William Oughtred_ A Great Seventeenth-Century Teacher of Mathematics Florian Cajori 584 downloads.pdf

The Project Gutenberg eBook of William Oughtred: A Great
Seventeenth-Century Teacher of Mathematics

This eBook is for the use of anyone anywhere in the United States and
most other parts of the world at no cost and with almost no restrictions
whatsoever. You may copy it, give it away or re-use it under the terms
of the Project Gutenberg License included with this eBook or online at
www.gutenberg.org. If you are not located in the United States, you will
have to check the laws of the country where you are located before
using this eBook.

Title: William Oughtred: A Great Seventeenth-Century Teacher of
Mathematics

Author: Florian Cajori

Release date: September 9, 2014 [eBook #46815]
Most recently updated: October 24, 2024

Language: English

Other information and formats: www.gutenberg.org/ebooks/46815

Credits: Produced by Brenda Lewis, Stephen Hutcheson, and the Online
Distributed Proofreading Canada Team at
http://www.pgdpcanada.net

*** START OF THE PROJECT GUTENBERG EBOOK WILLIAM
OUGHTRED: A GREAT SEVENTEENTH-CENTURY TEACHER OF
MATHEMATICS ***

WILLIAM OUGHTRED

WILLIAM OUGHTRED
A GREAT SEVENTEENTH-CENTURY
TEACHER OF
MATHEMATICS
BY
FLORIAN CAJORI, Ph.D.
Professor of Mathematics
Colorado College

CHICAGO LONDON
THE OPEN COURT PUBLISHING COMPANY
1916

Copyright 1916 By
The Open Court Publishing Co.

All Rights Reserved

Published September 1916

Composed and Printed By
The University of Chicago Press
Chicago, Illinois, U.S.A.

[v

TABLE OF CONTENTS

PAGE
Introduction 1
CHAPTER
I. Oughtred’s Life 3
At School and University 3
As Rector and Amateur Mathematician 6
His Wife 7
In Danger of Sequestration 8
His Teaching 9
Appearance and Habits 12
Alleged Travel Abroad 14
His Death 15
II. Principal Works 17
Clavis mathematicae 17
Circles of Proportion and Trigonometrie 35
Solution of Numerical Equations 39
Logarithms 46
Invention of the Slide Rule; Controversy on Priority of Invention 4
6
III. Minor Works 50
IV. Oughtred’s Influence upon Mathematical Progress and Teaching 57
Oughtred and Harriot 57
Oughtred’s Pupils 58
Oughtred, the “Todhunter of the Seventeenth Century” 60
Was Descartes Indebted to Oughtred? 69
The Spread of Oughtred’s Notations 73
V. Oughtred’s Ideas on the Teaching of Mathematics 84

General Statement 84
Mathematics, “a Science of the Eye” 85
Rigorous Thinking and the Use of Instruments 87
Newton’s Comments on Oughtred 94
Index 97

[1

INTRODUCTION

In the year 1660 the Royal Society was founded by royal favor in London,
although in reality its inception took place in 1645 when the Philosophical
Society (or, as Boyle called it, the “Invisible College”) came into being,
which held meetings at Gresham College in London and later in Oxford. It
was during the second half of the seventeenth century that Sir Isaac Newton,
surrounded by a group of great men—Wallis, Hooke, Barrow, Halley, Cotes
—carried on his epoch-making researches in mathematics, astronomy, and
physics. But it is not this half-century of science in England, nor any of its
great men, that especially engage our attention in this monograph. It is rather
the half-century preceding, an epoch of preparation, when in the early times
of the House of Stuart the sciences began to flourish in England. Says Dr. A.
E. Shipley: “Whatever were the political and moral deficiencies of the Stuart
kings, no one of them lacked intelligence in things artistic and scientific.” It
was at this time that mathematics, and particularly algebra, began to be
cultivated with greater zeal, when elementary algebra with its symbolism as
we know it now began to take its shape.

Biographers of Sir Isaac Newton make particular mention of five
mathematical books which he read while a young student at Cambridge,
namely, Euclid’s Elements, Descartes’s Géométrie, Vieta’s Works, Van
Schooten’s Miscellanies, and Oughtred’s Clavis mathematicae. The last of
these books has been receiving increasing attention from the historians of [2
algebra in recent years. We have prepared this sketch because we felt that
there were points of interest in the life and activity of Oughtred which have
not received adequate treatment. Historians have discussed his share in the
development of symbolic algebra, but some have fallen into errors, due to
inability to examine the original editions of Oughtred’s Clavis mathematicae,

which are quite rare and inaccessible to most readers. Moreover, historians
have failed utterly to recognize his inventions of mathematical instruments,
particularly the slide rule; they have completely overlooked his educational
views and his ideas on mathematical teaching. The modern reader may pause
with profit to consider briefly the career of this interesting man.

Oughtred was not a professional mathematician. He did not make his
livelihood as a teacher of mathematics or as a writer, nor as an engineer who
applies mathematics to the control and use of nature’s forces. Oughtred was
by profession a minister of the gospel. With him the study of mathematics
was a side issue, a pleasure, a recreation. Like the great French algebraist,
Vieta, from whom he drew much of his inspiration, he was an amateur
mathematician. The word “amateur” must not be taken here in the sense of
superficial or unthorough. Great Britain has had many men distinguished in
science who pursued science as amateurs. Of such men Oughtred is one of
the very earliest.

F. C.

[3

CHAPTER I
OUGHTRED’S LIFE

AT SCHOOL AND UNIVERSITY

William Oughtred, or, as he sometimes wrote his name, Owtred, was born at
Eton, the seat of Eton College, the year of his birth being variously given as
1573, 1574, and 1575. “His father,” says Aubrey, “taught to write at Eaton,
and was a scrivener; and understood common arithmetique, and ’twas no
small helpe and furtherance to his son to be instructed in it when a schoole-
[1]
boy.” He was a boy at Eton in the year of the Spanish Armada. At this
famous school, which prepared boys for the universities, young Oughtred
received thorough training in classical learning.

According to information received from F. L. Clarke, Bursar and Clerk of
King’s College, Cambridge, Oughtred was admitted at King’s a scholar from
Eton on September 1, 1592, at the age of seventeen. He was made Fellow at
King’s on September 1, 1595, while Elizabeth was still on the throne. He
received in 1596 the degree of Bachelor of Arts and in 1600 that of Master of
Arts. He vacated his fellowship about the beginning of August, 1603. His
career at the University of Cambridge we present in his own words. He says:

Next after Eaton schoole, I was bred up in Cambridge in Kings Colledge:
of which society I was a member about eleven or twelve yeares: wherein
how I behaved my selfe, going hand in hand with the rest of my ranke in
the ordinary Academicall studies and exercises, and with what [4
approbation, is well knowne and remembered by many: the time which
over and above those usuall studies I employed upon the Mathematicall

sciences, I redeemed night by night from my naturall sleep, defrauding my
body, and inuring it to watching, cold, and labour, while most others tooke
their rest. Neither did I therein seek only my private content, but the
benefit of many: and by inciting, assisting, and instructing others, brought
many into the love and study of those Arts, not only in our own, but in
some other Colledges also: which some at this time (men far better than
my selfe in learning, degree, and preferment) will most lovingly
[2]
acknowledge.

These words describe the struggles which every youth not endowed with the
highest genius must make to achieve success. They show, moreover, the
kindly feeling toward others and the delight he took throughout life in
assisting anyone interested in mathematics. Oughtred’s passion for this study
is the more remarkable as neither at Eton nor at Cambridge did it receive
emphasis. Even after his time at Cambridge mathematical studies and their
applications were neglected there. Jeremiah Horrox was at Cambridge in
1633-35, desiring to make himself an astronomer.

“But many impediments,” says Horrox, “presented themselves: the tedious
difficulty of the study itself deterred a mind not yet formed; the want of
means oppressed, and still oppresses, the aspirations of my mind: but that
which gave me most concern was that there was no one who could [5
instruct me in the art, who could even help my endeavours by joining
me in the study; such was the sloth and languor which had seized all. . . . .
[3]
I found that books must be used instead of teachers.”

Some attention was given to Greek mathematicians, but the works of Italian,
German, and French algebraists of the latter part of the sixteenth and
beginning of the seventeenth century were quite unknown at Cambridge in
Oughtred’s day. It was part of his life-work as a mathematician to make
algebra, as it was being developed in his time, accessible to English youths.

At the age of twenty-three Oughtred invented his Easy Way of Delineating
Sun-Dials by Geometry, which, though not published until about half a
century later, in the first English edition of Oughtred’s Clavis mathematicae
in 1647, was in the meantime translated into Latin by Christopher Wren, then
a Gentleman Commoner of Wadham College, Oxford, now best known

through his architectural creations. In 1600 Oughtred wrote a monograph on
the construction of sun-dials upon a plane of any inclination, but that paper
was withheld by him from publication until 1632. Sun-dials were interesting
objects of study, since watches and pendulum clocks were then still
unknown. All sorts of sun-dials, portable and non-portable, were used at that
time and long afterward. Several of the college buildings at Oxford and
Cambridge have sun-dials even at the present time.

[6

AS RECTOR AND AMATEUR MATHEMATICIAN

It was in 1604 that Oughtred entered upon his professional life-work as a
preacher, being instituted to the vicarage of Shalford in Surrey. In 1610 he
was made rector of Albury, where he spent the remainder of his long life.
Since the era of the Reformation two of the rectors of Albury obtained great
celebrity from their varied talents and acquirements—our William Oughtred
and Samuel Horsley. Oughtred continued to devote his spare time to
mathematics, as he had done in college. A great mathematical invention
made by a Scotchman soon commanded his attention—the invention of
logarithms. An informant writes as follows:

Lord Napier, in 1614, published at Edinburgh his Mirifici logarithmorum
canonis descriptio. . . . . It presently fell into the hands of Mr. Briggs, then
geometry-reader at Gresham College in London: and that gentleman,
forming a design to perfect Lord Napier’s plan, consulted Oughtred upon
it; who probably wrote his Treatise of Trigonometry about the same time,
[4]
since it is evidently formed upon the plan of Lord Napier’s Canon.

It will be shown later that Oughtred is very probably the author of an
“Appendix” which appeared in the 1618 edition of Edward Wright’s
translation into English of John Napier’s Descriptio. This “Appendix” relates
to logarithms and is an able document, containing several points of historical
interest. Mr. Arthur Hutchinson of Pembroke College informs me that in the
university library at Cambridge there is a copy of Napier’s Constructio
(1619) bound up with a copy of Kepler’s Chilias logarithmorum (1624), that
at the beginning of the Constructio is a blank leaf, and before this occurs [7
the title-page only of Napier’s Descriptio (1619), at the top of which

appears Oughtred’s autograph. The history of this interesting signature is
unknown.

HIS WIFE

In 1606 he married Christ’sgift Caryll, daughter of Caryll, Esq., of Tangley,
[5]
in an adjoining parish. We know very little about Oughtred’s family life.
[6]
The records at King’s College, Cambridge, mention a son, but it is certain
that there were more children. A daughter was married to Christopher
[7]
Brookes. But there is no confirmation of Aubrey’s statements, according to
which Oughtred had nine sons and four daughters. Reference to the wife and
children is sometimes made in the correspondence with Oughtred. In 1616 J.
Hales writes, “I pray let me be remembered, though unknown, to Mistress
[8]
Oughtred.”

As we shall see later, Oughtred had a great many young men who came to
his house and remained there free of charge to receive instruction in
mathematics, which was likewise gratuitous. This being the case, certainly
great appreciation was due to Mrs. Oughtred, upon whom the burden of
hospitality must have fallen. Yet chroniclers are singularly silent in regard to
her. Hers was evidently a life of obscurity and service. We greatly doubt the
accuracy of the following item handed down by Aubrey; it cannot be a [8
true characterization:

His wife was a penurious woman, and would not allow him to burne
candle after supper, by which meanes many a good notion is lost, and
many a probleme unsolved; so that Mr. [Thomas] Henshawe, when he was
[9]
there, bought candle, which was a great comfort to the old man.

IN DANGER OF SEQUESTRATION

Oughtred spent his years in “unremitted attention to his favourite study,”
sometimes, it has been whispered, to the neglect of his rectorial duties. Says
Aubrey:

I have heard his neighbour ministers say that he was a pittiful preacher;
the reason was because he never studyed it, but bent all his thoughts on the
mathematiques; but when he was in danger of being sequestred for a
royalist, he fell to the study of divinity, and preacht (they sayd) admirably
[10]
well, even in his old age.

This remark on sequestration brings to mind one of the political and religious
struggles of the time, the episcopacy against the independent movements.
Says Manning:

In 1646 he was cited before the Committee for Ecclesiastical Affairs,
where many articles had been deposed against him; but, by the favour of
Sir Bulstrode Whitlock and others, who, at the intercession of William
Lilye the Astrologer, appeared in great numbers on his behalf, he had a
[11]
majority on his side, and so escaped a sequestration.

Not without interest is the account of this matter given by Lilly himself:

About this Time, the most famous Mathematician of all Europe, (Mr.
William Oughtred, Parson of Aldbury in Surrey) was in Danger of [9
Sequestration by the Committee of or for plunder’d Ministers; (Ambo-
dexters they were;) several inconsiderable Articles were deposed and
sworn against him, material enough to have sequestred him, but that, upon
his Day of hearing, I applied my self to Sir Bolstrode Whitlock, and all my
own old Friends, who in such Numbers appeared in his Behalf, that
though the Chairman and many other Presbyterian Members were stiff
against him, yet he was cleared by the major Number. The truth is, he had
a considerable Parsonage, and that only was enough to sequester any
moderate Judgment: He was also well known to affect his Majesty
[Charles I]. In these Times many worthy Ministers lost their Livings or
[12]
Benefices, for not complying with the Three-penny Directory.

HIS TEACHING

Oughtred had few personal enemies. His pupils held him in highest esteem
and showed deep gratitude; only one pupil must be excepted, Richard
Delamain. Against him arose a bitter controversy which saddened the life of
Oughtred, then an old man. It involved, as we shall see later, the priority of
invention of the circular slide rule and of a horizontal instrument or portable
sun-dial. In defense of himself, Oughtred wrote in 1633 or 1634 the
Apologeticall Epistle, from which we quoted above. This document contains
biographical details, in part as follows:

Ever since my departure from the Vniversity, which is about thirty yeares,
I have lived neere to the Towne of Guildford in Surrey: where, whether I
have taken so much liberty to the losse of time, and the neglect of my
calling the whole Countrey thereabout, both Gentry and others, to whom I
am full well knowne, will quickely informe him; my house being not past
three and twenty miles from London: and yet I so hid my selve at home,
that I seldomly travelled so farre as London once in a yeare. Indeed [10
the life and mind of man cannot endure without some
interchangeablenesse of recreation, and pawses from the intensive actions
of our severall callings; and every man is drawne with his owne delight.
My recreations have been diversity of studies: and as oft as I was toyled
with the labour of my owne profession, I have allayed that tediousnesse by
walking in the pleasant and more then Elysian fields of the diverse and
various parts of humane learning, and not the Mathematics onely.

Even the opponents of Delamain must be grateful to him for having been the
means of drawing from Oughtred such interesting biographical details.
Oughtred proceeds to tell how, about 1628, he was induced to write his
Clavis mathematicae, upon which his reputation as a mathematician largely
rests:

About five yeares since, the Earle of Arundell my most honourable Lord
in a time of his private retiring to his house in the countrey then at West
Horsley, foure small miles from me (though since he hath a house in
Aldebury the parish where I live) hearing of me (by what meanes I know
not) was pleased to send for me: and afterward at London to appoint mee a
Chamber of his owne house: where, at such times, and in such manner as

it seemed him good to imploy me, and when I might not inconveniently be
spared from my charge, I have been most ready to present my selfe in all
humble and affectionate service: I hope also without the offence of God,
the transgression of the good Lawes of this Land, neglect of my calling, or
the deserved scandall of any good man. . . . .

And although I am no mercenary man, nor make profession to teach any
one in these arts for gaine and recompence, but as I serve at the Altar, so I
live onely of the Altar: yet in those interims that I am at London in my
Lords service, I have been still much frequented both by Natives and
Strangers, for my resolution and instruction in many difficult poynts of
Art; and have most freely and lovingly imparted my selfe and my [11
skill, such as I had, to their contentments, and much honourable
acknowledgement of their obligation to my Lord for bringing mee to
London, hath beene testifyed by many. Of which my liberallity and
unwearyed readinesse to doe good to all, scarce any one can give more
ample testimony then R. D. himselfe can: would he be but pleased to allay
the shame of this his hot and eager contention, blowne up onely with the
full bellowes of intended glory and gaine; . . . . they [the subjects in which
Delamain received assistance from Oughtred] were the first elements of
Astronomie concerning the second motions of the fixed starres, and of the
Sunne and Moone; they were the first elements of Conics, to delineate
those sections: they were the first elements of Optics, Catoptrics, and
Dioptrics: of all which you knew nothing at all.

These last passages are instructive as showing what topics were taken up for
study with some of his pupils. The chief subject of interest with most of them
was algebra, which at that time was just beginning to draw the attention of
English lovers of mathematics.

Oughtred carried on an extensive correspondence on mathematical subjects.
He was frequently called upon to assist in the solution of knotty problems—
sometimes to his annoyance, perhaps, as is shown by the following letter
which he wrote in 1642 to a stranger, named Price:

It is true that I have bestowed such vacant time, as I could gain from the
study of divinity, (which is my calling,) upon human knowledges, and,
amongst other, upon the mathematics, wherein the little skill I have

attained, being compared with others of my profession, who for the most
part contenting themselves only with their own way, refuse to tread these
salebrous and uneasy paths, may peradventure seem the more. But now
being in years and mindful of mine end, and having paid dearly for my
former delights both in my health and state, besides the prejudice of [12
such, who not considering what incessant labour may produce, reckon
so much wanting unto me in my proper calling, as they think I have
acquired in other sciences; by which opinion (not of the vulgar only) I
have suffered both disrespect, and also hinderance in some small
perferments I have aimed at. I have therefore now learned to spare myself,
and am not willing to descend again in arenam, and to serve such
ungrateful muses. Yet, sir, at your request I have perused your problem. . .
[13]
. . Your problem is easily wrought per Nicomedis conchoidem lineam.

APPEARANCE AND HABITS

Aubrey gives information about the appearance and habits of Oughtred:

He was a little man, had black haire, and blacke eies (with a great deal of
spirit). His head was always working. He would drawe lines and diagrams
on the dust. . . . .

He [his oldest son Benjamin] told me that his father did use to lye a bed
till eleaven or twelve a clock, with his doublet on, ever since he can
remember. Studyed late at night; went not to bed till 11 a clock; had his
tinder box by him; and on the top of his bed-staffe, he had his inke-horne
fix’t. He slept but little. Sometimes he went not to bed in two or three
nights, and would not come downe to meales till he had found out the
quaesitum.

He was more famous abroad for his learning, and more esteemed, then at
home. Severall great mathematicians came over into England on purpose
to converse with him. His countrey neighbours (though they understood
not his worth) knew that there must be extraordinary worth in him, that he
was so visited by foreigners. . . . .

When learned foreigners came and sawe how privately he lived, they did
admire and blesse themselves, that a person of so much worth and learning
should not be better provided for. . . . .

He has told bishop Ward, and Mr. Elias Ashmole (who was his [13
neighbour), that “on this spott of ground” (or “leaning against this
oake” or “that ashe”), “the solution of such or such a probleme came into
my head, as if infused by a divine genius, after I had thought on it without
successe for a yeare, two, or three.” . . . .

Nicolaus Mercator, Holsatus . . . . went to see him few yeares before he
dyed. . . . .
ble
The right hon Thomas Howard, earle of Arundel and Surrey, Lord High
Marshall of England, was his great patron, and loved him intirely. One
time they were like to have been killed together by the fall at Albury of a
[14]
grott, which fell downe but just as they were come out.

Oughtred’s friends convey the impression that, in the main, Oughtred
enjoyed a comfortable living at Albury. Only once appear indications of
financial embarrassment. About 1634 one of his pupils, W. Robinson, writes
as follows:

I protest unto you sincerely, were I as able as some, at whose hands you
have merited exceedingly, or (to speak more absolutely) as able as willing,
I would as freely give you 500 l. per ann. as 500 pence; and I cannot but
be astonished at this our age, wherein pelf and dross is made their
summum bonum, and the best part of man, with the true ornaments
[15]
thereof, science and knowledge, are so slighted. . . . .

In his letters Oughtred complains several times of the limitations for work
and the infirmities due to his advancing old age. The impression he made
upon others was quite different. Says one biographer:

He sometimes amused himself with archery, and sometimes practised as a
surveyor of land. . . . . He was sprightly and active, when more than eighty

[16]
years of age.

Another informant says that Oughtred was [14

as facetious in Greek and Latine as solid in Arithmetique, Astronomy, and
the sphere of all Measures, Musick, etc.; exact in his style as in his
judgment; handling his Cube, and other Instruments at eighty, as steadily,
as others did at thirty; owing this, he said, to temperance and Archery;
principling his people with plain and solid truths, as he did the world with
great and useful Arts; advancing new Inventions in all things but Religion.
Which in its old order and decency he maintained secure in his privacy,
[17]
prudence, meekness, simplicity, resolution, patience, and contentment.

ALLEGED TRAVEL ABROAD

According to certain sources of information, Oughtred traveled on the
European Continent and was invited to change his abode to the Continent.
We have seen no statement from Oughtred himself on this matter. He seldom
referred to himself in his books and letters. The autobiography contained in
his Apologeticall Epistle was written a quarter of a century before his death.
Aubrey gives the following:

In the time of the civill warres the duke of Florence invited him over, and
offered him 500 li. per annum; but he would not accept it, because of his
[18]
religion.

A portrait of Oughtred, painted in 1646 by Hollar and inserted in the English
edition of the Clavis of 1647, contains underneath the following lines:

“Haec est Oughtredi senio labantis imago
Itala quam cupiit, Terra Britanna tulit.”

