lowest point A in the circle to any other point in the
circumference just in the same time, as would be imployed by the body in
falling perpendicularly down through the diameter C A. But the time in
which the body will descend through the arch, is different from the time,
which it would take up in falling through the line A B.
60. I t has been thought by some, that because in very small arches this
correspondent straight line differs but little from the arch itself; therefore
the descent through this straight line would be performed in such small
arches nearly in the same time as through the arches themselves: so that if a
pendulum were to swing in small arches, half the time of a single swing
would be nearly equal to the time, in which a body would fall
perpendicularly through twice the length of the pendulum. That is, the
whole time of the swing, according to this opinion, will be four fold the
time required for the body to fall through half the length of the pendulum;
because the time of the body’s falling down twice the length of the
pendulum is half the time required for the fall through one quarter of this
space, that is through half the pendulum’s length. However there is here a
mistake; for the whole time of the swing, when the pendulum moves
through small arches, bears to the time required for a body to fall down
through half the length of the pendulum very nearly the same proportion, as
the circumference of a circle bears to its diameter; that is very nearly the
proportion of 355 to 113, or little more than the proportion of 3 to 1. If the
pendulum takes so great a swing, as to pass over an arch equal to one sixth
part of the whole circumference of the circle, it will swing 115 times, while
it ought according to this proportion to have swung 117 times; so that, when
it swings in so large an arch, it loses something less than two swings in an
hundred. If it swing through 1/10 only of the circle, it shall not lose above
one vibration in 160. If it swing in 1/20 of the circle, it shall lose about one
vibration in 690. If its swing be confined to 1/40 of the whole circle, it shall
lose very little more than one swing in 2600. And if it take no greater a
swing than through 1/60 of the whole circle, it shall not lose one swing in
5800.

61. N o w it follows from hence, that, when pendulums swing in small
arches, there is very nearly a constant proportion observed between the time
of their swing, and the time, in which a body would fall perpendicularly
down through half their length. And we have declared above, that the
spaces, through which bodies fall, are in a two fold proportion of the times,
which they take up in falling[63]. Therefore in pendulums of different
lengths, swinging through small arches, the lengths of the pendulums are in
a two fold or duplicate proportion of the times, they take in swinging; so
that a pendulum of four times the length of another shall take up twice the
time in each swing, one of nine times the length will make one swing only
for three swings of the shorter, and so on.
62. T h i s proportion in the swings of different pendulums not only
holds in small arches; but in large ones also, provided they be such, as the
geometers call similar; that is, if the arches bear the same proportion to the
whole circumferences of their respective circles. Suppose (in fig. 48.) A B,
C D to be two pendulums. Let the arch E F be described by the motion of
the pendulum A B, and the arch G H be described by the pendulum C D;
and let the arch E F bear the same proportion to the whole circumference,
which would be formed by turning the pendulum A B quite round about the
point A, as the arch G H bears to the whole circumference, that would be
formed by turning the pendulum C D quite round the point C. Then I say,
the proportion, which the length of the pendulum A B bears to the length of
the pendulum C D, will be two fold of the proportion, which the time taken
up in the description of the arch E F bears to the time employed in the
description of the arch G H.
63. T h u s pendulums, which swing in very small arches, are nearly an
equal measure of time. But as they are not such an equal measure to
geometrical exactness; the mathematicians have found out a method of
causing a pendulum so to swing, that, if its motion were not obstructed by
any resistance, it would always perform each swing in the same time,
whether it moved through a greater, or a lesser space. This was first
discovered by the great H u y g e n s, and is as follows. Upon the straight
line A B (in fig. 49.) let the circle C D E be so placed, as to touch the
straight line in the point C. Then let this circle roll along upon the straight
line A B, as a coach-wheel rolls along upon the ground. It is evident, that, as
soon as ever the circle begins to move, the point C in the circle will be lifted
off from the straight line A B; and in the motion of the circle will describe a

crooked course, which is represented by the line C F G H. Here the part
C H of the straight line included between the two extremities C and H of the
line C F G H will be equal to the whole circumference of the circle C D E;
and if C H be divided into two equal parts at the point I, and the straight
line I K be drawn perpendicular to C H, this line I K will be equal to the
diameter of the circle C D E. Now in this line if a body were to be let fall
from the point H, and were to be carried by its weight down the line H G K,
as far as the point K, which is the lowest point of the line C F G H; and if
from any other point G a body were to be let fall in the same manner; this
body, which falls from G, will take just the same time in coming to K, as
the body takes up, which falls from H. Therefore if a pendulum can be so
hung, that the ball shall move in the line A G F E, all its swings, whether
long or short, will be performed in the same time; for the time, in which the
ball will descend to the point K, is always half the time of the whole swing.
But the ball of a pendulum will be made to swing in this line by the
following means. Let K I (in fig. 52.) be prolonged upwards to L, till I L is
equal to I K. Then let the line L M H equal and like to K H be applied, as in
the figure between the points L and H, so that the point which in this line
L M H answers to the point H in the line K H shall be applied to the point
L, and the point answering to the point K shall be applied to the point H.
Also let such another line L N C be applied between L and C in the same
manner. This preparation being made; if a pendulum be hung at the point L
of such a length, that the ball thereof shall reach to K; and if the string shall
continually bend against the lines H M L and L N C, as the pendulum
swings to and fro; by this means the ball shall constantly keep in the line
C K H.
64. N o w in this pendulum, as all the swings, whether long or short, will
be performed in the same time; so the time of each will exactly bear the
same proportion to the time required for a body to fall perpendicularly
down, through half the length of the pendulum, that is from I to K, as the
circumference of a circle bears to its diameter.
65. I t may from hence be understood in some measure, why, when
pendulums swing in circular arches, the times of their swings are nearly
equal, if the arches are small, though those arches be of very unequal
lengths; for if with the semidiameter L K the circular arch O K P be
described, this arch in the lower part of it will differ very little from the line
C K H.

66. I t may not be amiss here to remark, that a body will fall in this line
C K H (fig. 53.) from C to any other point, as Q or R in a shorter space of
time, than if it moved through the straight line drawn from C to the other
point; or through any other line whatever, that can be drawn between these
two points.
67. B u t as I have observed, that the time, which a pendulum takes in
swinging, depends upon its length; I shall now say something concerning
the way, in which this length of the pendulum is to be estimated. If the
whole ball of the pendulum could be crouded into one point, this length, by
which the motion of the pendulum is to be computed, would be the length
of the string or rod. But the ball of the pendulum must have a sensible
magnitude, and the several parts of this ball will not move with the same
degree of swiftness; for those parts, which are farthest from the point,
whereon the pendulum is suspended, must move with the greatest velocity.
Therefore to know the time in which the pendulum swings, it is necessary
to find that point of the ball, which moves with the same degree of velocity,
as if the whole ball were to be contracted into that point.
68. T h i s point is not the center of gravity, as I shall now endeavour to
shew. Suppose the pendulum A B (in fig. 54.) composed of an inflexible rod
A C and ball C B, to be fixed on the point A, and lifted up into an horizontal
situation. Here if the rod were not fixed to the point A, the body C B would
descend directly with the whole force of its weight; and each part of the
body would move down with the same degree of swiftness. But when the
rod is fixed at the point A, the body must fall after another manner; for the
parts of the body must move with different degrees of velocity, the parts
more remote from A descending with a swifter motion, than the parts nearer
to A; so that the body will receive a kind of rolling motion while it
descends. But it has been observed above, that the effect of gravity upon
any body is the same, as if the whole force were exerted on the body’s
center of gravity[64].

Since therefore the power of gravity in drawing down the body must
also communicate to it the rolling motion just described; it seems evident,
that the center of gravity of the body cannot be drawn down as swiftly, as
when the power of gravity has no other effect to produce on the body, than
merely to draw it downward. If therefore the whole matter of the body C B
could be crouded into its center of gravity, so that being united into one
point, this rolling motion here mentioned might give no hindrance to its
descent; this center would descend faster, than it can now do. And the point,
which now descends as fast, as if the whole matter or the body C B were
crouded into it, will be farther removed from the point A, than the center of
gravity of the body C B.
69. A g a i n, suppose the pendulum A B (in fig. 55.) to hang obliquely.
Here the power of gravity will operate less upon the ball of the pendulum,

than before: but the line D E being drawn so, as to stand perpendicular to
the rod A C of the pendulum; the force of gravity upon the body C B, now it
is in this situation, will produce the same effect, as if the body were to glide
down an inclined plane in the position of D E. But here the motion of the
body, when the rod is fixed to the point A, will not be equal to the
uninterrupted descent of the body down this plane; for the body will here
also receive the same kind of rotation in its motion, as before; so that the
motion of the center of gravity will in like manner be retarded; and the
point, which here descends with that degree of swiftness, which the body
would have, if not hindered by being fixed to the point A; that is, the point,
which descends as fast, as if the whole body were crouded into it, will be as
far removed from the point A, as before.
70. T h i s point, by which the length of the pendulum is to be estimated,
is called the center of oscillation. And the mathematicians have laid down
general directions, whereby to find this center in all bodies. If the globe A B
(in fig. 56.) be hung by the string C D, whose weight need not be regarded,
the center of oscillation is found thus. Let the straight line drawn from C to
D be continued through the globe to F. That it will pass through the center
of the globe is evident. Suppose E to be this center of the globe; and take
the line G of such a length, that it shall bear the same proportion to E D, as
E D bears to E C. Then E H being made equal to ⅖ of G, the point H shall
be the center of oscillation[65]. If the weight of the rod C D is too
considerable to be neglected, divide C D (fig. 57) in I, that D I be equal to
⅓, part of C D; and take K in the same proportion to C I, as the weight of
the globe A B to the weight of the rod C D. Then having found H, the center
of oscillation of the globe, as before, divide I K in I, so that I L shall bear
the same proportion to L H, as the line C H bears to K; and L shall be the
center of oscillation of the whole pendulum.
71. T h i s computation is made upon supposition, that the center of
oscillation of the rod C D, if that were to swing alone without any other
weight annexed, would be the point I. And this point would be the true
center of oscillation, so far as the thickness of the rod is not to be regarded.
If any one chuses to take into consideration the thickness of the rod, he
must place the center of oscillation thereof so much below the point I, that
eight times the distance of the center from the point I shall bear the same

proportion to the thickness of the rod, as the thickness of the rod bears to its
length C D[66].
72. I t has been observed above, that when a pendulum swings in an
arch of a circle, as here in fig. 58, the pendulum A B swings in the circular
arch C D; if you draw an horizontal line, as E F, from the place whence the
pendulum is let fall, to the line A G, which is perpendicular to the horizon:
then the velocity, which the pendulum will acquire in coming to the point
G, will be the same, as any body would acquire in falling directly down
from F to G. Now this is to be understood of the circular arch, which is
described by the center of oscillation of the pendulum. I shall here farther
observe, that if the straight line E G be drawn from the point, whence the
pendulum falls, to the lowest point of the arch; in the same or in equal
pendulums the velocity, which the pendulum acquires in G, is proportional
to this line: that is, if the pendulum, after it has descended from E to G, be
taken back to H, and let fall from thence, and the line H G be drawn; the
velocity, which the pendulum shall acquire in G by its descent from H, shall
bear the same proportion to the velocity, which it acquires in falling from E
to G, as the straight line H G bears to the straight line E G.
73. W e may now proceed to those experiments upon the percussion of
bodies, which I observed above might be made with pendulums. This
expedient for examining the effects of percussion was first proposed by our
late great architect Sir C h r i s t o p h e r W r e n. And it is as follows. Two
balls, as A and B (in fig. 59.) either equal or unequal, are hung by two
strings from two points C and D, so that, when the balls hang down without
motion, they shall just touch each other, and the strings be parallel. Here if
one of these balls be removed to any distance from its perpendicular
situation, and then let fall to descend and strike against the other; by the last
preceding paragraph it will be known, with what velocity this ball shall
return into its first perpendicular situation, and consequently with what
force it shall strike against the other ball; and by the height to which this
other ball ascends after the stroke, the velocity communicated to this ball
will be discovered. For instance, let the ball A be taken up to E, and from
thence be let fall to strike against B, passing over in its descent the circular
arch E F. By this impulse let B fly up to G, moving through the circular arch
H G. Then E I and G K being drawn horizontally, the ball A will strike
against B with the velocity, which it would acquire in falling directly down
from I; and the ball B has received a velocity, wherewith, if it had been

thrown directly upward, it would have ascended up to K. Likewise if
straight lines be drawn from E to F and from H to G, the velocity of A,
wherewith it strikes, will bear the same proportion to the velocity, which B
has received by the blow, as the straight line E F bears to the straight line
H G. In the same manner by noting the place to which A ascends after the
stroke, its remaining velocity may be compared with that, wherewith it
struck against B. Thus may be experimented the effects of the body A
striking against B at rest. If both the bodies are lifted up, and so let fall as to
meet and impinge against each other just upon the coming of both into their
perpendicular situation; by observing the places into which they move after
the stroke, the effects of their percussion in all these cases may be found in
the same manner as before.
74. S i r I s a a c N e w t o n has described these experiments; and has
shewn how to improve them to a greater exactness by making allowance for
the resistance, which the air gives to the motion of the balls[67]. But as this
resistance is exceeding small, and the manner of allowing for it is delivered
by himself in very plain terms, I need not enlarge upon it here. I shall rather
speak to a discovery, which he made by these experiments upon the
elasticity of bodies. It has been explained above[68], that when two bodies
strike, if they be not elastic, they remain contiguous after the stroke; but that
if they are elastic, they separate, and that the degree of their elasticity
determines the proportion between the celerity wherewith they separate,
and the celerity wherewith they meet. Now our author found, that the
degree of elasticity appeared in the same bodies always the same, with
whatever degree of force they struck; that is, the celerity wherewith they
separated, always bore the same proportion to the celerity wherewith they
met: so that the elastic power in all the bodies, he made trial upon, exerted it
self in one constant proportion to the compressing force. Our author made
trial with balls of wool bound up very compact, and found the celerity with
which they receded, to bear about the proportion of 5 to 9 to the celerity
wherewith they met; and in steel he found nearly the same proportion; in
cork the elasticity was something less; but in glass much greater; for the
celerity, wherewith balls of that material separated after percussion, he
found to bear the proportion of 15 to 16 to the celerity wherewith they
met[69].

