(navigation image)
Home American Libraries | Canadian Libraries | Universal Library | Community Texts | Project Gutenberg | Children's Library | Biodiversity Heritage Library | Additional Collections
Search: Advanced Search
Anonymous User (login or join us)
Upload
See other formats

Full text of "William Oughtred, a great seventeenth-century teacher of mathematics"

V1LI 




CAJOR! 



ialh 



II! 



WILLIAM OUGHTRED 



ILLIAM 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 



<PA 



OSC3 



Composed and Printed By 

The University of Chicago Press 

Chicago, Illinois, U.S.A. 



TABLE OF CONTENTS 

PAGE 

INTRODUCTION . . . i 

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 

Clams mathematicae 17 

Circles of Proportion and Trigonometric . . .35 

Solution of Numerical Equations 39 

Logarithms 46 

Invention of the Slide Rule; Controversy on 

Priority of Invention 46 

III. MINOR WORKS 50 

IV. OUGHTRED'S INFLUENCE UPON MATHEMATICAL PROG- 

RESS 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 



vi Table of Contents 



V. OUGHTRED'S IDEAS ON THE TEACHING or MATHE- 
MATICS 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 



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 mathe- 
matics, 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 prepara- 
tion, 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 de- 
ficiencies of the Stuart kings, no one of them lacked intel- 
ligence in things artistic and scientific. " It was at this 
time that mathematics, and particularly algebra, began 
to be cultivated with greater zeal, when elementary alge- 
bra with its symbolism as we know it now began to take 
its shape. 

Biographers of Sir Isaac Newton make particular men- 
tion of five mathematical books which he read while a 
young student at Cambridge, namely, Euclid's Elements, 
Descartes's Geometric, Vieta's Works, Van Schooten's 
Miscellanies, and Oughtred's Clams mathematicae. The 
last of these books has been receiving increasing attention 



2 William Oughtred 

from the historians of 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 dis- 
cussed 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 Clams mathe- 
maticae, 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 over- 
looked 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 recrea- 
tion. Like the great French algebraist, Vieta, from whom 
he drew much of his inspiration, he was an amateur mathe- 
matician. 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. 



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-boy." 1 
He was a boy at Eton in the year of the Spanish Armada. 
At this famous school, which prepared boys for the uni- 
versities, 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 Septem- 
ber i, 1592, at the age of seventeen. He was made Fellow 
at King's on September i, 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 

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



4 William Oughtred 

studies and exercises, and with what 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 muring 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 prefer- 
ment) will most lovingly acknowledge. 1 

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 them- 
selves: the tedious difficulty of the study itself deterred a mind 
not yet formed; the want of means oppressed, and still op- 
presses, the aspirations of my mind: but that which gave me 

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



Oughtred' s Life 5 

most concern was that there was no one who could 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 I found that books must be used instead of 

teachers." 1 

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 mathe- 
maticae in 1647, was in the meantime translated into 
Latin by Christopher Wren, then a Gentleman Com- 
moner 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. 

1 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." 



6 William Oughtred 

AS RECTOR AND AMATEUR MATHEMATICAN 

It was in 1604 that Oughtred entered upon his profes- 
sional 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 mathe- 
matical 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, since it is evidently formed upon the plan of Lord Napier's 
Canon. 1 

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 Con- 
structio (1619) bound up with a copy of Kepler's Chilias 
logarithmorum (1624), that at the beginning of the Con- 

1 New and General Biographical Dictionary (John Nichols), Lon- 
don, 1784, art. "Oughtred." 



Oughtred's Life 7 

structio is a blank leaf, and before this occurs 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 Gary 11, daughter of 
Gary 11, Esq., of Tangley, in an adjoining parish. 1 We 
know very little about Oughtred's family life. The 
records at King's College, Cambridge, 2 mention a son, 
but it is certain that there were more children. A 
daughter was married to Christopher Brookes. But 
there is no confirmation of Aubrey's statements, 3 accord- 
ing 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 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 

1 Rev. Owen Manning, History of Antiquities in Surrey, Vol. II, 
p. 132. 

2 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." 

3 Aubrey, op. cit., Vol. II, p. 107. 

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



8 William Oughtred 

accuracy of the following item handed down by Aubrey; 
it cannot be a 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 there, bought candle, which 
was a great comfort to the old man. 1 

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 
well, even in his old age. 2 

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 majority on his side, 
and so escaped a sequestration.* 

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

About this Tune, the most famous Mathematician of all 
Europe, (Mr. William Oughtred, Parson of Aldbury in Surrey) 

1 Aubrey, op. cit., Vol. II, p. no. 

3 Ibid., p. in. 3 Op. cit., Vol. II, p. 132. 



Oughtred's Life 9 

was in Danger of Sequestration by the Committee of or for 
plunder'd Ministers; (Ambo-dexters they were;) several incon- 
siderable 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 Bene- 
fices, for not complying with the Three-penny Directory. 1 

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 Guild- 
ford in Surrey: where, whether / have taken so much liberty 
to the losse of time, and the neglect of my calling the whole Coun- 
trey 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 

1 Mr. William Lilly's History of His Life and Times, From the 
Year 1602 to 1681, London, 1715, p. 58. 



io William Oughtred 

once in a yeare. Indeed 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 tedious- 
nesse 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 Ought- 
red such interesting biographical details. Oughtred 
proceeds to tell how, about 1628, he was induced to write 
his Claris 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 affec- 
tionate 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 scandal! 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 



Oughtred's Life n 

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. himself e 
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 Astronomic concerning the second motions 
of the fixed starres, and of the Sunne and Moone; they were 
the first elements of Conies, 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 atten- 
tion 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 sale- 
brous 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, 



12 William Oughtred 

besides the prejudice of such, who not considering what inces- 
sant 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 Your problem is easily 

wrought per Nicomedis conchoidem lineam. 1 

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 1 1 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 

1 Rigaud, op. cit. t Vol. I, p. 60. 



Oughtred's Life 13 

He has told bishop Ward, and Mr. Elias Ashmole (who was 
his 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 

The right hon ble 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 grott, which fell 
downe but just as they were come out. 1 

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 abso- 
lutely) as able as willing, I would as freely give you 500 /. 
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 thereof, 
science and knowledge, are so slighted 3 

In his letters Oughtred complains several times of the 
limitations for work and the infirmities due to his advan- 
cing 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 years of age. 3 

1 Aubrey, op. cit., Vol. II, p. 107. 

3 Rigaud, op. cit., Vol. I, p. 16. 3 Owen Manning, op. cit., p. 132. 



14 William Oughtred 

Another informant says that Oughtred was 
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 main- 
tained secure in his privacy, prudence, meekness, simplicity, 
resolution, patience, and contentment. 1 

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 religion. 2 

A portrait of Oughtred, painted in 1646 by Hollar and 
inserted in the English edition of the Clams of 1647, con- 
tains 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 

1 New and General Biographical Dictionary (John Nichols), 
London, 1784, art. "Oughtred." 
3 Op. cit. t Vol. II, p. no. 



c 

Oughtred's Life 15 

possibly he might have been in Italy with his patron, the 
Earl of Arundel." 1 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 Con- 
tinent, 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 my self e 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 annota- 
tions 

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 2pth 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 
were then quite upon .the wing to fly away 2 

In this passage, as in others, due allowance must be 
made for Aubrey's lack of discrimination. He was not 

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

2 Op. cit., Vol. II, 1898, p. in. 



1 6 William Oughtred 

in the habit of sifting facts from mere 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 pub- 
lished at Oxford in 1677 under the title of Opuscula mathe- 
matica 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 Res- 
toration 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, by way of excuse, that he 
was eighty-six years old." 1 

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



CHAPTER II 

PRINCIPAL WORKS 

"CLAVIS MATHEMATICAE " 

Passing to the consideration of Oughtred's mathe- 
matical 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 Clams 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. 

