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






                            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]




                               CHAPTER II
                            PRINCIPAL WORKS


                         “CLAVIS MATHEMATICAE”

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

The Clavis mathematicae,[22] in its first edition of 1631, was a booklet
of only 88 small pages. Yet it contained in very 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 book underwent hardly any changes, except for the number of
tracts appended, and brief explanatory notes added at the close of the
chapters in the English editions of 1694 and 1702. The 1652 and 1667
editions were seen through the press by John Wallis; the 1698 impression
contains on the title-page the words: Ex Recognitione D. Johannis Wallis,
S.T.D. Geometriae Professoris Saviliani.

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

As compared with other contemporary works on algebra, Oughtred’s
distinguishes itself for the amount of symbolism used, particularly in
the treatment of geometric problems. Extraordinary emphasis was placed
upon what he called in the Clavis the “analytical art.”[24] 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.”[25] He meant to express by it condensed processes of rigid,
logical deduction expressed by appropriate symbols, as contrasted with
mere description or elucidation by passages fraught with verbosity. In
the preface to the first edition (1631) he says:

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

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

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

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

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

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

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

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

In the third and fourth cases are considered factors which appear as
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.”[26] This is in
harmony with etymological dictionaries which speak of a parabola as the
application of a given area to a given straight line. The dividend or
product is the area; the divisor or factor is the line.

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

                    17
                   3̸0̸3̸
                  2̸8̸0̸3̸
                1̸0̸9̸9̸3̸0̸
  3̣5̣7̣|0̣9̣2̣6425)  4̸6̸7̸0̸2̸3̸  (1307|80
                3̸5̸7̸0̸9̸3̸
                1̸0̸7̸1̸2̸7̸
                  2̸5̸0̸0̸
                   2̸8̸6̸


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

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

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

The characteristic in the Clavis that is most striking to a modern reader
is the total absence of indexes or exponents. There is much discussion in
the leading treatises of the latter part of the sixteenth and the early
part of the seventeenth century on the theory of indexes, but the modern
exponential notation, aⁿ, is of later date. The modern notation, for
positive integral exponents, first appears in Descartes’ Géométrie, 1637;
fractional and negative exponents were first used in the modern form by
Sir Isaac Newton, in his announcement of the binomial formula, in a
letter written in 1676. This total absence of our modern exponential
notation in Oughtred’s Clavis gives it a strange aspect. Like Vieta,
Oughtred uses ordinarily the capital letters, A, B, C, . . . . to
designate given numbers; A² is written Aq, A³ is written Ac; for A⁴, A⁵,
A⁶ he has, respectively, Aqq, Aqc, Acc. Only on rare occasions, usually
when some parallelism in notation is aimed at, does he use small
letters[27] to represent numbers or magnitudes. Powers of binomials or
polynomials 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 (). Parentheses had been used by
Girard, and by Clavius as early as 1609,[28] but did not come into
general use in mathematical language until the time of Leibniz and the
Bernoullis. Oughtred indicates aggregation by writing a colon (:) at both
ends. Thus, Q:A-E: means with him (A-E)². Similarly, √q:A+E: means
√(A+E). The two dots at the end are frequently omitted when the part
affected includes all the terms of the polynomial to the end. Thus,
C:A+B-E=.. means (A+B-E)³=.. There are still further departures from this
notation, but they occur so seldom that we incline to the interpretation
that they are simply printer’s errors. For proportion Oughtred uses the
symbol (::). The proportion a:b=c:d appears in his notation a·b::c·d.
Apparently, a proportion was not fully recognized in this day as being
the expression of an equality of ratios. That probably explains why he
did not use = here as in the notation of ordinary equations. Yet Oughtred
must have been very close to the interpretation of a proportion as an
equality; for he says in his Elementi decimi Euclidis declaratio,
“proportio, sive ratio aequalis ::” That he introduced this extra symbol
when the one for equality was sufficient is a misfortune. Simplicity
demands that no unnecessary symbols be introduced. However, Oughtred’s
symbolism is certainly superior to those which preceded. Consider the
notation of Clavius.[29] He wrote 20:60=4:x, x=12, thus: “20·60·4? fiunt
12.” The insufficiency of such a notation in the more involved
expressions frequently arising in algebra is readily seen. Hence
Oughtred’s notation (::) was early adopted by English mathematicians. It
was used by John Wallis at Oxford, by Samuel Foster at Gresham College,
by James Gregory of Edinburgh, by the translators into English of Rahn’s
algebra, and by many other early writers. Oughtred has been credited
generally with the introduction of St. Andrew’s cross × as the symbol for
multiplication in the Clavis of 1631. We have discovered that this
symbol, or rather the letter x which closely resembles it, occurs as the
sign of multiplication thirteen years earlier in an anonymous “Appendix
to the Logarithmes, shewing the practise of the Calculation of Triangles
etc.” to Edward Wright’s translation of John Napier’s Descriptio,
published in 1618.[30] Later we shall give our reasons for believing that
Oughtred is the author of that “Appendix.” The × has survived as a symbol
of multiplication.

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

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

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

The differences between the seven different editions of the Clavis lie
mainly in the special parts appended to some editions and dropped in the
latest editions. The part which originally constituted the Clavis was not
materially altered, except in two or three of the original twenty
chapters. These changes were made in the editions of 1647 and 1648. After
the first edition, great stress was laid upon the theory of indices upon
the very first page, as also in passages farther on. Of course, Oughtred
did not have our modern notation of indices or exponents, but their
theory had been a part of algebra and arithmetic for some time. Oughtred
incorporated this theory in his brief exposition of the Hindu-Arabic
notation and in his explanation of logarithms. As previously pointed out,
the last three chapters of the 1631 edition were considerably rearranged
in the later editions and combined into two chapters, so that the Clavis
proper had nineteen chapters instead of twenty in the additions after the
first. These chapters consisted of applications of algebra to geometry
and were so framed as to constitute a severe test of the student’s grip
of the subject. The very last problem deals with the division of angles
into equal parts. He derives the cubic equation upon which the trisection
depends algebraically, also the equations of the fifth degree and seventh
degree upon which the divisions of the angle into 5 and 7 equal parts
depend, respectively. The exposition was severely brief, yet accurate. He
did not believe in conducting the reader along level paths or along
slight inclines. He was a guide for mountain-climbers, and woe unto him
who lacked nerve.