75. I s h a l l finish my discourse on pendulums, with this farther
observation only, that the center of oscillation is also the center of another
force. If a body be fixed to any point, and being put in motion turns round
it; the body, if uninterrupted by the power of gravity or any other means,
will continue perpetually to move about with the same equable motion.
Now the force, with which such a body moves, is all united in the point,
which in relation to the power of gravity is called the center of oscillation.
Let the cylinder A B C D (in fig. 60.) whose axis is E F, be fixed to the
point E. And supposing the point E to be that on which the cylinder is
suspended, let the center of oscillation be found in the axis E F, as has been
explained above[70]. Let G be that center: then I say, that the force,
wherewith this cylinder turns round the point E, is so united in the point G,
that a sufficient force applied in that point shall stop the motion of the
cylinder, in such a manner, that the cylinder should immediately remain
without motion, though it were to be loosened from the point E at the same
instant, that the impediment was applied to G: whereas, if this impediment
had been applied to any other point of the axis, the cylinder would turn
upon the point, where the impediment was applied. If the impediment had
been applied between E and G, the cylinder would so turn on the point,
where the impediment was applied, that the end B C would continue to
move on the same way it moved before along with the whole cylinder; but
if the impediment were applied to the axis farther off from E than G, the
end A D of the cylinder would start out of its present place that way in
which the cylinder moved. From this property of the center of oscillation, it
is also called the center of percussion. That excellent mathematician, Dr.
Brook Taylor, has farther improved this doctrine concerning the center of
percussion, by shewing, that if through this point G a line, as G H I, be
drawn perpendicular to E F, and lying in the course of the body’s motion; a
sufficient power applied to any point of this line will have the same effect,
as the like power applied to G[71]: so that as we before shewed the center of
percussion within the body on its axis; by this means we may find this
center on the surface of the body also, for it will be where this line H I
crosses that surface.
76. I s h a l l now proceed to the last kind of motion, to be treated on in
this place, and shew what line the power of gravity will cause a body to
describe, when it is thrown forwards by any force. This was first discovered
by the great G a l i l e o, and is the principle, upon which engineers should

direct the shot of great guns. But as in this case bodies describe in their
motion one of those lines, which in geometry are called conic sections; it is
necessary here to premise a description of those lines. In which I shall be
the more particular, because the knowledge of them is not only necessary
for the present purpose, but will be also required hereafter in some of the
principal parts of this treatise.
77. T h e first lines considered by the ancient geometers were the
straight line and the circle. Of these they composed various figures, of
which they demonstrated many properties, and resolved divers problems
concerning them. These problems they attempted always to resolve by the
describing straight lines and circles. For instance, let a square A B C D (fig.
61.) be proposed, and let it be required to make another square in any
assigned proportion to this. Prolong one side, as D A, of this square to E, till
A E bear the same proportion to A D, as the new square is to bear to the
square A C. If the opposite side B C of the square A C be also prolonged to
F, till B F be equal to A E, and E F be afterwards drawn, I suppose my
readers will easily conceive, that the figure A B F E will bear to the square
A B C D the same proportion, as the line A E bears to the line A D.
Therefore the figure A B F E will be equal to the new square, which is to be
found, but is not it self a square, because the side A E is not of the same
length with the side E F. But to find a square equal to the figure A B F E
you must proceed thus. Divide the line D E into two equal parts in the point
G, and to the center G with the interval G D describe the circle D H E I;
then prolong the line A B, till it meets the circle in K; and make the square
A K L M, which square will be equal to the figure A B F E, and bear to the
square A B C D the same proportion, as the line A E bears to A D.
78. I s h a l l not proceed to the proof of this, having only here set it
down as a specimen of the method of resolving geometrical problems by
the description of straight lines and circles. But there are some problems,
which cannot be resolved by drawing straight lines or circles upon a plane.
For the management therefore of these they took into consideration solid
figures, and of the solid figures they found that, which is called a cone, to
be the most useful.
79. A c o n e is thus defined by Euclide in his elements of geometry[72].
If to the straight line A B (in fig. 62.) another straight line, as A C, be drawn
perpendicular, and the two extremities B and C be joined by a third straight

line composing the triangle A C B (for so every figure is called, which is
included under three straight lines) then the two points A and B being held
fixed, as two centers, and the triangle A C B being turned round upon the
line A B, as on an axis; the line A C will describe a circle, and the figure
A C B will describe a cone, of the form represented by the figure B C D E F
(fig. 63.) in which the circle C D E F is usually called the base of the cone,
and B the vertex.
80. N o w by this figure may several problems be resolved, which
cannot by the simple description of straight lines and circles upon a plane.
Suppose for instance, it were required to make a cube, which should bear
any assigned proportion to some other cube named. I need not here inform
my readers, that a cube is the figure of a dye. This problem was much
celebrated among the ancients, and was once inforced by the command of
an oracle. This problem may be performed by a cone thus. First make a
cone from a triangle, whose side A C shall be half the length of the side B C
Then on the plane A B C D (fig. 64.) let the line E F be exhibited equal in
length to the side of the cube proposed; and let the line F G be drawn
perpendicular to E F, and of such a length, that it bear the same proportion
to E F, as the cube to be sought is required to bear to the cube proposed.
Through the points E, F, and G let the circle F H I be described. Then let the
line E F be prolonged beyond F to K, that F K be equal to F E, and let the
triangle F K L, having all its sides F K, K L, L F equal to each other, be
hung down perpendicularly from the plane A B C D. After this, let another
plane M N O P be extended through the point L, so as to be equidistant
from the former plane A B C D, and in this plane let the line Q L R be
drawn so, as to be equidistant from the line E F K. All this being thus
prepared, let such a cone, as was above directed to be made, be so applied
to the plane M N O P, that it touch this plane upon the line Q R, and that the
vertex of the cone be applied to the point L. This cone, by cutting through
the first plane A B C D, will cross the circle F H I before described. And if
from the point S, where the surface of this cone intersects the circle, the line
S T be drawn so, as to be equidistant from the line E F; the line F T will be
equal to the side of the cube sought: that is, if there be two cubes or dyes
formed, the side of one being equal to E F, and the side of the other equal to
F T; the former of these cubes shall bear the same proportion to the latter, as
the line E F bears to F G.

81. I n d e e d this placing a cone to cut through a plane is not a
practicable method of resolving problems. But when the geometers had
discovered this use of the cone, they applied themselves to consider the
nature of the lines, which will be produced by the intersection of the surface
of a cone and a plane; whereby they might be enabled both to reduce these
kinds of solutions to practice, and also to render their demonstrations
concise and elegant.
82. W h e n e v e r the plane, which cuts the cone, is equidistant from
another plane, that touches the cone on the side; (which is the case of the
present figure;) the line, wherein the plane cuts the surface of the cone, is
called a parabola. But if the plane, which cuts the cone, be so inclined to
this other, that it will pass quite through the cone (as in fig. 65.) such a
plane by cutting the cone produces the figure called an ellipsis, in which we
shall hereafter shew the earth and other planets to move round the sun. If
the plane, which cuts the cone, recline the other way (as in fig. 66.) so as
not to be parallel to any plane, whereon the cone can lie, nor yet to cut quite
through the cone; such a plane shall produce in the cone a third kind of line,
which is called an hyperbola. But it is the first of these lines named the
parabola, wherein bodies, that are thrown obliquely, will be carried by the
force of gravity; as I shall here proceed to shew, after having first directed
my readers how to describe this sort of line upon a plane, by which the form
of it may be seen.
83. T o any straight line A B (fig. 67.) let a straight ruler C D be so
applied, as to stand against it perpendicularly. Upon the edge of this ruler let
another ruler E F be so placed, as to move along upon the edge of the first
ruler C D, and keep always perpendicular to it. This being so disposed, let
any point, as G, be taken in the line A B, and let a string equal in length to
the ruler E F be fastened by one end to the point G, and by the other to the
extremity F of the ruler E F. Then if the string be held down to the ruler E F
by a pin H, as is represented in the figure; the point of this pin, while the
ruler E F moves on the ruler C D, shall describe the line I K L, which will
be one part of the curve line, whose description we were here to teach: and
by applying the rulers in the like manner on the other side of the line A B,
we may describe the other part I M of this line. If the distance C G be equal
to half the line E F in fig. 64, the line M I L will be that very line, wherein
the plane A B C D in that figure cuts the cone.

84. T h e line A I is called the axis of the parabola M I L, and the point
G is called the focus.
85. N o w by comparing the effects of gravity upon falling bodies, with
what is demonstrated of this figure by the geometers, it is proved, that every
body thrown obliquely is carried forward in one of these lines, the axis
whereof is perpendicular to the horizon.
86. T h e geometers demonstrate, that if a line be drawn to touch a
parabola in any point, as the line A B (in fig. 68.) touches the parabola C D,
whose axis is Y Z, in the point E; and several lines F G, H I, K L be drawn
parallel to the axis of the parabola: then the line F G will be to H I in the
duplicate proportion of E F to E H, and F G to K L in the duplicate
proportion of E F to E K; likewise H I to K L in the duplicate proportion of
E H to E K. What is to be understood by duplicate or two-fold proportion,
has been already explained[73]. Accordingly I mean here, that if the line M
be taken to bear the same proportion to E H, as E H bears to E F, H I will
bear the same proportion to F G, as M bears to E F; and if the line N bears
the same proportion to E K, as E K bears to E F, K L will bear the same
proportion to F G, as N bears to E F; or if the line O bear the same
proportion to E K, as E K bears to E H, K L will bear the same proportion
to H I, as O bears to E H.
87. T h i s property is essential to the parabola, being so connected with
the nature of the figure, that every line possessing this property is to be
called by this name.
88. N o w suppose a body to be thrown from the point A (in fig. 69.)
towards B in the direction of the line A B. This body, if left to it self, would
move on with a uniform motion through this line A B. Suppose the eye of a
spectator to be placed at the point C just under the point A; and let us
imagine the earth to be so put into motion along with the body, as to carry
the spectator’s eye along the line C D parallel to A B; and that the eye
would move on with the same velocity, wherewith the body would proceed
in the line A B, if it were to be left to move without any disturbance from its
gravitation towards the earth. In this case if the body moved on without
being drawn towards the earth, it would appear to the spectator to be at rest.
But if the power of gravity exerted it self on the body, it would appear to the
spectator to fall directly down. Suppose at the distance of time, wherein the
body by its own progressive motion would have moved from A to E, it

should appear to the spectator to have fallen through a length equal to E F:
then the body at the end of this time will actually have arrived at the point
F. If in the space of time, wherein the body would have moved by its
progressive motion from A to G, it would have appeared to the spectator to
have fallen down the space G H: then the body at the end of this greater
interval of time will be arrived at the point H. Now if the line A F H I be
that, through which the body actually passes; from what has here been said,
it will follow, that this line is one of those, which I have been describing
under the name of the parabola. For the distances E F, G H, through which
the body is seen to fall, will increase in the duplicate proportion of the
times[74]; but the lines A E, A G will be proportional to the times wherein
they would have been described by the single progressive motion of the
body: therefore the lines E F, G H will be in the duplicate proportion of the
lines A F, A G; and the line A F H I possesses the property of the parabola.
89. I f the earth be not supposed to move along with the body, the case
will be a little different. For the body being constantly drawn directly
towards the center of the earth, the body in its motion will be drawn in a
direction a little oblique to that, wherein it would be drawn by the earth in
motion, as before supposed. But the distance to the center of the earth bears
so vast a proportion to the greatest length, to which we can throw bodies,
that this obliquity does not merit any regard. From the sequel of this
discourse it may indeed be collected, what line the body being thrown thus
would be found to describe, allowance being made for this obliquity of the
earth’s action[75]. This is the discovery of Sir Is. Newton; but has no use in
this place. Here it is abundantly sufficient to consider the body as moving in
a parabola.
90. T h e line, which a projected body describes, being thus known,
practical methods have been deduced from hence for directing the shot of
great guns to strike any object desired. This work was first attempted by
G a l i l e o, and soon after farther improved by his scholar T o r r i c e l l i;
but has lately been rendred more complete by the great Mr. C o t e s, whose
immature death is an unspeakable loss to mathematical learning. If it be
required to throw a body from the point A (in fig. 70.) so as to strike the
point B; through the points A, B draw the straight line C D, and erect the
line A E perpendicular to the horizon, and of four times the height, from
which a body must fall to acquire the velocity, wherewith the body is

intended to be thrown. Through the points A and E describe a circle, that
shall touch the line C D in the point A. Then from the point B draw the line
B F perpendicular to the horizon, intersecting the circle in the points G and
H. This being done, if the body be projected directly towards either of these
points G or H, it shall fall upon the point B; but with this difference, that, if
it be thrown in the direction A G, it shall sooner arrive at B, than if it were
projected in the direction A H. When the body is projected in the direction
A G; the time, it will take up in arriving at B, will bear the same proportion
to the time, wherein it would fall down through one fourth part of A E, as
A G bears to half A E. But when the body is thrown in the direction of A H,
the time of its passing to B will bear the same proportion to the time,
wherein it would fall through one fourth part of A E, as A H bears to half
A E.
91. I f the line A I be drawn so as to divide the angle under E A D in the
middle, and the line I K be drawn perpendicular to the horizon; this line will
touch the circle in the point I, and if the body be thrown in the direction A I,
it will fall upon the point K: and this point K is the farthest point in the line
A D, which the body can be made to strike, without increasing its velocity.
92. T h e velocity, wherewith the body every where moves, may be
found thus. Suppose the body to move in the parabola A B (fig. 71.) Erect
A C perpendicular to the horizon, and equal to the height, from which a
body must fall to acquire the velocity, wherewith the body sets out from A.
If you take any points as D and E in the parabola, and draw D F and E G
parallel to the horizon; the velocity of the body in D will be equal to what a
body will acquire in falling down by its own weight through C F, and in E
the velocity will be the same, as would be acquired in falling through C G.
Thus the body moves slowest at the highest point H of the parabola; and at
equal distances from this point will move with equal swiftness, and descend
from that highest point through the line H B altogether like to the line A H
in which it ascended; abating only the resistance of the air, which is not here
considered. If the line H I be drawn from the highest point H parallel to the
horizon, A I will be equal to ¼ of B G in fig. 70, when the body is projected
in the direction A G, and equal to ¼ of B H, when the body is thrown in the
direction A H provided A D be drawn horizontally.
93. T h u s I have recounted the principal discoveries, which had been
made concerning the motion of bodies by Sir I s a a c N e w t o n’s
predecessors; all these discoveries, by being found to agree with

experience, contributing to establish the laws of motion, from whence they
were deduced. I shall therefore here finish what I had to say upon those
laws; and conclude this chapter with a few words concerning the distinction
which ought to be made between absolute and relative motion. For some
have thought fit to confound them together; because they observe the laws
of motion to take place here on the earth, which is in motion, after the same
manner as if it were at rest. But Sir I s a a c N e w t o n has been careful to
distinguish between the relative and absolute consideration both of motion
and time[76]. The astronomers anciently found it necessary to make this
distinction in time. Time considered in it self passes on equably without
relation to any thing external, being the proper measure of the continuance
and duration of all things. But it is most frequently conceived of by us
under a relative view to some succession in sensible things, of which we
take cognizance. The succession of the thoughts in our own minds is that,
from whence we receive our first idea of time, but is a very uncertain
measure thereof; for the thoughts of some men flow on much more swiftly,
than the thoughts of others; nor does the same person think equally quick at
all times. The motions of the heavenly bodies are more regular; and the
eminent division of time into night and day, made by the sun, leads us to
measure our time by the motion of that luminary: nor do we in the affairs of
life concern our selves with any inequality, which there may be in that
motion; but the space of time which comprehends a day and night is rather
supposed to be always the same. However astronomers anciently found
these spaces of time not to be always of the same length, and have taught
how to compute their differences. Now the time, when so equated as to be
rendered perfectly equal, is the true measure of duration, the other not. And
therefore this latter, which is absolutely true time, differs from the other,
which is only apparent. And as we ordinarily make no distinction between
apparent time, as measured by the sun, and the true; so we often do not
distinguish in our usual discourse between the real, and the apparent or
relative motion of bodies; but use the same words for one, as we should for
the other. Though all things about us are really in motion with the earth; as
this motion is not visible, we speak of the motion of every thing we see, as
if our selves and the earth stood still. And even in other cases, where we
discern the motion of bodies, we often speak of them not in relation to the
whole motion we see, but with regard to other bodies, to which they are
contiguous. If any body were lying on a table; when that table shall be

carried along, we say the body rests upon the table, or perhaps absolutely,
that the body is at rest. However philosophers must not reject all distinction
between true and apparent motions, any more than astronomers do the
distinction between true and vulgar time; for there is as real a difference
between them, as will appear by the following consideration. Suppose all
the bodies of the universe to have their courses stopped, and reduced to
perfect rest. Then suppose their present motions to be again restored; this
cannot be done without an actual impression made upon some of them at
least. If any of them be left untouched, they will retain their former state,
that is, still remain at rest; but the other bodies, which are wrought upon,
will have changed their former state of rest, for the contrary state of motion.
Let us now suppose the bodies left at rest to be annihilated, this will make
no alteration in the state of the moving bodies; but the effect of the
impression, which was made upon them, will still subsist. This shews the
motion they received to be an absolute thing, and to have no necessary
dependence upon the relation which the body said to be in motion has to
any other body[77].
94. B e s i d e s absolute and relative motion are distinguishable by their
Effects. One effect of motion is, that bodies, when moved round any center
or axis, acquire a certain power, by which they forcibly press themselves
from that center or axis of motion. As when a body is whirled about in a
sling, the body presses against the sling, and is ready to fly out as soon as
liberty is given it. And this power is proportional to the true, not relative
motion of the body round such a center or axis. Of this Sir I s a a c
N e w t o n gives the following instance[78]. If a pail or such like vessel near
full of water be suspended by a string of sufficient length, and be turned
about till the string be hard twisted. If then as soon as the vessel and water
in it are become still and at rest, the vessel be nimbly turned about the
contrary way the string was twisted, the vessel by the strings untwisting it
self shall continue its motion a long time. And when the vessel first begins
to turn, the water in it shall receive little or nothing of the motion of the
vessel, but by degrees shall receive a communication of motion, till at last it
shall move round as swiftly as the vessel it self. Now the definition of
motion, which D e s C a r t e s has given us upon this principle of making
all motion meerly relative, is this: that motion, is a removal of any body
from its vicinity to other bodies, which were in immediate contact with it,
and are considered as at rest[79]. And if this be compared with what he soon

after says, that there is nothing real or positive in the body moved, for the
sake of which we ascribe motion to it, which is not to be found as well in
the contiguous bodies, which are considered as at rest[80]; it will follow
from thence, that we may consider the vessel as at rest and the water as
moving in it: and the water in respect of the vessel has the greatest motion,
when the vessel first begins to turn, and loses this relative motion more and
more, till at length it quite ceases. But now, when the vessel first begins to
turn, the surface of the water remains smooth and flat, as before the vessel
began to move; but as the motion of the vessel communicates by degrees
motion to the water, the surface of the water will be observed to change, the
water subsiding in the middle and rising at the edges: which elevation of the
water is caused by the parts of it pressing from the axis, they move about;
and therefore this force of receding from the axis of motion depends not
upon the relative motion of the water within the vessel, but on its absolute
motion; for it is least, when that relative motion is greatest, and greatest,
when that relative motion is least, or none at all.
95. T h u s the true cause of what appears in the surface of this water
cannot be assigned, without considering the water’s motion within the
vessel. So also in the system of the world, in order to find out the cause of
the planetary motions, we must know more of the real motions, which
belong to each planet, than is absolutely necessary for the uses of
astronomy. If the astronomer should suppose the earth to stand still, he
could ascribe such motions to the celestial bodies, as should answer all the
appearances; though he would not account for them in so simple a manner,
as by attributing motion to the earth. But the motion of the earth must of
necessity be considered, before the real causes, which actuate the planetary
system, can be discovered.