The Clams mathematicae* in its first edition of 1631, was 
a booklet of only 88 small pages. Yet it contained in very 

1 The full title of the Clams of 1631 is as follows: Arithmetics 

in numeris et speciebvs institvtio: Qvae tvm logisticae, tvm analyticae, 

17 



1 8 William Oughtred 

condensed form the 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 

atqve adeo totivs mathematicae, qvasi clavis est. Ad nobilissimvm 
spectatissimumque invenem Dn. Gvilelmvm Howard, Ordinis qui dicitur, 
Balnei Equitem, honoratissimi Dn. Thomae, Comitis Arvndeliae 6* 
Svrriae, Comitis Mareschalli Angliae, &Y filitim. Londini, Apud 
Thomam Harperom. M.DC.XXXI. 

In all there appeared five Latin editions, the second in 1648 at Lon- 
don, 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 trans- 
lation, the preface being written and the book recommended by the 
astronomer Edmund Halley. The 1694 and 1 702 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 a/ectarvm resolvtione in numeris. 
Eu.=Elementi decimi Eudidis declaratio. 
So. = De Solidis regular ibus, trac talus. 
An. = De Anatocismo, sive usura composita. 
Fa. = Regula falsae positionis. 

Ar. = Theorematum in libris Archimedis de Sphaera 6* cylindro 
declaratio. 

Ho. = Horologia sciolerica in piano, geometrice delincandi 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 



Principal Works 19 

book underwent hardly any changes, except for the num- 
ber 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 inter- 
est. When arranging for the printing of the 1667 edition 
of the Clams , 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 cfed. a book." 1 

As compared with other contemporary works on alge- 
bra, Oughtred's distinguishes itself for the amount of 
symbolism used, particularly in the treatment of geo- 
metric problems. Extraordinary emphasis was placed 
upon what he called in the Clams the "analytical art." 2 

editions, given in the order in which the tracts appear in each edition, 
are as follows: Clams of 1647, -^9-> An., Fa., Ho.; Clams of 1648, 
Eq., An., Fa., Eu., So.; Clams of 1652, Eq., Eu., So., An., Fa., 
Ar., Ho.; Clams of 1667, Eq., Eu., So., An., Fa., Ar., Ho.; Clams of 
1693 and 1698, Eq., Eu., So., An., Fa., Ar., Ho.; Clams of 1694 and 
1702, Eq. 

The title-page of the Clam's 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, she potius fabricata. Cum 
aliis quibusdam ejusdem commentationibus , quae in sequenti pagina 
recensentur. Editio tertia auctior 6* emendatior. Oxoniae, Excudebat 
Leon. Lichfield, Veneunt apud Tho. Robinson. 1652. 

1 Rigaud, op. cit., Vol. II, p. 476. 

2 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 ilia 
numerosa." 



2O William Oughtred 

By that term he did not mean our modern analysis or 
analytical geometry, but the art "in which by taking the 
thing sought as knowne, we finde out that we seeke." 1 
He meant to express by it condensed processes of rigid, 
logical deduction expressed by appropriate symbols, as 
contrasted with mere description or elucidation by pas- 
sages 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 publish- 
ing 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 instru- 
ments, 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 Clams 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 

1 Oughtred, The Key of the M athemalicks , London, 1647, p. 4. 



Principal Works 21 

English 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 o| 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 mathe- 
matical 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 inter- 
polation 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 



22 William Oughtred 

he took a position far in advance of his time. The part 
on abbreviated multiplication was rewritten in slightly 
enlarged form and with some unimportant alterations 
in the later edition of the Clams. We give it as it occurs 
in the revision. Four cases are given. In finding the 
product of 246 1 9 14 and 3 5 1 2 7 , " 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 Clams 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 7X4=28 
and carries 3 , then he finds 7X2+3=17 and writes down 1 7 . 
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 multi- 
plication of 246 [914 by 35 1 27 is now performed thus: 

2 4 6|g i 4 




74074200 

12345700 

493828 

172840 

8708)65 68 



Principal Works 23 

In the third and fourth cases are considered factors 
which appear as 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 congruous Latitude may be found." 1 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 Clams. 
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 multi- 
plication. 

1 Clams 1694, p. 19, and the Clams of 1631, p. 8. 



24 William Oughtred 

g 

17 




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 con- 
densed statement the mode of procedure, without further 
discussion. 

He does not attempt to establish the rules for the addi- 
tion, 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 multi- 
plication 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. 



Principal Works 25 

A word should be said on Oughtred's definition of + 
and . He recognizes their double function in algebra by 
saying (Claws, 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 Clams, thirty-four years after 
the death of Oughtred, these symbols are defined as signi- 
fying 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 Clams that is most striking 
to a modern reader is the total absence of indexes or expo- 
nents. 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, a n , is of later date. 
The modern notation, for positive integral exponents, first 
appears in Descartes' Geometric, 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 Clams 
gives it a strange aspect. Like Vieta, Oughtred uses ordi- 
narily the capital letters, A, B, C, .... to designate 
given numbers; A 2 is written Aq, A 3 is written Ac; for 
A 4 , A 5 , A 6 he has, respectively, Aqq, Aqc, Ace. Only on 
rare occasions, usually when some parallelism in notation 
is aimed at, does he use small letters 1 to represent numbers 
or magnitudes. Powers of binomials or polynomials 

1 See for instance, Oughtred's Elementi decimi Euclidis declaralio, 
1652, p. i, where he uses A and E, and also a and e. 



26 William Oughtred 

are marked by prefixing the capital letters Q (for square), 
C (for cube), QQ (for the fourth power), QC (for the fifth 
power), etc. 

Oughtred does not express aggregation by ( ). Par- 
entheses had been used by 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, i/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 dedaratio, 
"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. 2 He wrote 20:60=4::*;, x=i2, 

1 See Christophori Clavii Bambergensis Operum malhcmaticorum, 
lomus secundus, Moguntiae, M.DC.XI, algebra, p. 39. 

2 Christophori Clavii operum malhematicorum Towns Sccundiis, 
Moguntiae, M.DC.XI, Epitome arilhmelicae, p. 36. 



Principal Works 27 

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 mathema- 
ticians. It was used by John Wallis at Oxford, by Samuel 
Foster at Gresham College, by James Gregory of Edin- 
burgh, 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 X as the symbol for multiplication in the Clams 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 Napier's Descriptio, published in I6I8. 1 
Later we shall give our reasons for believing that Oughtred 
is the author of that " Appendix." The X 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 Z>, even when D is 
the larger number. 2 This symbol was used by John 
Wallis in 1657.3 

Oughtred represented in symbols also certain com- 
posite expressions, as for instance A+E=Z, A E=X, 
where A is greater than E. He represented by a sym- 
bol also each of the following: A 2 +E 2 , A*+E*, A 2 -E 2 , 



1 See F. Cajori, "The Cross X as a Symbol of Multiplication," 
in Nature, Vol. XCIV (1914), p. 363. 

2 See Elementi decimi Euclidis declaratio, 1652, p. 2. 

3 See Johannis Wallisii Operum mathematicorum pars prima, 
Oxonii, 1657, p. 247. 



28 William Oughtred 

Oughtred practically translated the tenth book of 
Euclid from its 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 Clams lie mainly in the special parts appended to some 
editions and dropped in the latest editions. The part 
which originally constituted the Clams 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 consider- 
ably rearranged in the later editions and combined into 
two chapters, so that the Clams 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 



Principal Works 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 Clams 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 con- 
verts 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. 

As a preliminary step 1 he lets 

Z = A+Esii\dA>E; 

he lets also X = A E. From these relations he obtains 
identities which, in modern notation, are \Z 2 AE = 
(%Z-E) 2 = IX 2 . Now, if we know Z and AE, we 
can find JX. Then $(Z+X)=A, and %(Z-X)=E, 
and 



Claris of 1631, chap, xix, sec. 5, p. 50. 



30 William Oughtred 

Having established these preliminaries, he proceeds 
thus: 

Datis igitur linea inaequaliter secta Z (10), & rectangulo 
sub segmentis AE (21) qui gnomon est: datur semidifferentia 
segmentorum %X: & per consequens ipsa segmenta. Nam 
ponatur alterutrum segmentum A : alterum erit Z A: Rec- 
tangulum auctem est ZA A q = AE. Et quia dantur Z & 
AE: estque \Z q -AE=lX q : & per 50. 18, %Z+%X=A: & 
\Z %X=E: Aequatio sic resoluetur: ^Z^Vq'.^ZqAE: 
_ . fmaius segment 
\minus segment. 