In the sketch of Oughtred by Owen Manning it is confessed that “it is not
known to what this alludes; but possibly he might have been in Italy with [15

[19]
his patron, the Earl of Arundel.” It would seem quite certain either that
Oughtred traveled in Europe or that he received some sort of an offer to
settle in Italy. In view of Aubrey’s explicit statement and of Oughtred’s well-
known habit of confining himself to his duties and studies in his own parish,
seldom going even as far as London, we strongly incline to the opinion that
he did not travel on the Continent, but that he received an offer from some
patron of the sciences—possibly some distinguished visitor—to settle in
Italy.

HIS DEATH

He died at Albury, June 30, 1660, aged about eighty-six years. Of his last
days and death, Aubrey speaks as follows:

Before he dyed he burned a world of papers, and sayd that the world was
not worthy of them; he was so superb. He burned also severall printed
bookes, and would not stirre, till they were consumed. . . . . I myselfe have
his Pitiscus, imbelished with his excellent marginall notes, which I
esteeme as a great rarity. I wish I could also have got his Bilingsley’s
Euclid, which John Collins sayes was full of his annotations. . . . .

Ralph Greatrex, his great friend, the mathematicall instrument-maker,
sayed he conceived he dyed with joy for the comeing-in of the king, which
was the 29th of May before. “And are yee sure he is restored?”—“Then
give me a glasse of sack to drinke his sacred majestie’s health.” His spirits
[20]
were then quite upon the wing to fly away. . . . .

In this passage, as in others, due allowance must be made for Aubrey’s lack
of discrimination. He was not in the habit of sifting facts from mere [16
gossip. That Oughtred should have declared that the world was not
worthy of his papers or manuscripts is not in consonance with the sweetness
of disposition ordinarily attributed to him. More probable was the feeling
that the papers he burned—possibly old sermons—were of no particular
value to the world. That he did not destroy a large mass of mathematical
manuscripts is evident from the fact that a considerable number of them

came after his death into the hands of Sir Charles Scarborough, M.D., under
whose supervision some of them were carefully revised and published at
Oxford in 1677 under the title of Opuscula mathematica hactenus inedita.

Aubrey’s story of Oughtred’s mode of death has been as widely circulated in
every modern biographical sketch as has his slander of Mrs. Oughtred by
claiming that she was so penurious that she would deny him the use of
candles to read by. Oughtred died on June 30; the Restoration occurred on
May 29. No doubt Oughtred rejoiced over the Restoration, but the story of
his drinking “a glass of sack” to his Majesty’s health, and then dying of joy
is surely apocryphal. De Morgan humorously remarks, “It should be added,
[21]
by way of excuse, that he was eighty-six years old.”

[17

CHAPTER II
PRINCIPAL WORKS

“CLAVIS MATHEMATICAE”

Passing to the consideration of Oughtred’s mathematical books, we begin
with the observation that he showed a marked disinclination to give his
writings to the press. His first paper on sun-dials was written at the age of
twenty-three, but we are not aware that more than one brief mathematical
manuscript was printed before his fifty-seventh year. In every instance,
publication in printed form seems to have been due to pressure exerted by
one or more of his patrons, pupils, or friends. Some of his manuscripts were
lent out to his pupils, who prepared copies for their own use. In some
instances they urged upon him the desirability of publication and assisted in
preparing copy for the printer. The earliest and best-known book of Oughtred
was his Clavis mathematicae, to which repeated allusion has already been
made. As he himself informs us, he was employed by the Earl of Arundel
about 1628 to instruct the Earl’s son, Lord William Howard (afterward
Viscount Stafford) in the mathematics. For the use of this young man
Oughtred composed a treatise on algebra which was published in Latin in the
year 1631 at the urgent request of a kinsman of the young man, Charles
Cavendish, a patron of learning.
[22]
The Clavis mathematicae, in its first edition of 1631, was a booklet of
only 88 small pages. Yet it contained in very condensed form the [18
essentials of arithmetic and algebra as known at that time.

Aside from the addition of four tracts, the 1631 edition underwent some
changes in the editions of 1647 and 1648, which two are much alike. The
twenty chapters of 1631 are reduced to nineteen in 1647 and in all the later
editions. Numerous minute alterations from the 1631 edition occur in all
parts of the books of 1647 and 1648. The material of the last three chapters
of the 1631 edition is rearranged, with some slight additions here and there.
The 1648 edition has no preface. In the print of 1652 there are only slight
alterations from the 1648 edition; after that the book underwent hardly [19
any changes, except for the number of tracts appended, and brief
explanatory notes added at the close of the chapters in the English editions of
1694 and 1702. The 1652 and 1667 editions were seen through the press by
John Wallis; the 1698 impression contains on the title-page the words: Ex
Recognitione D. Johannis Wallis, S.T.D. Geometriae Professoris Saviliani.

The cost of publishing may be a matter of some interest. When arranging for
the printing of the 1667 edition of the Clavis, Wallis wrote Collins: “I told
you in my last what price she [Mrs. Lichfield] expects for it, as I have
formerly understood from her, viz., £ 40 for the impression, which is about
[23]
9½d. a book.”

As compared with other contemporary works on algebra, Oughtred’s
distinguishes itself for the amount of symbolism used, particularly in the
treatment of geometric problems. Extraordinary emphasis was placed upon
[24] [20
what he called in the Clavis the “analytical art.” By that term he did
not mean our modern analysis or analytical geometry, but the art “in which
[25]
by taking the thing sought as knowne, we finde out that we seeke.” He
meant to express by it condensed processes of rigid, logical deduction
expressed by appropriate symbols, as contrasted with mere description or
elucidation by passages fraught with verbosity. In the preface to the first
edition (1631) he says:

In this little book I make known . . . . the rules relating to fundamentals,
collected together, just like a bundle, and adapted to the explanation of as
many problems as possible.

As stated in this preface, one of his reasons for publishing the book, is

. . . . that like Ariadne I might offer a thread to mathematical study by
which the mysteries of this science might be revealed, and direction given
to the best authors of antiquity, Euclid, Archimedes, the great geometrician
Apollonius of Perga, and others, so as to be easily and thoroughly
understood, their theorems being added, not only because to many they are
the height and depth of mathematical science (I ignore the would-be
mathematicians who occupy themselves only with the so-called practice,
which is in reality mere juggler’s tricks with instruments, the surface so to
speak, pursued with a disregard of the great art, a contemptible picture),
but also to show with what keenness they have penetrated, with what mass
of equations, comparisons, reductions, conversions and disquisitions these
heroes have ornamented, increased and invented this most beautiful
science.

The Clavis opens with an explanation of the Hindu-Arabic notation and of
decimal fractions. Noteworthy is the absence of the words “million,”
“billion,” etc. Though used on the Continent by certain mathematical writers
long before this, these words did not become current in English [21
mathematical books until the eighteenth century. The author was a great
admirer of decimal fractions, but failed to introduce the notation which in
later centuries came to be universally adopted. Oughtred wrote 0.56 in this
manner 0|56; the point he used to designate ratio. Thus 3:4 was written by
him 3·4. The decimal point (or comma) was first used by the inventor of
logarithms, John Napier, as early as 1616 and 1617. Although Oughtred had
mastered the theory of logarithms soon after their publication in 1614 and
was a great admirer of Napier, he preferred to use the dot for the designation
of ratio. This notation of ratio is used in all his mathematical books, except
in two instances. The two dots (:) occur as symbols of ratio in some parts of
Oughtred’s posthumous work, Opuscula mathematica hactenus inedita,
Oxford, 1677, but may have been due to the editors and not to Oughtred
himself. Then again the two dots (:) are used to designate ratio on the last
two pages of the tables of the Latin edition of Oughtred’s Trigonometria of
1657. In all other parts of that book the dot (·) is used. Probably someone
who supervised the printing of the tables introduced the (:) on the last two
pages, following the logarithmic tables, where methods of interpolation are
explained. The probability of this conjecture is the stronger, because in the
English edition of the Trigonometrie, brought out the same year (1657) but
after the Latin edition, the notation (:) at the end of the book is replaced by

the usual (·), except that in some copies of the English edition the
explanations at the end are omitted altogether.

Oughtred introduces an interesting, and at the same time new, feature of an
abbreviated multiplication and an abbreviated division of decimal fractions.
On this point he took a position far in advance of his time. The part on [22
abbreviated multiplication was rewritten in slightly enlarged form and
with some unimportant alterations in the later edition of the Clavis. We give
it as it occurs in the revision. Four cases are given. In finding the product of
246|914 and 35|27, “if you would have the Product without any Parts”
(without any decimal part), “set the place of Unity of the lesser under the
place of Unity in the greater: as in the Example,” writing the figures of the
lesser number in inverse order. From the example it will be seen that he
begins by multiplying by 3, the right-hand digit of the multiplier. In the first
edition of the Clavis he began with 7, the left digit. Observe also that he
“carries” the nearest tens in the product of each lower digit and the upper
digit one place to its right. For instance, he takes 7×4=28 and carries 3, then
he finds 7×2+3=17 and writes down 17.

2 4 6|9 1 4
7 2|5 3
-------
7 4 0 7
1 2 3 5
4 9
1 7
-------
8 7 0 8

The second case supposes that “you would have the Product with some
places of parts” (decimals), say 4: “Set the place of Unity of the lesser
Number under the Fourth place of the Parts of the greater.” The
multiplication of 246|914 by 35|27 is now performed thus:

2 4 6|9 1 4
7 2|5 3
---------------
7 4 0 7 4 2 0 0
1 2 3 4 5 7 0 0
4 9 3 8 2 8

1 7 2 8 4 0
---------------
8 7 0 8|6 5 6 8

In the third and fourth cases are considered factors which appear as [23
integers, but are in reality decimals; for instance, the sine of 54° is given
in the tables as 80902 when in reality it is .80902.

Of interest as regards the use of the word “parabola” is the following: “The
Number found by Division is called the Quotient, or also Parabola, because
it arises out of the Application of a plain Number to a given Longitude, that a
[26]
congruous Latitude may be found.” This is in harmony with etymological
dictionaries which speak of a parabola as the application of a given area to a
given straight line. The dividend or product is the area; the divisor or factor
is the line.

Oughtred gives two processes of long division. The first is identical with the
modern process, except that the divisor is written below every remainder,
each digit of the divisor being crossed out as soon as it has been used in the
partial multiplication. The second method of long division is one of the
several types of the old “scratch method.” This antiquated process held its
place by the side of the modern method in all editions of the Clavis. The
author divides 467023 by 357|0926425, giving the following instructions:
“Take as many of the first Figures of the Divisor as are necessary, for the
first Divisor, and then in every following particular Division drop one of the
Figures of the Divisor towards the Left Hand, till you have got a competent
Quotient.” He does not explain abbreviated division as thoroughly as
abbreviated multiplication.

[24
17
303
2803
109930
3̣5̣7̣|0̣9̣2̣6425) 467023 (1307|80
357093
107127
2500
286

Oughtred does not examine the degree of reliability or accuracy of his
processes of abbreviated multiplication and division. Here as in other places
he gives in condensed statement the mode of procedure, without further
discussion.

He does not attempt to establish the rules for the addition, subtraction,
multiplication, and division of positive and negative numbers. “If the Signs
are both alike, the Product will be affirmative, if unlike, negative”; then he
proceeds to applications. This attitude is superior to that of many writers of
the eighteenth and nineteenth centuries, on pedagogical as well as logical
grounds: pedagogically, because the beginner in the study of algebra is not in
a position to appreciate an abstract train of thought, as every teacher well
knows, and derives better intellectual exercise from the applications of the
rules to problems; logically, because the rule of signs in multiplication does
not admit of rigorous proof, unless some other assumption is first made
which is no less arbitrary than the rule itself. It is well known that the proofs
of the rule of signs given by eighteenth-century writers are invalid.
Somewhere they involve some surreptitious assumption. This criticism
applies even to the proof given by Laplace, which tacitly assumes the
distributive law in multiplication.

A word should be said on Oughtred’s definition of + and -. He [25
recognizes their double function in algebra by saying (Clavis, 1631, p.
2): “Signum additionis, sive affirmationis, est + plus” and “Signum
subductionis, sive negationis est - minus.” They are symbols which indicate
the quality of numbers in some instances and operations of addition or
subtraction in other instances. In the 1694 edition of the Clavis, thirty-four
years after the death of Oughtred, these symbols are defined as signifying
operations only, but are actually used to signify the quality of numbers as
well. In this respect the 1694 edition marks a recrudescence.

The characteristic in the Clavis that is most striking to a modern reader is the
total absence of indexes or exponents. There is much discussion in the
leading treatises of the latter part of the sixteenth and the early part of the
seventeenth century on the theory of indexes, but the modern exponential
notation, aⁿ, is of later date. The modern notation, for positive integral
exponents, first appears in Descartes’ Géométrie, 1637; fractional and
negative exponents were first used in the modern form by Sir Isaac Newton,

in his announcement of the binomial formula, in a letter written in 1676. This
total absence of our modern exponential notation in Oughtred’s Clavis gives
it a strange aspect. Like Vieta, Oughtred uses ordinarily the capital letters, A,
B, C, . . . . to designate given numbers; A² is written Aq, A³ is written Ac; for
A⁴, A⁵, A⁶ he has, respectively, Aqq, Aqc, Acc. Only on rare occasions,
usually when some parallelism in notation is aimed at, does he use small
[27]
letters to represent numbers or magnitudes. Powers of binomials or
polynomials are marked by prefixing the capital letters Q (for square), C [26
(for cube), QQ (for the fourth power), QC (for the fifth power), etc.

Oughtred does not express aggregation by (). Parentheses had been used by
[28]
Girard, and by Clavius as early as 1609, but did not come into general use
in mathematical language until the time of Leibniz and the Bernoullis.
Oughtred indicates aggregation by writing a colon (:) at both ends. Thus,
Q:A-E: means with him (A-E)². Similarly, √q:A+E: means √(A+E). The two
dots at the end are frequently omitted when the part affected includes all the
terms of the polynomial to the end. Thus, C:A+B-E=.. means (A+B-E)³=..
There are still further departures from this notation, but they occur so seldom
that we incline to the interpretation that they are simply printer’s errors. For
proportion Oughtred uses the symbol (::). The proportion a:b=c:d appears in
his notation a·b::c·d. Apparently, a proportion was not fully recognized in
this day as being the expression of an equality of ratios. That probably
explains why he did not use = here as in the notation of ordinary equations.
Yet Oughtred must have been very close to the interpretation of a proportion
as an equality; for he says in his Elementi decimi Euclidis declaratio,
“proportio, sive ratio aequalis ::” That he introduced this extra symbol when
the one for equality was sufficient is a misfortune. Simplicity demands that
no unnecessary symbols be introduced. However, Oughtred’s symbolism is
certainly superior to those which preceded. Consider the notation of Clavius.
[29] [27
He wrote 20:60=4:x, x=12, thus: “20·60·4? fiunt 12.” The
insufficiency of such a notation in the more involved expressions frequently
arising in algebra is readily seen. Hence Oughtred’s notation (::) was early
adopted by English mathematicians. It was used by John Wallis at Oxford,
by Samuel Foster at Gresham College, by James Gregory of Edinburgh, by
the translators into English of Rahn’s algebra, and by many other early
writers. Oughtred has been credited generally with the introduction of St.

Andrew’s cross × as the symbol for multiplication in the Clavis of 1631. We
have discovered that this symbol, or rather the letter x which closely
resembles it, occurs as the sign of multiplication thirteen years earlier in an
anonymous “Appendix to the Logarithmes, shewing the practise of the
Calculation of Triangles etc.” to Edward Wright’s translation of John
[30]
Napier’s Descriptio, published in 1618. Later we shall give our reasons
for believing that Oughtred is the author of that “Appendix.” The × has
survived as a symbol of multiplication.

Another symbol introduced by Oughtred and found in modern books is ~,
expressing difference; thus C~D signifies the difference between C and D,
[31]
even when D is the larger number. This symbol was used by John Wallis
[32]
in 1657.

Oughtred represented in symbols also certain composite expressions, as for
instance A+E=Z, A-E=X, where A is greater than E. He represented by a
symbol also each of the following: A²+E², A³+E³, A²-E², A³-E³.

Oughtred practically translated the tenth book of Euclid from its [28
ponderous rhetorical form into that of brief symbolism. An appeal to the
eye was a passion with Oughtred. The present writer has collected the
different mathematical symbols used by Oughtred and has found more than
one hundred and fifty of them.

The differences between the seven different editions of the Clavis lie mainly
in the special parts appended to some editions and dropped in the latest
editions. The part which originally constituted the Clavis was not materially
altered, except in two or three of the original twenty chapters. These changes
were made in the editions of 1647 and 1648. After the first edition, great
stress was laid upon the theory of indices upon the very first page, as also in
passages farther on. Of course, Oughtred did not have our modern notation
of indices or exponents, but their theory had been a part of algebra and
arithmetic for some time. Oughtred incorporated this theory in his brief
exposition of the Hindu-Arabic notation and in his explanation of logarithms.
As previously pointed out, the last three chapters of the 1631 edition were
considerably rearranged in the later editions and combined into two chapters,

so that the Clavis proper had nineteen chapters instead of twenty in the
additions after the first. These chapters consisted of applications of algebra to
geometry and were so framed as to constitute a severe test of the student’s
grip of the subject. The very last problem deals with the division of angles
into equal parts. He derives the cubic equation upon which the trisection
depends algebraically, also the equations of the fifth degree and seventh
degree upon which the divisions of the angle into 5 and 7 equal parts depend,
respectively. The exposition was severely brief, yet accurate. He did not [29
believe in conducting the reader along level paths or along slight
inclines. He was a guide for mountain-climbers, and woe unto him who
lacked nerve.

Oughtred lays great stress upon expansions of powers of a binomial. He
makes use of these expansions in the solution of numerical equations. To one
who does not specialize in the history of mathematics such expansions may
create surprise, for did not Newton invent the binomial theorem after the
death of Oughtred? As a matter of fact, the expansions of positive integral
powers of a binomial were known long before Newton, not only to
seventeenth-century but even to eleventh-century mathematicians.
Oughtred’s Clavis of 1631 gave the binomial coefficients for all powers up to
and including the tenth. What Newton really accomplished was the
generalization of the binomial expansion which makes it applicable to
negative and fractional exponents and converts it into an infinite series.

As a specimen of Oughtred’s style of writing we quote his solution of
quadratic equations, accompanied by a translation into English and into
modern mathematical symbols.
[33]
As a preliminary step he lets

Z=A+E and A>E;

he lets also X=A-E. From these relations he obtains identities which, in
modern notation, are ¼Z²-AE=(½Z-E)²=¼X². Now, if we know Z and AE, we
can find ½X. Then ½(Z+X)=A, and ½(Z-X)=E, and

A=½Z+√¼Z²-AE.

Having established these preliminaries, he proceeds thus: [30

Datis igitur linea inaequaliter secta Z (10), & rectangulo sub segmentis AE
(21) qui gnomon est: datur semidifferentia segmentorum ½X: & per
consequens ipsa segmenta. Nam ponatur alterutrum segmentum A: alterum
erit Z-A: Rectangulum auctem est ZA-Aq=AE. Et quia dantur Z & AE:
estque ¼Zq-AE=¼Xq: & per 5c. 18, ½Z+½X=A: & ½Z-½X=E: Aequatio
maius segment
sic resoluetur: ½Z±√q:¼Zq-AE:=A { minus segment.

Itaque proposita equatione, in qua sunt tres species aequaliter in ordine
tabellae adscendentes, altissima autem species ponitur negata: Magnitudo
data coefficiens mediam speciem est linea bisecanda: & magnitudo
absoluta data, ad quam sit aequatio, est rectangulum sub segmentis
inaequalibus, sine gnomon: vt ZA-Aq=AE: in numeris autem 10l-lq=21:
Estque A, vel 1l, alterutrum segmentum inaequale. Inuenitur autem sic:

Z
Dimidiata coefficiens median speciem est Z (5); cuius quadratum est q
2 4
Zq
(25): ex hoc tolle AE (21) absolutum: eritque -AE (4) quadratum
4
semidifferentiae segmentorum: latus huius quadratum (2) est
semidifferentia: quam si addas ad Z (5) semissem coefficientis, sive
2
lineae bisecandae, erit maius segment.; sin detrahas, erit minus segment:
Z
Dico Z ±√q: q -AE:=A { maius segmentum
2 4 minus segmentum.

We translate the Latin passage, using the modern exponential notation and
parentheses, as follows:

Given therefore an unequally divided line Z (10), and a rectangle beneath
the segments AE (21) which is a gnomon. Half the difference of the [31
segments ½X is given, and consequently the segment itself. For, if one
of the two segments is placed equal to A, the other will be Z-A. Moreover,
the rectangle is ZA-A²=AE. And because Z and AE are given, and there is

¼Z²-AE=¼X², and by 5c.18, ½Z+½X=A, and ½Z-½X=E, the equation will
major segment
be solved thus: ½Z±√(¼Z²-AE)=A {
minor segment.

And so an equation having been proposed in which three species (terms)
are in equally ascending powers, the highest species, moreover, being
negative, the given magnitude which constitutes the middle species is the
line to be bisected. And the given absolute magnitude to which it is equal
is the rectangle beneath the unequal segments, without gnomon. As ZA-
A²=AE, or in numbers, 10x-x²=21. And A or x is one of the two unequal
segments. It may be found thus:

The half of the middle species is Z (5), its square is Z² (25). From it
2 4
subtract the absolute term AE (21), and Z² -AE (4) will be the square of
4
half the difference of the segments. The square root of this,
Z² ²
√ [( )2 ]
-AE (2), is half the difference. If you add it to half the

coefficient Z (5), or half the line to be bisected, the longer segment is
2
obtained; if you subtract it, the smaller segment is obtained. I say:

Z
2
±√ ( Z²4 -AE ) =A { major segment
minor segment.