C h a p. III.
Of CENTRIPETAL FORCES.

W
E have just been describing in the preceding chapter the effects
produced on a body in motion, from its being continually acted
upon by a power always equal in strength, and operating in
parallel directions[81]. But bodies may be acted upon by powers,
which in different places shall have different degrees of force, and whose
several directions shall be variously inclined to each other. The most simple
of these in respect to direction is, when the power is pointed constantly to
one center. This is truly the case of that power, whose effects we described
in the foregoing chapter; though the center of that power is so far removed,
that the subject then before us is most conveniently to be considered in the
light, wherein we have placed it: But Sir Isaac Newton has considered very
particularly this other case of powers, which are constantly directed to the
same center. It is upon this foundation, that all his discoveries in the system
of the world are raised. And therefore, as this subject bears so very great a
share in the philosophy, of which I am discoursing, I think it proper in this
place to take a short view of some of the general effects of these powers,
before we come to apply them particularly to the system of the world.
2. T h e s e powers or forces are by Sir I s a a c N e w t o n called
centripetal; and their first effect is to cause the body, on which they act, to
quit the straight course, wherein it would proceed if undisturbed, and to
describe an incurvated line, which shall always be bent towards the center
of the force. It is not necessary, that such a power should cause the body to
approach that center. The body may continue to recede from the center of
the power, notwithstanding its being drawn by the power; but this property
must always belong to its motion, that the line, in which it moves, will

continually be concave towards the center, to which the power is directed.
Suppose A (in fig. 72.) to be the center of a force. Let a body in B be
moving in the direction of the straight line B C, in which line it would
continue to move, if undisturbed; but being attracted by the centripetal force
towards A, the body must necessarily depart from this line B C, and being
drawn into the curve line B D, must pass between the lines A B and B C. It
is evident therefore, that the body in B being gradually turned off from the
straight line B C, it will at first be convex toward the line B C, and
consequently concave towards the point A: for these centripetal powers are
supposed to be in strength proportional to the power of gravity, and, like
that, not to be able after the manner of an impulse to turn the body sensibly
out of its course into a different one in an instant, but to take up some space
of time in producing a visible effect. That the curve will always continue to
have its concavity towards A may thus appear. In the line B C near to B take
any point as E, from which the line E F G may be so drawn, as to touch the
curve line B D in some point as F. Now when the body is come to F, if the
centripetal power were immediately to be suspended, the body would no
longer continue to move in a curve line, but being left to it self would
forthwith reassume a straight course; and that straight course would be in
the line F G: for that line is in the direction of the body’s motion at the point
F. But the centripetal force continuing its energy, the body will be gradually
drawn from this line F G so as to keep in the line F D, and make that line
near the point F to be convex toward F G, and concave toward A. After the
same manner the body may be followed on in its course through the line
B D, and every part of that line be shewn to be concave toward the point A.
3. T h i s then is the constant character belonging to those motions,
which are carried on by centripetal forces; that the line, wherein the body
moves, is throughout concave towards the center of the force. In respect to
the successive distances of the body from the center there is no general rule
to be laid down; for the distance of the body from the center may either
increase, or decrease, or even keep always the same. The point A (in fig.
73.) being the center of a centripetal force, let a body at B set out in the
direction of the straight line B C perpendicular to the line A B drawn from
A to B. It will be easily conceived, that there is no other point in the line
B C so near to A, as the point B; that A B is the shortest of all the lines,
which can be drawn from A to any part of the line B C; all other lines, as
A D, or A E, drawn from A to the line B C being longer than A B. Hence it

follows, that the body setting out from B, if it moved in the line B C, it
would recede more and more from the point A. Now as the operation of a
centripetal force is to draw a body towards the center of the force: if such a
force act upon a resting body, it must necessarily put that body so into
motion, as to cause it to move towards the center of the force: if the body
were of it self moving towards that center, the centripetal force would
accelerate that motion, and cause it to move faster down: but if the body
were in such a motion, as being left to itself it would recede from this
center, it is not necessary, that the action of a centripetal power upon it
should immediately compel the body to approach the center, from which it
would otherwise have receded; the centripetal power is not without effect, if
it cause the body to recede more slowly from that center, than otherwise it
would have done. Thus in the case before us, the smallest centripetal power,
if it act on the body, will force it out of the line B C, and cause it to pass in a
bent line between B C and the point A, as has been before explained. When
the body, for instance, has advanced to the line A D, the effect of the
centripetal force discovers it self by having removed the body out of the
line B C, and brought it to cross the line A D somewhere between A and D:
suppose at F. Now A D being longer than A B, A F may also be longer than
A B. The centripetal power may indeed be so strong, that A F shall be
shorter than A B; or it may be so evenly balanced with the progressive
motion of the body, that A F and A B shall be just equal: and in this last
case, when the centripetal force is of that strength, as constantly to draw the
body as much toward the center, as the progressive motion would carry it
off, the body will describe a circle about the center A, this center of the
force being also the center of the circle.
4. I f the body, instead of setting out in the line B C perpendicular to
A B, had set out in another line B G more inclined towards the line A B,
moving in the curve line B H; then as the body, if it were to continue its
motion in the line B G, would for some time approach the center A; the
centripetal force would cause it to make greater advances toward that
center. But if the body were to set out in the line B I reclined the other way
from the perpendicular B C, and were to be drawn by the centripetal force
into the curve line B K; the body, notwithstanding any centripetal force,
would for some time recede from the center; since some part at least of the
curve line B K lies between the line B I and the perpendicular B C.

5. T h u s far we have explained such effects, as attend every centripetal
force. But as these forces may be very different in regard to the different
degrees of strength, wherewith they act upon bodies in different places; I
shall now proceed to make mention in general of some of the differences
attending these centripetal motions.
6. T o reassume the consideration of the last mentioned case. Suppose a
centripetal power directed toward the point A (in fig. 74.) to act on a body
in B, which is moving in the direction of the straight line B C, the line B C
reclining off from A B. If from A the straight lines A D, A E, A F are drawn
at pleasure to the line C B; the line C B being prolonged beyond B to G, it
appears that A D is inclined to the line G C more obliquely, than A B is
inclined to it, A E is inclined more obliquely than A D, and A F more than
A E. To speak more correctly, the angle under A D G is less than that under
A B G, the angle under A E G less than that under A D G, and the angle
under A F G less than that under A E G. Now suppose the body to move in
the curve line B H I K. Then it is here likewise evident, that the line
B H I K being concave towards A, and convex towards the line B C, it is
more and more turned off from the line B C; so that in the point H the line
A H will be less obliquely inclined to the curve line B H I K, than the same
line A H D is inclined to B C at the point D; at the point I the inclination of
the line A I to the curve line will be more different from the inclination of
the same line A I E to the line B C, at the point E; and in the points K and F
the difference of inclination will be still greater; and in both the inclination
at the curve will be less oblique, than at the straight line B C. But the
straight line A B is less obliquely inclined to B G, than A D is inclined
towards D G: therefore although the line A H be less obliquely inclined
towards the curve H B, than the same line A H D is inclined towards D G;
yet it is possible, that the inclination at H may be more oblique, than the
inclination at B. The inclination at H may indeed be less oblique than the
other, or they may be both the same. This depends upon the degree of
strength, wherewith the centripetal force exerts it self, during the passage of
the body from B to H. After the same manner the inclinations at I and K
depend entirely on the degree of strength, wherewith the centripetal force
acts on the body in its passage from H to K: if the centripetal force be weak
enough, the lines A H and A I drawn from the center A to the body at H and
at I shall be more obliquely inclined to the curve, than the line A B is
inclined towards B G. The centripetal force may be of that strength as to

render all these inclinations equal, or if stronger, the inclinations at I and K
will be less oblique than at B. Sir I s a a c N e w t o n has particularly
shewn, that if the centripetal power decreases after a certain manner with
the increase of distance, a body may describe such a curve line, that all the
lines drawn from the center to the body shall be equally inclined to that
curve line.[82] But I do not here enter into any particulars, my present
intention being only to shew, that it is possible for a body to be acted upon
by a force continually drawing it down towards a center, and yet that the
body shall continue to recede from that center; for here as long as the lines
A H, A I, &c drawn from the center A to the body do not become less
oblique to the curve, in which the body moves; so long shall those lines
perpetually increase, and consequently the body shall more and more recede
from the center.
7. B u t we may observe farther, that if the centripetal power, while the
body increases its distance from the center, retain sufficient strength to
make the lines drawn from the center to the body to become at length less
oblique to the curve; then if this diminution of the obliquity continue, till at
last the line drawn from the center to the body shall cease to be obliquely
inclined to the curve, and shall become perpendicular thereto; from this
instant the body shall no longer recede from the center, but in its following
motion it shall again descend, and shall describe a curve line in all respects
like to that, which it has described already; provided the centripetal power,
every where at the same distance from the center, acts with the same
strength. So we observed in the preceding chapter, that, when the motion of
a projectile became parallel to the horizon, the projectile no longer
ascended, but forthwith directed its course downwards, descending in a line
altogether like that, wherein it had before ascended[83].
8. T h i s return of the body may be proved by the following proposition:
that if the body in any place, suppose at I, were to be stopt, and be thrown
directly backward with the velocity, wherewith it was moving forward in
that point I; then the body, by the action of the centripetal force upon it,
would move back again over the path I H B, in which it had before
advanced forward, and would arrive again at the point B in the same space
of time, as was taken up in its passage from B to I; the velocity of the body
at its return to the point B being the same, as that wherewith it first set out
from that point. To give a full demonstration of this proposition, would

require that use of mathematics, which I here purpose to avoid; but, I
believe, it will appear in great measure evident from the following
considerations.
9. S u p p o s e (in fig. 75.) that a body were carried after the following
manner through the bent figure A B C D E F, composed of the straight lines
A B, B C, C D, D E, E F. First let it be moving in the line A B, from A
towards B, with any uniform velocity. At B let the body receive an impulse
directed toward some point, as G, taken within the concavity of the figure.
Now whereas this body, when once moving in the straight line A B, will
continue to move on in this line, so long as it shall be left to it self; but
being disturbed at the point B in its motion by the impulse, which there acts
upon it, it will be turned out of this line A B into some other straight line,
wherein it will afterwards continue to move, as long as it shall be left to
itself. Therefore let this impulse have strength sufficient to turn the body
into the line B C. Then let the body move on undisturbed from B to C, but
at C let it receive another impulse pointed toward the same point G, and of
sufficient strength to turn the body into the line C D. At D let a third
impulse, directed like the rest to the point G, turn the body into the line D E.
And at E let another impulse, directed likewise to the point G, turn the body
into the line E F. Now, I say, if the body while moving in the line E F be
stopt, and turned back again in this line with the same velocity, as that
wherewith it was moving forward in this line; then by the repetition of the
former impulse at E the body will be turned into the line E D, and move in
it from E to D with the same velocity as before it moved with from D to E;
by the repetition of the impulse at D, when the body shall have returned to
that point, it will be turned into the line D C; and by the repetition of the
other impulses at C and B the body will be brought back again into the line
B A, with the velocity, wherewith it first moved in that line.
10. T h i s I prove as follows. Let D E and F E be continued beyond E.
In D E thus continued take at pleasure the length E H, and let H I be so
drawn, as to be equidistant from the line G E. Then, by what has been
written upon the second law of motion[84], it follows, that after the impulse
on the body in E it will move through E I in the same time, as it would have
imployed in moving from E to H, with the velocity which it had in the line
D E. In F E prolonged take E K equal to E I, and draw K L equidistant from
G E. Then, because the body is thrown back in the line F E with the same
velocity as that wherewith it went forward in that line; if, when the body

was returned to E, it were permitted to go straight on, it would pass through
E K in the same time, as it took up in passing through E I, when it went
forward in the line E F. But, if at the body’s return to the point E, such an
impulse directed toward the point D were to be given it, whereby it should
be turned into the line D E; I say, that the impulse necessary to produce this
effect must be equal to that, which turned the body out of the line D E into
E F; and that the velocity, with which the body will return into the line E D,
is the same, as that wherewith it before moved through this line from D to
E. Because E K is equal to E I, and K L and H I, being each equidistant
from G E, are by consequence equidistant from each other; it follows, that
the two triangular figures I E H and K E L are altogether like and equal to
each other. If I were writing to mathematicians, I might refer them to some
proportions in the elements of Euclid for the proof of this[85] but as I do not
here address my self to such, so I think this assertion will be evident enough
without a proof in form; at least I must desire my readers to receive it as a
proposition true in geometry. But these two triangular figures being
altogether like each other and equal; as E K is equal to E I, so E L is equal
to E H, and K L equal to H I. Now the body after its return to E being
turned out of the line F E into E D by an impulse acting upon it in E, after
the manner above expressed; the body will receive such a velocity by this
impulse, as will carry it through E L in the same time, as it would have
imployed in passing through E K, if it had gone on in that line undisturbed.
And it has already been observed, that the time, in which the body would
pass over E K with the velocity wherewith it returns, is equal to the time it
took up in going forward from E to I; that is, equal to the time, in which it
would have gone through E H with the velocity, wherewith it moved from
D to E. Therefore the time, in which the body will pass through E L after its
return into the line E D, is the same, as would have been taken up by the
body in passing through E H with the velocity, wherewith the body first
moved in the line D E. Since therefore E L and E H are equal, the body
returns into the line D E with the velocity, which it had before in that line.
Again I say, the second impulse in E is equal to the first. By what has been
said on the second law of motion concerning the effect of oblique
impulses[86], it will be understood, that the impulse in E, whereby the body
was turned out of the line D E into the line E F, is of such strength, that if
the body had been at rest, when this impulse had acted upon it, this impulse
would have communicated so much motion to the body, as would have

carried it through a length equal to H I, in the time wherein the body would
have passed from E to H, or in the time wherein it passed from E to I. In the
same manner, on the return of the body, the impulse in E, whereby the body
is turned out of the line F E into E D, is of such strength, that if it had acted
on the body at rest, it would have caused the body to move through a length
equal to K L, in the same time, as the body would imploy in passing
through E K with the velocity, wherewith it returns in the line F E.
Therefore the second impulse, had it acted on the body at rest, would have
caused it to move through a length equal to K L in the same space of time,
as would be taken up by the body in passing through a length equal to H I,
were the first impulse to act on the body when at rest. That is, the effects of
the first and second impulse on the body when at rest would be the same;
for K L and H I are equal: consequently the second impulse is equal to the
first.
11. T h u s if the body be returned through F E with the velocity,
wherewith it moved forward; we have shewn how by the repetition of the
impulse, which acted on it at E, the body will return again into the line D E
with the velocity, which it had before in that line. By the same process of
reasoning it may be proved, that, when the body is returned back to D, the
impulse, which before acted on the body at that point, will throw the body
into the line D C with the velocity, which it first had in that line; and the
other impulses being successively repeated, the body will at length be
brought back again into the line B A with the velocity, wherewith it set out
in that line.
12. T h u s these impulses, by acting over again in an inverted order all
their operation on the body, bring it back again through the path, in which it
had proceeded forward. And this obtains equally, whatever be the number
of the straight lines, whereof this curve figure is composed. Now by a
method of reasoning, which Sir I s a a c N e w t o n makes great use of, and
which he introduced into geometry, thereby greatly inriching that
science[87]; we might make a transition from this figure composed of a
number of straight lines to a figure of one continued curvature, and from a
number of separate impulses repeated at distinct intervals to a continual
centripetal force, and shew, that, because what has been here advanced
holds universally true, whatever be the number of straight lines, whereof the
curve figure A C F is composed, and howsoever frequently the impulses at
the angles of this figure are repeated; therefore the same will still remain

