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.

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

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