The quadratic equation Aq+ZA=AE receives similar treatment. This and the
preceding equation, ZA-Aq=AE, constitute together a solution of the general
quadratic equation, x²+ax=b, provided that E or Z are not restricted to
positive values, but admit of being either positive or negative, a case not [32
adequately treated by Oughtred. Imaginary numbers and imaginary roots
receive no consideration whatever.

A notation suggested by Vieta and favored by Girard made vowels stand for
unknowns and consonants for knowns. This conventionality was adopted by
Oughtred in parts of his algebra, but not throughout. Near the beginning he

used Q to designate the unknown, though usually this letter stood with him
[34]
for the “square” of the expression after it.

It is of some interest that Oughtred used π to signify the ratio of the
δ
circumference to the diameter of a circle. Very probably this notation is the
forerunner of the π=3.14159 . . . . used in 1706 by William Jones. Oughtred
first used π in the 1647 edition of the Clavis mathematicae. In the 1652
δ
edition he says, “Si in circulo sit 7.22::δ·π::113.355:erit δ·π::2 R.P: periph.”
This notation was adopted by Isaac Barrow, who used it extensively. David
[35] [36]
Gregory used π in 1697, and De Moivre used c about 1697, to
ρ r
designate the ratio of the circumference to the radius.

We quote the description of the Clavis that was given by Oughtred’s [33
greatest pupil, John Wallis. It contains additional information of interest
to us. Wallis devotes chap. xv of his Treatise of Algebra, London, 1685, pp.
67-69, to Mr. Oughtred and his Clavis, saying:

Mr. William Oughtred (our Country-man) in his Clavis Mathematicae, (or
Key of Mathematicks,) first published in the Year 1631, follows Vieta (as
he did Diophantus) in the use of the Cossick Denominations; omitting (as
he had done) the names of Sursolids, and contenting himself with those of
Square and Cube, and the Compounds of these.

But he doth abridge Vieta’s Characters or Species, using only the letters q,
c, &c. which in Vieta are expressed (at length) by Quadrate, Cube, &c.
For though when Vieta first introduced this way of Specious Arithmetick,
it was more necessary (the thing being new,) to express it in words at
length: Yet when the thing was once received in practise, Mr. Oughtred
(who affected brevity, and to deliver what he taught as briefly as might be,
and reduce all to a short view,) contented himself with single Letters
instead of those words.

Thus what Vieta would have written

A Quadrate, into B Cube, Equal to FG Plane,
CDE Solid,

would with him be thus expressed

Aq Bc
=FG.
CDE

And the better to distinguish upon the first view, what quantities were
Known, and what Unknown, he doth (usually) denote the Known to
Consonants, and the Unknown by Vowels; as Vieta (for the same reason)
had done before him.

He doth also (to very great advantage) make use of several Ligatures, or
Compendious Notes, to signify Summs, Differences, and Rectangles of
several Quantities. As for instance, Of two Quantities A (the Greater), [34
and E (the Lesser), the Sum he calls Z, the Difference X, the
Rectangle AE. . . . .

Which being of (almost) a constant signification with him throughout, do
save a great circumlocution of words, (each Letter serving instead of a
Definition;) and are also made use of (with very great advantage) to
discover the true nature of divers intricate Operations, arising from the
various compositions of such Parts, Sums, Differences, and Rectangles;
(of which there is great plenty in his Clavis, Cap. 11, 16, 18, 19. and
elsewhere,) which without such Ligatures, or Compendious Notes, would
not be easily discovered or apprehended. . . . .

I know there are who find fault with his Clavis, as too obscure, because so
short, but without cause; for his words be always full, but not Redundant,
and need only a little attention in the Reader to weigh the force of every
word, and the Syntax of it; . . . . And this, when once apprehended, is
much more easily retained, than if it were expressed with the prolixity of
some other Writers; where a Reader must first be at the pains to weed out
a great deal of superfluous Language, that he may have a short prospect of
what is material; which is here contracted for him in a short Synopsis. . . . .

Mr. Oughtred in his Clavis, contents himself (for the most part) with the
solution of Quadratick Equations, without proceeding (or very sparingly)
to Cubick Equations, and those of Higher Powers; having designed that
Work for an Introduction into Algebra so far, leaving the Discussion of
Superior Equations for another work. . . . . He contents himself likewise in
Resolving Equations, to take notice of the Affirmative or Positive Roots;
omitting the Negative or Ablative Roots, and such as are called Imaginary
or Impossible Roots. And of those which, he calls Ambiguous Equations,
(as having more Affirmative Roots than one,) he doth not (that I
remember) any where take notice of more than Two Affirmative Roots:
(Because in Quadratick Equations, which are those he handleth, there are
indeed no more.) Whereas yet in Cubick Equations, there may be Three,
and in those of Higher Powers, yet more. Which Vieta was well aware [35
of, and mentioneth in some of his Writings; and of which Mr.
Oughtred could not be ignorant.

“CIRCLES OF PROPORTION” AND “TRIGONOMETRIE”

Oughtred wrote and had published three important mathematical books, the
[37] [38]
Clavis, the Circles of Proportion, and a Trigonometrie. This last
appeared in the year 1657 at London, in both Latin and English.

It is claimed that the trigonometry was “neither finished nor published by
himself, but collected out of his scattered papers; and though he connived at
the printing it, yet imperfectly done, as appears by his MSS.; and one of the
[39] [36
printed Books, corrected by his own Hand.” Doubtless more accurate
on this point is a letter of Richard Stokes who saw the book through the
press:

I have procured your Trigonometry to be written over in a fair hand, which
when finished I will send to you, to know if it be according to your mind;
for I intend (since you were pleased to give your assent) to endeavour to
print it with Mr. Briggs his Tables, and so soon as I can get the Prutenic
[40]
Tables I will turn those of the sun and moon, and send them to you.

In the preface to the Latin edition Stokes writes:

Since this trigonometry was written for private use without the intention of
having it published, it pleased the Reverend Author, before allowing it to
go to press, to expunge some things, to change other things and even to
make some additions and insert more lucid methods of exposition.

This much is certain, the Trigonometry bears the impress characteristic of
Oughtred. Like all his mathematical writings, the book was very condensed.
Aside from the tables, the text covered only 36 pages. Plane and spherical
triangles were taken up together. The treatise is known in the history of
trigonometry as among the very earliest works to adopt a condensed
symbolism so that equations involving trigonometric functions could be
easily taken in by the eye. In the work of 1657, contractions are given as
follows: s=sine, t=tangent, se=secant, s co=cosine (sine complement),
t co=cotangent, se co=cosecant, log=logarithm, Z cru=sum of the sides of a
rectangle or right angle, X cru=difference of these sides. It has been
generally overlooked by historians that Oughtred used the abbreviations of
trigonometric functions, named above, a quarter of a century earlier, in [37
his Circles of Proportion, 1632, 1633. Moreover, he used sometimes also
the abbreviations which are current at the present time, namely sin=sine,
tan=tangent, sec=secant. We know that the Circles of Proportion existed in
manuscript many years before they were published. The symbol sv for sinus
versus occurs in the Clavis of 1631. The great importance of well-chosen
symbols needs no emphasis to readers of the present day. With reference to
Oughtred’s trigonometric symbols. Augustus De Morgan said:

This is so very important a step, simple as it is, that Euler is justly held to
have greatly advanced trigonometry by its introduction. Nobody that we
know of has noticed that Oughtred was master of the improvement, and
[41]
willing to have taught it, if people would have learnt.

We find, however, that even Oughtred cannot be given the whole credit in
this matter. By or before 1631 several other writers used abbreviations of the
trigonometric functions. As early as 1624 the contractions sin for sine and
tan for tangent appear on the drawing representing Gunter’s scale, but
[42]
Gunter did not use them in his books, except in the drawing of his scale.

A closer competitor for the honor of first using these trigonometric
abbreviations is Richard Norwood in his Trigonometrie, London, 1631,
where s stands for sine, t for tangent, sc for sine complement (cosine), tc for
tangent complement (cotangent), and sec for secant. Norwood was a teacher
of mathematics in London and a well-known writer of books on navigation.
Aside from the abbreviations just cited Norwood did not use nearly as [38
much symbolism in his mathematics as did Oughtred.

Mention should be made of trigonometric symbols used even earlier than any
of the preceding, in “An Appendix to the Logarithmes, shewing the practise
of the Calculation of Triangles, etc.,” printed in Edward Wright’s edition of
Napier’s A Description of the Admirable Table of Logarithmes, London,
1618. We referred to this “Appendix” in tracing the origin of the sign ×. It
contains, on p. 4, the following passage: “For the Logarithme of an arch or
an angle I set before (s), for the antilogarithme or compliment thereof (s*)
and for the Differential (t).” In further explanation of this rather
unsatisfactory passage, the author (Oughtred?) says, “As for example:
sB+BC=CA. that is, the Logarithme of an angle B. at the Base of a plane
right-angled triangle, increased by the addition of the Logarithm of BC, the
hypothenuse thereof, is equall to the Logarithme of CA the cathetus.”

Here “logarithme of an angle B” evidently means “log sin B,” just as with
Napier, “Logarithms of the arcs” signifies really “Logarithms of the sines of
the angles.” In Napier’s table, the numbers in the column marked
“Differentiae” signify log sine minus log cosine of an angle; that is, the
logarithms of the tangents. This explains the contraction (t) in the
“Appendix.” The conclusion of all this is that as early as 1618 the signs s, s*,
t were used for sine, cosine, and tangent, respectively.

John Speidell, in his Breefe Treatise of Sphaericall Triangles, London, 1627,
uses Si. for sine, T. and Tan for tangent, Se. for secant, Si. Co. for cosine, Se.
Co. for cosecant, T. Co. for cotangent.

The innovation of designating the sides and angles of a triangle by A, B, C,
and a, b, c, so that A was opposite a, B opposite b, and C opposite c, is [39
attributed to Leonard Euler (1753), but was first used by Richard
Rawlinson of Queen’s College, Oxford, sometimes after 1655 and before
[43]
1668. Oughtred did not use Rawlinson’s notation.

In trigonometry English writers of the first half of the seventeenth century
used contractions more freely than their continental contemporaries; even
more freely, indeed, than English writers of a later period. Von Braunmühl,
the great historian of trigonometry, gives Oughtred much praise for his
trigonometry, and points out that half a century later the army of writers on
trigonometry had hardly yet reached the standard set by Oughtred’s analysis.
[44]
Oughtred must be credited also with the first complete proof that was
given to the first two of “Napier’s analogies.” His trigonometry contains
seven-place tables of sines, tangents, and secants, and six-place tables of
logarithmic sines and tangents; also seven-place logarithmic tables of
numbers. At the time of Oughtred there was some agitation in favor of a
wider introduction of decimal systems. This movement is reflected in those
tables which contain the centesimal division of the degree, a practice which
is urged for general adoption in our own day, particularly by the French.

SOLUTION OF NUMERICAL EQUATIONS

In the solution of numerical equations Oughtred does not mention the
sources from which he drew, but the method is substantially that of the great
French algebraist Vieta, as explained in a publication which appeared in [40
1600 in Paris under the title, De numerosa potestatum purarum atque
adfectarum ad exegesin resolutione tractatus. In view of the fact that Vieta’s
process has been described inaccurately by leading modern historians
[45] [46]
including H. Hankel and M. Cantor, it may be worth while to go into
[47]
some detail. By them it is made to appear as identical with the procedure
given later by Newton. The two are not the same. The difference lies in the
divisor used. What is now called “Newton’s method” is Newton’s method as
[48]
modified by Joseph Raphson. The Newton-Raphson method of
approximation to the roots of an equation f(x)=0 is usually given the form a-
[f(a)/f´(a)], where a is an approximate value of the required root. It will be
seen that the divisor is f´(a). Vieta’s divisor is different; it is

|f(a+s₁)-f(a)|-s₁ⁿ,

where f(x) is the left of the equation f(x)=k, n is the degree of equation, and s₁
is a unit of the denomination of the digit next to be found. Thus in
x³+420000x=247651713, it can be shown that 417 is approximately a root;
suppose that a has been taken to be 400, then s₁=10; but if, at the next step of
approximation, a is taken to be 410, then s₁=1. In this example, taking
a=400, Vieta’s divisor would have been 9120000; Newton’s divisor [41
would have been 900000.

A comparison of Vieta’s method with the Newton-Raphson method reveals
the fact that Vieta’s divisor is more reliable, but labors under the very great
disadvantage of requiring a much larger amount of computation. The latter
divisor is accurate enough and easier to compute. Altogether the Newton-
Raphson process marks a decided advance over that of Vieta.

As already stated, it is the method of Vieta that Oughtred explains. The
Englishman’s exposition is an improvement on that of Vieta, printed forty
years earlier. Nevertheless, Oughtred’s explanation is far from easy to follow.
The theory of equations was at that time still in its primitive stage of
development. Algebraic notation was not sufficiently developed to enable the
argument to be condensed into a form easily surveyed. So complicated does
Vieta’s process of approximation appear that M. Cantor failed to recognize
that Vieta possessed a uniform mode of procedure. But when one has in mind
the general expression for Vieta’s divisor which we gave above, one will
recognize that there was marked uniformity in Vieta’s approximations.

Oughtred allows himself twenty-eight sections in which to explain the
process and at the close cannot forbear remarking that 28 is a “perfect”
number (being equal to the sum of its divisors, 1, 2, 4, 7, 14).

The early part of his exposition shows how an equation may be transformed
m
so as to make its roots 10, 100, 1000, or 10 times smaller. This simplifies
the task of “locating a root”; that is, of finding between what integers the
root lies.

Taking one of Oughtred’s equations, x⁴-72x³+238600x=8725815, upon
dividing 72x³ by 10, 238600x by 1000, and 8725815 by 10,000, we [42
obtain x⁴-7·2x³+238·6x=872·5. Dividing both sides by x, we obtain
x³+238·6-7·2x²=x)872·5. Letting x=4, we have 64+238·6-115·2=187·4.

But 4)872·5(218·1; 4 is too small. Next let x=5, we have 125+238·6-
180=183·6.

But 5)872·5(174·5; 5 is too large. We take the lesser value, x=4, or in the
original equation, x=40. This method may be used to find the second digit in
the root. Oughtred divides both sides of the equation by x², and obtains
x²+x)238600-72x=x²)8725815. He tries x=47 and x=48, and finds that x=47.

He explains also how the last computation may be done by logarithms.
Thereby he established for himself the record of being the first to use
logarithms in the solution of affected equations.

As an illustration of Oughtred’s method of approximation after the root
sought has been located, we have chosen for brevity a cubic in preference to
a quartic. We selected the equation x³+420000x=247651713. By the process
explained above a root is found to lie between x=400 and x=500. From this
point on, the approximation as given by Oughtred is as shown on p. 43.

In further explanation of this process, observe that the given equation is of
the form Lc+CqL=Dc, where Lc is our x, Cq=420000, Dc=247651713. In the
first step of approximation, let L=A+E, where A=400 and E is, as yet,
undetermined. We have

Lc=(A+E)³=A³+3A²E+3AE²+E³

and

CqL=420000(A+E).

Subtract from 247651713 the sum of the known terms A³ (his Ac) and
420000 A (his CqA). This sum is 232000000 the remainder is 15651713.

[43
“Exemplum II

1c+42̣00̣00̣l=247̇651̇7̣1̣3̣̇

Hoc est, Lc+CqL=Dc

2 4 7̇ | 6 5 1̇ | 7̣ 1̣ 3̣̇ | ( 4 1 7
------+-------+-------+------------
4 2 | 0 0 0 | 0 | Cq
------+-------+-------+------------
6 4 | | | Ac
1 6 8 | 0 0 0 | 0 | Cq A
------+-------+-------+------------
2 3 2 | 0 0 0 | 0 | Ablatit.
===================================
R 1 5 | 6 5 1̇ | 7 1 3̣ |
------+-------+-------+------------
4 | 8 | | 3 Aq
| 1 2 | | 3 A
4 | 2 0 0 | 0 0 | Cq
------+-------+-------+------------
9 | 1 2 0 | 0 0 | Divisor.
------+-------+-------+------------
4 | 8 | | 3 Aq E
| 1 2 | | 3 A Eq
| 1 | | Ec
4 | 2 0 0 | 0 0 | Cq E
------+-------+-------+------------
9 | 1 2 1 | 0 0 | Ablatit.
===================================
R 6 | 5 3 0 | 7 1 3̣̇ | 4 | 1 |
------+-------+-------+------------ ----+-----+---
| 5 0 4 | 3 | 3 Aq | |
| 1 | 2 3 | 3 A 1 6 | 8 |
| 4 2 0 | 0 0 0 | Cq | 1 |
------+-------+-------+------------ ----+-----+---
| 9 2 5 | 5 3 0 | Divisor. 1 6 8 1
------+-------+-------+------------
3 | 5 3 0 | 1 | 3 Aq E
| 6 0 | 2 7 | 3 A Eq
| | 3 4 3 | Ec
2 | 9 4 0 | 0 0 0 | Cq E
------+-------+-------+------------
6 | 5 3 0 | 7 1 3 | Ablatit.”

Next, he evaluates the coefficients of E in 3A²E and 420000E, also 3A, [44
the coefficient of E². He obtains 3A²=480000, 3A=1200, Cq=420000. He
interprets 3A² and Cq as tens, 3A as hundreds. Accordingly, he obtains as
their sum 9120000, which is the divisor for finding the second digit in the

approximation. Observe that this divisor is the value of |f(a+s₁)-f(a)|-s₁ⁿ in
our general expression, where a=400, s₁=10, n=3, f(x)=x³+420000x.

Dividing the remainder 15651713 by 9120000, he obtains the integer 1 in
ten’s place; thus E=10, approximately. He now computes the terms 3A²E,
3AE² and E³ to be, respectively, 4800000, 120000, 1000. Their sum is
9121000. Subtracting it from the previous remainder, 15651713, leaves the
new remainder, 6530713.

From here on each step is a repetition of the preceding step. The new A is
410, the new E is to be determined. We have now in closer approximation,
L=A+E. This time we do not subtract A³ and CqA, because this subtraction is
already affected by the preceding work.

We find the second trial divisor by computing the sum of 3A², 3A and Cq;
that is, the sum of 504300, 1230, 420000, which is 925530. Again, this
divisor can be computed by our general expression for divisors, by taking
a=410, s₁=1, n=3.

Dividing 6530713 by 925530 yields the integer 7. Thus E=7. Computing
3A²E, 3AE², E³ and subtracting their sum, the remainder is 0. Hence 417 is an
exact root of the given equation.

Since the extraction of a cube root is merely the solution of a pure cubic
equation, x³=n, the process given above may be utilized in finding cube
roots. This is precisely what Oughtred does in chap. xiv of his Clavis. If [45
the foregoing computation is modified by putting Cq=0, the process will
yield the approximate cube root of 247651713.

Oughtred solves 16 examples by the process of approximation here
explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In all
cases he finds only one or two real roots. Of the roots sought, five are
irrational, the remaining are rational and are computed to their exact values.
Three of the computed roots have 2 figures each, 9 roots have 3 figures each,
4 roots have 4 figures each. While no attempt is made to secure all the roots
—methods of computing complex roots were invented much later—he
computes roots of equations which involve large coefficients and some of
them are of a degree as high as the fifth. In view of the fact that many

editions of the Clavis were issued, one impression as late as 1702, it
contributed probably more than any other book to the popularization of
Vieta’s method in England.

Before Oughtred, Thomas Harriot and William Milbourn are the only
Englishmen known to have solved numerical equations of higher degrees.
Milbourn published nothing. Harriot slightly modified Vieta’s process by
simplifying somewhat the formation of the trial divisor. This method of
approximation was the best in existence in Europe until the publication by
Wallis in 1685 of Newton’s method of approximation.

It should be stated that, before the time of Newton, the best method of
approximation to the roots of numerical equations existed, not in Europe, but
in China. As early as the thirteenth century the Chinese possessed a method
which is almost identical with what is known today as “Horner’s method.”

[46

LOGARITHMS

Oughtred’s treatment of logarithms is quite in accordance with the more
[49]
recent practice. He explains the finding of the “index” (our
“characteristic”); he states that “the sum of two Logarithms is the Logarithm
of the Product of their Valors; and their difference is the Logarithm of the
Quotient,” that “the Logarithm of the side [436] drawn upon the Index
number [2] of dimensions of any Potestas is the logarithm of the same
Potestas” [436²], that “the logarithm of any Potestas [436²] divided by the
number of its dimensions [2] affordeth the Logarithm of its Root [436].”
These statements of Oughtred occur for the first time in the Key of the
Mathematicks of 1647; the Clavis of 1631 contains no treatment of
logarithms.

If the characteristic of a logarithm is negative, Oughtred indicates this fact by
placing the - above the characteristic. He separates the characteristic and
mantissa by a comma, but still uses the sign |_ to indicate decimal fractions.
He uses the contraction “log.”