true, although this figure should be converted into one of a continued
curvature, and these distinct impulses should be changed into a continual
centripetal force. But as the explaining this method of reasoning is foreign
to my present design; so I hope my readers, after what has been said, will
find no difficulty in receiving the proposition laid down above: that, if the
body, which has moved through the curve line B H I (in fig. 74.) from B to
I, when it is come to I, be thrown directly back with the same velocity as
that, wherewith it proceeded forward, the centripetal force, by acting over
again all its operation on the body, shall bring the body back again in the
line I H B: and as the motion of the body in its course from B to I was every
where in such a manner oblique to the line drawn from the center to the
body, that the centripetal power acted in some degree against the body’s
motion, and gradually diminished it; so in the return of the body, the
centripetal power will every where draw the body forward, and accelerate
its motion by the same degrees, as before it retarded it.
13. T h i s being agreed, suppose the body in K to have the line A K no
longer obliquely inclined to its motion. In this case, if the body be turned
back, in the manner we have been considering, it must be directed back
perpendicularly to A K. But if it had proceeded forward, it would likewise
have moved in a direction perpendicular to A K; consequently, whether it
move from this point K backward or forward, it must describe the same
kind of course. Therefore since by being turned back it will go over again
the line K I H B; if it be permitted to go forward, the line K L, which it shall
describe, will be altogether similar to the line K H B.
14. I n like manner we may determine the nature of the motion, if the
line, wherein the body sets out, be inclined (as in fig. 76.) down toward the
line B A drawn between the body and the center. If the centripetal power so
much increases in strength, as the body approaches, that it can bend the
path, in which the body moves, to that degree, as to cause all the lines as
A H, A I, A K to remain no less oblique to the motion of the body, than A B
is oblique to B C; the body shall continually more and more approach the
center. But if the centripetal power increases in so much less a degree, as to
permit the line drawn from the center to the body, as it accompanies the
body in its motion, at length to become more and more erect to the curve
wherein the body moves, and in the end, suppose at K, to become
perpendicular thereto; from that time the body shall rise again. This is
evident from what has been said above; because for the very same reason

here also the body shall proceed from the point K to describe a line
altogether similar to the line, in which it has moved from B to K. Thus, as it
was observed of the pendulum in the preceding chapter[88], that all the time
it approaches towards being perpendicular to the horizon, it more and more
descends; but, as soon as it is come into that perpendicular situation, it
immediately rises again by the same degrees, as it descended by before: so
here the body more and more approaches the center all the time it is moving
from B to K; but thence forward it rises from the center again by the same
degrees, as it approached by before.
15. I f (in fig. 77.) the line B C be perpendicular to A B; then it has been
observed above[89], that the centripetal power may be so balanced with the
progressive motion of the body, that the body may keep moving round the
center A constantly at the same distance; as a body does, when whirled
about any point, to which it is tyed by a string. If the centripetal power be
too weak to produce this effect, the motion of the body will presently
become oblique to the line drawn from itself to the center, after the manner
of the first of the two cases, which we have been considering. If the
centripetal power be stronger, than what is required to carry the body in a
circle, the motion of the body will presently fall in with the second of the
cases, we have been considering.
16. I f the centripetal power so change with the change of distance, that
the body, after its motion has become oblique to the line drawn from itself
to the center, shall again become perpendicular thereto; which we have
shewn to be possible in both the cases treated of above; then the body shall
in its subsequent motion return again to the distance of A B, and from that
distance take a course similar to the former: and thus, if the body move in a
space free from all resistance, which has been here all along supposed; it
shall continue in a perpetual motion about the center, descending and
ascending alternately therefrom. If the body setting out from B (in fig. 78.)
in the line B C perpendicular to A B, describe the line B D E, which in D
shall be oblique to the line A D, but in E shall again become erect to A E
drawn from the body in E to the center A; then from this point E the body
shall describe the line E F G altogether like to the line B D E, and at G shall
be at the same distance from A, as it was at B. But likewise the line A G
shall be erect to the body’s motion. Therefore the body shall proceed to
describe from G the line G H I altogether similar to the line G F E, and at I

have the same distance from the center, as it had at E; and also have the line
A I erect to its motion: so that its following motion must be in the line I K L
similar to I H G, and the distance A L equal to A G. Thus the body will go
on in a perpetual round without ceasing, alternately inlarging and
contracting its distance from the center.

17. I f it so happen, that the point E fall upon the line B A continued
beyond A; then the point G will fall on B, I on E, and L also on B; so that
the body will describe in this case a simple curve line round the center A,
like the line B D E F in fig. 79, in which it will continually revolve from B
to E and from E to B without end.

18. I f A E in fig. 78 should happen to be perpendicular to A B, in this
case also a simple line will be described; for the point G will fall on the line
B A prolonged beyond A, the point I on the line A E prolonged beyond A,
and the point L on B: so that the body will describe a line like the curve line
B E G I in fig. 80, in which the opposite points B and G are equally distant
from A, and the opposite points E and I are also equally distant from the
same point A.
19. I n other cases the line described will have a more complex figure.
20. T h u s we have endeavoured to shew how a body, while it is
constantly attracted towards a center, may notwithstanding by its
progressive motion keep it self from falling down to that center; but
describe about it an endless circuit, sometimes approaching toward that
center, and at other times as much receding from the same.
21. B u t here we have supposed, that the centripetal power is of equal
strength every where at the same distance from the center. And this is the
case of that centripetal power, which will hereafter be shewn to be the
cause, that keeps the planets in their courses. But a body may be kept on in
a perpetual circuit round a center, although the centripetal power have not
this property. Indeed a body may by a centripetal force be kept moving in
any curve line whatever, that shall have its concavity turned every where
towards the center of the force.
22. T o make this evident I shall first propose the case of a body moving
through the incurvated figure A B C D E (in fig. 81.) which is composed of
the straight lines A B, B C, C D, D E, and E A; the motion being carried on
in the following manner. Let the body first move in the line A B with any
uniform velocity. When it is arrived at the point B, let it receive an impulse
directed toward any point F taken within the figure; and let the impulse be
of that strength as to turn the body out of the line A B into the line B C. The
body after this impulse, while left to itself, will continue moving in the line
B C. At C let the body receive another impulse directed towards the same
point F, of such strength, as to turn the body from the line B C into the line
C D. At D let the body by another impulse, directed likewise to the point F,
be turned out of the line C D into D E. And at E let another impulse,
directed toward the point F, turn the body from the line D E into E A. Thus
we see how a body may be carried through the figure A B C D E by certain

impulses directed always toward the same center, only by their acting on the
body at proper intervals, and with due degrees of strength.
23. B u t farther, when the body is come to the point A, if it there
receive another impulse directed like the rest toward the point F, and of
such a degree of strength as to turn the body into the line A B, wherein it
first moved; I say that the body shall return into this line with the same
velocity, as it had at first.
24. L e t A B be prolonged beyond B at pleasure, suppose to G; and
from G let G H be drawn, which if produced should always continue
equidistant from B F, or, according to the more usual phrase, let G H be
drawn parallel to B F. Then it appears, from what has been said upon the
second law of motion[90], that in the time, wherein the body would have
moved from B to G, had it not received a new impulse in B, by the means
of that impulse it will have acquired a velocity, which will carry it from B
to H. After the same manner, if C I be taken equal to B H, and I K be drawn
equidistant from or parallel to C F; the body will have moved from C to K
with the velocity, which it has in the line C D, in the same time, as it would
have employed in moving from C to I with the velocity, it had in the line
B C. Therefore since C I and B H are equal, the body will move through
C K in the same time, as it would have taken up in moving from B to G
with the original velocity, wherewith it moved through the line A B. Again,
D L being taken equal to C K and L M drawn parallel to D F; for the same
reason as before the body will move through D M with the velocity, which
it has in the line D E, in the same time, as it would imploy in moving
through B G with its original velocity. In the last place, if E N be taken
equal to D M, and N O be drawn parallel to E F; likewise if A P be taken
equal to E O, and P Q be drawn parallel to A F: then the body with the
velocity, wherewith it returns into the line A B, will pass through A Q in the
same time, as it would have imployed in passing through B G with its
original velocity. Now as all this follows directly from what has above been
delivered, concerning the effect of oblique impulses impressed upon bodies
in motion; so we must here observe farther, that it can be proved by
geometry, that A Q will always be equal to E G. The proof of this I am
obliged, from the nature of my present design, to omit; but this geometrical
proportion being granted, it follows, that the body has returned into the line
A B with the velocity, which it had, when it first moved in that line; for the
velocity, with which it returns into the line A B, will carry it over the line

A Q in the same time, as would have been taken up in its passing over an
equal line B G with the original velocity.
25. T h u s we have found, how a body may be carried round the figure
A B C D E by the action of certain impulses upon it which should all be
pointed toward one center. And we likewise see, that when the body is
brought back again to the point, whence it first set out; if it there meet with
an impulse sufficient to turn it again into the line, wherein it moved at first,
its original velocity will be again restored; and by the repetition of the same
impulses, the body will be carried again in the same round. Therefore if
these impulses, which act on the body at the points B, C, D, E, and A,
continue always the same, the body will make round this figure
innumerable revolutions.
26. T h e proof, which we have here made use of, holds the same in any
number of straight lines, whereof the figure A B D should be composed;
and therefore by the method of reasoning referred to above[91] we are to
conclude, that what has here been said upon this rectilinear figure, will
remain true, if this figure were changed into one of a continued curvature,
and instead of distinct impulses acting by intervals at the angles of this
figure, we had a continual centripetal force. We have therefore shewn, that a
body may be carried round in any curve figure A B C ( fig. 82.) which shall
every where be concave towards any one point as D, by the continual action
of a centripetal power directed to that point, and when it is returned to the
point, from whence it set out, it shall recover again the velocity, with which
it departed from that point. It is not indeed always necessary, that it should
return again into its first course; for the curve line may have some such
figure as the line A B C D B E in fig. 83. In this curve line, if the body set
out from B in the direction B F, and moved through the line B C D, till it
returned to B; here the body would not enter again into the line B C D,
because the two parts B D and B C of the curve line make an angle at the
point B: so that the centripetal power, which at the point B could turn the
body from the line B F into the curve, will not be able to turn the body into
the line B C from the direction, in which it returns to the point B; a
forceable impulse must be given the body in the point B to produce that
effect.
27. I f at the point B, whence the body sets out, the curve line return into
it self (as in fig. 82;) then the body, upon its arrival again at B, may return

into its former course, and thus make an endless circuit about the center of
the centripetal power.
28. W h a t has here been said, I hope, will in some measure enable my
readers to form a just idea of the nature of these centripetal motions.
29. I h a v e not attempted to shew, how to find particularly, what kind
of centripetal force is necessary to carry a body in any curve line proposed.
This is to be deduced from the degree of curvature, which the figure has in
each point of it, and requires a long and complex mathematical reasoning.
However I shall speak a little to the first proportion, which Sir I s a a c
N e w t o n lays down for this purpose. By this proposition, when a body is
found moving in a curve line, it may be known, whether the body be kept in
its course by a power always pointed toward the same center; and if it be so,
where that center is placed. The proposition is this: that if a line be drawn
from some fixed point to the body, and remaining by one extream united to
that point, it be carried round along with the body; then, if the power,
whereby the body is kept in its course, be always pointed to this fixed point
as a center, this line will move over equal spaces in equal portions of time.
Suppose a body were moving through the curve line A B C D (in fig. 84.)
and passed over the arches A B, B C, C D in equal portions of time; then if
a point, as E, can be found, from whence the line E A being drawn to the
body in A, and accompanying the body in its motion, it shall make the
spaces E A B, E B C, and E C D equal, over which it passes, while the body
describes the arches A B, B C, and C D: and if this hold the same in all
other arches, both great and small, of the curve line A B C D, that these
spaces are always equal, where the times are equal; then is the body kept in
this line by a power always pointed to E as a center.
30. T h e principle, upon which Sir I s a a c N e w t o n has
demonstrated this, requires but small skill in geometry to comprehend. I
shall therefore take the liberty to close the present chapter with an
explication of it; because such an example will give the clearest notion of
our author’s method of applying mathematical reasoning to these
philosophical subjects.
31. H e reasons thus. Suppose a body set out from the point A (in fig.
85.) to move in the straight line A B; and after it had moved for some time
in that line, it were to receive an impulse directed to some point as C. Let it
receive that impulse at D; and thereby be turned into the line D E; and let

the body after this impulse take the same length of time in passing from D
to E, as it imployed in the passing from A to D. Then the straight lines C A,
C D, and C E being drawn, Sir I s a a c N e w t o n proves, that the and
triangular spaces C A D and C D E are equal. This he does in the following
manner.
32. L e t E F be drawn parallel to C D. Then, from what has been said
upon the second law of motion[92], it is evident, that since the body was
moving in the line A B, when it received the impulse in the direction D C; it
will have moved after that impulse through the line D E in the same time, as
it would have taken up in moving through D F, provided it had received no
disturbance in D. But the time of the body’s moving from D to E is
supposed to be equal to the time of its moving through A D; therefore the
time, which the body would have imployed in moving through D F, had it
not been disturbed in D, is equal to the time, wherein it moved through A D:
consequently D F is equal in length to A D; for if the body had gone on to
move through the line A B without interruption, it would have moved
through all parts thereof with the same velocity, and have passed over equal
parts of that line in equal portions of time. Now C F being drawn, since A D
and D F are equal, the triangular space C D F is equal to the triangular
space C A D. Farther, the line E F being parallel to C D, it is proved by
Euclid, that the triangle C E D is equal to the triangle C F D[93]: therefore
the triangle C E D is equal to the triangle C A D.
33. A f t e r the same manner, if the body receive at E another impulse
directed toward the point C, and be turned by that impulse into the line E G;
if it move afterwards from E to G in the same space of time, as was taken
up by its motion from D to E, or from A to D; then C G being drawn, the
triangle C E G is equal to C D E. A third impulse at G directed as the two
former to C, whereby the body shall be turned into the line G H, will have
also the like effect with the rest. If the body move over G H in the same
time, as it took up in moving over E G, the triangle C G H will be equal to
the triangle C E G. Lastly, if the body at H be turned by a fresh impulse
directed toward C into the line H I, and at I by another impulse directed also
to C be turned into the line I K; and if the body move over each of the lines
H I, and I K in the same time, as it imployed in moving over each of the
preceding lines A D, D E, E G, and G H: then each of the triangles C H I,
and C I K will be equal to each of the preceding. Likewise as the time, in

which the body moves over A D E, is equal to the time of its moving over
E G H, and to the time of its moving over H I K; the space C A D E will be
equal to the space C E G H, and to the space C H I K. In the same manner
as the time, in which the body moved over A D E G is equal to the time of
its moving over G H I K, so the space C A D E G will be equal to the space
C G H I K.
34. F r o m this principle Sir I s a a c N e w t o n demonstrates the
proposition mentioned above, by that method of arguing introduced by him
into geometry, whereof we have before taken notice[94], by making
according to the principles of that method a transition from this incurvated
figure composed of straight lines, to a figure of continued curvature; and by
shewing, that since equal spaces are described in equal times in this present
figure composed of straight lines, the same relation between the spaces
described and the times of their description will also have place in a figure
of one continued curvature. He also deduces from this proposition the
reverse of it; and proves, that whenever equal spaces are continually
described; the body is acted upon by a centripetal force directed to the
center, at which the spaces terminate.

Chap. IV.
Of the RESISTANCE of FLUIDS.