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

As a specimen of Oughtred’s style of writing we quote his solution of
quadratic equations, accompanied by a translation into English and into
modern mathematical symbols.

As a preliminary step[33] he lets

  Z=A+E and A>E;

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

  A=½Z+√(¼Z²-AE).

Having established these preliminaries, he proceeds thus:

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

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

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

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

  Given therefore an unequally divided line Z (10), and a rectangle
  beneath the segments AE (21) which is a gnomon. Half the difference of
  the segments ½X is given, and consequently the segment itself. For, if
  one of the two segments is placed equal to A, the other will be Z-A.
  Moreover, the rectangle is ZA-A²=AE. And because Z and AE are given,
  and there is ¼Z²-AE=¼X², and by 5c.18, ½Z+½X=A, and ½Z-½X=E, the
  equation will be solved thus: ½Z±√(¼Z²-AE)=A {major segment/minor
  segment.

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

  The half of the middle species is Z/2 (5), its square is Z²/4 (25).
  From it subtract the absolute term AE (21), and Z²/4-AE (4) will be the
  square of half the difference of the segments. The square root of this,
  √[(Z²/2)²-AE] (2), is half the difference. If you add it to half the
  coefficient Z/2 (5), or half the line to be bisected, the longer
  segment is obtained; if you subtract it, the smaller segment is
  obtained. I say: Z/2±√(Z²/4-AE)=A {major segment/minor segment.

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

A notation suggested by Vieta and favored by Girard made vowels stand for
unknowns and consonants for knowns. This conventionality was adopted by
Oughtred in parts of his algebra, but not throughout. Near the beginning
he used Q to designate the unknown, though usually this letter stood with
him for the “square” of the expression after it.[34]

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

We quote the description of the Clavis 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 Clavis, saying:

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

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

  Thus what Vieta would have written

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

  would with him be thus expressed

    A_q B_c
    ------- = FG.
     C D E

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

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

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

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

  Mr. Oughtred in his Clavis, contents himself (for the most part) with
  the solution of Quadratick Equations, without proceeding (or very
  sparingly) to Cubick Equations, and those of Higher Powers; having
  designed that Work for an Introduction into Algebra so far, leaving the
  Discussion of Superior Equations for another work. . . . . He contents
  himself likewise in Resolving Equations, to take notice of the
  Affirmative or Positive Roots; omitting the Negative or Ablative Roots,
  and such as are called Imaginary or Impossible Roots. And of those
  which, he calls Ambiguous Equations, (as having more Affirmative Roots
  than one,) he doth not (that I remember) any where take notice of more
  than Two Affirmative Roots: (Because in Quadratick Equations, which are
  those he handleth, there are indeed no more.) Whereas yet in Cubick
  Equations, there may be Three, and in those of Higher Powers, yet more.
  Which Vieta was well aware 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,[37] and a Trigonometrie.[38] 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 Hand.”[39]
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.[40]

In the preface to the Latin edition Stokes writes:

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

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

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

We find, however, that even Oughtred cannot be given the whole credit in
this matter. By or before 1631 several other writers used abbreviations
of the trigonometric functions. As early as 1624 the contractions sin for
sine and tan for tangent appear on the drawing representing Gunter’s
scale, but Gunter did not use them in his books, except in the drawing of
his scale.[42] A closer competitor for the honor of first using these
trigonometric abbreviations is Richard Norwood in his Trigonometrie,
London, 1631, where s stands for sine, t for tangent, sc for sine
complement (cosine), tc for tangent complement (cotangent), and sec for
secant. Norwood was a teacher of mathematics in London and a well-known
writer of books on navigation. Aside from the abbreviations just cited
Norwood did not use nearly as much symbolism in his mathematics as did
Oughtred.

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

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

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

The innovation of designating the sides and angles of a triangle by A, B,
C, and a, b, c, so that A was opposite a, B opposite b, and C opposite c,
is 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.[43]

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


                    SOLUTION OF NUMERICAL EQUATIONS

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

                          |f(a+s₁)-f(a)|-s₁ⁿ,

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

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

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

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

The early part of his exposition shows how an equation may be transformed
so as to make its roots 10, 100, 1000, or 10^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, x⁴-72x³+238600x=8725815, upon
dividing 72x³ by 10, 238600x by 1000, and 8725815 by 10,000, we obtain
x⁴-7·2x³+238·6x=872·5. Dividing both sides by x, we obtain
x³+238·6-7·2x²=x)872·5. Letting x=4, we have 64+238·6-115·2=187·4.

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

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

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

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

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

  L_c=(A+E)³=A³+3A²E+3AE²+E³

and

  C_qL=420000(A+E).

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

                              “Exemplum II

                     1c+42̣00̣00̣l=247̇651̇7̣1̣3̣̇

                         Hoc est, L_c+C_qL=D_c

    2 4 7̇ | 6 5 1̇ | 7̣ 1̣ 3̣̇ | ( 4 1 7
    ------+-------+-------+------------
      4 2 | 0 0 0 | 0     | C_q
    ------+-------+-------+------------
      6 4 |       |       | A_c
    1 6 8 | 0 0 0 | 0     | C_q A
    ------+-------+-------+------------
    2 3 2 | 0 0 0 | 0     | Ablatit.
    ===================================
R     1 5 | 6 5 1̇ | 7 1 3̣ |
    ------+-------+-------+------------
        4 | 8     |       | 3 A_q
          | 1 2   |       | 3 A
        4 | 2 0 0 | 0 0   | C_q
    ------+-------+-------+------------
        9 | 1 2 0 | 0 0   | Divisor.
    ------+-------+-------+------------
        4 | 8     |       | 3 A_q E
          | 1 2   |       | 3 A E_q
          |     1 |       | E_c
        4 | 2 0 0 | 0 0   | C_q E
    ------+-------+-------+------------
        9 | 1 2 1 | 0 0   | Ablatit.
    ===================================
R       6 | 5 3 0 | 7 1 3̣̇ |                    4 | 1   |
    ------+-------+-------+------------      ----+-----+---
          | 5 0 4 | 3     | 3 A_q                |     |
          |     1 | 2 3   | 3 A              1 6 | 8   |
          | 4 2 0 | 0 0 0 | C_q                  |   1 |
    ------+-------+-------+------------      ----+-----+---
          | 9 2 5 | 5 3 0 | Divisor.         1 6   8 1
    ------+-------+-------+------------
        3 | 5 3 0 | 1     | 3 A_q E
          |   6 0 | 2 7   | 3 A E_q
          |       | 3 4 3 | E_c
        2 | 9 4 0 | 0 0 0 | C_q E
    ------+-------+-------+------------
        6 | 5 3 0 | 7 1 3 | Ablatit.”