INVENTION OF THE SLIDE RULE; CONTROVERSY ON
PRIORITY OF INVENTION

Oughtred’s most original line of scientific activity is the one least known to
the present generation. Augustus De Morgan, in speaking of Oughtred, who
was sometimes called “Oughtred Aetonensis,” remarks: “He is an animal of
extinct race, an Eton mathematician. Few Eton men, even of the minority
which knows what a sliding rule is, are aware that the inventor was of their
[50] [47
own school and college.” The invention of the slide rule has, until
[51]
recently, been a matter of dispute; it has been erroneously ascribed to
Edmund Gunter, Edmund Wingate, Seth Partridge, and others. We have been
able to establish that William Oughtred was the first inventor of slide rules,
though not the first to publish thereon. We shall see that Oughtred invented
slide rules about 1622, but the descriptions of his instruments were not put
into print before 1632 and 1633. Meanwhile one of his own pupils, Richard
Delamain, who probably invented the circular slide rule independently,
published a description in 1630, at London, in a pamphlet of 32 pages
entitled Grammelogia; or the Mathematicall Ring. In editions of this
pamphlet which appeared during the following three or four years, various
parts were added on, and some parts of the first and second editions
eliminated. Thus Delamain antedates Oughtred two years in the publication
of a description of a circular slide rule. But Oughtred had invented also a
rectilinear slide rule, a description of which appeared in 1633. To the
invention of this Oughtred has a clear title. A bitter controversy sprang up
between Delamain on one hand, and Oughtred and some of his pupils on the
other, on the priority and independence of invention of the circular slide rule.
Few inventors and scientific men are so fortunate as to escape contests. The
reader needs only to recall the disputes which have arisen, involving the
researches of Sir Isaac Newton and Leibniz on the differential and integral
calculus, of Thomas Harriot and René Descartes relating to the theory of
equations, of Robert Mayer, Hermann von Helmholtz, and Joule on the [48
principle of the conservation of energy, or of Robert Morse, Joseph
Henry, Gauss and Weber, and others on the telegraph, to see that questions of
priority and independence are not uncommon. The controversy between
Oughtred and Delamain embittered Oughtred’s life for many years. He refers
to it in print on more than one occasion. We shall confine ourselves at

present to the statement that it is by no means clear that Delamain stole the
invention from Oughtred; Delamain was probably an independent inventor.
Moreover, it is highly probable that the controversy would never have arisen,
had not some of Oughtred’s pupils urged and forced him into it. William
Forster stated in the preface to the Circles of Proportion of 1632 that while
he had been carefully preparing the manuscript for the press, “another to
whom the Author [Oughtred] in a louing confidence discouered this intent,
using more hast then good speed, went about to preocupate.” It was this
passage which started the conflagration. Another pupil, W. Robinson, wrote
to Oughtred, when the latter was preparing his Apologeticall Epistle as a
reply to Delamain’s countercharges: “Good sir, let me be beholden to you for
your Apology whensoever it comes forth, and (if I speak not too late) let me
entreat you, whip ignorance well on the blind side, and we may turn him
[52]
round, and see what part of him is free.” As stated previously, Oughtred’s
circular slide rule was described by him in his Circles of Proportion,
London, 1632, which was translated from Oughtred’s Latin manuscript and
then seen through the press by his pupil, William Forster. In 1633 appeared
An Addition vnto the Vse of the Instrvment called the Circles of [49
Proportion which contained at the end “The Declaration of the two
Rulers for Calculation,” giving a description of Oughtred’s rectilinear slide
rule. This Addition was bound with the Circles of Proportion as one volume.
About the same time Oughtred described a modified form of the rectilinear
[53]
slide rule, to be used in London for gauging.

[50

CHAPTER III
MINOR WORKS

Among the minor works of Oughtred must be ranked his booklet of forty
pages to which reference has already been made, entitled, The New Artificial
Gauging Line or Rod, London, 1633. His different designs of slide rules and
his inventions of sun-dials as well as his exposition of the making of watches
show that he displayed unusual interest and talent in the various
mathematical instruments. A short tract on watchmaking was brought out in
London as an appendix to the Horological Dialogues of a clock- and
watchmaker who signed himself “J. S.” (John Smith?). Oughtred’s tract
appeared with its own title-page, but with pagination continued from the
preceding part, as An Appendix wherein is contained a Method of
Calculating all Numbers for Watches. Written originally by that famous
Mathematician Mr. William Oughtred, and now made Publick. By J. S. of
London, Clock-maker. London, 1675.

“J. S.” says in his preface:

The method following was many years since Compiled by Mr. Oughtred
for the use of some Ingenious Gentlemen his friends, who for recreation at
the University, studied to find out the reason and Knowledge of Watch-
work, which seemed also to be a thing with which Mr. Oughtred himself
was much affected, as may in part appear by his putting out of his own
Son to the same Trade, for whose use (as I am informed) he did compile a
larger tract, but what became of it cannot be known.

Notwithstanding Oughtred’s marked activity in the design of [51
mathematical instruments, and his use of surveying instruments, he

always spoke in deprecating terms of their importance and their educational
value. In his epistle against Delamain he says:

The Instruments I doe not value or weigh one single penny. If I had been
ambitious of praise, or had thought them (or better then they) worthy, at
which to have taken my rise, out of my secure and quiet obscuritie, to
mount up into glory, and the knowledge of men: I could have done it many
yeares before. . . . .

Long agoe, when I was a young student of the Mathematicall Sciences, I
tryed many wayes and devices to fit my selve with some good Diall or
Instrument portable for my pocket, to finde the houre, and try other
conclusions by, and accordingly framed for that my purpose both
Quadrants, and Rings, and Cylinders, and many other composures. Yet not
to my full content and satisfaction; for either they performed but little, or
els were patched up with a diversity of lines by an unnaturall and forced
contexture. At last I . . . . found what I had before with much studie and
[54]
paines in vaine sought for.

Mention has been made in the previous pages of two of his papers on sun-
dials, prepared (as he says) when he was in his twenty-third year. The first
was published in the Clavis of 1647. The second paper appeared in his
Circles of Proportion.

Both before and after the time of Oughtred much was written on sun-dials.
Such instruments were set up against the walls of prominent buildings, much
as the faces of clocks in our time. The inscriptions that were put upon sun-
dials are often very clever: “I count only the hours of sunshine,” “Alas, how
fleeting.” A sun-dial on the grounds of Merchiston Castle, in Edinburgh, [52
where the inventor of logarithms, John Napier, lived for many years,
bears the inscription, “Ere time be tint, tak tent of time” (Ere time be lost,
take heed of time).

Portable sun-dials were sometimes carried in pockets, as we carry watches.
Thus Shakespeare, in As You Like It, Act II, sc. vii:

“And then he drew a diall from his poke.”

Watches were first made for carrying in the pocket about 1658.

Because of this literary, scientific, and practical interest in methods of
indicating time it is not surprising that Oughtred devoted himself to the
mastery and the advancement of methods of time-measurement.

Besides the accounts previously noted, there came from his pen: The
Description and Use of the double Horizontall Dyall: Whereby not onely the
hower of the day is shewne; but also the Meridian Line is found: And most
Astronomical Questions, which may be done by the Globe, are resolved.
Invented and written by W. O., London, 1636.

The “Horizontall Dyall” and “Horologicall Ring” appeared again as
appendixes to Oughtred’s translation from the French of a book on
mathematical recreations.

The fourth French edition of that work appeared in 1627 at Paris, under the
title of Recreations mathematiqve, written by “Henry van Etten,” a
pseudonym for the French Jesuit Jean Leurechon (1591-1690). English
editions appeared in 1633, 1653, and 1674. The full title of the 1653 edition
conveys an idea of the contents of the text: Mathematical Recreations, or, A
Collection of many Problemes, extracted out of the Ancient and Modern
Philosophers, as Secrets and Experiments in Arithmetick, Geometry, [53
Cosmographie, Horologiographie, Astronomie, Navigation, Musick,
Opticks, Architecture, Statick, Mechanicks, Chemistry, Water-works, Fire-
works, &c. Not vulgarly manifest till now. Written first in Greek and Latin,
lately compil’d in French, by Henry Van Etten, and now in English, with the
Examinations and Augmentations of divers Modern Mathematicians.
Whereunto is added the Description and Use of the Generall Horologicall
Ring. And The Double Horizontall Diall. Invented and written by William
Oughtred. London, Printed for William Leake, at the Signe of the Crown in
Fleet-street, between the two Temple-Gates. MDCLIII.

The graphic solution of spherical triangles by the accurate drawing of the
triangles on a sphere and the measurement of the unknown parts in the
drawing was explained by Oughtred in a short tract which was published by
his son-in-law, Christopher Brookes, under the following title: The Solution
of all Sphaerical Triangles both right and oblique By the Planisphaere:

Whereby two of the Sphaerical partes sought, are at one position most easily
found out. Published with consent of the Author, By Christopher Brookes,
Mathematique Instrument-maker, and Manciple of Wadham Colledge, in
Oxford.

Brookes says in the preface:

I have oftentimes seen my Reverend friend Mr. W. O. in his resolution of
all sphaericall triangles both right and oblique, to use a planisphaere,
without the tedious labour of Trigonometry by the ordinary Canons: which
planisphaere he had delineated with his own hands, and used in his
calculations more than Forty years before.

Interesting as one of our sources from which Oughtred obtained his
knowledge of the conic sections is his study of Mydorge. A tract which [54
he wrote thereon was published by Jonas Moore, in his Arithmetick in
two books . . . . [containing also] the two first books of Mydorgius his conical
sections analyzed by that reverend devine Mr. W. Oughtred, Englished and
completed with cuts. London, 1660. Another edition bears the date 1688.

To be noted among the minor works of Oughtred are his posthumous papers.
He left a considerable number of mathematical papers which his friend Sir
Charles Scarborough had revised under his direction and published at Oxford
in 1676 in one volume under the title, Gulielmi Oughtredi, Etonensis,
quondam Collegii Regalis in Cantabrigia Socii, Opuscula Mathematica
hactenus inedita. Its nine tracts are of little interest to a modern reader.

Here we wish to give our reasons for our belief that Oughtred is the author of
an anonymous tract on the use of logarithms and on a method of logarithmic
interpolation which, as previously noted, appeared as an “Appendix” to
Edward Wright’s translation into English of John Napier’s Descriptio, under
the title, A Description of the Admirable Table of Logarithmes, London,
1618. The “Appendix” bears the title, “An Appendix to the Logarithmes,
showing the practise of the Calculation of Triangles, and also a new and
ready way for the exact finding out of such lines and Logarithmes as are not
precisely to be found in the Canons.” It is an able tract. A natural guess is
that the editor of the book, Samuel Wright, a son of Edward Wright,
composed this “Appendix.” More probable is the conjecture which (Dr. J. W.

L. Glaisher informs me) was made by Augustus De Morgan, attributing the
authorship to Oughtred. Two reasons in support of this are advanced by Dr.
Glaisher, the use of x in the “Appendix” as the sign of multiplication (to [55
Oughtred is generally attributed the introduction of the cross × for
multiplication in 1631), and the then unusual designation “cathetus” for the
vertical leg of a right triangle, a term appearing in Oughtred’s books. We are
able to advance a third argument, namely, the occurrence in the “Appendix”
of (S*) as the notation for sine complement (cosine), while Seth Ward, an
early pupil of Oughtred, in his Idea trigonometriae demonstratae, Oxford,
1654, used a similar notation (S’). It has been stated elsewhere that Oughtred
claimed Seth Ward’s exposition of trigonometry as virtually his own.
Attention should be called also to the fact that, in his Trigonometria, p. 2,
Oughtred uses (’) to designate 180°-angle.

Dr. J. W. L. Glaisher is the first to call attention to other points of interest in
this “Appendix.” The interpolations are effected with the aid of a small table
containing the logarithms of 72 sines. Except for the omission of the decimal
point, these logarithms are natural logarithms—the first of their kind ever
published. In this table we find log 10=2302584; in modern notation, this is
stated, loge 10=2.302584. The first more extended table of natural logarithms
of numbers was published by John Speidell in the 1622 impression of his
New Logarithmes, which contains, besides trigonometric tables, the
logarithms of the numbers 1-1000.

The “Appendix” contains also the first account of a method of computing
logarithms, called the “radix method,” which is usually attributed to Briggs
who applied it in his Arithmetica logarithmica, 1624. In general, this method
consists in multiplying or dividing a number, whose logarithm is sought, by a
suitable factor and resolving the result into factors of the form 1±x/10ⁿ. [56
The logarithm of the number is then obtained by adding the previously
calculated logarithms of the factors. The method has been repeatedly
rediscovered, by Flower in 1771, Atwood in 1786, Leonelli in 1802,
Manning in 1806, Weddle in 1845, Hearn in 1847, and Orchard in 1848.

We conclude with the words of Dr. J. W. L. Glaisher:

The Appendix was an interesting and remarkable contribution to
mathematics, for in its sixteen small pages it contains (1) the first use of

the sign ×; (2) the first abbreviations, or symbols, for the sine, tangent,
cosine, and cotangent; (3) the invention of the radix method of calculating
[55]
logarithms; (4) the first table of hyperbolic logarithms.

[57

CHAPTER IV
OUGHTRED’S INFLUENCE UPON
MATHEMATICAL PROGRESS AND
TEACHING

OUGHTRED AND HARRIOT

Oughtred’s Clavis mathematicae was the most influential mathematical
publication in Great Britain which appeared in the interval between John
Napier’s Mirifici logarithmorum canonis descriptio, Edinburgh, 1614, and
the time, forty years later, when John Wallis began to publish his important
researches at Oxford. The year 1631 is of interest as the date of publication,
not only of Oughtred’s Clavis, but also of Thomas Harriot’s Artis analyticae
praxis. We have no evidence that these two mathematicians ever met.
Through their writings they did not influence each other. Harriot died ten
years before the appearance of his magnum opus, or ten years before the
publication of Oughtred’s Clavis. Strangely, Oughtred, who survived Harriot
thirty-nine years, never mentions him. There is no doubt that, of the two,
Harriot was the more original mind, more capable of penetrating into new
fields of research. But he had the misfortune of having a strong competitor in
René Descartes in the development of algebra, so that no single algebraic
achievement stands out strongly and conspicuously as Harriot’s own
contribution to algebraic science. As a text to serve as an introduction to
algebra, Harriot’s Artis analyticae praxis was inferior to Oughtred’s Clavis.
The former was a much larger book, not as conveniently portable, compiled
after the author’s death by others, and not prepared with the care in the [58
development of the details, nor with the coherence and unity and the

profound pedagogic insight which distinguish the work of Oughtred. Nor
was Harriot’s position in life such as to be surrounded by so wide a circle of
pupils as was Oughtred. To be sure, Harriot had such followers as Torporley,
William Lower, and Protheroe in Wales, but this group is small as compared
with Oughtred’s.

OUGHTRED’S PUPILS

There was a large number of distinguished men who, in their youth, either
visited Oughtred’s home and studied under his roof or else read his Clavis
and sought his assistance by correspondence. We permit Aubrey to
enumerate some of these pupils in his own gossipy style:

Seth Ward, M.A., a fellow of Sydney Colledge in Cambridge (now bishop
of Sarum), came to him, and lived with him halfe a yeare (and he would
not take a farthing for his diet), and learned all his mathematiques of him.
Sir Jonas More was with him a good while, and learn’t; he was but an
ordinary logist before. Sir Charles Scarborough was his scholar; so Dr.
John Wallis was his scholar; so was Christopher Wren his scholar, so was
Mr. . . . . Smethwyck, Regiae Societatis Socius. One Mr. Austin (a most
ingeniose man) was his scholar, and studyed so much that he became mad,
fell a laughing, and so dyed, to the great griefe of the old gentleman. Mr. .
. . . Stokes, another scholar, fell mad, and dream’t that the good old
gentleman came to him, and gave him good advice, and so he recovered,
and is still well. Mr. Thomas Henshawe, Regiae Societatis Socius, was his
scholar (then a young gentleman). But he did not so much like any as
those that tugged and tooke paines to worke out questions. He taught all
free.

He could not endure to see a scholar write an ill hand; he taught them [59
[56]
all presently to mend their hands.

Had Oughtred been the means of guiding the mathematical studies of only
John Wallis and Christopher Wren—one the greatest English mathematician
between Napier and Newton, the other one of the greatest architects of
England—he would have earned profound gratitude. But the foregoing list

embraces nine men, most of them distinguished in their day. And yet
Aubrey’s list is very incomplete. It is easy to more than double it by adding
the names of William Forster, who translated from Latin into English
Oughtred’s Circles of Proportion; Arthur Haughton, who brought out the
1660 Oxford edition of the Circles of Proportion; Robert Wood, an educator
and politician, who assisted Oughtred in the translation of the Clavis from
Latin into English for the edition of 1647; W. Gascoigne, a man of promise,
who fell in 1644 at Marston Moor; John Twysden, who was active as a
publisher; William Sudell, N. Ewart, Richard Shuttleworth, William
Robinson, and William Howard, the son of the Earl of Arundel, for whose
instruction Oughtred originally prepared the manuscript treatise that was
published in 1631 as the Clavis mathematicae.

Nor must we overlook the names of Lawrence Rooke (who “did admirably
well read in Gresham Coll. on the sixth chapt. of the said book,” the Clavis);
Christopher Brookes (a maker of mathematical instruments who married a
daughter of the famous mathematician); William Leech and William Brearly
(who with Robert Wood “have been ready and helpfull incouragers of me
[Oughtred] in this labour” of preparing the English Clavis of 1647), and [60
Thomas Wharton, who studied the Clavis and assisted in the editing of
the edition of 1647.

The devotion of these pupils offers eloquent testimony, not only of
Oughtred’s ability as a mathematician, but also of his power of drawing
young men to him—of his personal magnetism. Nor should we omit from the
list Richard Delamain, a teacher of mathematics in London, who
unfortunately had a bitter controversy with Oughtred on the priority and
independence of the invention of the circular slide rule and a form of sun-
dial. Delamain became later a tutor in mathematics to King Charles I, and
perished in the civil war, before 1645.

OUGHTRED, THE “TODHUNTER OF THE SEVENTEENTH
CENTURY”

To afford a clearer view of Oughtred as a teacher and mathematical expositor
we quote some passages from various writers and from his correspondence.

[57]
Anthony Wood gives an interesting account of how Seth Ward and
Charles Scarborough went from Cambridge University to the obscure home
of the country mathematician to be initiated into the mysteries of algebra:

Mr. Cha. Scarborough, then an ingenious young student and fellow of
Caius Coll. in the same university, was his [Seth Ward’s] great
acquaintance, and both being equally students in that faculty and desirous
to perfect themselves, they took a journey to Mr. Will. Oughtred living
then at Albury in Surrey, to be informed in many things in his Clavis
mathematica which seemed at that time very obscure to them. Mr.
Oughtred treated them with great humanity, being very much pleased to
see such ingenious young men apply themselves to these studies, and in
short time he sent them away well satisfied in their desires. When they
returned to Cambridge, they afterwards read the Clav. Math. to their [61
pupils, which was the first time that book was read in the said
university. Mr. Laur. Rook, a disciple of Oughtred, I think, and Mr. Ward’s
friend, did admirably well read in Gresham Coll. on the sixth chap. of the
said book, which obtained him great repute from some and greater from
Mr. Ward, who ever after had an especial favour for him.

Anthony Wood makes a similar statement about Thomas Henshaw:

While he remained in that coll. [University College, Oxford] which was
five years . . . . he made an excursion for about 9 months to the famous
mathematician Will. Oughtred parson of Aldbury in Surrey, by whom he
was initiated in the study of mathematics, and afterwards retiring to his
coll. for a time, he at length went to London, was entered a student in the
[58]
Middle Temple.

Extracts from letters of W. Gascoigne to Oughtred, of the years 1640 and
1641, throw some light upon mathematical teaching of the time:

Amongst the mathematical rarities these times have afforded, there are
none of that small number I (a late intruder into these studies) have yet
viewed, which so fully demonstrates their authors’ great abilities as your
Clavis, not richer in augmentations, than valuable for contraction; . . . .

Your belief that there is in all inventions aliquid divinum, an infusion
beyond human cogitations, I am confident will appear notably
strengthened, if you please to afford this truth belief, that I entered upon
these studies accidentally after I betook myself to the country, having
never had so much aid as to be taught addition, nor the discourse of an
artist (having left both Oxford and London before I knew what any
proposition in geometry meant) to inform me what were the best authors.
[59]

The following extracts from two letters by W. Robinson, written before [62
the appearance of the 1647 English edition of the Clavis, express the
feeling of many readers of the Clavis on its extreme conciseness and brevity
of explanation:

I shall long exceedingly till I see your Clavis turned into a pick-lock; and I
beseech you enlarge it, and explain it what you can, for we shall not need
to fear either tautology or superfluity; you are naturally concise, and your
clear judgment makes you both methodical and pithy; and your analytical
way is indeed the only way. . . . .

I will once again earnestly entreat you, that you be rather diffuse in the
setting forth of your English mathematical Clavis, than concise,
considering that the wisest of men noted of old, and said stultorum
infinitus est numerus, these arts cannot be made too easy, they are so
abstruse of themselves, and men either so lazy or dull, that their fastidious
wits take a loathing at the very entrance of these studies, unless it be
sweetened on with plainness and facility. Brevity may well argue a learned
author, that without any excess or redundance, either of matter or words,
can give the very substance and essence of the thing treated of; but it
seldom makes a learned scholar; and if one be capable, twenty are not; and
if the master sum up in brief the pith of his own long labours and travails,
it is not easy to imagine that scholars can with less labour than it cost their
[60]
masters dive into the depths thereof.

Here is the judgment of another of Oughtred’s friends:

. . . . with the character I received from your and my noble friend Sir
Charles Cavendish, then at Paris, of your second edition of the same piece,
made me at my return into England speedily to get, and diligently peruse
the same. Neither truly did I find my expectation deceived; having with
admiration often considered how it was possible (even in the hardest [63
things of geometry) to deliver so much matter in so few words, yet
with such demonstrative clearness and perspicuity: and hath often put me
in mind of learned Mersennus his judgment (since dead) of it, that there
was more matter comprehended in that little book than in Diophantus, and
[61]
all the ancients. . . . .

Oughtred’s own feeling was against diffuseness in textbook writing. In his
revisions of his Clavis the original character of that book was not altered. In
his reply to W. Robinson, Oughtred said:

. . . . But my art for all such mathematical inventions I have set down in
my Clavis Mathematica, which therefore in my title I say is tum logisticae
cum analyticae adeoque totius mathematicae quasi clavis, which if any
one of a mathematical genius will carefully study, (and indeed it must be
carefully studied,) he will not admire others, but himself do wonders. But I
(such is my tenuity) have enough fungi vice cotis, acutum reddere quae
ferrum valet, exsors ipsa secandi, or like the touchstone, which being but a
[62]
stone, base and little worth, can shew the excellence and riches of gold.

