Mathematicians From England

"Considering how few, and how simple the Principles are, upon which the whole Art of PERSPECTIVE depends, and withal how useful, nay how absolutely necessary this Art is to all forts of Designing; I have often wonder'd, that it has still been left in so low a degree of Perfection, as it is found to be, in the Books that have been hitherto wrote upon it."

"It seems that those, who have hitherto treated of this Subject, have been more conversant in the Practice of Designing, than in the Principles of Geometry... that might have enabled them to render the Principles of it more universal, and more convenient for Practice. In this Book I have endeavour'd to do this; and have done my utmost to render the Principles of the Art as general, and as universal as may be, and to devise such Constructions, as might be the most simple and useful in Practice."

"Seventeenth-century proponents of symbols frequently advanced the fecundity of their approach as evidence for its superiority over classical mathematics: Wallis’s Arithmetic of infinities displayed results 'neither discovered by nor known to others', and René Descartes’s 1637 Geometry announced its general solution to a family of problems that the Greeks had left mostly unsolved. ...Newton, inspired ...by these ...texts ...used symbols to invent an algorithmic version of the calculus."

"In order to this, I found it absolutely necessary to consider this Subject entirely anew, as if it had never been treated of before; the Principles of the old Perspective being so narrow, and so confined, that they could be of no use in my Design: And I was forced to invent new Terms of Art, those already in use being so peculiarly adapted to the imperfect Notions that have hitherto been had of this Art, that I could make no use of them in explaining those general Principles I intended to establish."

"I make no difference between the Plane of the , and any other Plane whatsoever; for since Planes, as Planes, are alike in Geometry, it is most proper to consider them as so, and to explain their Properties in general, leaving the Artist himself to apply them in particular Cases, as Occasion requires."

"When he is sufficiently perfect in these, I would have him learn Perspective. And when he has made some progress in this, so as to have prepared his Judgment with the right Notions of the Alterations that Figures must undergo, when they come to be drawn on a Flat, he may then be put to Drawing by View, and be exercised in this along with Perspective, till he comes to be sufficiently perfect in both."

"Brook Taylor... in his Methodus Incrementorum Directa et Inversa (1715), sought to clarify the ideas of the calculus but limited himself to algebraic functions and algebraic differential equations. ...Taylor's exposition, based on what we would call finite differences, failed to obtain many backers because it was arithmetical in nature when the British were trying to tie the calculus to geometry or to the physical notion of velocity."

"In 1668 he laid down the principles for determining the effects of the collision of imperfectly elastic bodies. This was followed in 1669 by a work on statics (centres of gravity) and in 1670 by one on dynamics: these provide a convenient synopsis of what was then known on the subject."

"Algebra appealed to Newton’s contemporaries... for its fecundity... [and] the... presentational qualities... In... his... Treatise on conic sections [dedication], Wallis contrasted his novel treatment of the s with the diagrammatic treatment in Apollonius’s Conics... 'neglected beyond measure' by Wallis’s contemporaries 'as though it were insurmountable and full of troublesome madness'. Wallis implied that geometers read it... superficially since they feared that it would drive them mad. He contrasted his own figures with Apollonius’s... to use 'schemata as simple as possible, lest intricate leadings of lines bring in confusion'."

"In 1665 Wallis published the first systematic treatise on Analytical conic sections. Analytical geometry was invented by Descartes and the first exposition of it was given in 1637: that exposition was both difficult and obscure, and to most of his contemporaries, to whom the method was new, it must have been incomprehensible. Wallis made the method intelligible to all mathematicians. This is the first book in which these curves are considered and defined as curves of the second degree and not as sections of a cone."

"Wallis saw Greek diagrams as intricate and confused. His own... clearly displayed the conic sections and the key lines characterizing... their ‘essential affections’... More importantly, he used the same diagram as often as possible and, when not... changed the diagram only minimally. ...[He] claim[ed]... his 1685 Treatise on algebra... considered the sections 'abstractly as Figures in plano, without the embranglings of the Cone'."

"As he was unacquainted with the he could not effect the quadrature of the circle, whose equation is y = (1 - x^2)^\frac{1}{2}, since he was unable to expand this in powers of x. He laid down however the principle of interpolation. He argued that as the ordinate of the circle is the geometrical mean between the ordinates of the curves y = (1 - x^2)^0 and y = (1 - x^2)^1, so as an approximation its area might be taken as the geometrical mean between 1 and \frac{2}{3}. This is equivalent to taking 4\sqrt{\frac{2}{3}} or 3.26... as the value of \pi. But, he continued, we have in fact a series 1, \frac{2}{3}, \frac{8}{15}, \frac{16}{35},... and thus the term interpolated between 1 and \frac{2}{3} ought to be so chosen as to obey the law of this series. This by an elaborate method leads to a value for the interpolated term which is equivalent to making\pi = 2\frac{2\cdot2\cdot4\cdot4\cdot6\cdot6\cdot8\cdot8...}{1\cdot3\cdot3\cdot5\cdot5\cdot7\cdot7\cdot9...}The subsequent mathematicians of the seventeenth century constantly used interpolation to obtain results which we should attempt to obtain by direct algebraic analysis."

