Florian Cajori

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 [73] than at home. Our study of seventeenth-century algebra has led us to the conviction that Oughtred deserves a higher place in the development of this science than is usually accorded to him ”

“ 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 Ac, and a3 or aaa. . . . . And the like [64] 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. ”

“ 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 [70] of mathematics to Descartes’ indebtedness to Oughtred. ”

“ 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 ”

“ 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. ”

“ If he invented sun-dials, he must have had a comprehension [93] of the cosmic motions involved; if he solved spherical triangles graphically by the aid of the planisphere, he must have understood the geometry of the sphere, so far as it relates to such triangles; if he invented slide rules, he had beforehand a thorough grasp of logarithms. The question at issue does not involve so much the invention of instruments, as the use by the pupil of instruments already constructed, before he fully understands the theory which is involved. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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. ”

“ 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 [29] believe in conducting the reader along level paths or along slight inclines. He was a guide for mountain-climbers, and woe unto him who lacked nerve.
Oughtred lays great stress upon expansions of powers of a binomial. ”

“ 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. ”

“ 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. ”

“ 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 [26] 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. ”

“ 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. ”

“ 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 [38] cited Norwood did not use nearly as much symbolism in his mathematics as did Oughtred. ”

“ 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. ”

“ 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. ”