Itaque proposita equatione, in qua sunt tres species aequa- 
liter in ordine tabellae adscendentes, altissima autem species 
ponitur negata: Magnitude data coemciens mediam speciem 
est linea bisecanda: & magnitude absoluta data, ad quam sit 
aequatio, est rectangulum sub segmentis inaequalibus, sine 
gnomon: vt ZAA q = AE: in numeris autem lol l q = 2i: 
Estque A, vel i/, alterutrum segmentum inaequale. Inuenitur 

autem sic: 

2 
Dimidiata coemciens median speciem est - (5); cuius 



y 



quadratum est (25): ex hoc tolle AE (21) absolutum: 
4 

eritque AE (4) quadratum semidifferentiae segmentorum : 
4 

7 

latus huius quadratum V q i AE (2) est semidifferentia: 

4 

9 

quam si addas ad - (5) semissem coefficients, sive 

lineae bisecandae, erit maius segment.; sin detrahas, erit 

Z , Zn . fmaius segmentum 

minus segment: Dico -+^-AE:-A{ minus segmentum . 

We translate the Latin passage^ nsirig the modern 
exponential notation and parenthesesVas follows: 

Given therefore an unequally divided line Z (10), and a 
rectangle beneath the segments AE (21) which is a gnomon. 



Principal Works 31 

Half the difference of the segments %X is given, and conse- 
quently the segment itself. For, if one of the two segments is 
placed equal to A, the other will be ZA. Moreover, the 
rectangle is ZA A 2 =AE. And because Z and AE are given, 
and there is \Z 2 -AE=IX 2 , and by 50.18, %Z+\X=A, and 
\Z-\X=E, the equation will be solved thus: 



mnor 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 rec- 
tangle beneath the unequal segments, without gnomon. As 
ZA A 2 =AE } or in numbers, lox x 2 = 2i. And A or x is 
one of the two unequal segments. It may be found thus: 

Z Z 2 

The half of the middle species is (5), its square is (25). 

Z 2 

From it subtract the absolute term A (21), and ^(4) 

will be the square of half the difference of the segments. 

The square root of this, T/[( ) -AE] (2), is half the differ- 

2f 

ence. If you add it to half the coefficient (5), or half the line 

to be bisected, the longer segment is obtained; if you subtract 
it, the smaller segment is obtained. I say : 

maj ' or se S ment 
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, x 2 +ax = b, provided that E or Z are not re- 
stricted to positive values, but admit of being either 



32 William Oughtred 

positive or negative, a case not adequately treated by 
Oughtred. Imaginary numbers and imaginary roots re- 
ceive 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 for the "square" of 
the expression after it. 1 

It is of some interest that Oughtred used -5 to signify 

the ratio of the circumference to the diameter of a circle. 
Very probably this notation is the forerunner of the TT = 
3.14159 .... used in 1706 by William Jones. Ought- 
red first used ^ in the 1647 edition of the Clams mathe- 

maticae. In the 1652 edition he says, "Si in circulo sit 
7.22: :8-7r: .-113. 355:erit S-7r::2 R.P: periph." This 
notation was adopted by Isaac Barrow, who used it exten- 
sively. David Gregory 2 used - in 1697, and De Moivre 3 

used - about 1697, to designate the ratio of the circumfer- 
ence to the radius. 

1 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. 

2 Philosophical Transactions, Vol. XIX, No. 231, London, p. 652. 
' Ibid., Vol. XIX, p. 56. 



Principal Works 33 

We quote the description of the Clams that was given 
by Oughtred's 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 Clams, 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 intro- 
duced 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, m 

CUE Solid, -- Ewl to FG Plane, 

would with him be thus expressed 
AB c 



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, Differ- 
ences, and Rectangles of several Quantities. As for instance, 



34 William Oughtred 

Of two Quantities A (the Greater), 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 composi- 
tions of such Parts, Sums, Differences, and Rectangles; (of 
which there is great plenty in his Clavis, Cap. n, 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 ob- 
scure, because so short, but without cause; for his words be 
always full, but not Redundant, and need only a little atten- 
tion 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 pro- 
lixity 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 pro- 
ceeding (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 Equa- 
tions for another work He contents himself likewise in 

Resolving Equations, to take notice of the Affirmative or Posi- 
tive 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 



Principal Works 35 

Vieta was well aware 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 Clavis, the Circles of Proportion, 1 
and a Trigonometric? 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 printed Books, corrected by his own 

1 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. 0. Translated into English: and set forth for the piiblique 
benefit by William For -ster. 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-ban. 1632. 
T. Cecill Sculp. 

In 1633 there was added the following, with a separate title-page: 
An addition vnto the Vse of the Instrument called the Circles of Propor- 
tion 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 imme- 
diately after Forster's dedication, instead of at the end of the volume. 

2 The complete title of the English edition is as follows : Trigono- 
metrie, 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. 



36 William Oughtred 

Hand." 1 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 Tables I will 
turn those of the sun and moon, and send them to you. 2 

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 im- 
press characteristic of Oughtred. Like all his mathemati- 
cal 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, con- 
tractions are given as follows: s = sine, / = tangent, se = 
secant, s co= cosine (sine complement), / 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, 

1 Jer. Collier, The Great Historical, Geographical, Genealogical and 
Poetical Dictionary, Vol. II, London, 1701, art. "Oughtred." 

2 Rigaud op. cit., Vol. I, p. 82. 



Principal Works 37 

in 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 Clams 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 willing to have 
taught it, if people would have learnt. 1 

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 trigo- 
nometric functions. As early as 1624 the contractions 
sin for sine and tan for tangent appear on the drawing 
representing Gunter's scale, but Gunter did not use them 
in his books, except in the drawing of his scale. 2 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 

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

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



38 William Oughtred 

cited Norwood did not use nearly as 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 "Ap- 
pendix" in tracing the origin of the sign X. 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 #," 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 (/) in the "Appendix." The 
conclusion of all this is that as early as 1618 the signs s } s*, 
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 



Principal Works 39 

a, B opposite b, and C opposite c, is attributed to Leonard 
Euler (1753), but was first used by Richard Rawlinson 
of Queen's College, Oxford, sometimes after 1655 and 
before 1668. Oughtred did not use Rawlinson's notation. 1 
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 Braunmuhl, 
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. 2 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 logarith- 
mic 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 alge- 
braist Vieta, as explained in a publication which appeared 

1 F. Cajori, "On the History of a Notation in Trigonometry," 
Nature, Vol. XCIV, 1915, pp. 642, 643. 

2 A. von Braunmuhl, Geschichte der Trigonometric, 2. Teil, Leipzig, 
1903, pp. 42, 91. 



40 William Oughtred 

in 1600 in Paris under the title, De numerosa poles tatum 
purarum atque adfectarum ad exegesin resolutione tractatus. 
In view of the fact that Vieta's process has been described 
inaccurately by leading modern historians including H. 
Hankel 1 and M. Cantor, 2 it may be worth while to go into 
some detail. 3 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 hi the divisor used. 
What is now called " Newton's method" is Newton's 
method as modified by Joseph Raphson. 4 The Newton- 
Raphson method of approximation to the roots of an 
equation f(x) = o 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 /'(#). Vieta's divisor 
is different; it is 

!/(+&)-/(<) l-r, 

where/0*;) is the left of the equation /(#) = k, n is the degree 
of equation, and Si is a unit of the denomination of the 
digit next to be found. Thus in ^+420000^ = 247651713, 
it can be shown that 417 is approximately a root; suppose 
that a has been taken to be 400, then Si = io; but if, at 
the next step of approximation, a is taken to be 410, then 
Si = i. In this example, taking a = 400, Vieta's divisor 

1 H. Hankel, Geschichte der Mathematik in Alterthum und Mit- 
telalter, Leipzig, 1874, pp. 369, 370. 

2 M. Cantor, Vorlesungen iiber Geschichte der Mathematik, II, 
1900, pp. 640, 641. 

3 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. Enestrom in Bibliotheca 
mathematica, 3. Folge, Vol. XI, 1911, pp. 234, 235. 

