WILLIAM OUGHTRED
WILLIAM OUGHTRED
A GREAT SEVENTEENTH-CENTURY
TEACHER OF
MATHEMATICS
BY
FLORIAN CAJORI, Ph.D.
Professor of Mathematics
Colorado College
CHICAGO LONDON
THE OPEN COURT PUBLISHING COMPANY
1916
Copyright 1916 By
The Open Court Publishing Co.
All Rights Reserved
Published September 1916
Composed and Printed By
The University of Chicago Press
Chicago, Illinois, U.S.A.
TABLE OF CONTENTS
PAGE
Introduction 1
CHAPTER
I. Oughtred’s Life 3
At School and University 3
As Rector and Amateur Mathematician 6
His Wife 7
In Danger of Sequestration 8
His Teaching 9
Appearance and Habits 12
Alleged Travel Abroad 14
His Death 15
II. Principal Works 17
Clavis mathematicae 17
Circles of Proportion and Trigonometrie 35
Solution of Numerical Equations 39
Logarithms 46
Invention of the Slide Rule; Controversy on Priority of Invention 46
III. Minor Works 50
IV. Oughtred’s Influence upon Mathematical Progress and Teaching 57
Oughtred and Harriot 57
Oughtred’s Pupils 58
Oughtred, the “Todhunter of the Seventeenth Century” 60
Was Descartes Indebted to Oughtred? 69
The Spread of Oughtred’s Notations 73
V. Oughtred’s Ideas on the Teaching of Mathematics 84
General Statement 84
Mathematics, “a Science of the Eye” 85
Rigorous Thinking and the Use of Instruments 87
Newton’s Comments on Oughtred 94
Index 97
INTRODUCTION
In the year 1660 the Royal Society was founded by royal favor in London,
although in reality its inception took place in 1645 when the
Philosophical Society (or, as Boyle called it, the “Invisible College”)
came into being, which held meetings at Gresham College in London and
later in Oxford. It was during the second half of the seventeenth century
that Sir Isaac Newton, surrounded by a group of great men—Wallis, Hooke,
Barrow, Halley, Cotes—carried on his epoch-making researches in
mathematics, astronomy, and physics. But it is not this half-century of
science in England, nor any of its great men, that especially engage our
attention in this monograph. It is rather the half-century preceding, an
epoch of preparation, when in the early times of the House of Stuart the
sciences began to flourish in England. Says Dr. A. E. Shipley: “Whatever
were the political and moral deficiencies of the Stuart kings, no one of
them lacked intelligence in things artistic and scientific.” It was at
this time that mathematics, and particularly algebra, began to be
cultivated with greater zeal, when elementary algebra with its symbolism
as we know it now began to take its shape.
Biographers of Sir Isaac Newton make particular mention of five
mathematical books which he read while a young student at Cambridge,
namely, Euclid’s Elements, Descartes’s Géométrie, Vieta’s Works, Van
Schooten’s Miscellanies, and Oughtred’s Clavis mathematicae. The last of
these books has been receiving increasing attention from the historians
of algebra in recent years. We have prepared this sketch because we felt
that there were points of interest in the life and activity of Oughtred
which have not received adequate treatment. Historians have discussed his
share in the development of symbolic algebra, but some have fallen into
errors, due to inability to examine the original editions of Oughtred’s
Clavis mathematicae, which are quite rare and inaccessible to most
readers. Moreover, historians have failed utterly to recognize his
inventions of mathematical instruments, particularly the slide rule; they
have completely overlooked his educational views and his ideas on
mathematical teaching. The modern reader may pause with profit to
consider briefly the career of this interesting man.
Oughtred was not a professional mathematician. He did not make his
livelihood as a teacher of mathematics or as a writer, nor as an engineer
who applies mathematics to the control and use of nature’s forces.
Oughtred was by profession a minister of the gospel. With him the study
of mathematics was a side issue, a pleasure, a recreation. Like the great
French algebraist, Vieta, from whom he drew much of his inspiration, he
was an amateur mathematician. The word “amateur” must not be taken here
in the sense of superficial or unthorough. Great Britain has had many men
distinguished in science who pursued science as amateurs. Of such men
Oughtred is one of the very earliest.