B
E F O R E the cause can be discovered, which keeps the planets in
motion, it is necessary first to know, whether the space, wherein
they move, is empty and void, or filled with any quantity of matter.
It has been a prevailing opinion, that all space contains in it matter
of some kind or other; so that where no sensible matter is found, there was
yet a subtle fluid substance by which the space was filled up; even so as to
make an absolute plenitude. In order to examine this opinion, Sir I s a a c
N e w t o n has largely considered the effects of fluids upon bodies moving
in them.
2. T h e s e effects he has reduced under these three heads. In the first
place he shews how to determine in what manner the resistance, which
bodies suffer, when moving in a fluid, gradually increases in proportion to
the space, they describe in any fluid; to the velocity, with which they
describe it; and to the time they have been in motion. Under the second
head he considers what degree of resistance different bodies moving in the
same fluid undergo, according to the different proportion between the
density of the fluid and the density of the body. The densities of bodies,
whether fluid or solid, are measured by the quantity of matter, which is
comprehended under the same magnitude; that body being the most dense
or compact, which under the same bulk contains the greatest quantity of
solid matter, or which weighs most, the weight of every body being
observed above to be proportional to the quantity of matter in it[95]. Thus
water is more dense than cork or wood, iron more dense than water, and
gold than iron. The third particular Sir I s . N e w t o n considers concerning

the resistance of fluids is the influence, which the diversity of figure in the
resisted body has upon its resistance.
3. F o r the more perfect illustration of the first of these heads, he
distinctly shews the relation between all the particulars specified upon three
different suppositions. The first is, that the same body be resisted more or
less in the simple proportion to its velocity; so that if its velocity be
doubled, its resistance shall become threefold. The second is of the
resistance increasing in the duplicate proportion of the velocity; so that, if
the velocity of a body be doubled, its resistance shall be rendered four
times; and if the velocity be trebled, nine times as great as at first. But what
is to be understood by duplicate proportion has been already explained[96].
The third supposition is, that the resistance increases partly in the single
proportion of the velocity, and partly in the duplicate proportion thereof.
4. I n all these suppositions, bodies are considered under two respects,
either as moving, and opposing themselves against the fluid by that power
alone, which is essential to them, of resisting to the change of their state
from rest to motion, or from motion to rest, which we have above called
their power of inactivity; or else, as descending or ascending, and so having
the power of gravity combined with that other power. Thus our author has
shewn in all those three suppositions, in what manner bodies are resisted in
an uniform fluid, when they move with the aforesaid progressive
motion[97]; and what the resistance is, when they ascend or descend
perpendicularly[98]. And if a body ascend or descend obliquely, and the
resistance be singly proportional to the velocity, it is shewn how the body is
resisted in a fluid of an uniform density, and what line it will describe[99],
which is determined by the measurement of the hyperbola, and appears to
be no other than that line, first considered in particular by Dr.
B a r r o w[100], which is now commonly known by the name of the
logarithmical curve. In the supposition that the resistance increases in the
duplicate proportion of the velocity, our author has not given us the line
which would be described in an uniform fluid; but has instead thereof
discussed a problem, which is in some sort the reverse; to find the density
of the fluid at all altitudes, by which any given curve line may be described;
which problem is so treated by him, as to be applicable to any kind of
resistance whatever[101]. But here not unmindful of practice, he shews that a
body in a fluid of uniform density, like the air, will describe a line, which

approaches towards an hyperbola; that is, its motion will be nearer to that
curve line than to the parabola. And consequent upon this remark, he shews
how to determine this hyperbola by experiment, and briefly resolves the
chief of those problems relating to projectiles, which are in use in the art of
gunnery, in this curve[102]; as T o r r i c e l l i and others have done in the
parabola[103], whose inventions have been explained at large above[104].
5. O u r author has also handled distinctly that particular sort of motion,
which is described by pendulums[105]; and has likewise considered some
few cases of bodies moving in resisting fluids round a center, to which they
are impelled by a centripetal force, in order to give an idea of those kinds of
motions[106].
6. T h e treating of the resistance of pendulums has given him an
opportunity of inserting into another part of his work some speculations
upon the motions of them without resistance, which have a very peculiar
elegance; where in he treats of them as moved by a gravitation acting in the
law, which he shews to belong to the earth below its surface[107];
performing in this kind of gravitation, where the force is proportional to the
distance from the center, all that Huygens had before done in the common
supposition of its being uniform, and acting in parallel lines[108].
7. H u y g e n s at the end of his treatise of the cause of gravity[109]
informs us, that he likewise had carried his speculations on the first of these
suppositions, of the resistance in fluids being proportional to the velocity of
the body, as far as our author. But finding by experiment that the second
was more conformable to nature, he afterwards made some progress in that,
till he was stopt, by not being able to execute to his wish what related to the
perpendicular descent of bodies; not observing that the measurement of the
curve line, he made use of to explain it by, depended on the hyperbola.
Which oversight may well be pardoned in that great man, considering that
our author had not been pleased at that time to communicate to the publick
his admirable discourse of the quadrature or measurement of curve lines,
with which he has since obliged the world: for without the use of that
treatise, it is I think no injury even to our author’s unparalleled abilities to
believe, it would not have been easy for himself to have succeeded so
happily in this and many other parts of his writings.
8. W h a t H u y g e n s found by experiment, that bodies were in reality
resisted in the duplicate proportion of their velocity, agrees with the

reasoning of our author[110], who distinguishes the resistance, which fluids
give to bodies by the tenacity of their parts, and the friction between them
and the body, from that, which arises from the power of inactivity, with
which the constituent particles of fluids are endued like all other portions of
matter, by which power the particles of fluids like other bodies make
resistance against being put into motion.
9. T h e resistance, which arises from the friction of the body against the
parts of the fluid, must be very inconsiderable; and the resistance, which
follows from the tenacity of the parts of fluids, is not usually very great, and
does not depend much upon the velocity of the body in the fluid; for as the
parts of the fluid adhere together with a certain degree of force, the
resistance, which the body receives from thence, cannot much depend upon
the velocity, with which the body moves; but like the power of gravity, its
effect must be proportional to the time of its acting. This the reader may
find farther explained by Sir I s a a c N e w t o n himself in the postscript to
a discourse published by me in the philosophical transactions, No 371.
The principal resistance, which most fluids give to bodies, arises from the
power of inactivity in the parts of the fluids, and this depends upon the
velocity, with which the body moves, on a double account. In the first place,
the quantity of the fluid moved out of place by the moving body in any
determinate space of time is proportional to the velocity, wherewith the
body moves; and in the next place, the velocity with which each particle of
the fluid is moved, will also be proportional to the velocity of the body:
therefore since the resistance, which any body makes against being put into
motion, is proportional both to the quantity of matter moved and the
velocity it is moved with; the resistance, which a fluid gives on this
account, will be doubly increased with the increase of the velocity in the
moving body; that is, the resistance will be in a two-fold or duplicate
proportion of the velocity, wherewith the body moves through the fluid.
10. F a r t h e r it is most manifest, that this latter kind of resistance
increasing with the increase of velocity, even in a greater degree than the
velocity it self increases, the swifter the body moves, the less proportion the
other species of resistance will bear to this: nay that this part of the
resistance may be so much augmented by a due increase of velocity, till the
former resistances shall bear a less proportion to this, than any that might be
assigned. And indeed experience shews, that no other resistance, than what

arises from the power of inactivity in the parts of the fluid, is of moment,
when the body moves with any considerable swiftness.
11. T h e r e is besides these yet another species of resistance, found
only in such fluids, as, like our air, are elastic. Elasticity belongs to no fluid
known to us beside the air. By this property any quantity of air may be
contracted into a less space by a forcible pressure, and as soon as the
compressing power is removed, it will spring out again to its former
dimensions. The air we breath is held to its present density by the weight of
the air above us. And as this incumbent weight, by the motion of the winds,
or other causes, is frequently varied (which appears by the barometer;) so
when this weight is greatest, we breath a more dense air than at other times.
To what degree the air would expand it self by its spring, if all pressure
were removed, is not known, nor yet into how narrow a compass it is
capable of being compressed. Mr. Boyle found it by experiment capable
both of expansion and compression to such a degree, that he could cause a
quantity of air to expand it self over a space some hundred thousand times
greater, than the space to which he could confine the same quantity[111]. But
I shall treat more fully of this spring in the air hereafter[112]. I am now only
to consider what resistance to the motion of bodies arises from it.
12. B u t before our author shews in what manner this cause of
resistance operates, he proposes a method, by which fluids may be rendered
elastic, demonstrating that if their particles be provided with a power of
repelling each other, which shall exert it self with degrees of strength
reciprocally proportional to the distances between the centers of the
particles; that then such fluids will observe the same rule in being
compressed, as our air does, which is this, that the space, into which it
yields upon compression, is reciprocally proportional to the compressing
weight[113]. The term reciprocally proportional has been explained
above[114]. And if the centrifugal force of the particles acted by other laws,
such fluids would yield in a different manner to compression[115].
13. W h e t h e r the particles of the air be endued with such a power, by
which they can act upon each other out of contact, our author does not
determine, but leaves that to future examination, and to be discussed by
philosophers. Only he takes occasion from hence to consider the resistance
in elastic fluids, under this notion; making remarks, as he passes along,
upon the differences, which will arise, if their elasticity be derived from any

other fountain[116]. And this, I think, must be confessed to be done by him
with great judgment; for this is far the most reasonable account, which has
been given of this surprizing power, as must without doubt be freely
acknowledged by any one, who in the least considers the insufficiency of all
the other conjectures, which have been framed; and also how little reason
there is to deny to bodies other powers, by which they may act upon each
other at a distance, as well as that of gravity; which we shall hereafter shew
to be a property universally belonging to all the bodies of the universe, and
to all their parts[117]. Nay we actually find in the loadstone a very apparent
repelling, as well as an attractive power. But of this more in the conclusion
of this discourse.
14. B y these steps our author leads the way to explain the resistance,
which the air and such like fluids will give to bodies by their elasticity;
which resistance he explains thus. If the elastic power of the fluid were to
be varied so, as to be always in the duplicate proportion of the velocity of
the resisted body, it is shewn that then the resistance derived from the
elasticity, would increase in the duplicate proportion of the velocity; in so
much that the whole resistance would be in that proportion, excepting only
that small part, which arises from the friction between the body and the
parts of the fluid. From whence it follows, that because the elastic power of
the same fluid does in truth continue the same, if the velocity of the moving
body be diminished, the resistance from the elasticity, and therefore the
whole resistance, will decrease in a less proportion, than the duplicate of the
velocity; and if the velocity be increased, the resistance from the elasticity
will increase in a less proportion, than the duplicate of the velocity, that is
in a less proportion, than the resistance made by the power of inactivity of
the parts of the fluid. And from this foundation is raised the proof of a
property of this resistance, given by the elasticity in common with the
others from the tenacity and friction of the parts of the fluid; that the
velocity may be increased, till this resistance from the fluid’s elasticity shall
bear no considerable proportion to that, which is produced by the power of
inactivity thereof[118]. From whence our author draws this conclusion; that
the resistance of a body, which moves very swiftly in an elastic fluid, is
near the same, as if the fluid were not elastic; provided the elasticity arises
from the centrifugal power of the parts of the medium, as before explained,
especially if the velocity be so great, that this centrifugal power shall want

time to exert it self[119]. But it is to be observed, that in the proof of all this
our author proceeds upon the supposition of this centrifugal power in the
parts of the fluid; but if the elasticity be caused by the expansion of the
parts in the manner of wool compressed, and such like bodies, by which the
parts of the fluid will be in some measure entangled together, and their
motion be obstructed, the fluid will be in a manner tenacious, and give a
resistance upon that account over and above what depends upon its
elasticity only[120]; and the resistance derived from that cause is to be
judged of in the manner before set down.
15. I t is now time to pass to the second part of this theory; which is to
assign the measure of resistance, according to the proportion between the
density of the body and the density of the fluid. What is here to be
understood by the word density has been explained above[121]. For this
purpose as our author before considered two distinct cases of bodies
moving in mediums; one when they opposed themselves to the fluid by
their power of inactivity only, and another when by ascending or
descending their weight was combined with that other power: so likewise,
the fluids themselves are to be regarded under a double capacity; either as
having their parts at rest, and disposed freely without restraint, or as being
compressed together by their own weight, or any other cause.
16. I n the first case, if the parts of the fluid be wholly disingaged from
one another, so that each particle is at liberty to move all ways without any
impediment, it is shewn, that if a globe move in such a fluid, and the globe
and particles of the fluid are endued with perfect elasticity; so that as the
globe impinges upon the particles of it, they shall bound off and separate
themselves from the globe, with the same velocity, with which the globe
strikes upon them; then the resistance, which the globe moving with any
known velocity suffers, is to be thus determined. From the velocity of the
globe, the time, wherein it would move over two third parts of its own
diameter with that velocity, will be known. And such proportion as the
density of the fluid bears to the density of the globe, the same the resistance
given to the globe will bear to the force, which acting, like the power of
gravity, on the globe without intermission during the space of time now
mentioned, would generate in the globe the same degree of motion, as that
wherewith it moves in the fluid[122]. But if neither the globe nor the
particles of the fluid be elastic, so that the particles, when the globe strikes

against them, do not rebound from it, then the resistance will be but half so
much[123]. Again, if the particles of the fluid and the globe are imperfectly
elastic, so that the particles will spring from the globe with part only of that
velocity wherewith the globe impinges upon them; then the resistance will
be a mean between the two preceding cases, approaching nearer to the first
or second, according as the elasticity is more or less[124].
17. T h e elasticity, which is here ascribed to the particles of the fluid, is
not that power of repelling one another, when out of contact, by which, as
has before been mentioned, the whole fluid may be rendred elastic; but such
an elasticity only, as many solid bodies have of recovering their figure,
whenever any forcible change is made in it, by the impulse of another body
or otherwise. Which elasticity has been explained above at large[125].
18. T h i s is the case of discontinued fluids, where the body, by pressing
against their particles, drives them before itself, while the space behind the
body is left empty. But in fluids which are compressed, so that the parts of
them removed out of place by the body resisted immediately retire behind
the body, and fill that space, which in the other case is left vacant, the
resistance is still less; for a globe in such a fluid which shall be free from all
elasticity, will be resisted but half as much as the least resistance in the
former case[126]. But by elasticity I now mean that power, which renders the
whole fluid so; of which if the compressed fluid be possessed, in the
manner of the air, then the resistance will be greater than by the foregoing
rule; for the fluid being capable in some degree of condensation, it will
resemble so far the case of uncompressed fluids[127]. But, as has been
before related, this difference is most considerable in slow motions.
19. I n the next place our author is particular in determining the degrees
of resistance accompanying bodies of different figures; which is the last of
the three heads, we divided the whole discourse of resistance into. And in
this disquisition he finds a very surprizing and unthought of difference,
between free and compressed fluids. He proves, that in the former kind, a
globe suffers but half the resistance, which the cylinder, that circumscribes
the globe, will do, if it move in the direction of its axis[128]. But in the latter
he proves, that the globe and cylinder are resisted alike[129]. And in general,
that let the shape of bodies be ever so different, yet if the greatest sections
of the bodies perpendicular to the axis of their motion be equal, the bodies
will be resisted equally[130].

20. P u r s u a n t to the difference found between the resistance of the
globe and cylinder in rare and uncompressed fluids, our author gives us the
result of some other inquiries of the same nature. Thus of all the frustums of
a cone, that can be described upon the same base and with the same
altitude, he shews how to find that, which of all others will be the least
resisted, when moving in the direction of its axis[131]. And from hence he
draws an easy method of altering the figure of any spheroidical solid, so
that its capacity may be enlarged, and yet the resistance of it
diminished[132]: a note which he thinks may not be useless to ship-wrights.
He concludes with determining the solid, which will be resisted the least
that is possible, in these discontinued fluids[133].
21. T h a t I may here be understood by readers unacquainted with
mathematical terms, I shall explain what I mean by a frustum of a cone, and
a spheroidical solid. A cone has been defined above. A frustum is what
remains, when part of the cone next the vertex is cut away by a section
parallel to the base of the cone, as in fig. 86. A spheroid is produced from an
ellipsis, as a sphere or globe is made from a circle. If a circle turn round on
its diameter, it describes by its motion a sphere; so if an ellipsis (which
figure has been defined above, and will be more fully explained
hereafter[134]) be turned round either upon the longest or shortest line, that
can be drawn through the middle of it, there will be described a kind of
oblong or flat sphere, as in fig. 87. Both these figures are called spheroids,
and any solid resembling these I here call spheroidical.
22. I f it should be asked, how the method of altering spheroidical
bodies, here mentioned, can contribute to the facilitating a ship’s motion,
when I just above affirmed, that the figure of bodies, which move in a
compressed fluid not elastic, has no relation to the augmentation or
diminution of the resistance; the reply is, that what was there spoken relates
to bodies deep immerged into such fluids, but not of those, which swim
upon the surface of them; for in this latter case the fluid, by the appulse of
the anterior parts of the body, is raised above the level of the surface, and
behind the body is sunk somewhat below; so that by this inequality in the
superficies of the fluid, that part of it, which at the head of the body is
higher than the fluid behind, will resist in some measure after the manner of
discontinued fluids[135], analogous to what was before observed to happen
in the air through its elasticity, though the body be surrounded on every side

by it[136]. And as far as the power of these causes extends, the figure of the
moving body affects its resistance; for it is evident, that the figure, which
presses least directly against the parts of the fluid, and so raises least the
surface of a fluid not elastic, and least compresses one that is elastic, will be
least resisted.
23. T h e way of collecting the difference of the resistance in rare fluids,
which arises from the diversity of figure, is by considering the different
effect of the particles of the fluid upon the body moving against them,
according to the different obliquity of the several parts of the body upon
which they respectively strike; as it is known, that any body impinging
against a plane obliquely, strikes with a less force, than if it fell upon it
perpendicularly; and the greater the obliquity is, the weaker is the force.
And it is the same thing, if the body be at rest, and the plane move against
it[137].
24. T h a t there is no connexion between the figure of a body and its
resistance in compressed fluids, is proved thus. Suppose A B C D (in fig.
88.) to be a canal, having such a fluid, water for instance, running through it
with an equable velocity; and let any body E, by being placed in the axis of
the canal, hinder the passage of the water. It is evident, that the figure of the
fore part of this body will have little influence in obstructing the water’s
motion, but the whole impediment will arise from the space taken up by the
body, by which it diminishes the bore of the canal, and straightens the
passage of the water[138]. But proportional to the obstruction of the water’s
motion, will be the force of the water upon the body E[139]. Now suppose
both orifices of the canal to be closed, and the water in it to remain at rest;
the body E to move, so that the parts of the water may pass by it with the
same degree of velocity, as they did before; it is beyond contradiction, that
the pressure of the water upon the body, that is, the resistance it gives to its
motion, will remain the same; and therefore will have little connexion with
the figure of the body[140].
25. B y a method of reasoning drawn from the same fountain is
determined the measure of resistance these compressed fluids give to
bodies, in reference to the proportion between the density of the body and
that of the fluid. This shall be explained particularly in my comment on Sir
I s . N e w t o n’s mathematical principles of natural philosophy; but is not a
proper subject to be insisted on farther in this place.