< See F. Cajori, op. cit., p. 193. 



Principal Works 41 

would have been 9120000; Newton's divisor 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 disad- 
vantage of requiring a much larger amount of computa- 
tion. 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 nota- 
tion was not sufficiently developed to enable the argument 
to be condensed into a form easily surveyed. So com- 
plicated 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 uni- 
formity 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, i, 2, 4, 7, 14). 

The early part of his exposition shows how an equation 
maybe transformed so as to make its roots 10, 100, 1000, or 
io m 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, x 4 723?+ 2386000; 
= 8725815, upon dividing J2X 3 by 10, 2386000: by 1000, 



42 William Oughtred 

and 8725815 by 10,000, we obtain x* 7-2.v 3 +238-6:x; = 
872-5. Dividing both sides by x, we obtain ^+238 6 
7 -2X? = x)Sj2 -5. Letting 2 = 4, we have 64+238-6 
ii5'2 = i87-4. 

But 4)872-5(218-1; 4 is too small. Next let # = 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, # = 40. This 
method may be used to find the second digit in the root. 
Oughtred divides both sides of the equation by af, and 
obtains x 2 -\-x) 238600 72^ = ^)8725815. He tries #=47 
and #=48, and finds that #=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 solu- 
tion of affected equations. 

As an illustration of Oughtred's method of approxima- 
tion after the root sought has been located, we have 
chosen for brevity a cubic in preference to a quartic. We 
selected the equation ^+420000^ = 24765 17 13. By the 
process explained above a root is found to lie between 
#=400 and # = 500. From this point on, the approxima- 
tion 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 L c -\-C q L=D c , where L c 
is our x, C g = 42oooo, ^ = 247651713. In the first step 
of approximation, let L=A+E, where ^=400 and E is, 
as yet, undetermined. We have 



and 

C q L= 420000(^4 +). 

Subtract from 247651713 the sum of the known terms A 3 



Principal Works 
"EXEMPLUM II 



43 



Hocest, L C +CJL=D C 



247 


65* 


7 i3 


(417 


42 


o o o 





C. 


6 4 








i68 


000 





C q A 


232 


o o o 





Ablatit. 


J 5 


651 


713 




4 


8 




2>A q 




I 2 


. 


3 A 


4 


200 


00 


C, 


9 


120 


00 


Divisor. 


4 


8 




$A q E 




I 2 




3 A E q 




I 




EC 


4 


200 


00 


C q E 


9 


121 


o o 


Ablatit. 


6 


530 


713 






S4 


3 


$A q 




i 


23 


3 A 




420 


000 


C q 




925 


530 


Divisor. 


3 


530 


I 


3 A q E 




60 


27 


3^ E q 






343 


EC 


2 


940 


o o o 


C q E 


6 


530 


7 13 


Ablatit." 



4 


i 




16 


8 






i 





16 8 i 



(his A c ) and 420000 A (his C q A). This sum is 232000000 
the remainder is 15651713. 



44 William Oughtred 

Next, he evaluates the coefficients of E in 3 A 2 E and 
420000 E, also 3^4, the coefficient of E 2 . He obtains 
3^4 2 = 480000, 3^4 = 1200, Q = 420000. He interprets 
$A* and C q as tens, $A 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-\-Si) f(a) \Si n in 
our general expression, where a = 400, Si = 10, n = ^,f(x) = 
x 3 +4 200002. 

Dividing the remainder 15651713 by 9120000, he ob- 
tains the integer i in ten's place; thus E=io, approxi- 
mately. He now computes the terms $A 2 E, ^AE* and 
E 3 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 pre- 
ceding 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 A 3 and C q A, 
because this subtraction is already affected by the pre- 
ceding work. 

We find the second trial divisor by computing the sum 
of 3^4 2 , 3^4 and C q ; that is, the sum of 504300, 1230, 
420000, which is 925530. Again, this divisor can be com- 
puted by our general expression for divisors, by taking 
= 410, Si = i t w = 3- 

Dividing 65307 13 by 925530 yields the integer 7 . Thus 
= 7. Computing 3 A 2 E, ^AE 2 , E 3 and subtracting their 
sum, the remainder is o. Hence 417 is an exact root of 
the given equation. 

Since the extraction of a cube root is merely the solu- 
tion of a pure cubic equation, x 3 = n, the process given 
above may be utilized in finding cube roots. This is 



Principal Works 45 

precisely what Oughtred does in chap, xiv of his Clams. 
If the foregoing computation is modified by putting C q = o, 
the process will yield the approximate cube root of 
247651713. 

Oughtred solves 16 examples by the process of approxi- 
mation 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 irra- 
tional, 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 Clams 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 Mil- 
bourn are the only Englishmen known to have solved 
numerical equations of higher degrees. Milbourn pub- 
lished 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 New- 
ton's method of approximation. 

It should be stated that, before the time of Newton, 
the best method of approximation to the roots of numeri- 
cal 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 William Oughtred 

LOGARITHMS 

Oughtred's treatment of logarithms is quite in accord- 
ance with the more recent practice. 1 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" 
[436 2 ], that "the logarithm of any Potestas [436 2 ] divided 
by the number of its dimensions [2] affordeth the Loga- 
rithm of its Root [436]." These statements of Oughtred 
occur for the first time in the Key of the Mathematicks of 
1647; the Clams of 1631 contains no treatment of 
logarithms. 

If the characteristic of a logarithm is negative, Ought- 
red indicates this fact by placing the above the char- 
acteristic. He separates the characteristic and mantissa 
by a comma, but still uses the sign L 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 

1 See William Oughtred's Key of the Mathemalicks, London, 1694, 
pp. 173-75, tract, "Of the Resolution of the Affected Equations," 
or any edition of the Clams after the first. 



Principal Works 47 

own school and college." 1 The invention of the slide 
rule has, until recently, 2 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 Mathematical 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 inven- 
tion of this Oughtred has a clear title. A bitter contro- 
versy 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 Rene Descartes 
relating to the theory of equations, of Robert Mayer, 

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

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



48 William Oughtred 

Hermann von Helmholtz, and Joule on the 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 con- 
troversy would never have arisen, had not some of Ought- 
red'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 Dela- 
main'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 round, and 
see what part of him is free." 1 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 Instrument called the Circles 

1 Rigaud, op. cit., Vol. I, p. 12. 



Principal Works 49 

of Proportion which contained at the end "The Declara- 
tion of the two Rulers for Calculation," giving a descrip- 
tion 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 slide rule, to be used in London for 
gauging. 1 

1 The New Artificial Gauging Line or Rod: together with rules con- 
cerning the use thereof: Invented and written by William Oughtred, 
London, 1633. 



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 instru- 
ments. 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 Public k. 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. 

So 



Minor Works 51 

Notwithstanding Oughtred's marked activity in the 
design of mathematical instruments, and his use of sur- 
veying 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 Mathemati- 
call Sciences, I tryed many waves 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 accord- 
ingly 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 paines in vaine sought 
for. 1 

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 Clams 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 

1 W. Oughtred, Apologeticall Epistle, p. 13. 



52 William Oughtred 

on the grounds of Merchiston Castle, in Edinburgh, 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 advance- 
ment 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 Philos- 



Minor Works 53 

ophers, as Secrets and Experiments in Arithmetick, 
Geometry, Cosmographie, H or ologio graphic, Astronomie, 
Navigation, Musick, Opticks, Architecture, Statick, Me- 
chanicks, Chemistry, Water-works, Fire-works, &c. Not 
vulgarly manifest till now. Written first in Greek and 
Latin, lately compiVd in French, by Henry Van Etten, 
and now in English, with the Examinations and Aug- 
mentations of divers Modern Mathematicians. W here- 
unto 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 accu- 
rate drawing of the triangles on a sphere and the measure- 
ment 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 paries 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 calcula- 
tions more than Forty years before. 