John Wallis held Oughtred’s Clavis in high regard. When in correspondence
with John Collins concerning plans for a new edition, Wallis wrote in 1666-
67, six years after the death of Oughtred:

. . . . But for the goodness of the book in itself, it is that (I confess) which I
look upon as a very good book, and which doth in as little room deliver as
much of the fundamental and useful part of geometry (as well as of
arithmetic and algebra) as any book I know; and why it should not be now
acceptable I do not see. It is true, that as in other things so in mathematics,
fashions will daily alter, and that which Mr. Oughtred designed by great
letters may be now by others be designed by small; but a mathematician
will, with the same ease and advantage, understand Ac, and a³ or aaa. . . . .
And the like I judge of Mr. Oughtred’s Clavis, which I look upon (as those

pieces of Vieta who first went in that way) as lasting books and classic [64
authors in this kind; to which, notwithstanding, every day may make
new additions. . . . .

But I confess, as to my own judgment, I am not for making the book
bigger, because it is contrary to the design of it, being intended for a
manual or contract; whereas comments, by enlarging it, do rather destroy
it. . . . . But it was by him intended, in a small epitome, to give the
[63]
substance of what is by others delivered in larger volumes. . . . .

That there continued to be a group of students and teachers who desired a
fuller exposition than is given by Oughtred is evident from the appearance,
over fifty years after the first publication of the Clavis, of a booklet by
Gilbert Clark, entitled Oughtredus Explicatus, London, 1682. A review of
this appeared in the Acta Eruditorum (Leipzig, 1684), on p. 168, wherein
Oughtred is named “clarissimus Angliae mathematicus.” John Collins wrote
Wallis in 1666-67 that Clark, “who lives with Sir Justinian Isham, within
seven miles of Northampton, . . . . intimates he wrote a comment on the
Clavis, which lay long in the hands of a printer, by whom he was abused,
[64]
meaning Leybourne.”

We shall have occasion below to refer to Oughtred’s inability to secure a
copy of a noted Italian mathematical work published a few years before. In
those days the condition of the book trade in England must have been
somewhat extraordinary. Dr. J. W. L. Glaisher throws some light upon this
[65] [65
subject. He found in the Calendar of State Papers, Domestic Series,
1637, a petition to Archbishop Laud in which it is set forth that when
Hooganhuysen, a Dutchman, “heretofore complained of in the High
Commission for importing books printed beyond the seas,” had been bound
“not to bring in any more,” one Vlacq (the computer and publisher of
logarithmic tables) “kept up the same agency and sold books in his stead. . . .
. Vlacq is now preparing to go beyond the seas to avoid answering his late
bringing over nine bales of books contrary to the decree of the Star
Chamber.” Judgment was passed that, “Considering the ill-consequence and
scandal that would arise by strangers importing and venting in this kingdom

books printed beyond the seas,” certain importations be prohibited, and
seized if brought over.

This want of easy intercommunication of results of scientific research in
Oughtred’s time is revealed in the following letter, written by Oughtred to
Robert Keylway, in 1645:

I speak this the rather, and am induced to a better confidence of your
performance, by reason of a geometric-analytical art or practice found out
by one Cavalieri, an Italian, of which about three years since I received
information by a letter from Paris, wherein was praelibated only a small
taste thereof, yet so that I divine great enlargement of the bounds of the
mathematical empire will ensue. I was then very desirous to see the
author’s own book while my spirits were more free and lightsome, but I
could not get it in France. Since, being more stept into years, daunted and
broken with the sufferings of these disastrous times, I must content myself
[66]
to keep home, and not put out to any foreign discoveries.

It was in 1655, when Oughtred was about eighty years old, that John Wallis,
the great forerunner of Newton in Great Britain, began to publish his [66
great researches on the arithmetic of infinites. Oughtred rejoiced over the
achievements of his former pupil. In 1655, Oughtred wrote John Wallis as
follows:

I have with unspeakable delight, so far as my necessary businesses, the
infirmness of my health, and the greatness of my age (approaching now to
an end) would permit, perused your most learned papers, of several choice
arguments, which you sent me: wherein I do first with thankfulness
acknowledge to God, the Father of lights, the great light he hath given
you; and next I congratulate you, even with admiration, the clearness and
perspicacity of your understanding and genius, who have not only gone,
but also opened a way into these profoundest mysteries of art, unknown
and not thought of by the ancients. With which your mysterious inventions
I am the more affected, because full twenty years ago, the learned patron
of learning, Sir Charles Cavendish, shewed me a paper written, wherein
were some few excellent new theorems, wrought by the way, as I suppose,
of Cavalieri, which I wrought over again more agreeably to my way. The
paper, wherein I wrought it, I shewed to many, whereof some took copies,

but my own I cannot find. I mention it for this, because I saw therein a
light breaking out for the discovery of wonders to be revealed to mankind,
in this last age of the world: which light I did salute as afar off, and now at
a nearer distance embrace in your prosperous beginnings. Sir, that you are
pleased to mention my name in your never dying papers, that is your noble
favour to me, who can add nothing to your glory, but only my applause. . .
[67]
..

The last sentence has reference to Wallis’ appreciative and eulogistic
reference to Oughtred in the preface. It is of interest to secure the opinion of
later English writers who knew Oughtred only through his books. John [67
Locke wrote in his journal under the date, June 24, 1681, “the best
[68]
algebra yet extant is Outred’s.” John Collins, who is known in the history
of mathematics chiefly through his very extensive correspondence with
nearly all mathematicians of his day, was inclined to be more critical. He
wrote Wallis about 1667:

It was not my intent to disparage the author, though I know many that did
lightly esteem him when living, some whereof are at rest, as Mr. Foster
and Mr. Gibson. . . . . You grant the author is brief, and therefore obscure,
and I say it is but a collection, which, if himself knew, he had done well to
have quoted his authors, whereto the reader might have repaired. You do
not like those words of Vieta in his theorems, ex adjunctione plano solidi,
plus quadrato quadrati, etc., and think Mr. Oughtred the first that abridged
those expressions by symbols; but I dissent, and tell you ’twas done before
[69]
by Cataldus, Geysius, and Camillus Gloriosus, who in his first decade
of exercises, (not the first tract,) printed at Naples in 1627, which was four
years before the first edition of the Clavis, proposeth this equation just as I
here give it you, viz. 1ccc+16qcc+41qqc-2304cc-18364qc-133000qq-
54505c+3728q+8064 N aequatur 4608, finds N or a root of it to be 24, and
composeth the whole out of it for proof, just in Mr. Oughtred’s symbols
and method. Cataldus on Vieta came out fifteen years before, and I cannot
quote that, as not having it by me.

. . . . And as for Mr. Oughtred’s method of symbols, this I say to it; it may
be proper for you as a commentator to follow it, but divers I know, men of

inferior rank that have good skill in algebra, that neither use nor approve
it. . . . . Is not A⁵ sooner wrote than Aqc? Let A be 2, the cube of 2 is 8,
which squared is 64: one of the questions between Maghet Grisio and [68
Gloriosus is whether 64=Acc or Aqc. The Cartesian method tells you it
[70]
is A⁶, and decides the doubt. . . . .

There is some ground for the criticisms passed by Collins. To be sure, the
first edition of the Clavis is dated 1631—six years before Descartes
suggested the exponential notation which came to be adopted as the
symbolism in our modern algebra. But the second edition of the Clavis,
1647, appeared ten years after Descartes’ innovation. Had Oughtred seen fit
to adopt the new exponential notation in 1647, the step would have been
epoch-making in the teaching of algebra in England. We have seen no
indication that Oughtred was familiar with Descartes’ Géométrie of 1637.

The year preceding Oughtred’s death Mr. John Twysden expressed himself
as follows in the preface to his Miscellanies:

It remains that I should adde something touching the beginning, and use of
these Sciences. . . . . I shall only, to their honours, name some of our own
Nation yet living, who have happily laboured upon both stages. That
succeeding ages may understand that in this of ours, there yet remained
some who were neither ignorant of these Arts, as if they had held them
vain, nor condemn them as superfluous. Amongst them all let Mr. William
Oughtred, of Aeton, be named in the first place, a Person of venerable
grey haires, and exemplary piety, who indeed exceeds all praise we can
bestow upon him. Who by an easie method, and admirable Key, hath
unlocked the hidden things of geometry. Who by an accurate
Trigonometry and furniture of Instruments, hath inriched, as well
geometry, as Astronomy. Let D. John Wallis, and D. Seth Ward, succeed in
the next place, both famous Persons, and Doctors in Divinity, the one of
geometry, the other of astronomy, Savilian Professors in the [69
[71]
University of Oxford.

The astronomer Edmund Halley, in his preface to the 1694 English edition of
the Clavis, speaks of this book as one of “so established a reputation, that it
were needless to say anything thereof,” though “the concise Brevity of the

author is such, as in many places to need Explication, to render it Intelligible
to the less knowing Mathematical matters.”

In closing this part of our monograph, we quote the testimony of Robert
Boyle, the experimental physicist, as given May 8, 1647, in a letter to Mr.
Hartlib:

The Englishing of, and additions to Oughtred’s Clavis mathematica does
much content me, I having formerly spent much study on the original of
that algebra, which I have long since esteemed a much more instructive
[72]
way of logic, than that of Aristotle.

WAS DESCARTES INDEBTED TO OUGHTRED?

This question first arose in the seventeenth century, when John Wallis, of
Oxford, in his Algebra (the English edition of 1685, and more particularly
the Latin edition of 1693), raised the issue of Descartes’ indebtedness to the
English scientists, Thomas Harriot and William Oughtred. In discussing
matters of priority between Harriot and Descartes, relating to the theory of
equations, Wallis is generally held to have shown marked partiality to
Harriot. Less attention has been given by historians of mathematics to [70
Descartes’ indebtedness to Oughtred. Yet this question is of importance
in tracing Oughtred’s influence upon his time.

On January 8, 1688-89, Samuel Morland addressed a letter of inquiry to John
Wallis, containing a passage which we translate from the Latin:

Some time ago I read in the elegant and truly precious book that you have
written on Algebra, about Descartes, this philosopher so extolled above all
for having arrived at a very perfect system by his own powers, without the
aid of others, this Descartes, I say, who has received in geometry very
great light from our Oughtred and our Harriot, and has followed their track
though he carefully suppressed their names. I stated this in a conversation
with a professor in Utrecht (where I reside at present). He requested me to
indicate to him the page-numbers in the two authors which justified this
accusation. I admitted that I could not do so. The Géométrie of Descartes

is not sufficiently familiar to me, although with Oughtred I am fairly
familiar. I pray you therefore that you will assume this burden. Give me at
least those references to passages of the two authors from the comparison
[73]
of which the plagiarism by Descartes is the most striking.

Following Morland’s letter in the De algebra tractatus, is printed Wallis’
reply, dated March 12, 1688 (“Stilo Angliae”), which is, in part, as follows:

I nowhere give him the name of a plagiarist; I would not appear so
impolite. However this I say, the major part of his algebra (if not all) is
found before him in other authors (notably in our Harriot) whom he does
not designate by name. That algebra may be applied to geometry, and that
it is in fact so applied, is nothing new. Passing the ancients in silence, we
state that this has been done by Vieta, Ghetaldi, Oughtred and others, [71
before Descartes. They have resolved by algebra and specious
arithmetic [literal arithmetic] many geometrical problems. . . . . But the
question is not as to application of algebra to geometry (a thing quite old),
but of the Cartesian algebra considered by itself.

Wallis then indicates in the 1659 edition of Descartes’ Géométrie where the
subjects treated on the first six pages are found in the writings of earlier
algebraists, particularly of Harriot and Oughtred. For example, what is found
on the first page of Descartes, relating to addition, subtraction,
multiplication, division, and root extraction, is declared by Wallis to be
drawn from Vieta, Ghetaldi, and Oughtred.

It is true that Descartes makes no mention of modern writers, except once of
Cardan. But it was not the purpose of Descartes to write a history of algebra.
To be sure, references to such of his immediate predecessors as he had read
would not have been out of place. Nevertheless, Wallis fails to show that
Descartes made illegitimate use of anything he may have seen in Harriot or
Oughtred.

The first inquiry to be made is, Did Descartes possess copies of the books of
Harriot and Oughtred? It is only in recent time that this question has been
answered as to Harriot. As to Oughtred, it is still unanswered. It is now
known that Descartes had seen Harriot’s Artis analyticae praxis (1631).

Descartes wrote a letter to Constantin Huygens in which he states that he is
[74]
sending Harriot’s book.

An able discussion of the question, what effect, if any, Oughtred’s Clavis
[75] [72
mathematicae of 1631 had upon Descartes’ Géométrie of 1637, is
given by H. Bosmans in a recent article. According to Bosmans no evidence
has been found that Descartes possessed a copy of Oughtred’s book, or that
he had examined it. Bosmans believes nevertheless that Descartes was
influenced by the Clavis, either directly or indirectly. He says:

If Descartes did not read it carefully, which is not proved, he was none the
less well informed with regard to it. No one denies his intimate knowledge
of the intellectual movement of his time. The Clavis mathematica enjoyed
a rapid success. It is impossible that, at least indirectly, he did not know
the more original ideas which it contained. Far from belittling Descartes,
[76]
as I much desire to repeat, this rather makes him the greater.

We ourselves would hardly go as far as does Bosmans. Unless Descartes
actually examined a copy of Oughtred it is not likely that he was influenced
by Oughtred in appreciable degree. Book reviews were quite unknown in
those days. No evidence has yet been adduced to show that Descartes
obtained a knowledge of Oughtred by correspondence. A most striking
feature about Oughtred’s Clavis is its notation. No trace of the Englishman’s
symbolism has been pointed out in Descartes’ Géométrie of 1637. Only six
years intervened between the publication of the Clavis and the Géométrie. It
took longer than this period for the Clavis to show evidence of its influence
upon mathematical books published in England; it is not probable that
abroad the contact was more immediate than at home. Our study of [73
seventeenth-century algebra has led us to the conviction that Oughtred
deserves a higher place in the development of this science than is usually
accorded to him; but that it took several decennia for his influence fully to
develop.

THE SPREAD OF OUGHTRED’S NOTATIONS

An idea of Oughtred’s influence upon mathematical thought and teaching
can be obtained from the spread of his symbolism. This study indicates that
the adoption was not immediate. The earliest use that we have been able to
find of Oughtred’s notation for proportion, A.B::C.D, occurs nineteen years
after the Clavis mathematicae of 1631. In 1650 John Kersey brought out in
London an edition of Edmund Wingates’ Arithmetique made easie, in which
this notation is used. After this date publications employing it became
frequent, some of them being the productions of pupils of Oughtred. We
[77] [78]
have seen it in Vincent Wing (1651), Seth Ward (1653), John Wallis
[79] [80]
(1655), in “R. B.,” a schoolmaster in Suffolk, Samuel Foster (1659),
[81] [82] [83]
Jonas Moore (1660), and Isaac Barrow (1657). In the latter part of
the seventeenth century Oughtred’s notation, A.B::C.D, became the [74
prevalent, though not universal, notation in Great Britain. A tremendous
impetus to their adoption was given by Seth Ward, Isaac Barrow, and
particularly by John Wallis, who was rising to international eminence as a
mathematician.

In France we have noticed Oughtred’s notation for proportion in Franciscus
[84] [85] [86]
Dulaurens (1667), J. Prestet (1675), R. P. Bernard Lamy (1684),
[87] [88] [89]
Ozanam (1691), De l’Hospital (1696), R. P. Petro Nicolas (1697).
[90]
In the Netherlands we have noticed it in R. P. Bernard Lamy (1680), and
[91]
in an anonymous work of 1690. In German and Italian works of the
seventeenth century we have not seen Oughtred’s notation for proportion.

In England a modified notation soon sprang up in which ratio was indicated
by two dots instead of a single dot, thus A:B::C:D. The reason for the change
lies probably in the inclination to use the single dot to designate decimal
fractions. W. W. Beman pointed out that this modified symbolism (:) for ratio
is found as early as 1657 in the end of the trigonometric and logarithmic [75
[92]
tables that were bound with Oughtred’s Trigonometria. It is not
probable, however, that this notation was used by Oughtred himself. The
Trigonometria proper has Oughtred’s A.B::C.D throughout. Moreover, in the

English edition of this trigonometry, which appeared the same year, 1657,
but subsequent to the Latin edition, the passages which contained the colon
as the symbol for ratio, when not omitted, are recast, and the regular
Oughtredian notation is introduced. In Oughtred’s posthumous work,
Opuscula mathematica hactenus inedita, 1677, the colon appears quite often
but is most likely due to the editor of the book.

We have noticed that the notation A:B::C:D antedates the year 1657. Vincent
Wing, the astronomer, published in 1651 in London the Harmonicon
coeleste, in which is found not only Oughtred’s notation A.B::C.D but also
the modified form of it given above. The two are used interchangeably. His
later works, the Logistica astronomica (1656), Doctrina spherica (1655), and
Doctrina theorica, published in one volume in London, all use the symbols
A:B::C:D exclusively. The author of a book entitled, An Idea of Arithmetick
at first designed for the use of the Free Schoole at Thurlow in Suffolk . . . . by
R. B., Schoolmaster there, London, 1655, writes A:a::C:c, though part of the
time he uses Oughtred’s unmodified notation.

We can best indicate the trend in England by indicating the authors of the
seventeenth century whom we have found using the notation A:B::C:D and
the authors of the eighteenth century whom we have found using A.B::C.D.
The former notation was the less common during the seventeenth but the [76
more common during the eighteenth century. We have observed the
symbols A:B::C:D (besides the authors already named) in John Collins
[93] [94] [95]
(1659), James Gregory (1663), Christopher Wren (1668-69),
[96] [97]
William Leybourn (1673), William Sanders (1686), John Hawkins
[98] [99] [100]
(1684), Joseph Raphson (1697), E. Wells (1698), and John Ward
[101]
(1698).

Of English eighteenth-century authors the following still clung to the
notation A.B::C.D: John Harris’ translation of F. Ignatius Gaston Pardies
[102] [103] [104]
(1701), George Shelley (1704), Sam Cobb (1709), J. Collins in
[105] [77
Commercium Epistolicum (1712), John Craig (1718), Jo. Wilson
[106]
(1724). The latest use of A.B::C.D which has come to our notice is in the

translation of the Analytical Institutions of Maria G. Agnesi, made by John
Colson sometime before 1760, but which was not published until 1801.
During the seventeenth century the notation A:B::C:D acquired almost
complete ascendancy in England.

In France Oughtred’s unmodified notation A.B::C.D, having been adopted
later, was also discarded later than in England. An approximate idea of the
situation appears from the following data. The notation A.B::C.D was used
[107] [108]
by M. Carré (1700), M. Guisnée (1705), M. de Fontenelle (1727),
[109] [110] [111]
M. Varignon (1725), M. Robillard (1753), M. Sebastien le Clerc
[112] [113] [114]
(1764), Clairaut (1731), M. L’Hospital (1781).

In Italy Oughtred’s modified notation a, b::c, d was used by Maria G. Agnesi
in her Instituzioni analitiche, Milano, 1748. The notation a:b::c:d found [78
entrance the latter part of the eighteenth century. In Germany the
[115]
symbolism a:b=c:d, suggested by Leibniz, found wider acceptance.

It is evident from the data presented that Oughtred proposed his notation for
ratio and proportion at a time when the need of a specific notation began to
be generally felt, that his symbol for ratio a.b was temporarily adopted in
England and France but gave way in the eighteenth century to the symbol
a:b, that Oughtred’s symbol for proportion :: found almost universal
adoption in England and France and was widely used in Italy, the
Netherlands, the United States, and to some extent in Germany; it has
survived to the present time but is now being gradually displaced by the sign
of equality =.

Oughtred’s notation to express aggregation of terms has received little
attention from historians but is nevertheless interesting. His books, as [79
well as those of John Wallis, are full of parentheses but they are not used
as symbols of aggregation in algebra; they are simply marks of punctuation
for parenthetical clauses. We have seen that Oughtred writes (a+b)² and √a+b
thus, Q:a+b:, √:a+b:, or Q:a+b, √:a+b, using on rarer occasions a single dot
in place of the colon. This notation did not originate with Oughtred, but, in
slightly modified form, occurs in writings from the Netherlands. In 1603 C.
Dibvadii in geometriam Evclidis demonstratio numeralis, Leyden, contains

many expressions of this sort, √·136+√2048, signifying √(136+√2048). The
dot is used to indicate that the root of the binomial (not of 136 alone) is
called for. This notation is used extensively in Ludolphi à Cevlen de circulo,
Leyden, 1619, and in Willebrordi Snellii De circuli dimensione, Leyden,
1621. In place of the single dot Oughtred used the colon (:), probably to [80
avoid confusion with his notation for ratio. To avoid further possibility of
uncertainty he usually placed the colon both before and after the algebraic
expression under aggregation. This notation was adopted by John Wallis and
Isaac Barrow. It is found in the writings of Descartes. Together with Vieta’s
horizontal bar, placed over two or more terms, it constituted the means used
almost universally for denoting aggregation of terms in algebra. Before
[116]
Oughtred the use of parentheses had been suggested by Clavius and
[117]
Girard. The latter wrote, for instance, √(2+√3). While parentheses never
became popular in algebra before the time of Leibniz and the Bernoullis they
were by no means lost sight of. We are able to point to the following authors
[118]
who made use of them: I. Errard de Bar-le-Duc (1619), Jacobo de Billy
[119]
(1643), one of whose books containing this notation was translated into
[120]
English, and also the posthumous works of Samuel Foster. J. W. L.
Glaisher points out that parentheses were used by Norwood in his
[121]
Trigonometrie (1631), p. 30.