26. W e have now gone through all the parts of this theory. There
remains nothing more, but in few words to mention the experiments, which
our author has made, both with bodies falling perpendicularly through
water, and the air[141], and with pendulums[142]: all which agree with the
theory. In the case of falling bodies, the times of their fall determined by the
theory come out the same, as by observation, to a surprizing exactness; in
the pendulums, the rod, by which the ball of the pendulum hangs, suffers
resistance as well as the ball, and the motion of the ball being reciprocal, it
communicates such a motion to the fluid, as increases the resistance, but the
deviation from the theory is no more, than what may reasonably follow
from these causes.
27. B y this theory of the resistance of fluids, and these experiments, our
author decides the question so long agitated among natural philosophers,
whether all space is absolutely full of matter. The Aristotelians and
Cartesians both assert this plenitude; the Atomists have maintained the
contrary. Our author has chose to determine this question by his theory of
resistance, as shall be explained in the following chapter.

B O O K I I.
Concerning the

SYSTEM of the WORLD.

Chap. I.
That the Planets move in a space empty of all
sensible matter.
I HAVE now gone through the first part of my design, and have
explained, as far as the nature of my undertaking would permit,
what Sir I s a a c N e w t o n has delivered in general
concerning the motion of bodies. It follows now to speak of the
discoveries, he has made in the system of the world; and to shew from him
what cause keeps the heavenly bodies in their courses. But it will be
necessary for the use of such, as are not skilled in astronomy, to premise a
brief description of the planetary system.
2. T h i s system is disposed in the following manner. In the middle is
placed the sun. About him six globes continually roll. These are the primary
planets; that which is nearest to the sun is called Mercury, the next Venus,
next to this is our earth, the next beyond is Mars, after him Jupiter, and the
outermost of all Saturn. Besides these there are discovered in this system
ten other bodies, which move about some of these primary planets in the
same manner, as they move round the sun. These are called secondary
planets. The most conspicuous of them is the moon, which moves round our
earth; four bodies move in like manner round Jupiter; and five round
Saturn. Those which move about Jupiter and Saturn, are usually called
satellites; and cannot any of them be seen without a telescope. It is not
impossible, but there may be more secondary planets, beside these; though
our instruments have not yet discovered any other. This disposition of the
planetary or solar system is represented in fig. 89.
3. T h e same planet is not always equally distant from the sun. But the
middle distance of Mercury is between ⅕ and ⅖ of the distance of the earth
from the sun; Venus is distant from the sun almost ¾ of the distance of the
earth; the middle distance of Mars is something more than half as much
again, as the distance of the earth; Jupiter’s middle distance exceeds five
times the distance of the earth, by between ⅕ and 1/6 part of this distance;
Saturn’s middle distance is scarce more than 9½ times the distance between

the earth and sun; but the middle distance between the earth and sun is
about 217⅛ times the sun’s semidiameter.

4. A l l these planets move one way, from west to east; and of the
primary planets the most remote is longest in finishing its course round the
sun. The period of Saturn falls short only sixteen days of 29 years and a
half. The period of Jupiter is twelve years wanting about 50 days. The
period of Mars falls short of two years by about 43 days. The revolution of
the earth constitutes the year. Venus performs her period in about 224½
days, and Mercury in about 88 days.
5. T h e course of each planet lies throughout in one plane or flat
surface, in which the sun is placed; but they do not all move in the same

plane, though the different planes, in which they move, cross each other in
very small angles. They all cross each other in lines, which pass through the
sun; because the sun lies in the plane of each orbit. This inclination of the
several orbits to each other is represented in fig. 90. The line, in which the
plane of any orbit crosses the plane of the earth’s motion, is called the line
of the nodes of that orbit.
6. E a c h planet moves round the sun in the line, which we have
mentioned above[143] under the name of ellipsis; which I shall here shew
more particularly how to describe. I have there said how it is produced in
the cone. I shall now shew how to form it upon a plane. Fix upon any plane
two pins, as at A and B in fig. 91. To these tye a string A C B of any length.
Then apply a third pin D so to the string, as to hold it strained; and in that
manner carrying this pin about, the point of it will describe an ellipsis. If
through the points A, B the straight line E A B F be drawn, to be terminated
at the ellipsis in the points E and F, this is the longest line of any, that can be
drawn within the figure, and is called the greater axis of the ellipsis. The
line G H, drawn perpendicular to this axis E F, so as to pass through the
middle of it, is called the lesser axis. The two points A and B are called
focus’s. Now each planet moves round the sun in a line of this kind, so that
the sun is found in one focus. Suppose A to be the place of the sun. Then E
is the point, wherein the planet will be nearest of all to the sun, and at F it
will be most remote. The point E is called the perihelion of the planet, and F
the aphelion. In G and H the planet is said to be in its middle or mean
distance; because the distance A G or A H is truly the middle between A E
the least, and A F the greatest distance. In fig. 92. is represented how the
greater axis of each orbit is situated in respect of the rest. The proportion
between the greatest and least distances of the planet from the sun is very
different in the different planets.

In Saturn the proportion of the greatest distance to the least is something
less, than the proportion of 9 to 8, but much nearer to this, than to the
proportion of 10 to 9. In Jupiter this proportion is a little greater, than that
of 11 to 10. In Mars it exceeds the proportion of 6 to 5. In the earth it is
about the proportion of 30 to 29. In Venus it is near to that of 70 to 69. And
in Mercury it comes not a great deal short of the proportion of 3 to 2.
7. E a c h of these planets so moves through its ellipsis, that the line
drawn from the sun to the planet, by accompanying the planet in its motion,
will describe about the sun equal spaces in equal times, after the manner
spoke of in the chapter of centripetal forces[144]. There is also a certain
relation between the greater axis’s of these ellipsis’s, and the times, in
which the planets perform their revolutions through them. Which relation
may be expressed thus. Let the period of one planet be denoted by the letter

A, the greater axis of its orbit by D; let the period of another
planet be denoted by B, and the greater axis of this planet’s
orbit by E. Then if C be taken to bear the same proportion to
B, as B bears to A; likewise if F be taken to bear the same
proportion to E, as E bears to D; and G taken to bear the
same proportion likewise to F, as E bears to D; then A shall
bear the same proportion to C, as D bears to G.
8. T h e secondary planets move round their respective
primary, much in the same manner as the primary do round
the sun. But the motions of these shall be more fully explained
hereafter[145]. And there is, besides the planets, another sort of bodies,
which in all probability move round the sun; I mean the comets. The farther
description of which bodies I also leave to the place, where they are to be
particularly treated on[146].
9. F a r without this system the fixed stars are placed. These are all so
remote from us, that we seem almost incapable of contriving any means to
estimate their distance. Their number is exceeding great. Besides two or
three thousand, which we see with the naked eye, telescopes open to our
view vast numbers; and the farther improved these instruments are, we still
discover more and more. Without doubt these are luminous globes, like our
sun, and ranged through the wide extent of space; each of which, it is to be
supposed, perform the same office, as our sun, affording light and heat to
certain planets moving about them. But these conjectures are not to be
pursued in this place.
10. I s h a l l therefore now proceed to the particular design of this
chapter, and shew, that there is no sensible matter lodged in the space where
the planets move.
11. T h a t they suffer no sensible resistance from any such matter, is
evident from the agreement between the observations of astronomers in
different ages, with regard to the time, in which the planets have been found
to perform their periods. But it was the opinion of Des Cartes[147], that the
planets might be kept in their courses by the means of a fluid matter, which
continually circulating round should carry the planets along with it. There is
one appearance that may seem to favour this opinion; which is, that the sun
turns round its own axis the same way, as the planets move. The earth also
turns round its axis the same way, as the moon moves round the earth. And

the planet Jupiter turns upon its axis the same way, as his satellites revolve
round him. It might therefore be supposed, that if the whole planetary
region were filled with a fluid matter, the sun, by turning round on its own
axis, might communicate motion first to that part of the fluid, which was
contiguous, and by degrees propagate the like motion to the parts more
remote. After the same manner the earth might communicate motion to this
fluid, to a distance sufficient to carry round the moon, and Jupiter
communicate the like to the distance of its satellites. Sir I s a a c
N e w t o n has particularly examined what might be the result of such a
motion as this[148]; and he finds, that the velocities, with which the parts of
this fluid will move in different distances from the center of the motion, will
not agree with the motion observed in different planets: for instance, that
the time of one intire circulation of the fluid, wherein Jupiter should swim,
would bear a greater proportion to the time of one intire circulation of the
fluid, where the earth is; than the period of Jupiter bears to the period of the
earth. But he also proves[149], that the planet cannot circulate in such a fluid,
so as to keep long in the same course, unless the planet and the contiguous
fluid are of the same density, and the planet be carried along with the same
degree of motion, as the fluid. There is also another remark made upon this
motion by our author; which is, that some vivifying force will be
continually necessary at the center of the motion[150]. The sun in particular,
by communicating motion to the ambient fluid, will lose from it self as
much motion, as it imparts to the fluid; unless some acting principle reside
in the sun to renew its motion continually. If the fluid be infinite, this
gradual loss of motion would continue till the whole should stop[151]; and if
the fluid were limited, this loss of motion would continue, till there would
remain no swifter a revolution in the sun, than in the utmost part of the
fluid; so that the whole would turn together about the axis of the sun, like
one solid globe[152].
12. I t is farther to be observed, that as the planets do not move in
perfect circles round the sun; there is a greater distance between their orbits
in some places, than in others. For instance, the distance between the orbit
of Mars and Venus is near half as great again in one part of their orbits, as in
the opposite place. Now here the fluid, in which the earth should swim,
must move with a less rapid motion, where there is this greater interval

between the contiguous orbits; but on the contrary, where the space is
straitest, the earth moves more slowly, than where it is widest[153].
13. F a r t h e r, if this our globe of earth swam in a fluid of equal
density with the earth it self, that is, in a fluid more dense than water; all
bodies put in motion here upon the earth’s surface must suffer a great
resistance from it; where as, by Sir I s a a c N e w t o n’s experiments
mentioned in the preceding chapter, bodies, that fell perpendicularly down
through the air, felt about 1/860 part only of the resistance, which bodies
suffered that fell in like manner through water.
14. Sir I s a a c N e w t o n applies these experiments yet farther, and
examines by them the general question concerning the absolute plenitude of
space. According to the Aristotelians, all space was full without any the
least vacuities whatever. DesCartes embraced the same opinion, and
therefore supposed a subtile fluid matter, which should pervade all bodies,
and adequately fill up their pores. The Atomical philosophers, who suppose
all bodies both fluid and solid to be composed of very minute but solid
atoms, assert that no fluid, how subtile soever the particles or atoms
whereof it is composed should be, can ever cause an absolute plenitude;
because it is impossible that any body can pass through the fluid without
putting the particles of it into such a motion, as to separate them, at least in
part, from one another, and so perpetually to cause small vacuities; by
which these Atomists endeavour to prove, that a vacuum, or some space
empty of all matter, is absolutely necessary to be in nature. Sir I s a a c
N e w t o n objects against the filling of space with such a subtile fluid, that
all bodies in motion must be unmeasurably resisted by a fluid so dense, as
absolutely to fill up all the space, through which it is spread. And lest it
should be thought, that this objection might be evaded by ascribing to this
fluid such very minute and smooth parts, as might remove all adhesion or
friction between them, whereby all resistance would be lost, which this
fluid might otherwise give to bodies moving in it; Sir I s a a c N e w t o n
proves, in the manner above related, that fluids resist from the power of
inactivity of their particles; and that water and the air resist almost entirely
on this account: so that in this subtile fluid, however minute and lubricated
the particles, which compose it, might be; yet if the whole fluid was as
dense as water, it would resist very near as much as water does; and
whereas such a fluid, whose parts are absolutely close together without any
intervening spaces, must be a great deal more dense than water, it must

resist more than water in proportion to its greater density; unless we will
suppose the matter, of which this fluid is composed, not to be endued with
the same degree of inactivity as other matter. But if you deprive any
substance of the property so universally belonging to all other matter,
without impropriety of speech it can scarce be called by this name.
15. Sir I s a a c N e w t o n made also an experiment to try in particular,
whether the internal parts of bodies suffered any resistance. And the result
did indeed appear to favour some small degree of resistance; but so very
little, as to leave it doubtful, whether the effect did not arise from some
other latent cause[154].

Chap. II.
Concerning the cause, which keeps in motion the
primary planets.

S
INCE the planets move in a void space and are free from resistance;
they, like all other bodies, when once in motion, would move on in a
straight line without end, if left to themselves. And it is now to be
explained what kind of action upon them carries them round the sun.
Here I shall treat of the primary planets only, and discourse of the
secondary apart in the next chapter. It has been just now declared, that these
primary planets move so about the sun, that a line extended from the sun to
the planet, will, by accompanying the planet in its motion, pass over equal
spaces in equal portions of time[155]. And this one property in the motion of
the planets proves, that they are continually acted on by a power directed
perpetually to the sun as a center. This therefore is one property of the
cause, which keeps the planets in their courses, that it is a centripetal power,
whose center is the sun.
2. A g a i n, in the chapter upon centripetal forces[156] it was observ’d,
that if the strength of the centripetal power was suitably accommodated
every where to the motion of any body round a center, the body might be
carried in any bent line whatever, whose concavity should be every where
turned towards the center of the force. It was farther remarked, that the
strength of the centripetal force, in each place, was to be collected from the
nature of the line, wherein the body moved[157]. Now since each planet
moves in an ellipsis, and the sun is placed in one focus; Sir I s a a c
N e w t o n deduces from hence, that the strength of this power is
reciprocally in the duplicate proportion of the distance from the sun. This is

deduced from the properties, which the geometers have discovered in the
ellipsis. The process of the reasoning is not proper to be enlarged upon here;
but I shall endeavour to explain what is meant by the reciprocal duplicate
proportion. Each of the terms reciprocal proportion, and duplicate
proportion, has been already defined[158]. Their sense when thus united is as
follows. Suppose the planet moved in the orbit A B C (in fig. 93.) about the
sun in S. Then, when it is said, that the centripetal power, which acts on the
planet in A, bears to the power acting on it in B a proportion, which is the
reciprocal of the duplicate proportion of the distance S A to the distance
S B; it is meant that the power in A bears to the power in B the duplicate of
the proportion of the distance S B to the distance S A. The reciprocal
duplicate proportion may be explained also by numbers as follows. Suppose
several distances to bear to each other proportions expressed by the
numbers 1, 2, 3, 4, 5; that is, let the second distance be double the first, the
third be three times, the fourth four times, and the fifth five times as great as
the first. Multiply each of these numbers by it self, and 1 multiplied by 1
produces still 1, 2 multiplied by 2 produces 4, 3 by 3 makes 9, 4 by 4 makes
16, and 5 by 5 gives 25. This being done, the fractions ¼, 1/9, 1/16, 1/25,
will respectively express the proportion, which the centripetal power in
each of the following distances bears to the power at the first distance: for
in the second distance, which is double the first, the centripetal power will
be one fourth part only of the power at the first distance; at the third
distance the power will be one ninth part only of the first power; at the
fourth distance, the power will be but one sixteenth part of the first; and at
the fifth distance, one twenty fifth part of the first power.
3. T h u s is found the proportion, in which this centripetal power
decreases, as the distance from the sun increases, within the compass of one
planet’s motion. How it comes to pass, that the planet can be carried about
the sun by this centripetal power in a continual round, sometimes rising
from the sun, then descending again as low, and from thence be carried up
again as far remote as before, alternately rising and falling without end;
appears from what has been written above concerning centripetal forces: for
the orbits of the planets resemble in shape the curve line proposed in § 17 of
the chapter on these forces[159].
4. B u t farther, in order to know whether this centripetal force extends
in the same proportion throughout, and consequently whether all the planets