Interesting as one of our sources from which Oughtred 
obtained his knowledge of the conic sections is his study 



54 William Oughtred 

of Mydorge. A tract which he wrote thereon was pub- 
lished 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 Canta- 
brigia 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 Loga- 
rithmes, showing the practise of the Calculation of Tri- 
angles, 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 



Minor Works 



55 



Oughtred is generally attributed the introduction of the 
cross X 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 com- 
plement (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 inter- 
polations are effected with the aid of a small table contain- 
ing the logarithms of 72 sines. Except for the omission 
of the decimal point, these logarithms are natural loga- 
rithms the first of their kind ever published. In this 
table we find log 10=2302584; in modern notation, this 
is stated, log e 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 Loga- 
rithmes, which contains, besides trigonometric tables, the 
logarithms of the numbers i-iooo. 

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 i = - . 



56 William Oughtred 

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 contribu- 
tion to mathematics, for in its sixteen small pages it contains 
(i) the first use of the sign X ; (2) the first abbreviations, or 
symbols, for the sine, tangent, cosine, and cotangent; (3) the 
invention of the radix method of calculating logarithms; 
(4) the first table of hyperbolic logarithms. 1 

1 Quarterly Journal of Pure and Applied Mathematics, Vol. XLVI, 
(1915), p. 169. In this article Glaisher republishes the "Appendix" 
in full. 



CHAPTER IV 

OUGHTRED'S INFLUENCE UPON MATHEMATICAL 
PROGRESS AND TEACHING 

OUGHTRED AND HARRIOT 

Oughtred's Clams mathematicae was the most influential 
mathematical publication in Great Britain which appeared 
in the interval between John Napier's Mirifici logarilh- 
morum 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 
Clams, but also of Thomas Harriot's Arils analylicae 
praxis. We have no evidence that these two mathe- 
maticians 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 Clams. Strangely, Ought- 
red, who survived Harriot thirty-nine years, never men- 
tions 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 Rene 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 analylicae praxis was inferior to Oughtred's Clams. 
The former was a much larger book, not as conveniently 
portable, compiled after the author's death by others, 

57 



58 William Oughtred 

and not prepared with the care in the 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 Clams 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 Cam- 
bridge (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 legist before. Sir Charles Scarborough was his 
scholar; so Dr. John Wallis was his scholar; so was Chris- 
topher 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. 



Influence on Mathematical Progress 59 

He could not endure to see a scholar write an ill hand; 
he taught them all presently to mend their hands. 1 

Had Oughtred been the means of guiding the mathe- 
matical studies of only John Wallis and Christopher 
Wren one the greatest English mathematician between 
Napier and Newton, the other one of the greatest archi- 
tects of England he would have earned profound grati- 
tude. 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 Clams 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 Clams 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 Clams); 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 Clams 

1 Aubrey, op. cit., Vol. II, 1898, p. ic8. 



60 William Oughtred 

of 1647), and Thomas Wharton, who studied the Clams 
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 Ought- 
red 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 

To afford a clearer view of Oughtred as a teacher and 
mathematical expositor we quote some passages from 
various writers and from his correspondence. Anthony 
Wood 1 gives an interesting account of how Seth Ward 
and Charles Scarborough went from Cambridge Uni- 
versity 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 Clams mathematica 
which seemed at that time very obscure to them. Mr. Ought- 
red 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 aftcr- 

1 Wood's Athenae Oxonienses (ed. P. Bliss), Vol. IV, 1820, p. 247. 



Influence on Mathematical Progress 61 

wards read the Clav. Math, to their 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 
Middle Temple. 1 

Extracts from letters of W. Gascoigne to Oughtred, 
of the years 1640 and 1641, throw some light upon mathe- 
matical 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. 2 

1 Wood, op. cit., Vol. II, p. 445- 

2 Rigaud, op. cit., Vol. I, pp. 33, 35. 



62 William Oughtred 

The following extracts from two letters by W. Robin- 
son, written before the appearance of the 1647 English 
edition of the Claws, express the feeling of many readers 
of the Clams on its extreme conciseness and brevity of 
explanation: 

I shall long exceedingly till I see your Clams 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 super- 
fluity; 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 Claris, 
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 
masters dive into the depths thereof. 1 

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 admira- 
tion often considered how it was possible (even in the hardest 

1 Rigaud, op. cit., Vol. I, pp. 16, 26. 



Influence on Mathematical Progress 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 all the ancients x 

Oughtred's own feeling was against diffuseness in text- 
book writing. In his revisions of his Clams 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 turn logisticae cum analyticae adeoque 
totius mathematicae quasi clavis, which if any one of a mathe- 
matical 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 stone, base and little 
worth, can shew the excellence and riches of gold. 2 

John Wallis held Oughtred's Clams 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 mathe- 
matics, 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 ad- 
vantage, understand A c , and a 3 or aaa And the like 

1 Rigaud, op. cit., Vol. I, p. 66. 2 Ibid., Vol. I, p. 9. 



64 William Oughtred 

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 authors hi 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 substance of what is 
by others delivered in larger volumes x 

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 Clams, of a booklet 
by Gilbert Clark, entitled Oughtredus Explicatus, London, 
1682. A review of this appeared in the Ada 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 Jus- 
tinian 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, 
meaning Leybourne." 2 

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 subject. 3 He found in the Calendar 

1 Rigaud, op. cit., Vol. II, p. 475. a Ibid., Vol. II, p. 471. 

3 J. W. L. Glaisher, "On Early Logarithmic Tables, and Their 
Calculators," Philosophical Magazine, 4th Ser., Vol. XLV (1873), 
PP- 378, 379- 



Influence on Mathematical Progress 65 

of State Papers, Domestic Series, 1637, a petition to Arch- 
bishop Laud in which it is set forth that when Hoogan- 
huysen, 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." Judg- 
ment was passed that, "Considering the ill-consequence and 
scandal that would arise by strangers importing and vent- 
ing 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 con- 
fidence 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 mathe- 
matical empire will ensue. I was then very desirous to see the 
author's own book while my spirits were more free and light- 
some, 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 to keep home, and not 
put out to any foreign discoveries. 1 

It was in 1655, when Oughtred was about eighty years 
old, that John Wallis, the great forerunner of Newton in 

1 Rigaud, op. cit., Vol. I, p. 65. 



66 William Oughtred 

Great Britain, began to publish his 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 clear- 
ness and perspicacity of your understanding and genius, who 
have not only gone, but also opened a way into these profound- 
est mysteries of art, unknown^angl^not thought of by the 
ancients. With whiclTyour 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 hi 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 x 

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 

1 Rigaud, op. cit., Vol. I, p. 87. 



Influence on Mathematical Progress 67 

Locke wrote in his journal under the date, June 24, 1681, 
" the best algebra yet extant is Outred's." 1 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 piano 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 by 
Cataldus, Geysius, and Camillus Gloriosus, 2 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. iccc-\- 
i6qcc+4iqqc 23040; 18364^ 133000^ 5455+3728g+ 
8064 N aequatur 4608, finds N or a root of it to be 24, and com- 
poseth 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 

AS sooner wrote than A qc ? Let A be 2, the cube of 2 is 8, 
which squared is 64: one of the questions between Magnet 

1 King's Life of John Locke, Vol. I, London, 1830, p. 227. 

2 Exercitationum Mathematicarum Decas prima, Naples, 1627, and 
probably Cataldus' Transformatio Geometrica, Bonon., 1612. 



68 William Oughtred 

Grisio and Gloriosus is whether 64=A CC or A qc . The Cartesian 
method tells you it is A 6 , and decides the doubt x 

There is some ground for the criticisms passed by 
Collins. To be sure, the first edition of the Clams 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 ClaviSj 1647, appeared ten years after Descartes' 
innovation. Had Oughtred seen fit to adopt the new expo- 
nential 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 Des- 
cartes' Geometric of 1637. 