"The Arithmetica infinitorum was followed in 1656 by a tract on the angle of contact; in 1657 by the Mathesis universalis; in 1658 by a correspondence with Fermat; and by a long controversy with Hobbes on the quadrature of the circle."

"By regularizing... symbols Wallis circumvented the typical role of diagrams in geometric argument. ...Wallis’s new lettering strategy meant that the reader would not have to constantly look back at the diagram to understand the argument... the reader might recognize from the letters which parts of the cone were signified."

"In 1659 Wallis published a tract on s in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of s: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the x^3 = ay^2. This was the first case in which the length of a curved line was determined by mathematics, and as all attempts to rectify the ellipse and hyperbola had (necessarily) been ineffectual, it had previously been generally supposed that no curves could be rectified."

"Wallis, whether by his own efforts or not, acquired sufficient mathematics at Cambridge to be ranked as the equal of mathematicians such as Descartes, Pascal, and Fermat."

"There was then no professorship in mathematics and no opening for a mathematician to a career at Cambridge; and so Wallis reluctantly left the university. In 1649 he was appointed to the Savilian chair of geometry at Oxford, where he lived until his death on Oct. 28, 1703. It was there that all his mathematical works were published. Besides those he wrote on theology, logic, and philosophy; and was the first to devise a system for teaching deaf mutes."

"The greatest of modern have been so far from adding any thing of importance to the discoveries of ancient mathematicians, that some of their most splendid inventions are either wholly erroneous or remarkable instances of the possibility of deducing true conclusions from unscientific and false principles. Strange, however as this assertion may seem, the following elementary treatise demonstrates it to be true; by showing that all the leading propositions of the Arithmetic of Infinites of Dr. Wallis are false, and that the Doctrine of Fluxions is a baseless fabric, and in the language of the ingenious Bishop Berkley, "must be considered only as a presumption, as a knack, an art, or rather an artifice, but not a scientific demonstration."

"The most notable of these [his mathematical works] was his Arithmetica infinitorum, which was published in 1656. It is prefaced by a short tract on conic sections which was subsequently expanded into a separate treatise. He then established the law of indices, and shewed that x^{-n} stood for the reciprocal of x^n and that x^\frac{p}{q} stood for the q^{th} root of x^p. He next proceeded to find by the method of indivisibles the area enclosed between the curve y = x^m, the axis of x, and any ordinate x = h; and he proved that this was to the parallelogram on the same base and of the same altitude in the ratio 1:m + 1. He apparently assumed that the same result would also be true for the curve y = ax^m, where a is any constant. In this result m may be any number positive or negative, and he considered in particular the case of the parabola in which m = 2, and that of the hyperbola in which m = -1: in the latter case his interpretation of the result is incorrect. He then shewed that similar results might be written down for any curve of the form y = \sum{ax^m}; so that if the ordinate y of a curve could be expanded in powers of the abscissa x, its quadrature could be determined. Thus he said that if the equation of a curve was y = x^0 + x^1 + x^2 +... its area would be y = x + \frac{1}{2}x^2 + \frac{1}{3}x^3 +... He then applied this to the quadrature of the curves y = (1 - x^2)^0, y = (1 - x^2)^1, y = (1 - x^2)^2, y = (1 - x^2)^3, &c. taken between the limits x = 0 and x = 1: and shewed that the areas are respectively1,\quad \frac{2}{3},\quad \frac{8}{15},\quad \frac{16}{35},\quad \&c."

"Before Newton and Leibniz, the man who did most to introduce analytical methods in the calculus was John Wallis. Though he did not begin to learn mathematics until he was about twenty—his university education at Cambridge was devoted to theology—he became professor of geometry at Oxford and the ablest British mathematician of the century, next to Newton. In his Arithmetica Infinitorum (1655), he applied analysis and the method of indivisibles to effect many quadratures and obtain broad and useful results."

"He next considered curves of the form y = x^\frac{1}{m} and established the theorem that the area bounded by the curve, the axis of x, and the ordinate x = 1 is to the area of the rectangle on the same base and of the same altitude as m:m + 1. This is equivalent to finding the value of \int_{0}^{1}x^\frac{1}{m}dx. He illustrated this by the parabola in which m = 2. He stated but did not prove the corresponding result for a curve of the form y = x^\frac{p}{q}."