are influenced by the very same power, our author proceeds thus. He
inquires what relation there ought to be between the periods of the different
planets, provided they were acted upon by the same power decreasing
throughout in the forementioned proportion; and he finds, that the period of
each in this case would have that very relation to the greater axis of its
orbit, as I have declared above[160] to be found in the planets by the
observations of astronomers. And this puts it beyond question, that the
different planets are pressed towards the sun, in the same proportion to their
distances, as one planet is in its several distances. And thence in the last
place it is justly concluded, that there is such a power acting towards the
sun in the foresaid proportion at all distances from it.
5. T h i s power, when referred to the planets, our author calls
centripetal, when to the sun attractive; he gives it likewise the name of
gravity, because he finds it to be of the same nature with that power of
gravity, which is observed in our earth, as will appear hereafter[161]. By all
these names he designs only to signify a power endued with the properties
before mentioned; but by no means would he have it understood, as if these
names referred any way to the cause of it. In particular in one place where
he uses the name of attraction, he cautions us expressly against implying
any thing but a power directing a body to a center without any reference to
the cause of it, whether residing in that center, or arising from any external
impulse[162].
6. B u t now, in these demonstrations some very minute inequalities in
the motion of the planets are neglected; which is done with a great deal of
judgment; for whatever be their cause, the effects are very inconsiderable,
they being so exceeding small, that some astronomers have thought fit
wholly to pass them by[163]. However the excellency of this philosophy,
when in the hands of so great a geometer as our author, is such, that it is
able to trace the least variations of things up to their causes. The only
inequalities, which have been observed common to all the planets, are the
motion of the aphelion and the nodes. The transverse axis of each orbit does
not always remain fixed, but moves about the sun with a very slow
progressive motion: nor do the planets keep constantly the same plane, but
change them, and the lines in which those planes intersect each other by
insensible degrees. The first of these inequalities, which is the motion of the
aphelion, may be accounted for, by supposing the gravitation of the planets

towards the sun to differ a little from the forementioned reciprocal duplicate
proportion of the distances; but the second, which is the motion of the
nodes, cannot be accounted for by any power directed towards the sun; for
no such can give the planet any lateral impulse to divert it from the plane of
its motion into any new plane, but of necessity must be derived from some
other center. Where that power is lodged, remains to be discovered. Now it
is proved, as shall be explained in the following chapter, that the three
primary planets Saturn, Jupiter, and the earth, which have satellites
revolving about them, are endued with a power of causing bodies, in
particular those satellites, to gravitate towards them with a force, which is
reciprocally in the duplicate proportion of their distances; and the planets
are in all respects, in which they come under our examination, so similar
and alike, that there is no reason to question, but they have all the same
property. Though it be sufficient for the present purpose to have it proved of
Jupiter and Saturn only; for these planets contain much greater quantities of
matter than the rest, and proportionally exceed the others in power[164]. But
the influence of these two planets being allowed, it is evident how the
planets come to shift continually their planes: for each of the planets
moving in a different plane, the action of Jupiter and Saturn upon the rest
will be oblique to the planes of their motion; and therefore will gradually
draw them into new ones. The same action of these two planets upon the
rest will cause likewise a progressive motion of the aphelion; so that there
will be no necessity of having recourse to the other cause for this motion,
which was before hinted at[165]; viz, the gravitation of the planets towards
the sun differing from the exact reciprocal duplicate proportion of the
distances. And in the last place, the action of Jupiter and Saturn upon each
other will produce in their motions the same inequalities, as their joint
action produces in the rest. All this is effected in the same manner, as the
sun produces the same kind of inequalities and many others in the motion of
the moon and the other secondary planets; and therefore will be best
apprehended by what shall be said in the next chapter. Those other
irregularities in the motion of the secondary planets have place likewise
here; but are too minute to be observable: because they are produced and
rectified alternately, for the most part in the time of a single revolution;
whereas the motion of the aphelion and nodes, which continually increase,
become sensible in a long series of years. Yet some of these other
inequalities are discernible in Jupiter and Saturn, in Saturn chiefly; for

when Jupiter, who moves faster than Saturn, approaches near to a
conjunction with him, his action upon Saturn will a little retard the motion
of that planet, and by the reciprocal action of Saturn he will himself be
accelerated. After conjunction, Jupiter will again accelerate Saturn, and be
likewise retarded in the same degree, as before the first was retarded and
the latter accelerated. Whatever inequalities besides are produced in the
motion of Saturn by the action of Jupiter upon that planet, will be
sufficiently rectified, by placing the focus of Saturn’s ellipsis, which should
otherwise be in the sun, in the common center of gravity of the sun and
Jupiter. And all the inequalities in the motion of Jupiter, caused by Saturn’s
action upon him, are much less considerable than the irregularities of
Saturn’s motion[166].
7. T h i s one principle therefore of the planets having a power, as well
as the sun, to cause bodies to gravitate towards them, which is proved by
the motion of the secondary planets to obtain in fact, explains all the
irregularities relating to the planets ever observed by astronomers.
8. Sir I s a a c N e w t o n after this proceeds to make an improvement
in astronomy by applying this theory to the farther correction of their
motions. For as we have here observed the planets to possess a principle of
gravitation, as well as the sun; so it will be explained at large hereafter, that
the third law of motion, which makes action and reaction equal, is to be
applied in this case[167]; and that the sun does not only attract each planet,
but is it self also attracted by them; the force, wherewith the planet is acted
on, bearing to the force, wherewith the sun it self is acted on at the same
time, the proportion, which the quantity of matter in the sun bears to the
quantity of matter in the planet. From the action between the sun and planet
being thus mutual Sir Isaac Newton proves that the sun and planet will
describe about their common center of gravity similar ellipsis’s; and then
that the transverse axis of the ellipsis described thus about the moveable
sun, will bear to the transverse axis of the ellipsis, which would be
described about the sun at rest in the same time, the same proportion as the
quantity of solid matter in the sun and planet together bears to the first of
two mean proportionals between this quantity and the quantity of matter in
the sun only[168].
9. A b o v e, where I shewed how to find a cube, that should bear any
proportion to another cube[169], the lines F T and T S are two mean

proportionals between E F and F G; and counting from E F, F T is called the
first, and F S the second of those means. In numbers these mean
proportionals are thus found.
Suppose A and B two numbers, and it be required to find
C the first, and D the second of the two mean proportionals
between them. First multiply A by it self, and the product
multiply by B; then C will be the number which in arithmetic
is called the cubic root of this last product; that is, the
number C being multiplied by it self, and the product again
multiplied by the same number C, will produce the product
above mentioned. In like manner D is the cubic root of the product of B
multiplied by it self, and the produce of that multiplication multiplied again
by A.
10. I t will be asked, perhaps, how this correction can be admitted,
when the cause of the motions of the planets was before found by supposing
the sun the center of the power, which acted upon them: for according to the
present correction this power appears rather to be directed to their common
center of gravity. But whereas the sun was at first concluded to be the
center, to which the power acting on the planets was directed, because the
spaces described round the sun in equal times were found to be equal; so Sir
I s a a c N e w t o n proves, that if the sun and planet move round their
common center of gravity, yet to an eye placed in the planet, the spaces,
which will appear to be described about the sun, will have the same relation
to the times of their description, as the real spaces would have, if the sun
were at rest[170]. I farther asserted, that, supposing the planets to move
round the sun at rest, and to be attracted by a power, which every where
should act with degrees of strength reciprocally in the duplicate proportion
of the distances; then the periods of the planets must observe the same
relation to their distances, as astronomers find them to do. But here it must
not be supposed, that the observations of astronomers absolutely agree
without any the least difference; and the present correction will not cause a
deviation from any one astronomer’s observations, so much as they differ
from one another. For in Jupiter, where this correction is greatest, it hardly
amounts to the 3000th part of the whole axis.
11. U p o n this head I think it not improper to mention a reflection made
by our excellent author upon these small inequalities in the planets motions;

which contains under it a very strong philosophical argument against the
eternity of the world. It is this, that these inequalities must continually
increase by slow degrees, till they render at length the present frame of
nature unfit for the purposes, it now serves[171]. And a more convincing
proof cannot be desired against the present constitution’s having existed
from eternity than this, that a certain period of years will bring it to an end.
I am aware this thought of our author has been represented even as impious,
and as no less than casting a reflection upon the wisdom of the author of
nature, for framing a perishable work. But I think so bold an assertion ought
to have been made with singular caution. For if this remark upon the
increasing irregularities of the heavenly motions be true in fact, as it really
is, the imputation must return upon the asserter, that this does detract from
the divine wisdom. Certainly we cannot pretend to know all the omniscient
Creator’s purposes in making this world, and therefore cannot undertake to
determine how long he designed it should last. And it is sufficient, if it
endure the time intended by the author. The body of every animal shews the
unlimited wisdom of its author no less, nay in many respects more, than the
larger frame of nature; and yet we see, they are all designed to last but a
small space of time.
12. T h e r e need nothing more be said of the primary planets; the
motions of the secondary shall be next considered.

Chap. III.
Of the motion of the MOON and the other
SECONDARY PLANETS.

T
HE excellency of this philosophy sufficiently appears from its
extending in the manner, which has been related, to the minutest
circumstances of the primary planets motions; which nevertheless
bears no proportion to the vast success of it in the motions of the
secondary; for it not only accounts for all the irregularities, by which their
motions were known to be disturbed, but has discovered others so
complicated, that astronomers were never able to distinguish them, and
reduce them under proper heads; but these were only to be found out from
their causes, which this philosophy has brought to light, and has shewn the
dependence of these inequalities upon such causes in so perfect a manner,
that we not only learn from thence in general, what those inequalities are,
but are able to compute the degree of them. Of this Sir I s . N e w t o n has
given several specimens, and has moreover found means to reduce the
moon’s motion so completely to rule, that he has framed a theory, from
which the place of that planet may at all times be computed, very nearly or
altogether as exactly, as the places of the primary planets themselves, which
is much beyond what the greatest astronomers could ever effect.
2. T h e first thing demonstrated of these secondary planets is, that they
are drawn towards their respective primary in the same manner as the
primary planets are attracted by the sun. That each secondary planet is kept
in its orbit by a power pointed towards the center of the primary planet,
about which the secondary revolves; and that the power, by which the
secondaries of the same primary are influenced, bears the same relation to
the distance from the primary, as the power, by which the primary planets

are guided, does in regard to the distance from the sun[172]. This is proved
in the satellites of Jupiter and Saturn, because they move in circles, as far as
we can observe, about their respective primary with an equable course, the
respective primary being the center of each orbit: and by comparing the
times, in which the different satellites of the same primary perform their
periods, they are found to observe the same relation to the distances from
their primary, as the primary planets observe in respect of their mean
distances from the sun[173]. Here these bodies moving in circles with an
equable motion, each satellite passes over equal parts of its orbit in equal
portions of time; consequently the line drawn from the center of the orbit,
that is, from the primary planet, to the satellite, will pass over equal spaces
along with the satellite in equal portions of time; which proves the power,
by which each satellite is held in its orbit, to be pointed towards the primary
as a center[174]. It is also manifest that the centripetal power, which carries a
body in a circle concentrical with the power, acts upon the body at all times
with the same strength. But Sir I s a a c N e w t o n demonstrates that, when
bodies are carried in different circles by centripetal powers directed to the
centers of those circles, then, the degrees of strength of those powers are to
be compared by considering the relation between the times, in which the
bodies perform their periods through those circles[175]; and in particular he
shews, that if the periodical times bear that relation, which I have just now
asserted the satellites of the same primary to observe; then the centripetal
powers are reciprocally in the duplicate proportion of the semidiameters of
the circles, or in that proportion to the distances of the bodies from the
centers[176]. Hence it follows that in the planets Jupiter and Saturn, the
centripetal power in each decreases with the increase of distance, in the
same proportion as the centripetal power appertaining to the sun decreases
with the increase of distance. I do not here mean that this proportion of the
centripetal powers holds between the power of Jupiter at any distance
compared with the power of Saturn at any other distance; but only in the
change of strength of the power belonging to the same planet at different
distances from him. Moreover what is here discovered of the planets Jupiter
and Saturn by means of the different satellites, which revolve round each of
them, appears in the earth by the moon alone; because she is found to move
round the earth in an ellipsis after the same manner as the primary planets
do about the sun; excepting only some small irregularities in her motion, the

cause of which will be particularly explained in what follows, whereby it
will appear, that they are no objection against the earth’s acting on the moon
in the same manner as the sun acts on the primary planets; that is, as the
other primary planets Jupiter and Saturn act upon their satellites. Certainly
since these irregularities can be otherwise accounted for, we ought not to
depart from that rule of induction so necessary in philosophy, that to like
bodies like properties are to be attributed, where no reason to the contrary
appears. We cannot therefore but ascribe to the earth the same kind of
action upon the moon, as the other primary planets Jupiter and Saturn have
upon their satellites; which is known to be very exactly in the proportion
assigned by the method of comparing the periodical times and distances of
all the satellites which move about the same planet; this abundantly
compensating our not being near enough to observe the exact figure of their
orbits. For if the little deviation of the moon’s orbit orbit from a true
permanent ellipsis arose from the action of the earth upon the moon not
being in the exact reciprocal duplicate proportion of the distance, were
another moon to revolve about the earth, the proportion between the
periodical times of this new moon, and the present, would discover the
deviation from the mentioned proportion much more manifestly.
3. B y the number of satellites, which move round Jupiter and Saturn,
the power of each of these planets is measured in a great diversity of
distance; for the distance of the outermost satellite in each of these planets
exceeds several times the distance of the innermost. In Jupiter the
astronomers have usually placed the innermost satellite at a distance from
the center of that planet equal to about 5⅔ of the semidiameters of Jupiter’s
body, and this satellite performs its revolution in about 1 day 18½ hours.
The next satellite, which revolves round Jupiter in about 3 days 13⅕ hours,
they place at the distance from Jupiter of about 9 of that planet’s
semidiameters. To the third satellite, which performs its period nearly in 7
days 3¾ hours, they assign the distance of about 14⅖ semidiameters. But
the outermost satellite they remove to 25⅓ semidiameters, and this satellite
makes its period in about 16 days 16½ hours[177]. In Saturn there is still a
greater diversity in the distance of the several satellites. By the observations
of the late C a s s i n i, a celebrated astronomer in France, who first
discovered all these satellites, except one known before, the innermost is
distant about 4½ of Saturn’s semidiameters from his center, and revolves

round in about 1 day 21⅓ hours. The next satellite is distant about 5¾
semidiameters, and makes its period in about 2 days 17⅔ hours. The third is
removed to the distance of about 8 semidiameters, and performs its
revolution in near 4 days 12½ hours. The fourth satellite discovered first by
the great Huygens, is near 18⅔ semidiameters, and moves round Saturn in
about 15 days 22⅔ hours. The outermost is distant 56 semidiameters, and
makes its revolution in about 79 days 7 ⅘ hours[178]. Besides these
satellites, there belongs to the planet Saturn another body of a very singular
kind. This is a shining, broad, and flat ring, which encompasses the planet
round. The diameter of the outermost verge of this ring is more than double
the diameter of Saturn. H u y g e n s, who first described this ring, makes the
whole diameter thereof to bear to the diameter of Saturn the proportion of 9
to 4. The late reverend Mr. Pound makes the proportion something greater,
viz. that of 7 to 3. The distances of the satellites of this planet Saturn are
compared by C a s s i n i to the diameter of the ring. His numbers I have
reduced to those above, according to Mr. Pound’s proportion between the
diameters of Saturn and of his ring. As this ring appears to adhere no where
to Saturn, so the distance of Saturn from the inner edge of the ring seems
rather greater than the breadth of the ring. The distances, which have here
been given, of the several satellites, both for Jupiter and Saturn, may be
more depended on in relation to the proportion, which those belonging to
the same primary planet bear one to another, than in respect to the very
numbers, that have been here set down, by reason of the difficulty there is
in measuring to the greatest exactness the diameters of the primary planets;
as will be explained hereafter, when we come to treat of telescopes[179]. By
the observations of the forementioned Mr. Pound, in Jupiter the distance of
the innermost satellite should rather be about 6 semidiameters, of the
second 9-½, of the third 15, and of the outermost 26⅔[180]; and in Saturn the
distance of the innermost satellite 4 semidiameters, of the next 6¼, of the
third 8¾, of the fourth 20⅓, and of the fifth 59[181]. However the proportion
between the distances of the satellites in the same primary is the only thing
necessary to the point we are here upon.
4. B u t moreover the force, wherewith the earth acts in different
distances, is confirmed from the following consideration, yet more expresly
than by the preceding analogical reasoning. It will appear, that if the power
of the earth, by which it retains the moon in her orbit, be supposed to act at