F. C.
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
universities, young Oughtred received thorough training in classical
learning.
According to information received from F. L. Clarke, Bursar and Clerk of
King’s College, Cambridge, Oughtred was admitted at King’s a scholar from
Eton on September 1, 1592, at the age of seventeen. He was made Fellow at
King’s on September 1, 1595, while Elizabeth was still on the throne. He
received in 1596 the degree of Bachelor of Arts and in 1600 that of
Master of Arts. He vacated his fellowship about the beginning of August,
1603. His career at the University of Cambridge we present in his own
words. He says:
Next after Eaton schoole, I was bred up in Cambridge in Kings Colledge:
of which society I was a member about eleven or twelve yeares: wherein
how I behaved my selfe, going hand in hand with the rest of my ranke in
the ordinary Academicall studies and exercises, and with what
approbation, is well knowne and remembered by many: the time which over
and above those usuall studies I employed upon the Mathematicall
sciences, I redeemed night by night from my naturall sleep, defrauding
my body, and inuring it to watching, cold, and labour, while most
others tooke their rest. Neither did I therein seek only my private
content, but the benefit of many: and by inciting, assisting, and
instructing others, brought many into the love and study of those Arts,
not only in our own, but in some other Colledges also: which some at
this time (men far better than my selfe in learning, degree, and
preferment) will most lovingly acknowledge.[2]
These words describe the struggles which every youth not endowed with the
highest genius must make to achieve success. They show, moreover, the
kindly feeling toward others and the delight he took throughout life in
assisting anyone interested in mathematics. Oughtred’s passion for this
study is the more remarkable as neither at Eton nor at Cambridge did it
receive emphasis. Even after his time at Cambridge mathematical studies
and their applications were neglected there. Jeremiah Horrox was at
Cambridge in 1633-35, desiring to make himself an astronomer.
“But many impediments,” says Horrox, “presented themselves: the tedious
difficulty of the study itself deterred a mind not yet formed; the want
of means oppressed, and still oppresses, the aspirations of my mind:
but that which gave me most concern was that there was no one who could
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.”[3]
Some attention was given to Greek mathematicians, but the works of
Italian, German, and French algebraists of the latter part of the
sixteenth and beginning of the seventeenth century were quite unknown at
Cambridge in Oughtred’s day. It was part of his life-work as a
mathematician to make algebra, as it was being developed in his time,
accessible to English youths.
At the age of twenty-three Oughtred invented his Easy Way of Delineating
Sun-Dials by Geometry, which, though not published until about half a
century later, in the first English edition of Oughtred’s Clavis
mathematicae in 1647, was in the meantime translated into Latin by
Christopher Wren, then a Gentleman Commoner of Wadham College, Oxford,
now best known through his architectural creations. In 1600 Oughtred
wrote a monograph on the construction of sun-dials upon a plane of any
inclination, but that paper was withheld by him from publication until
1632. Sun-dials were interesting objects of study, since watches and
pendulum clocks were then still unknown. All sorts of sun-dials, portable
and non-portable, were used at that time and long afterward. Several of
the college buildings at Oxford and Cambridge have sun-dials even at the
present time.
AS RECTOR AND AMATEUR MATHEMATICIAN
It was in 1604 that Oughtred entered upon his professional life-work as a
preacher, being instituted to the vicarage of Shalford in Surrey. In 1610
he was made rector of Albury, where he spent the remainder of his long
life. Since the era of the Reformation two of the rectors of Albury
obtained great celebrity from their varied talents and acquirements—our
William Oughtred and Samuel Horsley. Oughtred continued to devote his
spare time to mathematics, as he had done in college. A great
mathematical invention made by a Scotchman soon commanded his
attention—the invention of logarithms. An informant writes as follows:
Lord Napier, in 1614, published at Edinburgh his Mirifici logarithmorum
canonis descriptio. . . . . It presently fell into the hands of Mr.
