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.