"Newton opposed entangled and tedious algebraic calculations to simple, elegant geometric constructions; Wallis opposed difficult, embrangled geometric diagrams to simplified, rationally lettered diagrams and the symbolic expressions they enabled."

"[E]arly analytic geometers—Descartes in particular—did not accept that geometry could be based on numbers or algebra. Perhaps the first to take the idea of arithmetizing geometry seriously was Wallis... [(1657) Mathesis universalis. Opera 1, 11-228.] Chs. XXIII and XXV, gave the first arithmetic treatment of Euclid's Books II and V, and he had earlier given purely algebraic treatment of s [(1655) De sectionibus conicus. Opera 1, 291-364.]. He initially derived equations from classical definitions by sections of the cone but then proceeded to derive their properties from the equations, "without the embranglings of the cone," as he put it."

"Wallis did not become interested in mathematics till the age of thirty-one, but devoted himself to the subject for the rest of his life. One of the earliest and most important books on algebra ever written in English was his treatise published in 1685. It contains a brief historical sketch of the subject which is unfortunately not entirely accurate, but his treatment of the theory and practice of arithmetic and algebra has made the book a standard work for reference ever since."

"In that part... of my book where I treat of geometry, I thought it necessary in my definitions to express those motions by which lines, superficies, solids, and figures were drawn and described, little expecting that any professor of geometry should find fault therewith, but on the contrary supposing I might thereby not only avoid the cavils of the sceptics, but also demonstrate divers propositions which on other principles are indemonstrable. And truly, if you shall find those my principles of motion made good, you shall find also that I have added something to that which was formerly extant in geometry. For first, from the seventh chapter of my book De Corpore, to the thirteenth, I have rectified and explained the principles of the science; id est, I have done that business for which Dr. Wallis receives the wages."

"During the wars between Charles I and Cromwell, Wallis's sympathies were with Cromwell, and he was of great service in reading royalist dispatches written in cipher. In fact, he was one of the most famous cryptologists of his day."

"You can see without admonition, what effect this false ground of yours will produce in the whole structure of your Arithmetica Infinitorum; and how it makes all that you have said unto the end of your thirty-eighth proposition, undemonstrated, and much of it false. The thirty-ninth is this other lemma: "In a series of quantities beginning with a point or cypher and proceeding according to the series of the cubic numbers as 0.1.8.27.64, &c. to find the proportion of the sum of the cubes to the sum of the greatest cube, so many times taken as there be terms." And you conclude that "they have a proportion of 1 to 4;" which is false. ... And yet there is grounded upon it all that which you have of comparing parabolas and paraboloeides with the parallelograms wherein they are accommodated. ... Besides, any man may perceive that without these two lemmas (which are mingled with all your compounded series with their excesses) there is nothing demonstrated to the end of your book: which to prosecute particularly, were but a vain expense of time. Truly, were it not that I must defend my reputation, I should not have showed the world how little there is of sound doctrine in any of your books. For when I think how dejected you will be for the future, and how the grief of so much time irrecoverably lost, together with the conscience of taking so great a stipend, for mis-teaching the young men of the University, and the consideration of how much your friends will be ashamed of you, will accompany you for the rest of your life, I have more compassion for you than you have deserved. Your treatise of the Angle of Contact, I have before confuted in a very few leaves. And for that of your Conic Sections, it is so covered over with the scab of symbols, that I had not the patience to examine whether it be well or ill demonstrated."

"It is customary to consider Chebyshev, Gauss, Jacobi, and Legendre as the main creators of the theory of orthogonal polynomials. However, their contributions were directly influenced by Brouncker and Wallis who, in March of 1655, made discoveries which influenced the development of analysis for the next hundred years. Namely, Wallis found an infinite product of rational numbers converging to 4/π and Brouncker gave a remarkable continued fraction for this quantity. ...The only mathematician who understood the importance of these discoveries was Euler. ...he felt that the recovery of the original Brouncker's proof could open up new perspectives for analysis. As usual, Euler was right."

"By March of 1655 John Wallis had almost completed his Arithmetica Infinitorum in which he promoted an important method of interpolation. This was a great work. ...Wallis discovered that analytic formulas can be interpolated by their values at integer numbers. ...Wallis successfully applied his interpolation to find formulas for the areas under many curves. Only one curve remained uncovered. It was the unit circle. In 1593 Viète had found the formula \frac{2}{\pi} = \frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdot\cdots. Since the multipliers in Viète's formula are algebraic irrationalities of increasing order, it was not the formula which could meet Wallis' requirements. Finally in March of 1655, Wallis obtained his now well-known formula \frac{2}{\pi} = \frac{1\cdot3}{2\cdot2}\cdot\frac{3\cdot5}{4\cdot4}\cdot\frac{5\cdot7}{6\cdot6}\cdot\cdots\frac{(2n-1)\cdot(2n+1)}{2n\cdot2n}\cdot\cdots."