Briggs, then geometry-reader at Gresham College in London: and that
gentleman, forming a design to perfect Lord Napier’s plan, consulted
Oughtred upon it; who probably wrote his Treatise of Trigonometry about
the same time, since it is evidently formed upon the plan of Lord
Napier’s Canon.[4]
It will be shown later that Oughtred is very probably the author of an
“Appendix” which appeared in the 1618 edition of Edward Wright’s
translation into English of John Napier’s Descriptio. This “Appendix”
relates to logarithms and is an able document, containing several points
of historical interest. Mr. Arthur Hutchinson of Pembroke College informs
me that in the university library at Cambridge there is a copy of
Napier’s Constructio (1619) bound up with a copy of Kepler’s Chilias
logarithmorum (1624), that at the beginning of the Constructio is a blank
leaf, and before this occurs the title-page only of Napier’s Descriptio
(1619), at the top of which appears Oughtred’s autograph. The history of
this interesting signature is unknown.
HIS WIFE
In 1606 he married Christ’sgift Caryll, daughter of Caryll, Esq., of
Tangley, in an adjoining parish.[5] We know very little about Oughtred’s
family life. The records at King’s College, Cambridge,[6] 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,[7] according to which Oughtred had nine sons and four
daughters. Reference to the wife and children is sometimes made in the
correspondence with Oughtred. In 1616 J. Hales writes, “I pray let me be
remembered, though unknown, to Mistress Oughtred.”[8]
As we shall see later, Oughtred had a great many young men who came to
his house and remained there free of charge to receive instruction in
mathematics, which was likewise gratuitous. This being the case,
certainly great appreciation was due to Mrs. Oughtred, upon whom the
burden of hospitality must have fallen. Yet chroniclers are singularly
silent in regard to her. Hers was evidently a life of obscurity and
service. We greatly doubt the accuracy of the following item handed down
by Aubrey; it cannot be a 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.[9]
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.[10]
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.[11]
Not without interest is the account of this matter given by Lilly
himself:
About this Time, the most famous Mathematician of all Europe, (Mr.
William Oughtred, Parson of Aldbury in Surrey) was in Danger of
Sequestration by the Committee of or for plunder’d Ministers;
(Ambo-dexters they were;) several inconsiderable Articles were deposed
and sworn against him, material enough to have sequestred him, but
that, upon his Day of hearing, I applied my self to Sir Bolstrode
Whitlock, and all my own old Friends, who in such Numbers appeared in
his Behalf, that though the Chairman and many other Presbyterian
Members were stiff against him, yet he was cleared by the major Number.
The truth is, he had a considerable Parsonage, and that only was enough
to sequester any moderate Judgment: He was also well known to affect
his Majesty [Charles I]. In these Times many worthy Ministers lost
their Livings or Benefices, for not complying with the Three-penny
Directory.[12]
HIS TEACHING
Oughtred had few personal enemies. His pupils held him in highest esteem
and showed deep gratitude; only one pupil must be excepted, Richard
Delamain. Against him arose a bitter controversy which saddened the life
of Oughtred, then an old man. It involved, as we shall see later, the
priority of invention of the circular slide rule and of a horizontal
instrument or portable sun-dial. In defense of himself, Oughtred wrote in
1633 or 1634 the Apologeticall Epistle, from which we quoted above. This
document contains biographical details, in part as follows:
Ever since my departure from the Vniversity, which is about thirty
yeares, I have lived neere to the Towne of Guildford in Surrey: where,
whether I have taken so much liberty to the losse of time, and the
neglect of my calling the whole Countrey thereabout, both Gentry and
others, to whom I am full well knowne, will quickely informe him; my
house being not past three and twenty miles from London: and yet I so
hid my selve at home, that I seldomly travelled so farre as London once
in a yeare. Indeed the life and mind of man cannot endure without some
interchangeablenesse of recreation, and pawses from the intensive
actions of our severall callings; and every man is drawne with his owne
delight. My recreations have been diversity of studies: and as oft as I
was toyled with the labour of my owne profession, I have allayed that
tediousnesse by walking in the pleasant and more then Elysian fields of
the diverse and various parts of humane learning, and not the
Mathematics onely.
Even the opponents of Delamain must be grateful to him for having been
the means of drawing from Oughtred such interesting biographical details.