The year preceding Oughtred's death Mr. John Twys- 
den expressed himself as follows in the preface to his 
Miscellanies: 

It remains that I should adde something touching the begin- 
ning, 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 accu- 
rate Trigonometry and furniture of Instruments, hath in- 
riched, 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 

1 Rigaud, op. tit., Vol. II, pp. 477~8o 



Influence on Mathematical Progress 69 

other of astronomy, Savilian Professors in the University 
of Oxford. 1 

The astronomer Edmund Halley, in his preface to the 
1694 English edition of the Clams, 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 Explica- 
tion, to render it Intelligible to the less knowing Mathe- 
matical 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 Clams 
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 way of logic, than that 
of Aristotle. 2 

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 

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

2 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. 



70 William Oughtred 

of mathematics to 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 philos- 
opher 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 accusa- 
tion. I admitted that I could not do so. The Geometric 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 of which 
the plagiarism by Descartes is the most striking. 1 

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 

1 The letter is printed in John Wallis' De algebra tractatus, 1693, 
p. 206. 



Influence on Mathematical Progress 71 

and others, 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' 
Geometric where the subjects treated on the first six pages 
are found in the writings of earlier algebraists, particu- 
larly 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 pur- 
pose 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. Neverthe- 
less, Wallis fails to show that Descartes made illegiti- 
mate 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 analy- 
ticae praxis (1631). Descartes wrote a letter to Con- 
stantin Huygens in which he states that he is sending 
Harriot's book. 1 

An able discussion of the question, what effect, if 
any, Oughtred's Clams mathematicae of 1631 had upon 

1 See La Correspondence de Descartes, published by Charles Adam 
and Paul Tannery, Vol. II, Paris, 1898, pp. 456 and 457. 



72 William Oughtred 

Descartes' 1 Geometric of 1637, is given by H. Bosnians in 
a recent article. According to Bosnians no evidence has 
been found that Descartes possessed a copy of Oughtred's 
book, or that he had examined it. Bosnians 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 move- 
ment 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, as I much desire to repeat, this rather 
makes him the greater. 2 

We ourselves would hardly go as far as does Bosnians. 
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 Ought- 
red's Clams is its notation. No trace of the Englishman's 
symbolism has been pointed out in Descartes' Geometric 
of 1637. Only six years intervened between the publica- 
tion of the Clavis and the Geometric. It took longer than 
this period for the Clams to show evidence of its influence 
upon mathematical books published in England; it is 
not probable that abroad the contact was more immediate 

1 H. Bosmans, S.J., "La premiere Edition de la Clavis Mathematica 
d'Oughtred. Son influence sur la G6omtrie de Descartes," Annales 
de la societe scientifique de Bruxelles, 35th year, 1910-11, Part II, 
pp. 24-78. 

a Ibid., p. 78. 



Influence on Mathematical Progress 73 

than at home. Our study of 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 Clams 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 publica- 
tions employing it became frequent, some of them being 
the productions of pupils of Oughtred. We have seen it in 
Vincent Wing (1651),' Seth Ward (i653), 2 John Wallis 
(i655), 3 in "R. B.," a schoolmaster in Suffolk/ Samuel 
Foster (i659), s Jonas Moore (i66o), 6 and Isaac Barrow 
(i657). 7 In the latter part of the seventeenth century 

1 Vincent Wing, Harmonicon coeleste, London, 1651, p. 5. 

2 Seth Ward, In Ismaelis Bullialdi astronomiae philolaicae funda- 
menta inquisitio brevis, Oxford, 1653, p. 7. 

3 John Wallis, Elenchus geometriae Hobbianae, Oxford, 1655, p. 48. 

4 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. 

5 The Miscellanies: or Mathematical Lucubrations, of Mr. Samuel 
Foster .... by John Twysden, London, 1659, p. i. 

6 Moor's Arithmetick in two Books, London, 1660, p. 89. 

7 Isaac Barrow, Euclidis data, Cambridge, 1657, p. 2. 



74 William Oughtred 

Oughtred's notation, A.B::C.D, became the prevalent, 
though not universal, notation in Great Britain. A tre- 
mendous 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 Dulaurens (1667), I J. Prestet 
(i675), 3 R - P- Bernard Lamy (i684), 3 Ozanam (1691),* 
De 1'Hospital (1696,)* R. P. Petro Nicolas (i697). 6 

In the Netherlands we have noticed it in R. P. Bernard 
Lamy (i68o), 7 and in an anonymous work of i690. 8 
In German and Italian works of the seventeenth cen- 
tury we have not seen Oughtred's notation for propor- 
tion. 

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 desig- 
nate 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 

1 Francisci Dulaurens Specima mathematica, Paris, 1667, p. i. 

2 Elemens des mathematiques , Paris, 1675, Preface signed "J. P." 

3 Nouveaux elemens de geometric, Paris, 1692 (permission to print 
1684). 

4 Ozanam, Dictionnaire mathematique, Paris, 1691, p. 12. 
s Analyse des infiniment petits, Paris, 1696, p. n. 

6 Petro Nicolas, De conchoidibus et cissoidibus exercitationes 
geometricae, Toulouse, 1697, p. 17. 

7 R. P. Bernard Lamy, Elemens des mathematiques, Amsterdam, 
1692 (permission to print 1680). 

8 Nouveaux elemens de geometrie, 2d ed., The Hague, 1690, 
p. 304. 



Influence on Mathematical Progress 75 

tables that were bound with Oughtred's Trigonometria. 1 
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 ante- 
dates 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 Arithmetic^, 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 

1 W. W. Beman in U inter mediaire des mathematiciens , Paris, Vol. 
IX, 1902, p. 229, question 2424. 



76 William Oughtred 

the seventeenth but the more common during the eight- 
eenth century. We have observed the symbols A : B : : 
C : D (besides the authors already named) in John Collins 
(1659),' James Gregory (i663), 2 Christopher Wren (1668- 
69) ,3 William Leybourn (1673),^ William Sanders (i686), 5 
John Hawkins (i684), 6 Joseph Raphson (1697),? E. Wells 
(i698), 8 and John Ward (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 (i7oi), 10 George 
Shelley (1704)," Sam Cobb (1709)," J. Collins in Com- 
mercium Epistolicum (1712), John Craig (i7i8), 13 Jo. 

1 John Collins, The Mariner's Plain Scale New Plain'd, London, 
1659, p. 25. 

2 James Gregory, Optica promota, London, 1663, pp. 19, 48. 

3 Philosophical Transactions, Vol. Ill, London, p. 868. 

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

5 Elementa geometriae .... a Gulielmo Sanders, Glasgow, 1686, 
P- 3- 

6 Cocker's Decimal Arithmetick, .... perused by John Hawkins, 
London, 1695 (preface dated 1684), p. 41. 

7 Joseph Raphson, Analysis Aequationum unvoer sails, London, 

1697, p. 26. 

8 E. Wells, Elementa arithmeticae numerosae et speciosae, Oxford 

1698, p. 107. 

'John Ward, A Compendium of Algebra, 2d ed., London, 1698, 
p. 62. 

10 Plain Elements of Geometry and Plain Trigonometry, London, 
1701, p. 63. 

" George Shelley, Wingate's Arithmetick, London, 1704, p. 343. 

" A Synopsis of Algebra, Being a posthumous work of John Alex- 
ander of Bern, Swisserland Done from the Latin by Sam. 

Cobb, London, 1709, p. 16. 

13 John Craig, De Calculo fluenlium, London, 1718, p. 35. The 
notation A :B: :C:D is given also. 



Influence on Mathematical Progress 77 

Wilson (I724). 1 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 by M. Carre (i7oo), a M. Guisnee (i705), 3 M. de 
Fontenelle (i727), 4 M. Varignon (i725), s M. Robillard 
(1753), 6 M. Sebastien le Clerc (i764), 7 Clairaut (i73i), 8 
M. L'Hospital (i78i). 9 

In Italy Oughtred's modified notation a, b::c, d was 
used by Maria G. Agnesi in her Institmioni analitiche, 

1 Trigonometry, 26. ed., Edinburgh, 1724, p. n. 

2 Methode pour la mesure des surfaces, la dimension des solides 
.... par M. Carre de V academic r. des sciences, 1700, p. 59. 