Next, he evaluates the coefficients of E in 3A²E and 420000E, also 3A,
the coefficient of E². He obtains 3A²=480000, 3A=1200, C_q=420000. He
interprets 3A² and C_q as tens, 3A as hundreds. Accordingly, he obtains
as their sum 9120000, which is the divisor for finding the second digit
in the approximation. Observe that this divisor is the value of
|f(a+s₁)-f(a)|-s₁ⁿ in our general expression, where a=400, s₁=10, n=3,
f(x)=x³+420000x.

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

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

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

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

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

Oughtred solves 16 examples by the process of approximation here
explained. Of these, 9 are cubics, 5 are quartics, and 2 are quintics. In
all cases he finds only one or two real roots. Of the roots sought, five
are irrational, the remaining are rational and are computed to their
exact values. Three of the computed roots have 2 figures each, 9 roots
have 3 figures each, 4 roots have 4 figures each. While no attempt is
made to secure all the roots—methods of computing complex roots were
invented much later—he computes roots of equations which involve large
coefficients and some of them are of a degree as high as the fifth. In
view of the fact that many editions of the Clavis were issued, one
impression as late as 1702, it contributed probably more than any other
book to the popularization of Vieta’s method in England.

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

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


                               LOGARITHMS

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

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


   INVENTION OF THE SLIDE RULE; CONTROVERSY ON PRIORITY OF INVENTION

Oughtred’s most original line of scientific activity is the one least
known to the present generation. Augustus De Morgan, in speaking of
Oughtred, who was sometimes called “Oughtred Aetonensis,” remarks: “He is
an animal of extinct race, an Eton mathematician. Few Eton men, even of
the minority which knows what a sliding rule is, are aware that the
inventor was of their own school and college.”[50] The invention of the
slide rule has, until recently,[51] been a matter of dispute; it has been
erroneously ascribed to Edmund Gunter, Edmund Wingate, Seth Partridge,
and others. We have been able to establish that William Oughtred was the
first inventor of slide rules, though not the first to publish thereon.
We shall see that Oughtred invented slide rules about 1622, but the
descriptions of his instruments were not put into print before 1632 and
1633. Meanwhile one of his own pupils, Richard Delamain, who probably
invented the circular slide rule independently, published a description
in 1630, at London, in a pamphlet of 32 pages entitled Grammelogia; or
the Mathematicall Ring. In editions of this pamphlet which appeared
during the following three or four years, various parts were added on,
and some parts of the first and second editions eliminated. Thus Delamain
antedates Oughtred two years in the publication of a description of a
circular slide rule. But Oughtred had invented also a rectilinear slide
rule, a description of which appeared in 1633. To the invention of this
Oughtred has a clear title. A bitter controversy sprang up between
Delamain on one hand, and Oughtred and some of his pupils on the other,
on the priority and independence of invention of the circular slide rule.
Few inventors and scientific men are so fortunate as to escape contests.
The reader needs only to recall the disputes which have arisen, involving
the researches of Sir Isaac Newton and Leibniz on the differential and
integral calculus, of Thomas Harriot and René Descartes relating to the
theory of equations, of Robert Mayer, Hermann von Helmholtz, and Joule on
the principle of the conservation of energy, or of Robert Morse, Joseph
Henry, Gauss and Weber, and others on the telegraph, to see that
questions of priority and independence are not uncommon. The controversy
between Oughtred and Delamain embittered Oughtred’s life for many years.
He refers to it in print on more than one occasion. We shall confine
ourselves at present to the statement that it is by no means clear that
Delamain stole the invention from Oughtred; Delamain was probably an
independent inventor. Moreover, it is highly probable that the
controversy would never have arisen, had not some of Oughtred’s pupils
urged and forced him into it. William Forster stated in the preface to
the Circles of Proportion of 1632 that while he had been carefully
preparing the manuscript for the press, “another to whom the Author
[Oughtred] in a louing confidence discouered this intent, using more hast
then good speed, went about to preocupate.” It was this passage which
started the conflagration. Another pupil, W. Robinson, wrote to Oughtred,
when the latter was preparing his Apologeticall Epistle as a reply to
Delamain’s countercharges: “Good sir, let me be beholden to you for your
Apology whensoever it comes forth, and (if I speak not too late) let me
entreat you, whip ignorance well on the blind side, and we may turn him
round, and see what part of him is free.”[52] As stated previously,
Oughtred’s circular slide rule was described by him in his Circles of
Proportion, London, 1632, which was translated from Oughtred’s Latin
manuscript and then seen through the press by his pupil, William Forster.
In 1633 appeared An Addition vnto the Vse of the Instrvment called the
Circles of Proportion which contained at the end “The Declaration of the
two Rulers for Calculation,” giving a description of Oughtred’s
rectilinear slide rule. This Addition was bound with the Circles of
Proportion as one volume. About the same time Oughtred described a
modified form of the rectilinear slide rule, to be used in London for
gauging.[53]




                              CHAPTER III
                              MINOR WORKS


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

“J. S.” says in his preface:

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

Notwithstanding Oughtred’s marked activity in the design of mathematical
instruments, and his use of surveying instruments, he always spoke in
deprecating terms of their importance and their educational value. In his
epistle against Delamain he says:

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

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

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

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

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

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

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

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

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

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

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

The graphic solution of spherical triangles by the accurate drawing of
the triangles on a sphere and the measurement of the unknown parts in the
drawing was explained by Oughtred in a short tract which was published by
his son-in-law, Christopher Brookes, under the following title: The
Solution of all Sphaerical Triangles both right and oblique By the
Planisphaere: Whereby two of the Sphaerical partes sought, are at one
position most easily found out. Published with consent of the Author, By
Christopher Brookes, Mathematique Instrument-maker, and Manciple of
Wadham Colledge, in Oxford.