The symbol for the arithmetical difference between two numbers, ~, is [81
usually attributed to John Wallis, but it occurs in Oughtred’s Clavis
mathematicae of 1652, in the tract on Elementi decimi Euclidis declaratio, at
an earlier date than in any of Wallis’ books. As Wallis assisted in putting this
edition through the press it is possible, though not probable, that the symbol
was inserted by him. Were the symbol Wallis’, Oughtred would doubtless
have referred to its origin in the preface. During the eighteenth century the
[122]
symbol found its way into foreign texts even in far-off Italy. It is one of
three symbols presumably invented by Oughtred and which are still used at
the present time. The others are × and ::.

The curious and ill-chosen symbols, for “greater than,” and for “less
than,” were certain to succumb in their struggle for existence against
Harriot’s admirably chosen > and <. Yet such was the reputation of Oughtred
that his symbols were used in England quite extensively during the
seventeenth and the beginning of the eighteenth century. Considerable
confusion has existed among algebraists and also among historians as to
what Oughtred’s symbols really were. Particularly is this true of the sign for
“less than” which is frequently written . Oughtred’s symbols, or these
[123]
symbols turned about in some way, have been used by Seth Ward, John
[124] [125] [126] [127] [82
Wallis, Isaac Barrow, John Kersey, E. Wells, John
[128] [129] [130] [131]
Hawkins, Tho. Baker, Richard Sault, Richard Rawlinson,
[132] [133] [134]
Franciscus Dulaurens, James Milnes, George Cheyne, John
[135] [136] [137]
Craig, Jo. Wilson, and J. Collins.

General acceptance has been accorded to Oughtred’s symbol ×. The first
printed appearance of this symbol for multiplication in 1618 in the form of
the letter x hardly explains its real origin. The author of the “Appendix” (be
he Oughtred or someone else) may not have used the letter x at all, but may
have written the cross ×, called the St. Andrew’s cross, while the printer, in
the absence of any type accurately representing that cross, may have
substituted the letter x in its place. The hypothesis that the symbol × of
multiplication owes its origin to the old habit of using directed bars to

indicate that two numbers are to be combined, as for instance in the [83
multiplication of 23 and 34, thus,

2 3

|×|
3 4

7 8 2

[138] [139]
has been advanced by two writers, C. Le Paige and Gravelaar.
Bosmans is more inclined to the belief that Oughtred adopted the symbol
somewhat arbitrarily, much as he did the numerous symbols in his Elementi
[140]
decimi Euclidis declaratio.

Le Paige’s and Gravelaar’s theory finds some support in the fact that the
cross ×, without the two additional vertical lines shown above, occurs in a
[141]
commentary published by Oswald Schreshensuchs in 1551, where the
sign is written between two factors placed one above the other.

[84

CHAPTER V
OUGHTRED’S IDEAS ON THE TEACHING OF
MATHEMATICS

GENERAL STATEMENT

Nowhere has Oughtred given a full and systematic exposition of his views
on mathematical teaching. Nevertheless, he had very pronounced and clear-
cut ideas on the subject. That a man who was not a teacher by profession
should have mature views on teaching is most interesting. We gather his
ideas from the quality of the books he published, from his prefaces, and from
passages in his controversial writing against Delamain. As we proceed to
give quotations unfolding Oughtred’s views, we shall observe that three
points receive special emphasis: (1) an appeal to the eye through suitable
symbolism; (2) emphasis upon rigorous thinking; (3) the postponement of
the use of mathematical instruments until after the logical foundations of a
subject have been thoroughly mastered.

The importance of these tenets is immensely reinforced by the conditions of
the hour. This voice from the past speaks wisdom to specialists of today.
Recent methods of determining educational values and the modern cult of
utilitarianism have led some experts to extraordinary conclusions.
Laboratory methods of testing, by the narrowness of their range, often
mislead. Thus far they have been inferior to the word of a man of experience,
insight, and conviction.

[85

MATHEMATICS, “A SCIENCE OF THE EYE”

Oughtred was a great admirer of the Greek mathematicians—Euclid,
Archimedes, Apollonius of Perga, Diophantus. But in reading their works he
experienced keenly what many modern readers have felt, namely, that the
almost total absence of mathematical symbols renders their writings
unnecessarily difficult to read. Statements that can be compressed into a few
well-chosen symbols which the eye is able to survey as a whole are
expressed in long-drawn-out sentences. A striking illustration of the
importance of symbolism is afforded by the history of the formula

ix=log(cos x+i sin x).

It was given in Roger Cotes’ Harmonia mensurarum, 1722, not in symbols,
but expressed in rhetorical form, destitute of special aids to the eye. The
result was that the theorem remained in the book undetected for 185 years
and was meanwhile rediscovered by others. Owing to the prominence of
Cotes as a mathematician it is very improbable that such a thing could have
happened had the theorem been thrust into view by the aid of mathematical
symbols.

In studying the ancient authors Oughtred is reported to have written down on
the margin of the printed page some of the theorems and their proofs,
expressed in the symbolic language of algebra.

In the preface of his Clavis of 1631 and of 1647 he says:

Wherefore, that I might more clearly behold the things themselves, I
uncasing the Propositions and Demonstrations out of their covert of
words, designed them in notes and species appearing to the very eye. After
that by comparing the divers affections of Theorems, inequality, [86
proportion, affinity, and dependence, I tryed to educe new out of them.

It was this motive which led him to introduce the many abbreviations in
algebra and trigonometry to which reference has been made in previous
pages. The pedagogical experience of recent centuries has indorsed
Oughtred’s view, provided of course that the pupil is carefully taught the
exact meaning of the symbols. There have been and there still are those who

oppose the intensive use of symbolism. In our day the new symbolism for all
mathematics, suggested by the school of Peano in Italy, can hardly be said to
be received with enthusiasm. In Oughtred’s day symbolism was not yet the
fashion. To be convinced of this fact one need only open a book of Edmund
Gunter, with whom Oughtred came in contact in his youth, or consult the
Principia of Sir Isaac Newton, who flourished after Oughtred. The
mathematical works of Gunter and Newton, particularly the former, are
surprisingly destitute of mathematical symbols. The philosopher Hobbes, in
a controversy with John Wallis, criticized the latter for that “Scab of
Symbols,” whereupon Wallis replied:

I wonder how you durst touch M. Oughtred for fear of catching the Scab.
For, doubtlesse, his book is as much covered over with the Scab of
Symbols, as any of mine. . . . . As for my Treatise of Conick Sections, you
say, it is covered over with the Scab of Symbols, that you had not the
[142]
patience to examine whether it is well or ill demonstrated.

Oughtred maintained his view of the importance of symbols on many [87
different occasions. Thus, in his Circles of Proportion, 1632, p. 20:

This manner of setting downe Theoremes, whether they be Proportions, or
Equations, by Symboles or notes of words, is most excellent, artificiall,
and doctrinall. Wherefore I earnestly exhort every one, that desireth
though but to looke into these noble Sciences Mathematicall, to accustome
themselves unto it: and indeede it is easie, being most agreeable to reason,
yea even to sence. And out of this working may many singular
consectaries be drawne: which without this would, it may be, for ever lye
hid.

RIGOROUS THINKING AND THE USE OF INSTRUMENTS

The author’s elevated concept of mathematical study as conducive to
rigorous thinking shines through the following extract from his preface to the
1647 Clavis:

. . . . Which Treatise being not written in the usuall synthetical manner, nor
with verbous expressions, but in the inventive way of Analitice, and with
symboles or notes of things instead of words, seemed unto many very
hard; though indeed it was but their owne diffidence, being scared by the
newnesse of the delivery; and not any difficulty in the thing it selfe. For
this specious and symbolicall manner, neither racketh the memory with
multiplicity of words, nor chargeth the phantasie with comparing and
laying things together; but plainly presenteth to the eye the whole course
and processe of every operation and argumentation.

Now my scope and intent in the first Edition of that my Key was, and in
this New Filing, or rather forging of it, is, to reach out to the ingenious
lovers of these Sciences, as it were Ariadnes thread, to guide them through
the intricate Labyrinth of these studies, and to direct them for the more
easie and full understanding of the best and antientest Authors. . . . . [88
That they may not only learn their propositions, which is the highest
point of Art that most Students aime at; but also may perceive with what
solertiousnesse, by what engines of aequations, Interpretations,
Comparations, Reductions, and Disquisitions, those antient Worthies have
beautified, enlarged, and first found out this most excellent Science. . . . .
Lastly, by framing like questions problematically, and in a way of
Analysis, as if they were already done, resolving them into their
principles, I sought out reasons and means whereby they might be
effected. And by this course of practice, not without long time, and much
industry, I found out this way for the helpe and facilitation of Art.

Still greater emphasis upon rigorous thinking in mathematics is laid in the
preface to the Circles of Proportion and in some parts of his Apologeticall
Epistle against Delamain. In that preface William Forster quotes the reply of
Oughtred to the question how he (Oughtred) had for so many years
concealed his invention of the slide rule from himself (Forster) whom he had
taught so many other things. The reply was:

That the true way of Art is not by Instruments, but by Demonstration: and
that it is a preposterous course of vulgar Teachers, to begin with
Instruments, and not with the Sciences, and so in-stead of Artists, to make
their Scholers only doers of tricks, and as it were Iuglers: to the despite of
Art, losse of previous time, and betraying of willing and industrious wits,

vnto ignorance, and idlenesse. That the vse of Instruments is indeed
excellent, if a man be an Artist: but contemptible, being set and opposed to
Art. And lastly, that he meant to commend to me, the skill of Instruments,
but first he would haue me well instructed in the Sciences.”

Delamain took a different view, arguing that instruments might very well be
placed in the hands of pupils from the start. At the time of this controversy
Delamain supported himself by teaching mathematics in London and he [89
advertised his ability to give instruction in mathematics, including the
use of instruments. Delamain brought the charge against Oughtred of
unjustly calling “many of the [British] Nobility and Gentry doers of trickes
and juglers.” To this Oughtred replies:

As I did to Delamain and to some others, so I did to William Forster: I
freely gave him my helpe and instruction in these faculties: only this was
the difference, I had the very first moulding (as I may say) of this latter:
But Delamain was already corrupted with doring upon Instruments, and
quite lost from ever being made an Artist: I suffered not William Forster
for some time so much as speake of any Instrument, except only the Globe
it selfe; and to explicate, and worke the questions of the Sphaere, by the
way of the Analemma: which also himselfe did describe for the present
occasion. And this my restraint from such pleasing avocations, and
holding him to the strictnesse of percept, brought forth this fruit, that in
short time, even by his owne skill, he could not onely use any Instrument
he should see, but also was able to delineate the like, and devise others.
[143]

As representing Delamain’s views, we make the following selection from his
Grammelogia (London, about 1633), the part near the end of the book and
bearing the title, “In the behalfe of vulgar Teachers and others,” where
Delamain refers to Oughtred’s charge that the scholars of “vulgar” teachers
are “doers of tricks, as it were iuglers.” Delamain says:

. . . . Which words are neither cautelous, nor subterfugious, but are as
downe right in their plainnesse, as they are touching, and pernitious, by
two much derogating from many, and glancing upon many noble
personages, with too grosse, if not too base an attribute, in tearming them
doers of tricks, as it were to iuggle: because they perhaps make use of a

necessitie in the furnishing of themselves with such knowledge by [90
Practicall Instrumentall operation, when their more weighty
negotiations will not permit them for Theoreticall figurative
demonstration; those that are guilty of the aspertion, and are touched
therewith may answer for themselves, and studie to be more Theoreticall,
than Practicall: for the Theory, is as the Mother that produceth the
daughter, the very sinewes and life of Practise, the excellencie and
highest degree of true Mathematicall Knowledge: but for those that would
make but a step as it were into that kind of Learning, whose onely desire
is expedition, and facilitie, both which by the generall consent of all are
best effected with Instrument, rather then with tedious regular
demonstrations, it was ill to checke them so grosly, not onely in what they
have Practised, but abridging them also of their liberties with what they
may Practise, which aspertion may not easily be slighted off by any glosse
or Apologie, without an Ingenuous confession, or some mentall
reservation: To which vilification, howsoever, in the behalfe of my selfe,
and others, I answer; That Instrumentall operation is not only the
Compendiating, and facilitating of Art, but even the glory of it, whole
demonstration both of the making, and operation is soly in the science,
and to an Artist or disputant proper to be knowne, and so to all, who would
truly know the cause of the Mathematicall operations in their originall;
But, for none to know the use of a Mathematicall Instrumen[t], except he
knowes the cause of its operation, is somewhat too strict, which would
keepe many from affecting the Art, which of themselves are ready enough
every where, to conceive more harshly of the difficultie, and impossibilitie
of attayning any skill therein, then it deserves, because they see nothing
but obscure propositions, and perplex and intricate demonstrations before
their eyes, whose unsavoury tartnes, to an unexperienced palate like bitter
pills is sweetned over, and made pleasant with an Instrumentall
compendious facilitie, and made to goe downe the more readily, and yet to
retaine the same vertue, and working; And me thinkes in this queasy [91
age, all helpes may bee used to procure a stomacke, all bates and
invitations to the declining studie of so noble a Science, rather then by
rigid Method and generall Lawes to scarre men away. All are not of like
disposition, neither all (as was sayd before) propose the same end, some
resolve to wade, others to put a finger in onely, or wet a hand: now thus to
tye them to an obscure and Theoricall forme of teaching, is to crop their
hope, even in the very bud. . . . . The beginning of a mans knowledge even

in the use of an Instrument, is first founded on doctrinal precepts, and
these precepts may be conceived all along in its use: and are so farre from
being excluded, that they doe necessarily concomitate and are contained
therein: the practicke being better understood by the doctrinall part, and
this later explained by the Instrumentall, making precepts obvious unto
sense, and the Theory going along with the Instrument, better informing
and inlightning the understanding, etc. vis vnita fortior, so as if that in
Phylosophy bee true, Nihil est [in] intellectu quod non prius fuit in sensu.

The difference between Oughtred and Delamain as to the use of
mathematical instruments raises important questions. Should the slide rule be
placed in the hands of a boy before, or after, he has mastered the theory of
logarithms? Should logarithmic tables be withheld from him until the
theoretical foundation is laid in the mind of the pupil? Is it a good thing to let
a boy use a surveying instrument unless he first learns trigonometry? Is it
advisable to permit a boy to familiarize himself with the running of a
dynamo before he has mastered the underlying principles of electricity?
Does the use of instruments ordinarily discourage a boy from mastery of the
theory? Or does such manipulation constitute a natural and pleasing
approach to the abstract? On this particular point, who showed the
profounder psychological insight, Oughtred or Delamain?

In July, 1914, there was held in Edinburgh a celebration of the three- [92
hundredth anniversary of the invention of logarithms. On that occasion
there was collected at Edinburgh university one of the largest exhibits ever
seen of modern instruments of calculation. The opinion was expressed by an
experienced teacher that “weapons as those exhibited there are for men and
not for boys, and such danger as there may be in them is of the same
character as any form of too early specialization.”

It is somewhat of a paradox that Oughtred, who in his student days and
during his active years felt himself impelled to invent sun-dials,
planispheres, and various types of slide rules—instruments which represent
the most original contributions which he handed down to posterity—should
discourage the use of such instruments in teaching mathematics to beginners.
That without the aid of instruments he himself should have succeeded so
well in attracting and inspiring young men constitutes the strongest evidence
of his transcendent teaching ability. It may be argued that his pedagogic

dogma, otherwise so excellent, here goes contrary to the course he himself
followed instinctively in his self-education along mathematical lines. We
read that Sir Isaac Newton, as a child, constructed sun-dials, windmills,
kites, paper lanterns, and a wooden clock. Should these activities have been
suppressed? Ordinary children are simply Isaac Newtons on a smaller
intellectual scale. Should their activities along these lines be encouraged or
checked?

On the other hand, it may be argued that the paradox alluded to above admits
of explanation, like all paradoxes, and that there is no inconsistency between
Oughtred’s pedagogic views and his own course of development. If he
invented sun-dials, he must have had a comprehension of the cosmic [93
motions involved; if he solved spherical triangles graphically by the aid
of the planisphere, he must have understood the geometry of the sphere, so
far as it relates to such triangles; if he invented slide rules, he had beforehand
a thorough grasp of logarithms. The question at issue does not involve so
much the invention of instruments, as the use by the pupil of instruments
already constructed, before he fully understands the theory which is
involved. Nor does Sir Isaac Newton’s activity as a child establish
Delamain’s contention. Of course, a child should not be discouraged from
manual activity along the line of producing interesting toys in imitation of
structures and machines that he sees, but to introduce him to the realm of
abstract thought by the aid of instruments is a different proposition, fraught
with danger. A boy may learn to use a slide rule mechanically and, because
of his ability to obtain practical results, feel justified in foregoing the mastery
of underlying theory; or he may consider the ability of manipulating a
surveying instrument quite sufficient, even though he be ignorant of
geometry and trigonometry; or he may learn how to operate a dynamo and an
electric switchboard and be altogether satisfied, though having no grasp of
electrical science. Thus instruments draw a youth aside from the path leading
to real intellectual attainments and real efficiency; they allure him into lanes
which are often blind alleys. Such were the views of Oughtred.

Who was right, Oughtred or Delamain? It may be claimed that there is a
middle ground which more nearly represents the ideal procedure in teaching.
Shall the slide rule be placed in the student’s hands at the time when he is
engaged in the mastery of principles? Shall there be an alternate study of the
theory of logarithms and of the slide rule—on the idea of one hand washing

the other—until a mastery of both the theory and the use of the [94
instrument has been attained? Does this method not produce the best and
most lasting results? Is not this Delamain’s actual contention? We leave it to
the reader to settle these matters from his own observation, knowledge, and
experience.

NEWTON’S COMMENTS ON OUGHTRED

Oughtred is an author who has been found to be of increasing interest to
modern historians of mathematics. But no modern writer has, to our
knowledge, pointed out his importance in the history of the teaching of
mathematics. Yet his importance as a teacher did receive recognition in the
seventeenth century by no less distinguished a scientist than Sir Isaac
Newton. On May 25, 1694, Sir Isaac Newton wrote a long letter in reply to a
request for his recommendation on a proposed new course of study in
mathematics at Christ’s Hospital. Toward the close of his letter, Newton
says:

And now I have told you my opinion in these things, I will give you Mr.
Oughtred’s, a Man whose judgment (if any man’s) may be safely relyed
upon. For he in his book of the circles of proposition, in the end of what
he writes about Navigation (page 184) has this exhortation to Seamen.
“And if,” saith he, “the Masters of Ships and Pilots will take the pains in
the Journals of their Voyages diligently and faithfully to set down in
severall columns, not onely the Rumb they goe on and the measure of the
Ships way in degrees, and the observation of Latitude and variation of
their compass; but alsoe their conjectures and reason of their correction
they make of the aberrations they shall find, and the qualities and
condition of their ship, and the diversities and seasons of the winds, and
the secret motions or agitations of the Seas, when they begin, and how [95
long they continue, how farr they extend and with what inequality;
and what else they shall observe at Sea worthy consideration, and will be
pleased freely to communicate the same with Artists, such as are indeed
skilfull in the Mathematicks and lovers and enquirers of the truth: I doubt
not but that there shall be in convenient time, brought to light many
e
necessary precepts which may tend to y perfecting of Navigation, and the

help and safety of such whose Vocations doe inforce them to commit their
lives and estates in the vast Ocean to the providence of God.” Thus farr
that very good and judicious man Mr. Oughtred. I will add, that if instead
of sending the Observations of Seamen to able Mathematicians at Land,
the Land would send able Mathematicians to Sea, it would signify much
t
more to the improvem of Navigation and safety of Mens lives and estates
[144]
on that element.

May Oughtred prove as instructive to the modern reader as he did to
Newton!

[96

Footnotes

[1]
Aubrey’s Brief Lives, ed. A. Clark, Vol. II, Oxford, 1898, p. 106.

[2]
“To the English Gentrie, and all others studious of the Mathematicks,
which shall bee Readers hereof. The just Apologie of Wil: Ovghtred,
against the slaunderous insimulations of Richard Delamain, in a
Pamphlet called Grammelogia, or the Mathematicall Ring, or Mirifica
logarithmorum projectio circularis” [1633?], p. 8. Hereafter we shall
refer to this pamphlet as the Apologeticall Epistle, this name appearing
on the page-headings.

[3]
Companion to the [British] Almanac of 1837, p. 28, in an article by
Augustus De Morgan on “Notices of English Mathematical and
Astronomical Writers between the Norman Conquest and the Year 1600.”

[4]
New and General Biographical Dictionary (John Nichols), London, 1784,
art. “Oughtred.”

[5]
Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, p. 132.

[6]
Skeleton Collegii Regalis Cantab.: Or A Catalogue of All the Provosts,
Fellows and Scholars, of the King’s College . . . . since the Foundation
Thereof, Vol. II, “William Oughtred.”

[7]
Aubrey, op. cit., Vol. II, p. 107.

[8]
Rigaud, Correspondence of Scientific Men of the Seventeenth Century,
Oxford, Vol. I, 1841, p. 5.

[9]
Aubrey, op. cit., Vol. II, p. 110.

[10]
Ibid., p. 111.

[11]
Op. cit., Vol. II, p. 132.

[12]
Mr. William Lilly’s History of His Life and Times, From the Year 1602 to
1681, London, 1715, p. 58.

[13]
Rigaud, op. cit., Vol. I, p. 60.

[14]
Aubrey, op. cit., Vol. II, p. 107.

[15]
Rigaud, op. cit., Vol. I, p. 16.

[16]
Owen Manning, op. cit., p. 132.

[17]
New and General Biographical Dictionary (John Nichols), London, 1784,
art. “Oughtred.”

[18]
Op. cit., Vol. II, p. 110.

[19]
Rev. Owen Manning, The History and Antiquities of Surrey, Vol. II,
London, 1809, p. 132.

[20]
Op. cit., Vol. II, 1898, p. 111.

[21]
Budget of Paradoxes, London, 1872, p. 451; 2d ed., Chicago and London,
1915, Vol. II, p. 303.