Oughtred proceeds to tell how, about 1628, he was induced to write his
Clavis mathematicae, upon which his reputation as a mathematician largely
rests:
About five yeares since, the Earle of Arundell my most honourable Lord
in a time of his private retiring to his house in the countrey then at
West Horsley, foure small miles from me (though since he hath a house
in Aldebury the parish where I live) hearing of me (by what meanes I
know not) was pleased to send for me: and afterward at London to
appoint mee a Chamber of his owne house: where, at such times, and in
such manner as it seemed him good to imploy me, and when I might not
inconveniently be spared from my charge, I have been most ready to
present my selfe in all humble and affectionate service: I hope also
without the offence of God, the transgression of the good Lawes of this
Land, neglect of my calling, or the deserved scandall of any good man.
. . . .
And although I am no mercenary man, nor make profession to teach any
one in these arts for gaine and recompence, but as I serve at the
Altar, so I live onely of the Altar: yet in those interims that I am at
London in my Lords service, I have been still much frequented both by
Natives and Strangers, for my resolution and instruction in many
difficult poynts of Art; and have most freely and lovingly imparted my
selfe and my skill, such as I had, to their contentments, and much
honourable acknowledgement of their obligation to my Lord for bringing
mee to London, hath beene testifyed by many. Of which my liberallity
and unwearyed readinesse to doe good to all, scarce any one can give
more ample testimony then R. D. himselfe can: would he be but pleased
to allay the shame of this his hot and eager contention, blowne up
onely with the full bellowes of intended glory and gaine; . . . . they
[the subjects in which Delamain received assistance from Oughtred] were
the first elements of Astronomie concerning the second motions of the
fixed starres, and of the Sunne and Moone; they were the first elements
of Conics, to delineate those sections: they were the first elements of
Optics, Catoptrics, and Dioptrics: of all which you knew nothing at
all.
These last passages are instructive as showing what topics were taken up
for study with some of his pupils. The chief subject of interest with
most of them was algebra, which at that time was just beginning to draw
the attention of English lovers of mathematics.
Oughtred carried on an extensive correspondence on mathematical subjects.
He was frequently called upon to assist in the solution of knotty
problems—sometimes to his annoyance, perhaps, as is shown by the
following letter which he wrote in 1642 to a stranger, named Price:
It is true that I have bestowed such vacant time, as I could gain from
the study of divinity, (which is my calling,) upon human knowledges,
and, amongst other, upon the mathematics, wherein the little skill I
have attained, being compared with others of my profession, who for the
most part contenting themselves only with their own way, refuse to
tread these salebrous and uneasy paths, may peradventure seem the more.
But now being in years and mindful of mine end, and having paid dearly
for my former delights both in my health and state, besides the
prejudice of such, who not considering what incessant labour may
produce, reckon so much wanting unto me in my proper calling, as they
think I have acquired in other sciences; by which opinion (not of the
vulgar only) I have suffered both disrespect, and also hinderance in
some small perferments I have aimed at. I have therefore now learned to
spare myself, and am not willing to descend again in arenam, and to
serve such ungrateful muses. Yet, sir, at your request I have perused
your problem. . . . . Your problem is easily wrought per Nicomedis
conchoidem lineam.[13]
APPEARANCE AND HABITS
Aubrey gives information about the appearance and habits of Oughtred:
He was a little man, had black haire, and blacke eies (with a great
deal of spirit). His head was always working. He would drawe lines and
diagrams on the dust. . . . .
He [his oldest son Benjamin] told me that his father did use to lye a
bed till eleaven or twelve a clock, with his doublet on, ever since he
can remember. Studyed late at night; went not to bed till 11 a clock;
had his tinder box by him; and on the top of his bed-staffe, he had his
inke-horne fix’t. He slept but little. Sometimes he went not to bed in
two or three nights, and would not come downe to meales till he had
found out the quaesitum.
He was more famous abroad for his learning, and more esteemed, then at
home. Severall great mathematicians came over into England on purpose
to converse with him. His countrey neighbours (though they understood
not his worth) knew that there must be extraordinary worth in him, that
he was so visited by foreigners. . . . .