Brookes says in the preface:

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

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

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

Here we wish to give our reasons for our belief that Oughtred is the
author of an anonymous tract on the use of logarithms and on a method of
logarithmic interpolation which, as previously noted, appeared as an
“Appendix” to Edward Wright’s translation into English of John Napier’s
Descriptio, under the title, A Description of the Admirable Table of
Logarithmes, London, 1618. The “Appendix” bears the title, “An Appendix
to the Logarithmes, showing the practise of the Calculation of Triangles,
and also a new and ready way for the exact finding out of such lines and
Logarithmes as are not precisely to be found in the Canons.” It is an
able tract. A natural guess is that the editor of the book, Samuel
Wright, a son of Edward Wright, composed this “Appendix.” More probable
is the conjecture which (Dr. J. W. L. Glaisher informs me) was made by
Augustus De Morgan, attributing the authorship to Oughtred. Two reasons
in support of this are advanced by Dr. Glaisher, the use of x in the
“Appendix” as the sign of multiplication (to Oughtred is generally
attributed the introduction of the cross × for multiplication in 1631),
and the then unusual designation “cathetus” for the vertical leg of a
right triangle, a term appearing in Oughtred’s books. We are able to
advance a third argument, namely, the occurrence in the “Appendix” of
(S*) as the notation for sine complement (cosine), while Seth Ward, an
early pupil of Oughtred, in his Idea trigonometriae demonstratae, Oxford,
1654, used a similar notation (S’). It has been stated elsewhere that
Oughtred claimed Seth Ward’s exposition of trigonometry as virtually his
own. Attention should be called also to the fact that, in his
Trigonometria, p. 2, Oughtred uses (’) to designate 180°-angle.

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

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

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

  The Appendix was an interesting and remarkable contribution to
  mathematics, for in its sixteen small pages it contains (1) the first
  use of the sign ×; (2) the first abbreviations, or symbols, for the
  sine, tangent, cosine, and cotangent; (3) the invention of the radix
  method of calculating logarithms; (4) the first table of hyperbolic
  logarithms.[55]




                               CHAPTER IV
      OUGHTRED’S INFLUENCE UPON MATHEMATICAL PROGRESS AND TEACHING


                          OUGHTRED AND HARRIOT

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


                           OUGHTRED’S PUPILS

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

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

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

Had Oughtred been the means of guiding the mathematical studies of only
John Wallis and Christopher Wren—one the greatest English mathematician
between Napier and Newton, the other one of the greatest architects of
England—he would have earned profound gratitude. But the foregoing list
embraces nine men, most of them distinguished in their day. And yet
Aubrey’s list is very incomplete. It is easy to more than double it by
adding the names of William Forster, who translated from Latin into
English Oughtred’s Circles of Proportion; Arthur Haughton, who brought
out the 1660 Oxford edition of the Circles of Proportion; Robert Wood, an
educator and politician, who assisted Oughtred in the translation of the
Clavis from Latin into English for the edition of 1647; W. Gascoigne, a
man of promise, who fell in 1644 at Marston Moor; John Twysden, who was
active as a publisher; William Sudell, N. Ewart, Richard Shuttleworth,
William Robinson, and William Howard, the son of the Earl of Arundel, for
whose instruction Oughtred originally prepared the manuscript treatise
that was published in 1631 as the Clavis mathematicae.

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

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


              OUGHTRED, THE “TODHUNTER OF THE SEVENTEENTH
                                CENTURY”

To afford a clearer view of Oughtred as a teacher and mathematical
expositor we quote some passages from various writers and from his
correspondence. Anthony Wood[57] gives an interesting account of how Seth
Ward and Charles Scarborough went from Cambridge University to the
obscure home of the country mathematician to be initiated into the
mysteries of algebra:

  Mr. Cha. Scarborough, then an ingenious young student and fellow of
  Caius Coll. in the same university, was his [Seth Ward’s] great
  acquaintance, and both being equally students in that faculty and
  desirous to perfect themselves, they took a journey to Mr. Will.
  Oughtred living then at Albury in Surrey, to be informed in many things
  in his Clavis mathematica which seemed at that time very obscure to
  them. Mr. Oughtred treated them with great humanity, being very much
  pleased to see such ingenious young men apply themselves to these
  studies, and in short time he sent them away well satisfied in their
  desires. When they returned to Cambridge, they afterwards read the
  Clav. Math. to their 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.[58]

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

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

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

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

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

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

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

  . . . . with the character I received from your and my noble friend Sir
  Charles Cavendish, then at Paris, of your second edition of the same
  piece, made me at my return into England speedily to get, and
  diligently peruse the same. Neither truly did I find my expectation
  deceived; having with admiration often considered how it was possible
  (even in the hardest 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. . . . .[61]

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

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

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

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

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

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

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.[65] He found in the Calendar of State Papers, Domestic Series,
1637, a petition to Archbishop Laud in which it is set forth that when
Hooganhuysen, a Dutchman, “heretofore complained of in the High
Commission for importing books printed beyond the seas,” had been bound
“not to bring in any more,” one Vlacq (the computer and publisher of
logarithmic tables) “kept up the same agency and sold books in his stead.
. . . . Vlacq is now preparing to go beyond the seas to avoid answering
his late bringing over nine bales of books contrary to the decree of the
Star Chamber.” Judgment was passed that, “Considering the ill-consequence
and scandal that would arise by strangers importing and venting in this
kingdom books printed beyond the seas,” certain importations be
prohibited, and seized if brought over.

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

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

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

  I have with unspeakable delight, so far as my necessary businesses, the
  infirmness of my health, and the greatness of my age (approaching now
  to an end) would permit, perused your most learned papers, of several
  choice arguments, which you sent me: wherein I do first with
  thankfulness acknowledge to God, the Father of lights, the great light
  he hath given you; and next I congratulate you, even with admiration,
  the clearness and perspicacity of your understanding and genius, who
  have not only gone, but also opened a way into these profoundest
  mysteries of art, unknown and not thought of by the ancients. With
  which your mysterious inventions I am the more affected, because full
  twenty years ago, the learned patron of learning, Sir Charles
  Cavendish, shewed me a paper written, wherein were some few excellent
  new theorems, wrought by the way, as I suppose, of Cavalieri, which I
  wrought over again more agreeably to my way. The paper, wherein I
  wrought it, I shewed to many, whereof some took copies, but my own I
  cannot find. I mention it for this, because I saw therein a light
  breaking out for the discovery of wonders to be revealed to mankind, in
  this last age of the world: which light I did salute as afar off, and
  now at a nearer distance embrace in your prosperous beginnings. Sir,
  that you are pleased to mention my name in your never dying papers,
  that is your noble favour to me, who can add nothing to your glory, but
  only my applause. . . . .[67]

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

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

  . . . . And as for Mr. Oughtred’s method of symbols, this I say to it;
  it may be proper for you as a commentator to follow it, but divers I
  know, men of inferior rank that have good skill in algebra, that
  neither use nor approve it. . . . . Is not A⁵ sooner wrote than A_qc?
  Let A be 2, the cube of 2 is 8, which squared is 64: one of the
  questions between Maghet Grisio and Gloriosus is whether 64=A_cc or
  A_qc. The Cartesian method tells you it is A⁶, and decides the doubt. .
  . . .[70]

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

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

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

The astronomer Edmund Halley, in his preface to the 1694 English edition
of the Clavis, speaks of this book as one of “so established a
reputation, that it were needless to say anything thereof,” though “the
concise Brevity of the author is such, as in many places to need
Explication, to render it Intelligible to the less knowing Mathematical
matters.”