[22]
The full title of the Clavis of 1631 is as follows: Arithmeticae in numeris
et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae, atqve adeo
totivs mathematicae, qvasi clavis est.—Ad nobilissimvm
spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur,
Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae &
Svrriae, Comitis Mareschalli Angliae, &c filium.—Londini, Apud
Thomam Harpervm. M.DC.XXXI.

In all there appeared five Latin editions, the second in 1648 at
London, the third in 1652 at Oxford, the fourth in 1667 at Oxford, the
fifth in 1693 and 1698 at Oxford. There were two independent
English editions: the first in 1647 at London, translated in greater part
by Robert Wood of Lincoln College, Oxford, as is stated in the
preface to the 1652 Latin edition; the second in 1694 and 1702 is a
new translation, the preface being written and the book recommended
by the astronomer Edmund Halley. The 1694 and 1702 impressions
labored under the defect of many sense-disturbing errors due to
careless reading of the proofs. All the editions of the Clavis, after the
first edition, had one or more of the following tracts added on:

Eq.=De Aequationum affectarvm resolvtione in numeris.
Eu.=Elementi decimi Euclidis declaratio.
So.=De Solidis regularibus, tractatus.
An.=De Anatocismo, sive usura composita.
Fa.=Regula falsae positionis.
Ar.=Theorematum in libris Archimedis de Sphaera & cylindro
declaratio.

Ho.=Horologia scioterica in plano, geometricè delineandi modus.

The abbreviated titles given here are, of course, our own. The lists of
tracts added to the Clavis mathematicae of 1631 in its later editions,
given in the order in which the tracts appear in each edition, are as
follows: Clavis of 1647, Eq., An., Fa., Ho.; Clavis of 1648, Eq., An.,
Fa., Eu., So.; Clavis of 1652, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis
of 1667, Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1693 and 1698,
Eq., Eu., So., An., Fa., Ar., Ho.; Clavis of 1694 and 1702, Eq.

The title-page of the Clavis was considerably modified after the first
edition. Thus, the 1652 Latin edition has this title-page: Guilelmi
Oughtred Aetonensis, quondam Collegii Regalis in Cantabrigia Socii,
Clavis mathematicae denvo limata, sive potius fabricata. Cum aliis
quibusdam ejusdem commentationibus, quae in sequenti pagina
recensentur. Editio tertia auctior & emendatior. Oxoniae, Excudebat
Leon. Lichfield, Veneunt apud Tho. Robinson. 1652.

[23]
Rigaud, op. cit., Vol. II, p. 476.

[24]
See, for instance, the Clavis mathematicae of 1652, where he expresses
himself thus (p. 4): “Speciosa haec Arithmetica arti Analyticae (per quam
ex sumptione quaesiti, tanquam noti, investigatur quaesitum) multo
accommodatior est, quam illa numerosa.”

[25]
Oughtred, The Key of the Mathematicks, London, 1647, p. 4.

[26]
Clavis 1694, p. 19, and the Clavis of 1631, p. 8.

[27]
See for instance, Oughtred’s Elementi decimi Euclidis declaratio, 1652, p.
1, where he uses A and E, and also a and e.

[28]
See Christophori Clavii Bambergensis Operum mathematicorum, tomus
secundus, Moguntiae, M.DC.XI, algebra, p. 39.

[29]
Christophori Clavii operum mathematicorum Tomus Secundus,
Moguntiae, M.DC.XI, Epitome arithmeticae, p. 36.

[30]
See F. Cajori, “The Cross × as a Symbol of Multiplication,” in Nature,
Vol. XCIV (1914), p. 363.

[31]
See Elementi decimi Euclidis declaratio, 1652, p. 2.

[32]
See Johannis Wallisii Operum mathematicorum pars prima, Oxonii,
1657, p. 247.

[33]
Clavis of 1631, chap. xix, sec. 5, p. 50.

[34]
We have noticed the representation of known quantities by consonants
and the unknown by vowels in Wingate’s Arithmetick made easie, edited
by John Kersey, London, 1650, algebra, p. 382; and in the second part,
section 19, of Jonas Moore’s Arithmetick in two parts, London, 1660,
Moore suggests as an alternative the use of z, y, x, etc., for the unknowns.
The practice of representing unknowns by vowels did not spread widely
in England.

[35]
Philosophical Transactions, Vol. XIX, No. 231, London, p. 652.

[36]
Ibid., Vol. XIX, p. 56.

[37]
There are two title-pages to the edition of 1632. The first title-page is as
follows: The Circles of Proportion and The Horizontall Instrument. Both
invented, and the vses of both Written in Latine by Mr. W. O. Translated
into English: and set forth for the publique benefit by William Forster.
London. Printed for Elias Allen maker of these and all other
mathematical Instruments, and are to be sold at his shop over against St.
Clements church with out Temple-barr. 1632. T. Cecill Sculp.

In 1633 there was added the following, with a separate title-page: An
addition vnto the Vse of the Instrvment called the Circles of
Proportion. . . . . London, 1633, this being followed by Oughtred’s To
the English Gentrie etc. In the British Museum there is a copy of
another impression of the Circles of Proportion, dated 1639, with the
Addition vnto the Vse of the Instrument etc., bearing the original date,
1633, and with the epistle, To the English Gentrie, etc., inserted
immediately after Forster’s dedication, instead of at the end of the
volume.

[38]
The complete title of the English edition is as follows: Trigonometrie, or,
The manner of calculating the Sides and Angles of Triangles, by the
Mathematical Canon, demonstrated. By William Oughtred Etonens. And
published by Richard Stokes Fellow of Kings Colledge in Cambridge,
and Arthur Haughton Gentleman. London, Printed by R. and W.
Leybourn, for Thomas Johnson at the Golden Key in St. Pauls Church-
yard. M.DC.LVII.

[39]
Jer. Collier, The Great Historical, Geographical, Genealogical and
Poetical Dictionary, Vol. II, London, 1701, art. “Oughtred.”

[40]
Rigaud op. cit., Vol. I, p. 82.

[41]
A. De Morgan, Budget of Paradoxes, London, 1872, p. 451; 2d ed.,
Chicago, 1915, Vol. II, p. 303.

[42]
E. Gunter, Description and Use of the Sector, the Crosse-staffe and other
Instruments, London, 1624, second book, p. 31.

[43]
F. Cajori, “On the History of a Notation in Trigonometry,” Nature, Vol.
XCIV, 1915, pp. 642, 643.

[44]
A. von Braunmühl, Geschichte der Trigonometrie, 2. Teil, Leipzig, 1903,
pp. 42, 91.

[45]
H. Hankel, Geschichte der Mathematik in Alterthum und Mittelalter,
Leipzig, 1874, pp. 369, 370.

[46]
M. Cantor, Vorlesungen über Geschichte der Mathematik, II, 1900, pp.
640, 641.

[47]
This matter has been discussed in a paper by F. Cajori, “A History of the
Arithmetical Methods of Approximation, etc., Colorado College
Publication, General Series No. 51, 1910, pp. 182-84. Later this subject
was again treated by G. Eneström in Bibliotheca mathematica, 3. Folge,
Vol. XI, 1911, pp. 234, 235.

[48]
See F. Cajori, op. cit., p. 193.

[49]
See William Oughtred’s Key of the Mathematicks, London, 1694, pp. 173-
75, tract, “Of the Resolution of the Affected Equations,” or any edition of
the Clavis after the first.

[50]
A. De Morgan, op. cit., p. 451; 2d ed., Vol. II, p. 303.

[51]
See F. Cajori, History of the Logarithmic Slide Rule, New York, 1909, pp.
7-14, Addenda, p. ii.

[52]
Rigaud, op. cit., Vol. I, p. 12.

[53]
The New Artificial Gauging Line or Rod: together with rules concerning
the use thereof: Invented and written by William Oughtred, London,
1633.

[54]
W. Oughtred, Apologeticall Epistle, p. 13.

[55]
Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, (1915),
p. 169. In this article Glaisher republishes the “Appendix” in full.

[56]
Aubrey, op. cit., Vol. II, 1898, p. 108.

[57]
Wood’s Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247.

[58]
Wood, op. cit., Vol. II, p. 445.

[59]
Rigaud, op. cit., Vol. I, pp. 33, 35.

[60]
Rigaud, op. cit., Vol. I, pp. 16, 26.

[61]
Rigaud, op. cit., Vol. I, p. 66.

[62]
Ibid., Vol. I, p. 9.

[63]
Rigaud, op. cit., Vol. II, p. 475.

[64]
Ibid., Vol. II, p. 471.

[65]
J. W. L. Glaisher, “On Early Logarithmic Tables, and Their Calculators,”
Philosophical Magazine, 4th Ser., Vol. XLV (1873), pp. 378, 379.

[66]
Rigaud, op. cit., Vol. I, p. 65.

[67]
Rigaud, op. cit., Vol. I, p. 87.

[68]
King’s Life of John Locke, Vol. I, London, 1830, p. 227.

[69]
Exercitationum Mathematicarum Decas prima, Naples, 1627, and
probably Cataldus’ Transformatio Geometrica, Bonon., 1612.

[70]
Rigaud, op. cit., Vol. II, pp. 477-80.

[71]
Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster,
Sometimes publike Professor of Astronomie in Gresham Colledge in
London, by John Twysden, London, 1659.

[72]
The Works of the Honourable Robert Boyle in five volumes, to which is
prefixed the Life of the Author, Vol. I, London, 1744, p. 24.

[73]
The letter is printed in John Wallis’ De algebra tractatus, 1693, p. 206.

[74]
See La Correspondance de Descartes, published by Charles Adam and
Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457.

[75]
H. Bosmans, S.J., “La première édition de la Clavis Mathematica
d’Oughtred. Son influence sur la Géométrie de Descartes,” Annales de la
société scientifique de Bruxelles, 35th year, 1910-11, Part II, pp. 24-78.

[76]
Ibid., p. 78.

[77]
Vincent Wing, Harmonicon coeleste, London, 1651, p. 5.

[78]
Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae fundamenta
inquisitio brevis, Oxford, 1653, p. 7.

[79]
John Wallis, Elenchus geometriae Hobbianae, Oxford, 1655, p. 48.

[80]
An Idea of Arithmetick, at first designed for the use of the Free Schoole at
Thurlow in Suffolk. . . . . By R. B., Schoolmaster there, London, 1655, p.
6.

[81]
The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster . .
. . by John Twysden, London, 1659, p. 1.

[82]
Moor’s Arithmetick in two Books, London, 1660, p. 89.

[83]
Isaac Barrow, Euclidis data, Cambridge, 1657, p. 2.

[84]
Francisci Dulaurens Specima mathematica, Paris, 1667, p. 1.

[85]
Elémens des mathématiques, Paris, 1675, Preface signed “J. P.”

[86]
Nouveaux élémens de géométrie, Paris, 1692 (permission to print 1684).

[87]
Ozanam, Dictionnaire mathématique, Paris, 1691, p. 12.

[88]
Analyse des infiniment petits, Paris, 1696, p. 11.

[89]
Petro Nicolas, De conchoidibus et cissoidibus exercitationes geometricae,
Toulouse, 1697, p. 17.

[90]
R. P. Bernard Lamy, Elémens des mathématiques, Amsterdam, 1692
(permission to print 1680).

[91]
Nouveaux élémens de géométrie, 2d ed., The Hague, 1690, p. 304.

[92]
W. W. Beman in L’intermédiaire des mathématiciens, Paris, Vol. IX,
1902, p. 229, question 2424.

[93]
John Collins, The Mariner’s Plain Scale New Plain’d, London, 1659, p.
25.

[94]
James Gregory, Optica promota, London, 1663, pp. 19, 48.

[95]
Philosophical Transactions, Vol. III, London, p. 868.

[96]
William Leybourn, The Line of Proportion, London, 1673, p. 14.

[97]
Elementa geometriae . . . . a Gulielmo Sanders, Glasgow, 1686, p. 3.

[98]
Cocker’s Decimal Arithmetick, . . . . perused by John Hawkins, London,
1695 (preface dated 1684), p. 41.

[99]
Joseph Raphson, Analysis Aequationum universalis, London, 1697, p. 26.

[100]
E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, 1698,
p. 107.

[101]
John Ward, A Compendium of Algebra, 2d ed., London, 1698, p. 62.

[102]
Plain Elements of Geometry and Plain Trigonometry, London, 1701, p.
63.

[103]
George Shelley, Wingate’s Arithmetick, London, 1704, p. 343.

[104]
A Synopsis of Algebra, Being a posthumous work of John Alexander of
Bern, Swisserland. . . . . Done from the Latin by Sam. Cobb, London,
1709, p. 16.

[105]
John Craig, De Calculo fluentium, London, 1718, p. 35. The notation
A:B::C:D is given also.

[106]
Trigonometry, 2d ed., Edinburgh, 1724, p. 11.

[107]
Méthode pour la mésure des surfaces, la dimension des solides . . . . par
M. Carré de l’académie r. des sciences, 1700, p. 59.

[108]
Application de l’algèbre à géométrie . . . . Paris, 1705.

[109]
Elémens de la géométrie de l’infini, by M. de Fontenelle, Paris, 1727, p.
110.

[110]
Eclaircissemens sur l’analyse des infiniment petits, by M. Varignon,
Paris, 1725, p. 87.

[111]
Application de la géométrie ordinaire et des calculs différentiel et
intégral, by M. Robillard, Paris, 1753.

[112]
Traité de géométrie théorique et pratique, new ed., Paris, 1764, p. 15.

[113]
Recherches sur les courbes à double courbure, Paris, 1731, p. 13.

[114]
Analyse des infiniment petits, by the Marquis de L’Hospital. New ed. by
M. Le Fèvre, Paris, 1781, p. 41. In this volume passages in fine print,
probably supplied by the editor, contain the notation a:b::c:d; the parts in
large type give Oughtred’s original notation.

[115]
The tendency during the eighteenth century is shown in part by the
following data: Jacobi Bernoulli Opera, Tomus primus, Geneva, 1744,
gives B.A::D.C on p. 368, the paper having been first published in 1688;
on p. 419 is given GE:AG=LA:ML, the paper having been first published
in 1689. Bernhardi Nieuwentiit, Considerationes circa analyseos ad
quantitates infinitè parvas applicatae principia, Amsterdam, 1694, p. 20,
and Analysis infinitorum, Amsterdam, 1695, on p. 276, have x:c::s:r. Paul
Halcken’s Deliciae mathematicae, Hamburg, 1719, gives a:b::c:d.
Johannis Baptistae Caraccioli, Geometria algebraica universa, Rome,
1759, p. 79, has a.b::c.d. Delle corde ouverto fibre elastiche schediasmi
fisico-matematici del conte Giordano Riccati, Bologna, 1767, p. 65,

gives P:b::r:ds. “Produzioni mathematiche” del Conte Giulio Carlo de
Fagnano, Vol. I, Pesario, 1750, p. 193, has a.b::c.d. L. Mascheroni,
Géométrie du compas, translated by A. M. Carette, Paris, 1798, p. 188,
gives √3:2::√2:Lp. Danielis Melandri and Paulli Frisi, De theoria lunae
commentarii, Parma, 1769, p. 13, has a:b::c:d. Vicentio Riccato and
Hieronymo Saladino, Institutiones analyticae, Vol. I, Bologna, 1765, p.
47, gives x:a::m:n+m. R. G. Boscovich, Opera pertinentia ad opticam et
astronomiam, Bassani, 1785, p. 409, uses a:b::c:d. Jacob Bernoulli, Ars
Conjectandi, Basel, 1713, has n-r.n-1::c.d. Pavlini Chelvicii, Institutiones
analyticae, editio post tertiam Romanam prima in Germania, Vienna,
1761, p. 2, a.b::c.d. Christiani Wolfii, Elementa matheseos universae,
Vol. III, Geneva, 1735, p. 63, has AB:AE=1:q. Johann Bernoulli, Opera
omnia, Vol. I, Lausanne and Geneva, 1742, p. 43, has a:b=c:d. D. C.
Walmesley, Analyse des mesures des rapports et des angles, Paris, 1749,
uses extensively a.b::c.d, later a:b::c:d. G. W. Krafft, Institutiones
geometriae sublimoris, Tübingen, 1753, p. 194, has a:b=c:d. J. H.
Lambert, Photometria, 1760, p. 104, has C:π=BC²:MH². Meccanica
sublime del Dott. Domenico Bartaloni, Naples, 1765, has a:b::c:d.
Occasionally ratio is not designated by a.b, nor by a:b, but by a, b, as for
instance in A. de Moivre’s Doctrine of Chance, London, 1756, p. 34,
where he writes a, b::1, q. A further variation in the designation of ratio is
found in James Atkinson’s Epitome of the Art of Navigation, London,
1718, p. 24, namely, 3..2::72..48. Curious notations are given in Rich.
Balam’s Algebra, London, 1653.

[116]
Chr. Clavii Operum mathematicorum tomus secundus, Mayence, 1611,
Algebra, p. 39.

[117]
Invention nouvelle en l’algèbre, by Albert Girard, Amsterdam, 1629, p.
17.

[118]
La géométrie et pratique générale d’icelle, par I. Errard de Bar-le-Duc,
Ingénieur ordinaire de sa Majesté, 3d ed., revised by D. H. P. E. M.,
Paris, 1619, p. 216.

[119]
Novae geometriae clavis algebra, authore P. Jacobo de Billy, Paris,
1643, p. 157; also an Abridgement of the Precepts of Algebra. Written in
French by James de Billy, London, 1659, p. 346.

[120]
Miscellanies: or Mathematical Lucubrations, of Mr. Samuel Foster,
Sometime publike Professor of Astronomie in Gresham Colledge in
London, London, 1659, p. 7.

[121]
Quarterly Jour. of Pure and Applied Math., Vol. XLVI (London, 1915),
p. 191.

[122]
Pietro Cossali, Origine, trasporto in Italia primi progressi in essa dell’
algebra, Vol. I, Parmense, 1797, p. 52.

[123]
In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio brevis,
Auctore Setho Wardo, Oxford, 1653, p. 1.

[124]
John Wallis, Algebra, London, 1685, p. 321, and in some of his other
works. He makes greater use of Harriot’s symbols.

[125]
Euclidis data, 1657, p. 1; also Euclidis elementorum libris XV, London,
1659, p. 1.

[126]
John Kersey, Algebra, London, 1673, p. 321.

[127]
E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford, 1698,
p. 142.

[128]
Cocker’s Decimal Arithmetick, perused by John Hawkins, London, 1695
(preface dated 1684), p. 278.

[129]
Th. Baker, The Geometrical Key, London, 1684, p. 15.

[130]
Richard Sault, A New Treatise of Algebra, London (no date).

[131]
Richard Rawlinson in a pamphlet without date, issued sometime between
1655 and 1668, containing trigonometric formulas. There is a copy in the
British Museum.

[132]
F. Dulaurens, Specima mathematica, Paris, 1667, p. 1.

[133]
J. Milnes, Sectionum conicarum elementa, Oxford, 1702, p. 42.

[134]
Cheyne, Philosophical Principles of Natural Religion, London, 1705, p.
55.

[135]
J. Craig, De calculo fluentium, London, 1718, p. 86.

[136]
Jo. Wilson, Trigonometry, 2d ed., Edinburgh, 1724, p. v.

[137]
Commercium Epistolicum, 1712, p. 20.

[138]
C. Le Paige, “Sur l’origine de certains signes d’opération,” Annales de la
société scientifique de Bruxelles, 16th year, 1891-92, Part II, pp. 79-82.

[139]
Gravelaar, “Over den oorsprong van ons maalteeken (×),” Wiskundig
Tijdschrift, 6th year. We have not had access to this article.

[140]
H. Bosmans, op. cit., p. 40.

[141]
Claudii Ptolemaei . . . . annotationes, Bâle, 1551. This reference is taken
from the Encyclopédie des sciences mathématiques, Tome I, Vol. I, Fasc.
1, p. 40.

[142]
Due Correction for Mr. Hobbes. Or Schoole Discipline, for not saying
his Lessons right. In answer to his Six Lessons, directed to the Professors
of Mathematicks. By the Professor of Geometry. Oxford, 1656, pp. 7, 47,
50.

[143]
Oughtred, Apologeticall Epistle, p. 27.