When learned foreigners came and sawe how privately he lived, they did
admire and blesse themselves, that a person of so much worth and
learning should not be better provided for. . . . .
He has told bishop Ward, and Mr. Elias Ashmole (who was his 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.[14]
Oughtred’s friends convey the impression that, in the main, Oughtred
enjoyed a comfortable living at Albury. Only once appear indications of
financial embarrassment. About 1634 one of his pupils, W. Robinson,
writes as follows:
I protest unto you sincerely, were I as able as some, at whose hands
you have merited exceedingly, or (to speak more absolutely) as able as
willing, I would as freely give you 500 l. per ann. as 500 pence; and I
cannot but be astonished at this our age, wherein pelf and dross is
made their summum bonum, and the best part of man, with the true
ornaments thereof, science and knowledge, are so slighted. . . . .[15]
In his letters Oughtred complains several times of the limitations for
work and the infirmities due to his advancing old age. The impression he
made upon others was quite different. Says one biographer:
He sometimes amused himself with archery, and sometimes practised as a
surveyor of land. . . . . He was sprightly and active, when more than
eighty years of age.[16]
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
maintained secure in his privacy, prudence, meekness, simplicity,
resolution, patience, and contentment.[17]
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.[18]
A portrait of Oughtred, painted in 1646 by Hollar and inserted in the
English edition of the Clavis of 1647, contains underneath the following
lines:
“Haec est Oughtredi senio labantis imago
Itala quam cupiit, Terra Britanna tulit.”
In the sketch of Oughtred by Owen Manning it is confessed that “it is not
known to what this alludes; but possibly he might have been in Italy with
his patron, the Earl of Arundel.”[19] It would seem quite certain either
that Oughtred traveled in Europe or that he received some sort of an
offer to settle in Italy. In view of Aubrey’s explicit statement and of
Oughtred’s well-known habit of confining himself to his duties and
studies in his own parish, seldom going even as far as London, we
strongly incline to the opinion that he did not travel on the Continent,
but that he received an offer from some patron of the sciences—possibly
some distinguished visitor—to settle in Italy.
HIS DEATH
He died at Albury, June 30, 1660, aged about eighty-six years. Of his
last days and death, Aubrey speaks as follows:
Before he dyed he burned a world of papers, and sayd that the world was
not worthy of them; he was so superb. He burned also severall printed
bookes, and would not stirre, till they were consumed. . . . . I
myselfe have his Pitiscus, imbelished with his excellent marginall
notes, which I esteeme as a great rarity. I wish I could also have got
his Bilingsley’s Euclid, which John Collins sayes was full of his
annotations. . . . .
Ralph Greatrex, his great friend, the mathematicall instrument-maker,
sayed he conceived he dyed with joy for the comeing-in of the king,
which was the 29th of May before. “And are yee sure he is
restored?”—“Then give me a glasse of sack to drinke his sacred
majestie’s health.” His spirits were then quite upon the wing to fly
away. . . . .[20]
In this passage, as in others, due allowance must be made for Aubrey’s
lack of discrimination. He was not in the habit of sifting facts from
mere gossip. That Oughtred should have declared that the world was not
worthy of his papers or manuscripts is not in consonance with the
sweetness of disposition ordinarily attributed to him. More probable was
the feeling that the papers he burned—possibly old sermons—were of no
particular value to the world. That he did not destroy a large mass of
mathematical manuscripts is evident from the fact that a considerable
number of them came after his death into the hands of Sir Charles
Scarborough, M.D., under whose supervision some of them were carefully
revised and published at Oxford in 1677 under the title of Opuscula
mathematica hactenus inedita.
Aubrey’s story of Oughtred’s mode of death has been as widely circulated
in every modern biographical sketch as has his slander of Mrs. Oughtred
by claiming that she was so penurious that she would deny him the use of
candles to read by. Oughtred died on June 30; the Restoration occurred on
May 29. No doubt Oughtred rejoiced over the Restoration, but the story of
his drinking “a glass of sack” to his Majesty’s health, and then dying of
joy is surely apocryphal. De Morgan humorously remarks, “It should be
added, by way of excuse, that he was eighty-six years old.”[21]