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

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


                  WAS DESCARTES INDEBTED TO OUGHTRED?

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

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

  Some time ago I read in the elegant and truly precious book that you
  have written on Algebra, about Descartes, this philosopher so extolled
  above all for having arrived at a very perfect system by his own
  powers, without the aid of others, this Descartes, I say, who has
  received in geometry very great light from our Oughtred and our
  Harriot, and has followed their track though he carefully suppressed
  their names. I stated this in a conversation with a professor in
  Utrecht (where I reside at present). He requested me to indicate to him
  the page-numbers in the two authors which justified this accusation. I
  admitted that I could not do so. The Géométrie of Descartes is not
  sufficiently familiar to me, although with Oughtred I am fairly
  familiar. I pray you therefore that you will assume this burden. Give
  me at least those references to passages of the two authors from the
  comparison of which the plagiarism by Descartes is the most
  striking.[73]

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

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

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

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

The first inquiry to be made is, Did Descartes possess copies of the
books of Harriot and Oughtred? It is only in recent time that this
question has been answered as to Harriot. As to Oughtred, it is still
unanswered. It is now known that Descartes had seen Harriot’s Artis
analyticae praxis (1631). Descartes wrote a letter to Constantin Huygens
in which he states that he is sending Harriot’s book.[74]

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

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

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


                   THE SPREAD OF OUGHTRED’S NOTATIONS

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

In France we have noticed Oughtred’s notation for proportion in
Franciscus Dulaurens (1667),[84] J. Prestet (1675),[85] R. P. Bernard
Lamy (1684),[86] Ozanam (1691),[87] De l’Hospital (1696),[88] R. P. Petro
Nicolas (1697).[89]

In the Netherlands we have noticed it in R. P. Bernard Lamy (1680),[90]
and in an anonymous work of 1690.[91] In German and Italian works of the
seventeenth century we have not seen Oughtred’s notation for proportion.

In England a modified notation soon sprang up in which ratio was
indicated by two dots instead of a single dot, thus A:B::C:D. The reason
for the change lies probably in the inclination to use the single dot to
designate decimal fractions. W. W. Beman pointed out that this modified
symbolism (:) for ratio is found as early as 1657 in the end of the
trigonometric and logarithmic tables that were bound with Oughtred’s
Trigonometria.[92] It is not probable, however, that this notation was
used by Oughtred himself. The Trigonometria proper has Oughtred’s
A.B::C.D throughout. Moreover, in the English edition of this
trigonometry, which appeared the same year, 1657, but subsequent to the
Latin edition, the passages which contained the colon as the symbol for
ratio, when not omitted, are recast, and the regular Oughtredian notation
is introduced. In Oughtred’s posthumous work, Opuscula mathematica
hactenus inedita, 1677, the colon appears quite often but is most likely
due to the editor of the book.

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

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

Of English eighteenth-century authors the following still clung to the
notation A.B::C.D: John Harris’ translation of F. Ignatius Gaston Pardies
(1701),[102] George Shelley (1704),[103] Sam Cobb (1709),[104] J. Collins
in Commercium Epistolicum (1712), John Craig (1718),[105] Jo. Wilson
(1724).[106] 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. Carré (1700),[107] M. Guisnée (1705),[108] M. de Fontenelle
(1727),[109] M. Varignon (1725),[110] M. Robillard (1753),[111] M.
Sebastien le Clerc (1764),[112] Clairaut (1731),[113] M. L’Hospital
(1781).[114]

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

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

Oughtred’s notation to express aggregation of terms has received little
attention from historians but is nevertheless interesting. His books, as
well as those of John Wallis, are full of parentheses but they are not
used as symbols of aggregation in algebra; they are simply marks of
punctuation for parenthetical clauses. We have seen that Oughtred writes
(a+b)² and √(a+b) thus, Q:a+b:, √:a+b:, or Q:a+b, √:a+b, using on rarer
occasions a single dot in place of the colon. This notation did not
originate with Oughtred, but, in slightly modified form, occurs in
writings from the Netherlands. In 1603 C. Dibvadii in geometriam Evclidis
demonstratio numeralis, Leyden, contains many expressions of this sort,
√·136+√2048, signifying √(136+√2048). The dot is used to indicate that
the root of the binomial (not of 136 alone) is called for. This notation
is used extensively in Ludolphi à Cevlen de circulo, Leyden, 1619, and in
Willebrordi Snellii De circuli dimensione, Leyden, 1621. In place of the
single dot Oughtred used the colon (:), probably to avoid confusion with
his notation for ratio. To avoid further possibility of uncertainty he
usually placed the colon both before and after the algebraic expression
under aggregation. This notation was adopted by John Wallis and Isaac
Barrow. It is found in the writings of Descartes. Together with Vieta’s
horizontal bar, placed over two or more terms, it constituted the means
used almost universally for denoting aggregation of terms in algebra.
Before Oughtred the use of parentheses had been suggested by Clavius[116]
and Girard.[117] The latter wrote, for instance, √(2+√3). While
parentheses never became popular in algebra before the time of Leibniz
and the Bernoullis they were by no means lost sight of. We are able to
point to the following authors who made use of them: I. Errard de
Bar-le-Duc (1619),[118] Jacobo de Billy (1643),[119] one of whose books
containing this notation was translated into English, and also the
posthumous works of Samuel Foster.[120] J. W. L. Glaisher points out that
parentheses were used by Norwood in his Trigonometrie (1631), p. 30.[121]