[144]
J. Edleston, Correspondence of Sir Isaac Newton and Professor Cotes,
London, 1850, pp. 279-92.
[97

INDEX

Adam, Charles, 71
Agnesi, Maria G., 77
Alexander, J., 76
Allen, E., 35
Analysis, 19, 20
Apollonius of Perga, 20, 85
Archimedes, 18, 20, 85
Aristotle, 69
Ashmole, E., 13
Atkinson, J., 79
Atwood, 56
Aubrey, 3, 7, 8, 12-16, 58, 59
Austin, 58

Baker, T., 82
Balam, R., 79
Bar-le-Duc, de, 80
Barrow, S., 1, 32, 73, 74, 80, 81
Bartaloni, D., 79
Beman, W. W., 74, 75
Bernoulli, Jakob, 78-80
Bernoulli, John, 79, 80
Billingsley’s Euclid, 15
Billion, 20
Billy, Jacobo de, 80
Binomial formula, 25, 29
Bliss, P., 60

Boscovich, R. G., 78
Bosmans, H., 72, 83
Boyle, R., 1, 69
Braunmühl, von, 39
Brearly, W., 59
Briggs, 6, 36, 55
Brookes, Christopher, 7, 53, 59

Cajori, F., 27, 39, 40, 47
Cantor, M., 40, 41
Caraccioli, J. B., 78
Cardan, 71
Carré, 77
Carrete, N. M., 78
Caryll, C., 7
Cataldi, 67
Cavalieri, 65, 66
Cavendish, Charles, 17, 62, 66
Charles I, 9, 60
Chelvicius, P., 79
Cheyne, G., 82
Circles of Proportion, 35, 37, 48, 49, 51, 59, 87, 88
Clairaut, 77
Clark, A., 3
Clark, G., 63
Clarke, F. L., 3
Clavis mathematicae, 1, 5, 10, 14, 17-35, 45, 46, 51, 57-63, 68-73, 81, 85, 87
Clavius, 26, 80
Clerc, le, 77
Cobb, S., 76
Cocker, 76, 82
Collins, John, 15, 19, 63, 64, 67, 68, 76, 82
Colson, J., 77
Conchoid, 12
Conic sections, 11, 53
Cossali, P., 81
Cotes, R., 1, 85
Craig, J., 76, 82

Cross, symbol of multiplication, 27, 38, 55, 56, 82, 83
Cubic equations, 28, 34, 42, 45

Decimal fractions, notation of, 21
Degree, centesimal division, 39
Delamain, R., 4, 9, 10, 11, 47, 48, 51, 60, 84, 88, 89, 91, 93, 94
De Moivre, 32, 79
De Morgan, A., 5, 16, 37, 46, 47, 54
Descartes, R., 1, 25, 47, 57, 68-72, 80
Dibuadius, 79
Difference, symbol for, 27, 81
Diophantus, 63, 85
Division, abbreviated, 21, 23, 24
Dulaurens, F., 74, 82

Earl of Arundel, 10, 13, 15, 17
Edleston, J., 95
Eneström, G., 40
Equations, solution of, 18, 28, 29, 31, 34, 39-45, 87
Errard de Bar-le-Duc, 80
Eton College, 3, 4
Euclid, 1, 15, 18, 20, 25, 27, 28, 79, 81, 83, 85
Euler, L., 37, 39
Ewart, 59
Exponents, 25, 28, 29

Fagnano, de, 78
Flower, 56
Fontenelle, de, 77
Forster, W., 35, 48, 59, 88
Foster, S., 27, 67, 69, 73, 80, 89
Frisi, P., 78

Gascoigne, 59, 61
Gauss, C. F., 48
Geysius, 67
Ghetaldi, 70, 71
Gibson, 67

Girard, A., 32, 80
Glaisher, J. W. L., 54-56, 64, 80
Glorioso, 67, 68
Grammelogia, 4, 47, 89
Gravelaar, 83
Greater than, symbol for, 81
Greatrex, R., 15
Gregory, D., 32
Gregory, J., 27, 76
Gresham College, 1, 6, 27, 59, 61, 80
Guisnée, 77
Gunter, E., 37, 47, 86
Gunter’s scale, 37

Halcken, P., 78
Hales, J., 7
Halley, E., 1, 18, 69
Hankel, H., 40
Harper, T., 18
Harriot, T., 45, 47, 57, 58, 69-71, 81
Harris, J., 76
Hartlib, 69
Haughton, A., 35, 59
Hawkins, J., 76, 82
Hearn, 56
Helmholtz, 48
Henry, J., 48
Henry van Etten, 52, 53
Henshaw, T., 8, 58, 61
Hobbes, 73, 86
Hollar, 14
Holsatus, 13
Hooganhuysen, 64
Hooke, Rb., 1
Horner’s method, 45
Horology, 18, 50
Horrox, J., 4
Hospital, de l’, 74, 77

Howard, Th. See Earl of Arundel.
Howard, W., 17, 18, 59
Hutchinson, A., 6

Invisible college, 1

Joule, 48

Kepler, J., 6
Kersey, J., 32, 73, 82
Keylway, R., 65
King, 67
Kings College, Cambridge, 3, 35
Krafft, G. W., 79

Lambert, J. H., 79
Lamy, R. P. B., 74
Laud, Archbishop, 65
Leake, W., 53
Le Clerc, 77
Leech, W., 59
Le Fèvre, 77
Leibniz, 47, 78, 80
Leonelli, 56
Le Paige, de, 83
Less than, symbol for, 81
Leurechon, 52
Leybourn, 35, 64, 76
Lichfield, Mrs., 19
Lilly, W., 8, 9
Locke, J., 67
Logarithms, 6, 21, 27, 28, 38, 39, 42, 46, 54-56, 65, 92, 93;
natural, 55;
radix method of computing, 55, 56
Lower, W., 58
Ludolph à Ceulen, 79

Manning, 56

Manning, O., 7, 8, 13-15
Mascheroni, L., 78
Mayer, R., 47
Melandri, D., 78
Mercator, N., 13
Mersenne, 63
Milbourn, W., 45
Million, 20
Milnes, J., 82
Moivre, de, 32, 79
Moore, Jonas, 32, 54, 58, 73
Moreland, S., 70
Morse, R., 48
Multiplication, abbreviated, 21, 22, 24;
symbol for, 27, 82, 83
Mydorge, 54

Napier, J., 6, 7, 21, 27, 38, 39, 52, 54, 57, 59
Napier’s analogies, 39
Newton, Sir Isaac, 1, 25, 29, 40, 41, 45, 47, 59, 65, 86, 92-95
Nichols, J., 6, 14
Nicolas, R. P. P., 74
Nieuwentiit, B., 78
Norwood, R., 37, 38, 80

Opuscula mathematica hactenus inedita, 16, 21, 75
Orchard, 56
Oughtredus explicatus, 64
Ozanam, 74

π, symbol for, 32
Paige, C. de, 83
Pardies, 76
Parentheses, 26, 79, 80
Partridge, S., 47
Peano, 86
Perfect number, 41
Pitiscus, 15

Planisphere, 53, 92, 93
Prestet, J., 74
Price, 11
Proportion, notation for, 26, 27, 73-79
Protheroe, 58
Ptolemy, 83

Quadratic equation, 29, 31, 34

Radix method, 55, 56
Rahn, 27
Raphson, J., 40, 41, 76
Ratio, notation of, 21, 73-80
Rawlinson, R., 39, 82
Regula falsa, 18
Regular solids, 18
Riccati, G., 78
Riccati, V., 78
Rigaud, 7, 12, 13, 19, 48, 61-66, 68
Robillard, 77
Robinson, W., 13, 48, 59, 62, 63
Rooke, L., 59, 61

Saladini, H., 78
Sanders, W., 76
Sault, R., 82
Scarborough, Charles, 16, 54, 58, 60
Schooten, Van, 1
Schreshensuchs, O., 83
Scratch method, 23
Shakespeare, 52
Shelley, G., 76
Shipley, A. E., 1
Shuttleworth, 59
Slide rule, 9, 46-49, 50, 60, 88, 93
Smethwyck, 58
Smith, J., 50
Snellius, W., 79

Solids, regular, 18
Speidell, John, 38, 55
Spherical triangles, 53, 54, 93
Stokes, R., 35, 36, 58
Sudell, 59
Sun dials, 5, 9, 50, 51, 52, 60, 92

Tannery, P., 71
Todhunter, 60
Torporley, 58
Triangles, spherical, 53, 54, 93
Trigonometria, 21, 36, 55, 75
Trigonometric functions, symbols for, 36, 37, 55, 56
Trigonometrie, 21, 35, 39
Trisection of angles, 28
Twysden, 59, 68, 69, 73

Varignon, 77
Vieta, 1, 2, 25, 32, 33, 35, 39-41, 45, 63, 67, 70, 71
Vlack, 65
Von Braunmühl, 39

Wadham College, 5, 53
Wallis, John, 1, 19, 27, 33, 45, 57-59, 63, 64, 66-74, 79-81, 86
Walmesley, D. C., 79
Ward, Bishop, 13
Ward, John, 76
Ward, Seth, 55, 58, 60, 68, 73, 74, 81
Watch-making, 18, 50
Weber, W. E., 48
Weddle, 56
Wells, E., 76, 82
Wharton, 60
Whitlock, B., 8, 9
Wilson, J., 77, 82
Wing, V., 73, 75
Wingate, E., 32, 47, 73
Wolf, Christian, 79

Wood, A., 60, 61
Wood, R., 18, 59
Wren, Christopher, 5, 58, 59, 76
Wright, E., 6, 27, 38, 54
Wright, S., 54

Transcriber’s Notes

A handful of typos, mostly misplaced punctuation, were silently corrected.

HTML and UTF text versions make heavy use of mathematical symbols:
particularly superscripts, subscripts, and combining characters. Some
viewers may require user assistance to find fonts containing these characters.

*** END OF THE PROJECT GUTENBERG EBOOK WILLIAM
OUGHTRED: A GREAT SEVENTEENTH-CENTURY TEACHER OF
MATHEMATICS ***

Updated editions will replace the previous one—the old editions will
be renamed.

Creating the works from print editions not protected by U.S. copyright
law means that no one owns a United States copyright in these works,
so the Foundation (and you!) can copy and distribute it in the United
States without permission and without paying copyright royalties.
Special rules, set forth in the General Terms of Use part of this license,
apply to copying and distributing Project Gutenberg™ electronic
works to protect the PROJECT GUTENBERG™ concept and
trademark. Project Gutenberg is a registered trademark, and may not
be used if you charge for an eBook, except by following the terms of
the trademark license, including paying royalties for use of the Project
Gutenberg trademark. If you do not charge anything for copies of this
eBook, complying with the trademark license is very easy. You may
use this eBook for nearly any purpose such as creation of derivative
works, reports, performances and research. Project Gutenberg eBooks
may be modified and printed and given away—you may do practically
ANYTHING in the United States with eBooks not protected by U.S.
copyright law. Redistribution is subject to the trademark license,
especially commercial redistribution.

START: FULL LICENSE

THE FULL PROJECT GUTENBERG™ LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg™ mission of promoting the free
distribution of electronic works, by using or distributing this work (or
any other work associated in any way with the phrase “Project
Gutenberg”), you agree to comply with all the terms of the Full Project
Gutenberg License available with this file or online at
www.gutenberg.org/license.

Section 1. General Terms of Use and Redistributing
Project Gutenberg electronic works

1.A. By reading or using any part of this Project Gutenberg electronic
work, you indicate that you have read, understand, agree to and accept
all the terms of this license and intellectual property
(trademark/copyright) agreement. If you do not agree to abide by all
the terms of this agreement, you must cease using and return or destroy
all copies of Project Gutenberg electronic works in your possession. If
you paid a fee for obtaining a copy of or access to a Project Gutenberg
electronic work and you do not agree to be bound by the terms of this
agreement, you may obtain a refund from the person or entity to whom
you paid the fee as set forth in paragraph 1.E.8.

1.B. “Project Gutenberg” is a registered trademark. It may only be
used on or associated in any way with an electronic work by people
who agree to be bound by the terms of this agreement. There are a few
things that you can do with most Project Gutenberg electronic works
even without complying with the full terms of this agreement. See
paragraph 1.C below. There are a lot of things you can do with Project
Gutenberg electronic works if you follow the terms of this agreement
and help preserve free future access to Project Gutenberg electronic
works. See paragraph 1.E below.

1.C. The Project Gutenberg Literary Archive Foundation (“the
Foundation” or PGLAF), owns a compilation copyright in the

collection of Project Gutenberg electronic works. Nearly all the
individual works in the collection are in the public domain in the
United States. If an individual work is unprotected by copyright law in
the United States and you are located in the United States, we do not
claim a right to prevent you from copying, distributing, performing,
displaying or creating derivative works based on the work as long as
all references to Project Gutenberg are removed. Of course, we hope
that you will support the Project Gutenberg mission of promoting free
access to electronic works by freely sharing Project Gutenberg works
in compliance with the terms of this agreement for keeping the Project
Gutenberg name associated with the work. You can easily comply with
the terms of this agreement by keeping this work in the same format
with its attached full Project Gutenberg License when you share it
without charge with others.

1.D. The copyright laws of the place where you are located also govern
what you can do with this work. Copyright laws in most countries are
in a constant state of change. If you are outside the United States,
check the laws of your country in addition to the terms of this
agreement before downloading, copying, displaying, performing,
distributing or creating derivative works based on this work or any
other Project Gutenberg work. The Foundation makes no
representations concerning the copyright status of any work in any
country other than the United States.

1.E. Unless you have removed all references to Project Gutenberg:

1.E.1. The following sentence, with active links to, or other immediate
access to, the full Project Gutenberg License must appear prominently
whenever any copy of a Project Gutenberg work (any work on which
the phrase “Project Gutenberg” appears, or with which the phrase
“Project Gutenberg” is associated) is accessed, displayed, performed,
viewed, copied or distributed:

This eBook is for the use of anyone anywhere in the United States
and most other parts of the world at no cost and with almost no
restrictions whatsoever. You may copy it, give it away or re-use it
under the terms of the Project Gutenberg™ License included with

this eBook or online at www.gutenberg.org. If you are not located in
the United States, you will have to check the laws of the country
where you are located before using this eBook.

1.E.2. If an individual Project Gutenberg electronic work is derived
from texts not protected by U.S. copyright law (does not contain a
notice indicating that it is posted with permission of the copyright
holder), the work can be copied and distributed to anyone in the United
States without paying any fees or charges. If you are redistributing or
providing access to a work with the phrase “Project Gutenberg”
associated with or appearing on the work, you must comply either with
the requirements of paragraphs 1.E.1 through 1.E.7 or obtain
permission for the use of the work and the Project Gutenberg
trademark as set forth in paragraphs 1.E.8 or 1.E.9.

1.E.3. If an individual Project Gutenberg electronic work is posted
with the permission of the copyright holder, your use and distribution
must comply with both paragraphs 1.E.1 through 1.E.7 and any
additional terms imposed by the copyright holder. Additional terms
will be linked to the Project Gutenberg License for all works posted
with the permission of the copyright holder found at the beginning of
this work.

1.E.4. Do not unlink or detach or remove the full Project Gutenberg
License terms from this work, or any files containing a part of this
work or any other work associated with Project Gutenberg.

1.E.5. Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph 1.E.1 with
active links or immediate access to the full terms of the Project
Gutenberg License.

1.E.6. You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including
any word processing or hypertext form. However, if you provide
access to or distribute copies of a Project Gutenberg work in a format
other than “Plain Vanilla ASCII” or other format used in the official

version posted on the official Project Gutenberg website
(www.gutenberg.org), you must, at no additional cost, fee or expense
to the user, provide a copy, a means of exporting a copy, or a means of
obtaining a copy upon request, of the work in its original “Plain
Vanilla ASCII” or other form. Any alternate format must include the
full Project Gutenberg License as specified in paragraph 1.E.1.

1.E.7. Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg works
unless you comply with paragraph 1.E.8 or 1.E.9.

1.E.8. You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg electronic works provided
that:

• You pay a royalty fee of 20% of the gross profits you derive from the
use of Project Gutenberg works calculated using the method you
already use to calculate your applicable taxes. The fee is owed to the
owner of the Project Gutenberg trademark, but he has agreed to donate
royalties under this paragraph to the Project Gutenberg Literary
Archive Foundation. Royalty payments must be paid within 60 days
following each date on which you prepare (or are legally required to
prepare) your periodic tax returns. Royalty payments should be clearly
marked as such and sent to the Project Gutenberg Literary Archive
Foundation at the address specified in Section 4, “Information about
donations to the Project Gutenberg Literary Archive Foundation.”

• You provide a full refund of any money paid by a user who notifies
you in writing (or by e-mail) within 30 days of receipt that s/he does
not agree to the terms of the full Project Gutenberg™ License. You
must require such a user to return or destroy all copies of the works
possessed in a physical medium and discontinue all use of and all
access to other copies of Project Gutenberg™ works.

• You provide, in accordance with paragraph 1.F.3, a full refund of any
money paid for a work or a replacement copy, if a defect in the
electronic work is discovered and reported to you within 90 days of
receipt of the work.

• You comply with all other terms of this agreement for free distribution
of Project Gutenberg™ works.

1.E.9. If you wish to charge a fee or distribute a Project Gutenberg™
electronic work or group of works on different terms than are set forth
in this agreement, you must obtain permission in writing from the
Project Gutenberg Literary Archive Foundation, the manager of the
Project Gutenberg™ trademark. Contact the Foundation as set forth in
Section 3 below.

1.F.

1.F.1. Project Gutenberg volunteers and employees expend
considerable effort to identify, do copyright research on, transcribe and
proofread works not protected by U.S. copyright law in creating the
Project Gutenberg™ collection. Despite these efforts, Project
Gutenberg™ electronic works, and the medium on which they may be
stored, may contain “Defects,” such as, but not limited to, incomplete,
inaccurate or corrupt data, transcription errors, a copyright or other
intellectual property infringement, a defective or damaged disk or
other medium, a computer virus, or computer codes that damage or
cannot be read by your equipment.

1.F.2. LIMITED WARRANTY, DISCLAIMER OF DAMAGES -
Except for the “Right of Replacement or Refund” described in
paragraph 1.F.3, the Project Gutenberg Literary Archive Foundation,
the owner of the Project Gutenberg™ trademark, and any other party
distributing a Project Gutenberg™ electronic work under this
agreement, disclaim all liability to you for damages, costs and
expenses, including legal fees. YOU AGREE THAT YOU HAVE NO
REMEDIES FOR NEGLIGENCE, STRICT LIABILITY, BREACH
OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
PROVIDED IN PARAGRAPH 1.F.3. YOU AGREE THAT THE
FOUNDATION, THE TRADEMARK OWNER, AND ANY
DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT,
CONSEQUENTIAL, PUNITIVE OR INCIDENTAL DAMAGES

EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
DAMAGE.

1.F.3. LIMITED RIGHT OF REPLACEMENT OR REFUND - If you
discover a defect in this electronic work within 90 days of receiving it,
you can receive a refund of the money (if any) you paid for it by
sending a written explanation to the person you received the work
from. If you received the work on a physical medium, you must return
the medium with your written explanation. The person or entity that
provided you with the defective work may elect to provide a
replacement copy in lieu of a refund. If you received the work
electronically, the person or entity providing it to you may choose to
give you a second opportunity to receive the work electronically in lieu
of a refund. If the second copy is also defective, you may demand a
refund in writing without further opportunities to fix the problem.

1.F.4. Except for the limited right of replacement or refund set forth in
paragraph 1.F.3, this work is provided to you ‘AS-IS’, WITH NO
OTHER WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED,
INCLUDING BUT NOT LIMITED TO WARRANTIES OF
MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.

1.F.5. Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of damages. If
any disclaimer or limitation set forth in this agreement violates the law
of the state applicable to this agreement, the agreement shall be
interpreted to make the maximum disclaimer or limitation permitted by
the applicable state law. The invalidity or unenforceability of any
provision of this agreement shall not void the remaining provisions.

1.F.6. INDEMNITY - You agree to indemnify and hold the
Foundation, the trademark owner, any agent or employee of the
Foundation, anyone providing copies of Project Gutenberg™
electronic works in accordance with this agreement, and any
volunteers associated with the production, promotion and distribution
of Project Gutenberg™ electronic works, harmless from all liability,
costs and expenses, including legal fees, that arise directly or indirectly
from any of the following which you do or cause to occur: (a)

distribution of this or any Project Gutenberg work, (b) alteration,
modification, or additions or deletions to any Project Gutenberg work,
and (c) any Defect you cause.

Section 2. Information about the Mission of Project
Gutenberg

Project Gutenberg is synonymous with the free distribution of
electronic works in formats readable by the widest variety of
computers including obsolete, old, middle-aged and new computers. It
exists because of the efforts of hundreds of volunteers and donations
from people in all walks of life.

Volunteers and financial support to provide volunteers with the
assistance they need are critical to reaching Project Gutenberg’s goals
and ensuring that the Project Gutenberg collection will remain freely
available for generations to come. In 2001, the Project Gutenberg
Literary Archive Foundation was created to provide a secure and
permanent future for Project Gutenberg and future generations. To
learn more about the Project Gutenberg Literary Archive Foundation
and how your efforts and donations can help, see Sections 3 and 4 and
the Foundation information page at www.gutenberg.org.

Section 3. Information about the Project Gutenberg
Literary Archive Foundation

The Project Gutenberg Literary Archive Foundation is a non-profit
501(c)(3) educational corporation organized under the laws of the state
of Mississippi and granted tax exempt status by the Internal Revenue
Service. The Foundation’s EIN or federal tax identification number is
64-6221541. Contributions to the Project Gutenberg Literary Archive
Foundation are tax deductible to the full extent permitted by U.S.
federal laws and your state’s laws.

The Foundation’s business office is located at 41 Watchung Plaza
#516, Montclair NJ 07042, USA, +1 (862) 621-9288. Email contact

links and up to date contact information can be found at the
Foundation’s website and official page at www.gutenberg.org/contact

Section 4. Information about Donations to the Project
Gutenberg Literary Archive Foundation

Project Gutenberg™ depends upon and cannot survive without
widespread public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can be
freely distributed in machine-readable form accessible by the widest
array of equipment including outdated equipment. Many small
donations ($1 to $5,000) are particularly important to maintaining tax
exempt status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United States.
Compliance requirements are not uniform and it takes a considerable
effort, much paperwork and many fees to meet and keep up with these
requirements. We do not solicit donations in locations where we have
not received written confirmation of compliance. To SEND
DONATIONS or determine the status of compliance for any particular
state visit www.gutenberg.org/donate.

While we cannot and do not solicit contributions from states where we
have not met the solicitation requirements, we know of no prohibition
against accepting unsolicited donations from donors in such states who
approach us with offers to donate.

International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States. U.S. laws alone swamp our small staff.

Please check the Project Gutenberg web pages for current donation
methods and addresses. Donations are accepted in a number of other
ways including checks, online payments and credit card donations. To
donate, please visit: www.gutenberg.org/donate.

Section 5. General Information About Project Gutenberg
electronic works

Professor Michael S. Hart was the originator of the Project Gutenberg
concept of a library of electronic works that could be freely shared
with anyone. For forty years, he produced and distributed Project
Gutenberg eBooks with only a loose network of volunteer support.

Project Gutenberg eBooks are often created from several printed
editions, all of which are confirmed as not protected by copyright in
the U.S. unless a copyright notice is included. Thus, we do not
necessarily keep eBooks in compliance with any particular paper
edition.

Most people start at our website which has the main PG search facility:
www.gutenberg.org.

This website includes information about Project Gutenberg, including
how to make donations to the Project Gutenberg Literary Archive
Foundation, how to help produce our new eBooks, and how to
subscribe to our email newsletter to hear about new eBooks.