3 Application de Valgebre d geometrie .... Paris, 7705. 

4 Elemens de la geometrie de Mnfini, by M. de Fontenelle, Paris, 
1727, p. no. 

5 Eclaircissemens sur V analyse des infiniment petits, by M. Varig- 
non, Paris, 1725, p. 87. 

6 Application de la geometrie ordinaire et des calculs di/erentiel et 
integral, by M. Robillard, Paris, 1753. 

i Traite de geometrie theorique et pratique, new ed., Paris, 1764, 
P- IS- 

8 Recherches sur les courbes d double courbure, Paris, 1731, p. 13. 

9 Analyse des infiniment petits, by the Marquis de L'Hospital. 
New ed. by M. Le Fevre, 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. 



78 William Oughtred 

Milano, 1748. The notation a:b::c:d found entrance 
the latter part of the eighteenth century. In Germany 
the symbolism a:b = c:d, suggested by Leibniz, found 
wider acceptance. 1 

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 never- 

1 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 pub- 
lished in 1688; on p. 419 is given GE:AG=LA :ML, the paper having 
been first published in 1689. Bernhardi Nieuwentiit, Consider ationes 
circa analyseos ad quantitates infinite parvas applicatae principia, 
Amsterdam, 1694, p. 20, and Analysis infinitorum, Amsterdam, 
1695, on p. 276, have x:c::s:r. Paul Halcken's Deliciae mathe- 
malicae, 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, 
Geometric du compas, translated by A. M. Carette, Paris, 1798, p. 
188, gives i/3 :2: 'i/'z'.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, 



Influence on Mathematical Progress 79 

theless interesting. His books, as 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) a and V~o+b thus, Q:a+b:, 
l/:a-\-b:, or Q:a-\-b, i/: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, 1/-I36+ 
1/2048, signifying i/ (136-}- 1/2048). 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 
a 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 

Opera pertinentia ad opticam et astronomiam, Bassani, 1785, p. 409 ? 
uses a:b::c:d. Jacob Bernoulli, Ars Conjectandi, Basel, 1713, has 
nr.ni:'.c.d. Pavlini Chelvicii, Institutiones analyticae, editio 
post tertiam Romanam prima in Germania, Vienna, 1761, p. 2, a.b: : 
c.d. Christian! Wolfii, Elementa matheseos universae, Vol. Ill, 
GeneVa, 1735, p. 63, has AB:AE=i: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 suUimoris, Tubingen, 1753, p. 194, 
has a:b = c:d. J. H. Lambert, Photometria, 1760, p. 104, has C\ir = 
BC 2 :MH 2 . Meccanica sublime del Dott. Domenico Bar taloni, 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::i,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. 



8o William Oughtred 

to 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 Des- 
cartes. 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 Oughtred the use of parentheses had 
been suggested by Clavius 1 and Girard. 2 The latter 
wrote, for instance, i/(2+i/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 who made 
use of them: I. Errard de Bar-le-Duc (i6i9), 3 Jacobo 
de Billy (i643), 4 one f whose books containing this 
notation was translated into English, and also the post- 
humous works of Samuel Foster. 5 J. W. L. Glaisher 
points out that parentheses were used by Norwood in his 
Trigonometric (1631), p. 3P. 6 

1 Chr. Clavii Operum mathematicorum tomus secundus, Mayence, 
1611, Algebra, p. 39. 

2 Invention nouvelle en Valgebre, by Albert Girard, Amsterdam, 
1629, p. 17. 

3 La geometrie el pratique generate d'icelle, par I. Errard de Bar-le- 
Duc, Ingenieur ordinaire de sa Majeste, 3d ed., revised by D. H. P. 
E. M., Paris, 1619, p. 216. 

* 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- 34 6 - 

5 Miscellanies: or Mathematical Lucubrations, of Mr. Samuel 
Foster, Sometime publike Professor of Astronomic in Gresham C 'oiled ge 
in London, London, 1659, p. 7. 

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



Influence on Mathematical Progress 81 

The symbol for the arithmetical difference between 
two numbers, ^, is usually attributed to John Wallis, 
but it occurs in Oughtred's Clams mathematicae of 1652, 
in the tract on Elementi decimi Eudidis 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. Dur- 
ing the eighteenth century the symbol found its way into 
foreign texts even in far-off Italy. 1 It is one of three 
symbols presumably invented by Oughtred and which are 
still used at the present time. The others are X and : : . 

The curious and ill-chosen symbols, CT~ for " greater 
than," and 1 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 alge- 
braists 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 ~D. Oughtred's 
symbols, or these symbols turned about in some way, have 
been used by Seth Ward, 2 John Wallis, 3 Isaac Barrow, 4 

1 Pietro Cossali, Origine, trasporto in Italia primi progressi in 
essa dell' algebra, Vol. I, Parmense, 1797, p. 52. 

3 In Is. Bullialdi astronomiae philolaicae fundamenta inquisitio 
brews, Auctore Setho Wardo, Oxford, 1653, P- * 

3 John Wallis, Algebra, London, 1685, p. 321, and in some of his 
other works. He makes greater use of Harriot's symbols. 

* Eudidis data, 1657, p. i; also Eudidis elementorum libris XV, 
London, 1659, p. i. 



82 William Oughtred 

John Kersey, 1 E. Wells, 2 John Hawkins,' Tho. Baker/ 
Richard Sault, 5 Richard Rawlinson, 6 Franciscus Dulau- 
rens, 7 James Milnes, 8 George Cheyne, 9 John Craig, 10 Jo. 
Wilson," and J. Collins. 12 

General acceptance has been accorded to Oughtred's 
symbol X. 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 X, 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 X of multiplication owes its origin to 
the old habit of using directed bars to indicate that two 

1 John Kersey, Algebra, London, 1673, P- 3 21 - 

2 E. Wells, Elementa arithmeticae numerosae el speciosae, Oxford, 
1698, p. 142. 

3 Cocker's Decimal Arithmetick, perused by John Hawkins, 
London, 1695 (preface dated 1684), p. 278. 

4 Th. Baker, The Geometrical Key, London, 1684, p. 15. 

5 Richard Sault, A New Treatise of Algebra, London (no date). 

6 Richard Rawlinson in a pamphlet without date, issued some- 
time between 1655 and 1668, containing trigonometric formulas. 
There is a copy in the British Museum. 

"i F. Dulaurens, Specima mathematica, Pars, 1667, p. i. 

8 J. Milnes, Sectionum eonicarum elementa, Oxford, 1702, p. 42. 

9 Cheyne, Philosophical Principles of Natural Religion, London, 
1705, P- 55- 

10 J. Craig, De calculo fluentium, London, 1718, p. 86. 

11 Jo. Wilson, Trigonometry, 2d ed., Edinburgh, 1724, p. v. 
" Commercium Epistolicum, 1712, p. 20. 



Influence on Mathematical Progress 83 

numbers are to be combined, as for instance in the multi- 
plication of 23 and 34, thus, 




has been advanced by two writers, C. Le Paige 1 and 
Gravelaar. 2 Bosnians is more inclined to the belief that 
Oughtred adopted the symbol somewhat arbitrarily, 
much as he did the numerous symbols in his Elementi 
decimi Euclidis declaration 

Le Paige's and Gravelaar's theory finds some support 
in the fact that the cross X, without the two additional 
vertical lines shown above, occurs in a commentary 
published by Oswald Schreshensuchs 4 in 1551, where the 
sign is written between two factors placed one above the 
other. 

1 C. Le Paige, "Sur Porigine de certains signes d'operation," 
Annales de la societe scientifique de Bruxelles, i6th year, 1891-92, 
Part II, pp. 79-82. 

2 Gravelaar, "Over den oorsprong van ons maalteeken (X)," 
Wiskundig Tijdschrift, 6th year. We have not had access to this 
article. 