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

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

General acceptance has been accorded to Oughtred’s symbol ×. The first
printed appearance of this symbol for multiplication in 1618 in the form
of the letter x hardly explains its real origin. The author of the
“Appendix” (be he Oughtred or someone else) may not have used the letter
x at all, but may have written the cross ×, called the St. Andrew’s
cross, while the printer, in the absence of any type accurately
representing that cross, may have substituted the letter x in its place.
The hypothesis that the symbol × of multiplication owes its origin to the
old habit of using directed bars to indicate that two numbers are to be
combined, as for instance in the multiplication of 23 and 34, thus,

   2   3
   |\ /|
   | x |
   |/ \|
   3   4
  -------
   7 8 2

has been advanced by two writers, C. Le Paige[138] and Gravelaar.[139]
Bosmans 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 declaratio.[140]

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




                               CHAPTER V
            OUGHTRED’S IDEAS ON THE TEACHING OF MATHEMATICS


                           GENERAL STATEMENT

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

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


                  MATHEMATICS, “A SCIENCE OF THE EYE”

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

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

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

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

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

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

It was this motive which led him to introduce the many abbreviations in
algebra and trigonometry to which reference has been made in previous
pages. The pedagogical experience of recent centuries has indorsed
Oughtred’s view, provided of course that the pupil is carefully taught
the exact meaning of the symbols. There have been and there still are
those who oppose the intensive use of symbolism. In our day the new
symbolism for all mathematics, suggested by the school of Peano in Italy,
can hardly be said to be received with enthusiasm. In Oughtred’s day
symbolism was not yet the fashion. To be convinced of this fact one need
only open a book of Edmund Gunter, with whom Oughtred came in contact in
his youth, or consult the Principia of Sir Isaac Newton, who flourished
after Oughtred. The mathematical works of Gunter and Newton, particularly
the former, are surprisingly destitute of mathematical symbols. The
philosopher Hobbes, in a controversy with John Wallis, criticized the
latter for that “Scab of Symbols,” whereupon Wallis replied:

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

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 earnestly exhort every one,
  that desireth though but to looke into these noble Sciences
  Mathematicall, to accustome themselves unto it: and indeede it is
  easie, being most agreeable to reason, yea even to sence. And out of
  this working may many singular consectaries be drawne: which without
  this would, it may be, for ever lye hid.


              RIGOROUS THINKING AND THE USE OF INSTRUMENTS

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

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

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

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

  That the true way of Art is not by Instruments, but by Demonstration:
  and that it is a preposterous course of vulgar Teachers, to begin with
  Instruments, and not with the Sciences, and so in-stead of Artists, to
  make their Scholers only doers of tricks, and as it were Iuglers: to
  the despite of Art, losse of previous time, and betraying of willing
  and industrious wits, vnto ignorance, and idlenesse. That the vse of
  Instruments is indeed excellent, if a man be an Artist: but
  contemptible, being set and opposed to Art. And lastly, that he meant
  to commend to me, the skill of Instruments, but first he would haue me
  well instructed in the Sciences.”

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

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

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

  . . . . Which words are neither cautelous, nor subterfugious, but are
  as downe right in their plainnesse, as they are touching, and
  pernitious, by two much derogating from many, and glancing upon many
  noble personages, with too grosse, if not too base an attribute, in
  tearming them doers of tricks, as it were to iuggle: because they
  perhaps make use of a necessitie in the furnishing of themselves with
  such knowledge by Practicall Instrumentall operation, when their more
  weighty negotiations will not permit them for Theoreticall figurative
  demonstration; those that are guilty of the aspertion, and are touched
  therewith may answer for themselves, and studie to be more
  Theoreticall, than Practicall: for the Theory, is as the Mother that
  produceth the daughter, the very sinewes and life of Practise, the
  excellencie and highest degree of true Mathematicall Knowledge: but for
  those that would make but a step as it were into that kind of Learning,
  whose onely desire is expedition, and facilitie, both which by the
  generall consent of all are best effected with Instrument, rather then
  with tedious regular demonstrations, it was ill to checke them so
  grosly, not onely in what they have Practised, but abridging them also
  of their liberties with what they may Practise, which aspertion may not
  easily be slighted off by any glosse or Apologie, without an Ingenuous
  confession, or some mentall reservation: To which vilification,
  howsoever, in the behalfe of my selfe, and others, I answer; That
  Instrumentall operation is not only the Compendiating, and facilitating
  of Art, but even the glory of it, whole demonstration both of the
  making, and operation is soly in the science, and to an Artist or
  disputant proper to be knowne, and so to all, who would truly know the
  cause of the Mathematicall operations in their originall; But, for none
  to know the use of a Mathematicall Instrumen[t], except he knowes the
  cause of its operation, is somewhat too strict, which would keepe many
  from affecting the Art, which of themselves are ready enough every
  where, to conceive more harshly of the difficultie, and impossibilitie
  of attayning any skill therein, then it deserves, because they see
  nothing but obscure propositions, and perplex and intricate
  demonstrations before their eyes, whose unsavoury tartnes, to an
  unexperienced palate like bitter pills is sweetned over, and made
  pleasant with an Instrumentall compendious facilitie, and made to goe
  downe the more readily, and yet to retaine the same vertue, and
  working; And me thinkes in this queasy age, all helpes may bee used to
  procure a stomacke, all bates and invitations to the declining studie
  of so noble a Science, rather then by rigid Method and generall Lawes
  to scarre men away. All are not of like disposition, neither all (as
  was sayd before) propose the same end, some resolve to wade, others to
  put a finger in onely, or wet a hand: now thus to tye them to an
  obscure and Theoricall forme of teaching, is to crop their hope, even
  in the very bud. . . . . The beginning of a mans knowledge even in the
  use of an Instrument, is first founded on doctrinal precepts, and these
  precepts may be conceived all along in its use: and are so farre from
  being excluded, that they doe necessarily concomitate and are contained
  therein: the practicke being better understood by the doctrinall part,
  and this later explained by the Instrumentall, making precepts obvious
  unto sense, and the Theory going along with the Instrument, better
  informing and inlightning the understanding, etc. vis vnita fortior, so
  as if that in Phylosophy bee true, Nihil est [in] intellectu quod non
  prius fuit in sensu.