all distances between the earth and moon, according to the forementioned
rule; this power will be sufficient to produce upon bodies, near the surface
of the earth, all the effects ascribed to the principle of gravity. This is
discovered by the following method. Let A (in fig. 94.) represent the earth,
B the moon, B C D the moon’s orbit, which differs little from a circle, of
which A is the center. If the moon in B were left to it self to move with the
velocity, it has in the point B, it would leave the orbit, and proceed right
forward in the line B E, which touches the orbit in B. Suppose the moon
would upon this condition move from B to E in the space of one minute of
time. By the action of the earth upon the moon, whereby it is retained in its
orbit, the moon will really be found at the end of this minute in the point F,
from whence a straight line drawn to A shall make the space B F A in the
circle equal to the triangular space B E A; so that the moon in the time
wherein it would have moved from B to E, if left to it self, has been
impelled towards the earth from E to F. And when the time of the moon’s
passing from B to F is small, as here it is only one minute, the distance
between E and F scarce differs from the space, through which the moon
would descend in the same time, if it were to fall directly down from B
toward A without any other motion. A B the distance of the earth and moon
is about 60 of the earth’s semidiameters, and the moon completes her
revolution round the earth in about 27 days 7 hours and 43 minutes:
therefore the space E F will here be found by computation to be about 16⅛
feet. Consequently, if the power, by which the moon is retained in its orbit,
be near the surface of the earth greater, than at the distance of the moon in
the duplicate proportion of that distance; the number of feet, a body would
descend near the surface of the earth by the action of this power upon it in
one minute of time, would be equal to 16⅛ multiplied twice into the
number 60, that is, equal to 58050. But how fast bodies fall near the surface
of the earth may be known by the pendulum[182]; and by the exactest
experiments they are found to descend the space of 16⅛ feet in a second of
time; and the spaces described by falling bodies being in the duplicate
proportion of the times of their fall[183], the number of feet, a body would
describe in its fall near the surface of the earth in one minute of time, will
be equal to 16⅛ twice multiplied by 60, the same as would be caused by the
power which acts upon the moon.
5. I n this computation the earth is supposed to be at rest, whereas it
would have been more exact to have supposed it to move, as well as the

moon, about their common center of gravity; as will easily be understood,
by what has been said in the preceding chapter, where it was shewn, that the
sun is subjected to the like motion about the common center of gravity of it
self and the planets. The action of the sun upon the moon, which is to be
explain’d in what follows, is likewise here neglected: and Sir Isaac Newton
shews, if you take in both these considerations, the present computation will
best agree to a somewhat greater distance of the moon and earth, viz. to
60½ semidiameters of the earth, which distance is more conformable to
astronomical observations.
6. T h e s e computations afford an additional proof, that the action of
the earth observes the same proportion to the distance, which is here
contended for. Before I said, it was reasonable to conclude so by induction
from the planets Jupiter and Saturn; because they act in that manner. But
now the same thing will be evident by drawing no other consequence from
what is seen in those planets, than that the power, by which the primary
planets act on their secondary, is extended from the primary through the
whole interval between, so that it would act in every part of the
intermediate space. In Jupiter and Saturn this power is so far from being
confined to a small extent of distance, that it not only reaches to several
satellites at very different distances, but also from one planet to the other,
nay even through the whole planetary system[184]. Consequently there is no
appearance of reason, why this power should not act at all distances, even at
the very surfaces of these planets as well as farther off. But from hence it
follows, that the power, which retains the moon in her orbit, is the same, as
causes bodies near the surface of the earth to gravitate. For since the power,
by which the earth acts on the moon, will cause bodies near the surface of
the earth to descend with all the velocity they are found to do, it is certain
no other power can act upon them besides; because if it did, they must of
necessity descend swifter. Now from all this it is at length very evident, that
the power in the earth, which we call gravity, extends up to the moon, and
decreases in the duplicate proportion of the increase of the distance from
the earth.
7. T h i s finishes the discoveries made in the action of the primary
planets upon their secondary. The next thing to be shewn is, that the sun
acts upon them likewise: for this purpose it is to be observed, that if to the
motion of the satellite, whereby it would be carried round its primary at
rest, be superadded the same motion both in regard to velocity and

direction, as the primary it self has, it will describe about the primary the
same orbit, with as great regularity, as if the primary was indeed at rest. The
cause of this is that law of motion, which makes a body near the surface of
the earth, when let fall, to descend perpendicularly, though the earth be in
so swift a motion, that if the falling body did not partake of it, its descent
would be remarkably oblique; and that a body projected describes in the
most regular manner the same parabola, whether projected in the direction,
in which the earth moves, or in the opposite direction, if the projecting force
be the same[185]. From this we learn, that if the satellite moved about its
primary with perfect regularity, besides its motion about the primary, it
would participate of all the motion of its primary; have the same
progressive velocity, with which the primary is carried about the sun; and
be impelled with the same velocity as the primary towards the sun, in a
direction parallel to that impulse of its primary. And on the contrary, the
want of either of these, in particular of the impulse towards the sun, will
occasion great inequalities in the motion of the secondary planet. The
inequalities, which would arise from the absence of this impulse towards
the sun are so great, that by the regularity, which appears in the motion of
the secondary planets, it is proved, that the sun communicates, the same
velocity to them by its action, as it gives to their primary at the same
distance. For Sir I s a a c N e w t o n informs us, that upon examination he
found, that if any of the satellites of Jupiter were attracted by the sun more
or less, than Jupiter himself at the same distance, the orbit of that satellite,
instead of being concentrical to Jupiter, must have its center at a greater or
less distance, than the center of Jupiter from the sun, nearly in the
subduplicate proportion of the difference between the sun’s action upon the
satellite, and upon Jupiter; and therefore if any satellite were attracted by
the sun but 1/1000 part more or less, than Jupiter is at the same distance, the
center of the orbit of that satellite would be distant from the center of
Jupiter no less than a fifth part of the distance of the outermost satellite
from Jupiter[186]; which is almost the whole distance of the innermost
satellite. By the like argument the satellites of Saturn gravitate towards the
sun, as much as Saturn it self at the same distance; and the moon as much as
the earth.
8. T h u s is proved, that the sun acts upon the secondary planets, as
much as upon the primary at the same distance: but it was found in the last
chapter, that the action of the sun upon bodies is reciprocally in the

duplicate proportion of the distance; therefore the secondary planets being
sometimes nearer to the sun than the primary, and sometimes more remote,
they are not alway acted upon in the same degree with their primary, but
when nearer to the sun, are attracted more, and when farther distant, are
attracted less. Hence arise various inequalities in the motion of the
secondary planets[187].
9. S o m e of these inequalities would take place, though the moon, if
undisturbed by the sun, would have moved in a circle concentrical to the
earth, and in the plane of the earth’s motion; others depend on the elliptical
figure, and the oblique situation of the moon’s orbit. One of the first kind is,
that the moon is caused so to move, as not to describe equal spaces in equal
times, but is continually accelerated, as she passes from the quarter to the
new or full, and is retarded again by the like degrees in returning from the
new and full to the next quarter. Here we consider not so much the absolute,
as the apparent motion of the moon in respect to us.
10. T h e principles of astronomy teach how to distinguish these two
motions. Let S (in fig. 95.) represent the sun, A the earth moving in its orbit
B C, D E F G the moon’s orbit, the place of the moon H. Suppose the earth
to have moved from A to I. Because it has been shewn, that the moon
partakes of all the progressive motion of the earth; and likewise that the sun
attracts both the earth and moon equally, when they are at the same distance
from it, or that the mean action of the sun upon the moon is equal to its
action upon the earth: we must therefore consider the earth as carrying
about with it the moon’s orbit; so that when the earth is removed from A to
I, the moon’s orbit shall likewise be removed from its former situation into
that denoted by K L M N. But now the earth being in I, if the moon were
found in O, so that O I should be parallel to H A, though the moon would
really have moved from H to O, yet it would not have appeared to a
spectator upon the earth to have moved at all, because the earth has moved
as much it self; so that the moon would still appear in the same place with
respect to the fixed stars. But if the moon be observed in P, it will then
appear to have moved, its apparent motion being measured by the angle
under O I P. And if the angle under P I S be less than the angle under H A S,
the moon will have approached nearer to its conjunction with the sun.
11. T o come now to the explication of the mentioned inequality in the
moon’s motion: let S (in fig. 96.) represent the sun, A the earth, B C D E the

moon’s orbit, C the place of the moon, when in the latter quarter. Here it
will be nearly at the same distance from the sun, as the earth is. In this case
therefore they will both be equally attracted, the earth in the direction A S,
and the moon in the direction C S. Whence as the earth in moving round the
sun is continually descending toward it, so the moon in this situation must
in any equal portion of time descend as much; and therefore the position of
the line A C in respect of A S, and the change, which the moon’s motion
produces in the angle under C A S, will not be altered by the sun.
12. B u t now as soon as ever the moon is advanced from the quarter
toward the new or conjunction, suppose to G, the action of the sun upon it
will have a different effect. Here, were the sun’s action upon the moon to be
applied in the direction G H parallel to A S, if its action on the moon were
equal to its action on the earth, no change would be wrought by the sun on
the apparent motion of the moon round the earth. But the moon receiving a
greater impulse in G than the earth receives in A, were the sun to act in the
direction G H, yet it would accelerate the description of the space D A G,
and cause the angle under G A D to decrease faster, than otherwise it
would. The sun’s action will have this effect upon account of the obliquity
of its direction to that, in which the earth attracts the moon. For the moon
by this means is drawn by two forces oblique to each other, one drawing
from G toward A, the other from G toward H, therefore the moon must
necessarily be impelled toward D. Again, because the sun does not act in
the direction G H parallel to S A, but in the direction G S oblique to it, the
sun’s action on the moon will by reason of this obliquity farther contribute
to the moon’s acceleration. Suppose the earth in any short space of time
would have moved from A to I, if not attracted by the sun; the point I being
in the straight line C E, which touches the earth’s orbit in A. Suppose the
moon in the same time would have moved in her orbit from G to K, and
besides have partook of all the progressive motion of the earth. Then if K L
be drawn parallel to A I, and taken equal to it, the moon, if not attracted by
the sun, would be found in L. But the earth by the sun’s action is removed
from I. Suppose it were moved down to M in the line I M N parallel to S A,
and if the moon were attracted but as much, and in the same direction, as
the earth is here supposed to be attracted, so as to have descended during
the same time in the line L O, parallel also to A S, down as far as P, till L P
were equal to I M; the angle under P M N would be equal to that under
L I N, that is, the moon will appear advanced no farther forward, than if

neither it nor the earth had been subject to the sun’s action. But this is upon
the supposition, that the action of the sun upon the moon and earth were
equal; whereas the moon being acted upon more than the earth, did the
sun’s action draw the moon in the line L O parallel to A S, it would draw it
down so far as to make L P greater than I M; whereby the angle under
P M N will be rendred less, than that under L I N. But moreover, as the sun
draws the earth in a direction oblique to I N, the earth will be found in its
orbit somewhat short of the point M; however the moon is attracted by the
sun still more out of the line L O, than the earth is out of the line I N;
therefore this obliquity of the sun’s action will yet farther diminish the
angle under P M N.
13. T h u s the moon at the point G receives an impulse from the sun,
whereby her motion is accelerated. And the sun producing this effect in
every place between the quarter and the conjunction, the moon will move
from the quarter with a motion continually more and more accelerated; and
therefore by acquiring from time to time additional degrees of velocity in its
orbit, the spaces, which are described in equal times by the line drawn from
the earth to the moon, will not be every where equal, but those toward the
conjunction will be greater, than those toward the quarter. But now in the
moon’s passage from the conjunction D to the next quarter the sun’s action
will again retard the moon, till at the next quarter in E it be restored to the
first velocity, which it had in C.
14. A g a i n as the moon moves from E to the full or opposition to the
sun in B, it is again accelerated, the deficiency of the sun’s action upon the
moon, from what it has upon the earth, producing here the same effect as
before the excess of its action. Consider the moon in Q, moving from E
towards B. Here if the moon were attracted by the sun in a direction parallel
to A S, yet being acted on less than the earth, as the earth descends toward
the sun, the moon will in some measure be left behind. Therefore Q F being
drawn parallel to S B, a spectator on the earth would see the moon move, as
if attracted from the point Q in the direction Q F with a degree of force
equal to that, whereby the sun’s action on the moon falls short of its action
on the earth. But the obliquity of the sun’s action has also here an effect. In
the time the earth would have moved from A to I without the influence of
the sun, let the moon have moved in its orbit from Q to R. Drawing
therefore R T parallel to A I, and equal to the same, for the like reason as
before, the moon by the motion of its orbit, if not at all attracted by the sun,

must be found in T; and therefore, if attracted in a direction parallel to S A,
would be in the line T V parallel to A S; suppose in W. But the moon in Q
being farther off the sun than the earth, it will be less attracted, that is, T W
will be less than I M, and if the line S M be prolonged toward X, the angle
under X M W will be less than that under X I T. Thus by the sun’s action
the moon’s passage from the quarter to the full would be accelerated, if the
sun were to act on the earth and moon in a direction parallel to A S: and the
obliquity of the sun’s action will still more increase this acceleration. For
the action of the sun on the moon is oblique to the line S A the whole time
of the moon’s passage from Q to T, and will carry the moon out of the line
T V toward the earth. Here I suppose the time of the moon’s passage from
Q to T so short, that it shall not pass beyond the line S A. The earth also
will come a little short of the line I N, as was said before. From these causes
the angle under X M W will be still farther lessened.
15. T h e moon in passing from the opposition B to the next quarter will
be retarded again by the same degrees, as it is accelerated before its appulse
to the opposition. Because this action of the sun, which in the moon’s
passage from the quarter to the opposition causes it to be extraordinarily
accelerated, and diminishes the angle, which measures its distance from the
opposition; will make the moon slacken its pace afterwards, and retard the
augmentation of the same angle in its passage from the opposition to the
following quarter; that is, will prevent that angle from increasing so fast, as
otherwise it would. And thus the moon, by the sun’s action upon it, is twice
accelerated and twice restored to its first velocity, every circuit it makes
round the earth. This inequality of the moon’s motion about the earth is
called by astronomers its variation.
16. T h e next effect of the sun upon the moon is, that it gives the orbit
of the moon in the quarters a greater degree of curvature, than it would
receive from the action of the earth alone; and on the contrary in the
conjunction and opposition the orbit is less inflected.
17. W h e n the moon is in conjunction with the sun in the point D, the
sun attracting the moon more forcibly than it does the earth, the moon by
that means is impelled less toward the earth, than otherwise it would be, and
so the orbit is less incurvated; for the power, by which the moon is impelled
toward the earth, being that, by which it is inflected from a rectilinear
course, the less that power is, the less it will be inflected. Again, when the
moon is in the opposition in B, farther removed from the sun than the earth

is; it follows then, though the earth and moon are both continually
descending to the sun, that is, are drawn by the sun toward it self out of the
place they would otherwise move into, yet the moon descends with less
velocity than the earth; insomuch that the moon in any given space of time
from its passing the point of opposition will have less approached the earth,
than otherwise it would have done, that is, its orbit in respect of the earth
will approach nearer to a straight line. In the last place, when the moon is in
the quarter in F, and equally distant from the sun as the earth, we observed
before, that the earth and moon would descend with equal pace toward the
sun, so as to make no change by that descent in the angle under F A S; but
the length of the line F A must of necessity be shortned. Therefore the moon
in moving from F toward the conjunction with the sun will be impelled
more toward the earth by the sun’s action, than it would have been by the
earth alone, if neither the earth nor moon had been acted on by the sun; so
that by this additional impulse the orbit is rendred more curve, than it would
otherwise be. The same effect will also be produced in the other quarter.
18. A n o t h e r effect of the sun’s action, consequent upon this we have
now explained, is, that though the moon undisturbed by the sun might move
in a circle having the earth for its center; by the sun’s action, if the earth
were to be in the very middle or center of the moon’s orbit, yet the moon
would be nearer the earth at the new and full, than in the quarters. In this
probably will at first appear some difficulty, that the moon should come
nearest to the earth, where it is least attracted to it, and be farthest off when
most attracted. Which yet will appear evidently to follow from that very
cause, by considering what was last shewn, that the orbit of the moon in the
conjunction and opposition is rendred less curve; for the less curve the orbit
of the moon is, the less will the moon have descended from the place it
would move into, without the action of the earth. Now if the moon were to
move from any place without farther disturbance from that action, since it
would proceed in the line, which would touch its orbit in that place, it
would recede continually from the earth; and therefore if the power of the
earth upon the moon, be sufficient to retain it at the same distance, this
diminution of that power will cause the distance to increase, though in a
less degree. But on the other hand in the quarters, the moon, being pressed
more towards the earth than by the earth’s single action, will be made to
approach it; so that in passing from the conjunction or opposition to the
quarters the moon ascends from the earth, and in passing from the quarters