3 H. Bosnians, op. tit., p. 40. 

Claudii Ptolemaei .... annotationes , Bale, 1551. This refer- 
ence is taken from the Encyclopedic des sciences mathematiques , 
Tome I, Vol. I, Fasc. i, p. 40. 



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. Never- 
theless, 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 pub- 
lished, from his prefaces, and from passages in his con- 
troversial writing against Delamain. As we proceed to 
give quotations unfolding Oughtred's views, we shall 
observe that three points receive special emphasis: (i) an 
appeal to the eye through suitable symbolism; (2) em- 
phasis 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 nar- 
rowness of their range, often mislead. Thus far they have 
been inferior to the word of a man of experience, insight, 
and conviction. 

84 



Ideas on Teaching Mathematics 85 

MATHEMATICS, "A SCIENCE OF THE EYE" 

Oughtred was a great admirer of the Greek mathe- 
maticians 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, j 
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 illus- I 
tration of the importance of symbolism is afforded by the 
history of the formula 

c = 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 Clams 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 



86 William Oughtred 

affections of Theorems, inequality, 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 peda- 
gogical experience of recent centuries has indorsed Ought- 
red'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," where- 
upon Wallis replied: 

I wonder how you durst touch M. Oughtred for fear of catch- 
ing 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 patience to examine 
whether it is well or ill demonstrated. 1 

1 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 Mat hematic ks. By the Professor of Geometry. Oxford, 
1656, pp. 7, 47, 50. 



Ideas on Teaching Mathematics 87 

Oughtred maintained his view of the importance of 
symbols on many 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 earn- 
estly 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 fol- 
lowing extract from his preface to the 1647 Clams: 

.... Which Treatise being not written in the usuall syn- 
thetical manner, nor with verbous expressions, but in the inven- 
tive 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 pre- 
senteth to the eye the whole course and processe of every opera- 
tion 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 William Oughtred 

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 aequa- 
tions, Interpretations, Comparations, Reductions, and Dis- 
quisitions, 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 
tune, and much industry, I found out this way for the helpe 
and facilitation of Art. 

Still greater emphasis upon rigorous thinking in mathe- 
matics 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 luglers: to the despite of Art, losse 
of previous time, and betraying of willing and industrious 
wits, vnto ignorance, and idlenesse. That the vse of Instru- 
ments is indeed excellent, if a man be an Artist: but contemp- 
tible, 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 instru- 
ments might very well be placed in the hands of pupils 
from the start. At the time of this controversy Delamain * 



Ideas on Teaching Mathematics 89 

supported himself by teaching mathematics in London 
and he advertised his ability to give instruction in mathe- 
matics, 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. 1 

As representing Delamain's views, we make the fol- 
lowing selection from his Grammelogia (London, about 
1633), the part near the end of the book and bearing the 
title, "In the behalf e 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 . " D elamain say s : 

.... 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 glan- 
cing upon many noble personages, with too grosse, if not too 
base an attribute, in tearming them doers of tricks, as it were to 

1 Oughtred, Apologeticdl Epistle, p. 27. 



go William Oughtred 

iuggle: because they perhaps make use of a necessitie in the 
furnishing of themselves with such knowledge by Practicall 
Instrumental 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 excel- 
lencie 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 behalf e of my selfe, and others, I answer; That Instru- 
mentall operation is not only the Compendia ting, and facilitat- 
ing 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 Mathe- 
maticall Instrument], except he knowes the cause of its opera- 
tion, 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 difncultie, and impos- 
sibilitie 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 



Ideas on Teaching Mathematics 91 

queasy 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 f arre 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 loga- 
rithms ? 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 under- 
lying principles of electricity? Does the use of instru- 
ments ordinarily discourage a boy from mastery of the 
theory ? Or does such manipulation constitute a natural 
and pleasing approach to the abstract ? On this particu- 
lar point, who showed the profounder psychological in- 
sight, Oughtred or Delamain? 



92 William Oughtred 

In July, 1914, there was held in Edinburgh a celebra- 
tion of the three-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 mathe- 
matical 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 



Ideas on Teaching Mathematics 93 

of the cosmic 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 



94 William Oughtred 

the slide rule on the idea of one hand washing the other 
until a mastery of both the theory and the use of the 
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, knowl- 
edge, and experience. 



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 mathe- 
matics. Yet his importance as a teacher did receive 
recognition in the seventeenth century by no less distin- 
guished 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 con- 
jectures and reason of their correction they make of the aber- 
rations they shall find, and the qualities and condition of I heir 
ship, and the diversities and seasons of the winds, and the 
secret motions or agitations of the Seas, when they begin, and 



Ideas on Teaching Mathematics 95 

how 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 Mathe- 
ma ticks and lovers and enquirers of the truth: I doubt not 
but that there shall be in convenient time, brought to light 
many necessary precepts which may tend to y e 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 more to the improvem* of Navigation and 
safety of Mens lives and estates on that element. 1 

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

1 J. Edleston, Correspondence of Sir Isaac Newton and Professor 
Cotes, London, 1850, pp. 279-92. 



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., i, 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 

Bosnians, H., 72, 83 

Boyle, R., i, 69 

Braunmiihl, 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 
Carre, 77 

Carrete, N. M., 78 
Caiyll, 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, Si, 59, 87, 88 
Clairaut, 77 
Clark, A., 3 
Clark, G., 63 
Clarke, F. L., 3 
Clavis mathematicae, i, 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, n, 53 
Cossali, P., 81 
Cotes, R., i, 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, n, 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., i, 25, 47, 57, 68- 

72,80 

Dibuadius, 79 
Difference, symbol for, 27, 81 



97 



William Oughtred 



Diophantus, 63, 85 

Division, abbreviated, 21, 23, 24 

Dulaurens, F., 74, 82 

Earl of Arundel, 10, 13, 15, 17 

Edleston, J., 95 

Enestrom, G., 40 

Equations, solution of, 18, 28, 

29, 3i, 34, 39-45, 87 
Errard de Bar-le-Duc, 80 
Eton College, 3, 4 
Euclid, i, 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, i, 6, 27, 59, 61, 

80 

Guisn6e, 77 
Gunter, E., 37, 47, 86 
Gunter's scale, 37 

Halcken, P., 78 
Hales, J., 7 
Halley, E., i, 18, 69 
Hankel, H., 40 
Harper, T., 18 



Harriot, T., 45, 47, 57, 58, 69- 

7i,8i 

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., i 
Horner's method, 45 
Horology, 18, 50 
Horrox, J., 4 
Hospital, de 1', 74, 77 
Howard, Th. See Earl of 

Arundel. 

Howard, W., 17, 18, 59 
Hutchinson, A., 6 

Invisible college, i 
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 Fevre, 77 
Leibniz, 47, 78, 80 
Leonelli, 56 
Le Paige, de, 83 
Less than, symbol for, 81 
Leurechon, 52 
Leybourn, 35, 64, 76 



Index 



99 



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 a 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, i, 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 

TT, 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, ii 

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, i 
Schreshensuchs, O., 83 
Scratch method, 23 
Shakespeare, 52 
Shelley, G., 76 
Shipley, A. E., i 
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 



100 



William Oughtred 



Tannery, P., 71 
Todhunter, 60 
Torporley, 58 

Triangles, spherical, 53, 54, 93 
Trigonometric,, 21, 36, 55, 75 
Trigonometric functions, sym- 
bols for, 36, 37, 55, 56 
Trigonometric, 21, 35, 39 
Trisection of angles, 28 
Twysden, 59, 68, 69, 73 

Varignon, 77 

Vieta, i, 2, 25, 32, 33, 35, 39- 

41, 45, 63, 67, 70, 71 
Vlack, 65 
Von Braunmiihl, 39 

Wadham College, 5, 53 
Wallis, John, i, 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, 6 1 
Wood, R., 18, 59 
Wren, Christopher, 5, 58, 59, 76 
Wright, E., 6, 27, 38, 54 
Wright, S., 54 



QA Cajori, Florian 

29 William Oughtred 

08C3 



PLEASE DO NOT REMOVE 
CARDS OR SLIPS FROM THIS POCKET 

UNIVERSITY OF TORONTO LIBRARY