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

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

It is somewhat of a paradox that Oughtred, who in his student days and
during his active years felt himself impelled to invent sun-dials,
planispheres, and various types of slide rules—instruments which
represent the most original contributions which he handed down to
posterity—should discourage the use of such instruments in teaching
mathematics to beginners. That without the aid of instruments he himself
should have succeeded so well in attracting and inspiring young men
constitutes the strongest evidence of his transcendent teaching ability.
It may be argued that his pedagogic dogma, otherwise so excellent, here
goes contrary to the course he himself followed instinctively in his
self-education along mathematical lines. We read that Sir Isaac Newton,
as a child, constructed sun-dials, windmills, kites, paper lanterns, and
a wooden clock. Should these activities have been suppressed? Ordinary
children are simply Isaac Newtons on a smaller intellectual scale. Should
their activities along these lines be encouraged or checked?

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

Who was right, Oughtred or Delamain? It may be claimed that there is a
middle ground which more nearly represents the ideal procedure in
teaching. Shall the slide rule be placed in the student’s hands at the
time when he is engaged in the mastery of principles? Shall there be an
alternate study of the theory of logarithms and of the slide rule—on the
idea of one hand washing the other—until a mastery of both the theory and
the use of the instrument has been attained? Does this method not produce
the best and most lasting results? Is not this Delamain’s actual
contention? We leave it to the reader to settle these matters from his
own observation, knowledge, and experience.


                     NEWTON’S COMMENTS ON OUGHTRED

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

  And now I have told you my opinion in these things, I will give you Mr.
  Oughtred’s, a Man whose judgment (if any man’s) may be safely relyed
  upon. For he in his book of the circles of proposition, in the end of
  what he writes about Navigation (page 184) has this exhortation to
  Seamen. “And if,” saith he, “the Masters of Ships and Pilots will take
  the pains in the Journals of their Voyages diligently and faithfully to
  set down in severall columns, not onely the Rumb they goe on and the
  measure of the Ships way in degrees, and the observation of Latitude
  and variation of their compass; but alsoe their conjectures and reason
  of their correction they make of the aberrations they shall find, and
  the qualities and condition of their ship, and the diversities and
  seasons of the winds, and the secret motions or agitations of the Seas,
  when they begin, and how long they continue, how farr they extend and
  with what inequality; and what else they shall observe at Sea worthy
  consideration, and will be pleased freely to communicate the same with
  Artists, such as are indeed skilfull in the Mathematicks and lovers and
  enquirers of the truth: I doubt not but that there shall be in
  convenient time, brought to light many 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^t of Navigation and safety of Mens lives and estates on that
  element.[144]

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




                               Footnotes


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

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

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

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

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

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

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

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

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

[10]Ibid., p. 111.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[76]Ibid., p. 78.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[115]The tendency during the eighteenth century is shown in part by the
   following data: Jacobi Bernoulli Opera, Tomus primus, Geneva, 1744,
   gives B.A::D.C on p. 368, the paper having been first published in
   1688; on p. 419 is given GE:AG=LA:ML, the paper having been first
   published in 1689. Bernhardi Nieuwentiit, Considerationes circa
   analyseos ad quantitates infinitè parvas applicatae principia,
   Amsterdam, 1694, p. 20, and Analysis infinitorum, Amsterdam, 1695, on
   p. 276, have x:c::s:r. Paul Halcken’s Deliciae mathematicae, Hamburg,
   1719, gives a:b::c:d. Johannis Baptistae Caraccioli, Geometria
   algebraica universa, Rome, 1759, p. 79, has a.b::c.d. Delle corde
   ouverto fibre elastiche schediasmi fisico-matematici del conte
   Giordano Riccati, Bologna, 1767, p. 65, gives P:b::r:ds. “Produzioni
   mathematiche” del Conte Giulio Carlo de Fagnano, Vol. I, Pesario,
   1750, p. 193, has a.b::c.d. L. Mascheroni, Géométrie du compas,
   translated by A. M. Carette, Paris, 1798, p. 188, gives
   √(3):2::√(2):Lp. Danielis Melandri and Paulli Frisi, De theoria lunae
   commentarii, Parma, 1769, p. 13, has a:b::c:d. Vicentio Riccato and
   Hieronymo Saladino, Institutiones analyticae, Vol. I, Bologna, 1765,
   p. 47, gives x:a::m:n+m. R. G. Boscovich, Opera pertinentia ad opticam
   et astronomiam, Bassani, 1785, p. 409, uses a:b::c:d. Jacob Bernoulli,
   Ars Conjectandi, Basel, 1713, has n-r.n-1::c.d. Pavlini Chelvicii,
   Institutiones analyticae, editio post tertiam Romanam prima in
   Germania, Vienna, 1761, p. 2, a.b::c.d. Christiani Wolfii, Elementa
   matheseos universae, Vol. III, Geneva, 1735, p. 63, has AB:AE=1:q.
   Johann Bernoulli, Opera omnia, Vol. I, Lausanne and Geneva, 1742, p.
   43, has a:b=c:d. D. C. Walmesley, Analyse des mesures des rapports et
   des angles, Paris, 1749, uses extensively a.b::c.d, later a:b::c:d. G.
   W. Krafft, Institutiones geometriae sublimoris, Tübingen, 1753, p.
   194, has a:b=c:d. J. H. Lambert, Photometria, 1760, p. 104, has C:π
   =BC²:MH². Meccanica sublime del Dott. Domenico Bartaloni, Naples,
   1765, has a:b::c:d. Occasionally ratio is not designated by a.b, nor
   by a:b, but by a, b, as for instance in A. de Moivre’s Doctrine of
   Chance, London, 1756, p. 34, where he writes a, b::1, q. A further
   variation in the designation of ratio is found in James Atkinson’s
   Epitome of the Art of Navigation, London, 1718, p. 24, namely,
   3..2::72..48. Curious notations are given in Rich. Balam’s Algebra,
   London, 1653.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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




                                 INDEX


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

  Baker, T., 82
  Balam, R., 79
  Bar-le-Duc, de, 80
  Barrow, S., 1, 32, 73, 74, 80, 81
  Bartaloni, D., 79
  Beman, W. W., 74, 75
  Bernoulli, Jakob, 78-80
  Bernoulli, John, 79, 80
  Billingsley’s Euclid, 15
  Billion, 20
  Billy, Jacobo de, 80
  Binomial formula, 25, 29
  Bliss, P., 60
  Boscovich, R. G., 78
  Bosmans, H., 72, 83
  Boyle, R., 1, 69
  Braunmühl, von, 39
  Brearly, W., 59
  Briggs, 6, 36, 55
  Brookes, Christopher, 7, 53, 59

  Cajori, F., 27, 39, 40, 47
  Cantor, M., 40, 41
  Caraccioli, J. B., 78
  Cardan, 71
  Carré, 77
  Carrete, N. M., 78
  Caryll, C., 7
  Cataldi, 67
  Cavalieri, 65, 66
  Cavendish, Charles, 17, 62, 66
  Charles I, 9, 60
  Chelvicius, P., 79
  Cheyne, G., 82
  _Circles of Proportion_, 35, 37, 48, 49, 51, 59, 87, 88
  Clairaut, 77
  Clark, A., 3
  Clark, G., 63
  Clarke, F. L., 3
  _Clavis mathematicae_, 1, 5, 10, 14, 17-35, 45, 46, 51, 57-63,
          68-73, 81, 85, 87
  Clavius, 26, 80
  Clerc, le, 77
  Cobb, S., 76
  Cocker, 76, 82
  Collins, John, 15, 19, 63, 64, 67, 68, 76, 82
  Colson, J., 77
  Conchoid, 12
  Conic sections, 11, 53
  Cossali, P., 81
  Cotes, R., 1, 85
  Craig, J., 76, 82
  Cross, symbol of multiplication, 27, 38, 55, 56, 82, 83
  Cubic equations, 28, 34, 42, 45

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

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

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

  Gascoigne, 59, 61
  Gauss, C. F., 48
  Geysius, 67
  Ghetaldi, 70, 71
  Gibson, 67
  Girard, A., 32, 80
  Glaisher, J. W. L., 54-56, 64, 80
  Glorioso, 67, 68
  _Grammelogia_, 4, 47, 89
  Gravelaar, 83
  Greater than, symbol for, 81
  Greatrex, R., 15
  Gregory, D., 32
  Gregory, J., 27, 76
  Gresham College, 1, 6, 27, 59, 61, 80
  Guisnée, 77
  Gunter, E., 37, 47, 86
  Gunter’s scale, 37

  Halcken, P., 78
  Hales, J., 7
  Halley, E., 1, 18, 69
  Hankel, H., 40
  Harper, T., 18
  Harriot, T., 45, 47, 57, 58, 69-71, 81
  Harris, J., 76
  Hartlib, 69
  Haughton, A., 35, 59
  Hawkins, J., 76, 82
  Hearn, 56
  Helmholtz, 48
  Henry, J., 48
  Henry van Etten, 52, 53
  Henshaw, T., 8, 58, 61
  Hobbes, 73, 86
  Hollar, 14
  Holsatus, 13
  Hooganhuysen, 64
  Hooke, Rb., 1
  Horner’s method, 45
  Horology, 18, 50
  Horrox, J., 4
  Hospital, de l’, 74, 77
  Howard, Th. _See_ Earl of Arundel.
  Howard, W., 17, 18, 59
  Hutchinson, A., 6

  Invisible college, 1

  Joule, 48

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

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

  Manning, 56
  Manning, O., 7, 8, 13-15
  Mascheroni, L., 78
  Mayer, R., 47
  Melandri, D., 78
  Mercator, N., 13
  Mersenne, 63
  Milbourn, W., 45
  Million, 20
  Milnes, J., 82
  Moivre, de, 32, 79
  Moore, Jonas, 32, 54, 58, 73
  Moreland, S., 70
  Morse, R., 48
  Multiplication, abbreviated, 21, 22, 24;
      symbol for, 27, 82, 83
  Mydorge, 54

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

  _Opuscula mathematica hactenus inedita_, 16, 21, 75
  Orchard, 56
  _Oughtredus explicatus_, 64
  Ozanam, 74

  π, symbol for, 32
  Paige, C. de, 83
  Pardies, 76
  Parentheses, 26, 79, 80
  Partridge, S., 47
  Peano, 86
  Perfect number, 41
  Pitiscus, 15
  Planisphere, 53, 92, 93
  Prestet, J., 74
  Price, 11
  Proportion, notation for, 26, 27, 73-79
  Protheroe, 58
  Ptolemy, 83

  Quadratic equation, 29, 31, 34

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

  Saladini, H., 78
  Sanders, W., 76
  Sault, R., 82
  Scarborough, Charles, 16, 54, 58, 60
  Schooten, Van, 1
  Schreshensuchs, O., 83
  Scratch method, 23
  Shakespeare, 52
  Shelley, G., 76
  Shipley, A. E., 1
  Shuttleworth, 59
  Slide rule, 9, 46-49, 50, 60, 88, 93
  Smethwyck, 58
  Smith, J., 50
  Snellius, W., 79
  Solids, regular, 18
  Speidell, John, 38, 55
  Spherical triangles, 53, 54, 93
  Stokes, R., 35, 36, 58
  Sudell, 59
  Sun dials, 5, 9, 50, 51, 52, 60, 92

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

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

  Wadham College, 5, 53
  Wallis, John, 1, 19, 27, 33, 45, 57-59, 63, 64, 66-74, 79-81, 86
  Walmesley, D. C., 79
  Ward, Bishop, 13
  Ward, John, 76
  Ward, Seth, 55, 58, 60, 68, 73, 74, 81
  Watch-making, 18, 50
  Weber, W. E., 48
  Weddle, 56
  Wells, E., 76, 82
  Wharton, 60
  Whitlock, B., 8, 9
  Wilson, J., 77, 82
  Wing, V., 73, 75
  Wingate, E., 32, 47, 73
  Wolf, Christian, 79
  Wood, A., 60, 61
  Wood, R., 18, 59
  Wren, Christopher, 5, 58, 59, 76
  Wright, E., 6, 27, 38, 54
  Wright, S., 54




                          Transcriber’s Notes


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

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

The text versions miss much of the formatting, especially in mathematical
formulas:


--Several arithmetic examples must be viewed in a monospaced font (which
  recognizes combining characters) to be legible.

--Formulas under the horizontal line of a square root symbol are
  parenthesized.

--Subscripts are preceded by “_”.

--Superscripts are preceded by “^”.

--Italics, used primarily in formulas and bibliographical entries, are
  not indicated in the text.

--Italics in the index are delimited by “_”.

--Underlines are not indicated (in particular, in the fractional part of
  a decimal number in Oughtred's notation).

--The idiosyncratic “greater than” and “less than” symbols are indicated
  as {symbol} in the ASCII version.

--Overdots, underdots, and slashmarks around digits (in the long division
  example) are not indicated in the ASCII version.

--Greek letters are spelled out within {curly brackets} in the ASCII
  version.