"The true "principle of number," for Wallis as for Stevin, is the "nought". It is the sole numerical analogue of the geometric point (just as the instant is the temporary analogue... Wallis expressly rejects the accusation that he is relinquishing the unanimous opinion of the ancients and the moderns, who all saw the unit as the element of number. ...the traditional opinion can be brought into accord with his own if the following distinction is taken account of: Something can be a "principle" of something (1) which is the "first which is such" (primum quod sic) as to be of the same nature as the thing itself and (2) which is the last which is not" (ultimum quod non) such as to be of the same nature of the thing itself. In the first sense the unit may indeed be called the "principle of number," while the nought is a "principle" in the second sense. ...The ancients... overlooked the fact that the analogy which exists is not between the "point" and the "unit," but between the point and the "nought." For this reason they were able to develop their algebra only for "geometric magnitudes"..."

"Wallis' mathematical work, most notably his Arithmetica Infinitorum, was the polemic target of Pierre de Fermat and Thomas Hobbes. ...the letters of the French mathematician were reproduced in Wallis' Commercium Epistolicum (1658) ...One of the criticisms leveled at Wallis concerned the validity of induction. The fact that a proposition is proven true for a few numbers... does not imply that it is true for all... as Fermat, a master of number theory, knew too well. Fermat invited Wallis to devote himself to number theory, but Wallis found it of little interest. Number theory struck him as something of little use in applications, in other words, as a useless inquiry. ...Wallis ...claimed that induction methods were not his invention but had been employed both recently by Henry Briggs and Viète and in the ancient world by Euclid."

"Vieta died in 1603, Porta died in 1615, and Dr. Wallis was born in 1617. So that, in all Probability, here is a great Deduction to be made from the many hundred Years, in which we were to have understood that the Art of Decyphering had been in being before Dr. Wallis was born."

"How the Dr. first came to apply himself to this Art, we shall have from himself; what use he made of it, we have had, tho unfairly, from other Hands. But as his Skill in the Art of Decyphering has no Relation to his Political Principles, he might be very sagacious in one Respect, and very erroneous in the other. I wish, both for his own Credit and the publick Good, that he had employed his Skill in the Service of the King; but it is too Well known, that he did not so. He was publickly charged in his Life-time by Henry Stubbe, and from him by Anthony Wood, with "having decyphered (besides others, to the Ruin of many loyal Persons) the King's Cabinet taken at Naseby and as a Monument of his noble Performances, depositing the Original, with the Decyphering, in the publick Library at Oxford."

"Accountants eventually became comfortable with using negative numbers... but for a long time mathematicians remained wary... the negatives were known as absurd numbers—numeri absurdi...Consider this equation:\frac{-1}{\quad 1} = \frac{\quad 1}{-1}...it states that the ratio of a smaller number, -1, to a larger number, 1, is equal to the ratio of a larger number, 1, to a smaller one, -1. The paradox was much discussed... To make sense of negative numbers, many mathematicians, including Leonhard Euler, came to the bizarre conclusion that they were larger than infinity. ...One voice of clarity among the confusion belonged to... John Wallis, who devised a powerful visual interpretation for the negative numbers. In his 1685 work A Treatise of Algebra, he first described the ","... By replacing the idea of quantity with the idea of position, Wallis argued that negative numbers were neither "Unuseful [nor] Absurd,"...It took a few years for Wallis' idea to hit the mainstream, but... it is the most successful explicatory diagram of all time."

"It is not that I do not approve it, but all his propositions could be proved in the usual, regular Archimedian way in many fewer words than this book [Arithmetica Infinitorum] contains. I do not know why he has preferred this method with algebraic notation to the older way which is both more convincing and more elegant."

"[W]e advise that you would lay aside (for some time at least) the Notes, Symbols, or Analytick Species (now since Vieta's time, in frequent use,) in the construction and demonstration of Geometrick Problems, and perform them in such method as Euclide and Apollonius were wont to do; that the neatness and elegance of Construction and Demonstrations, by them so much affected, do not by any degrees grow into disuse."

"Paralleling what happened in France, an English group centered about John Wallis began in 1645 to hold meetings in Gresham College, London, These men emphasized mathematics and astronomy. The group was given a formal charter by Charles II in 1662 and adopted the name of the Royal Society of London for the Promotion of Natural Knowledge."

"[I]n On conic sections, Wallis... claimed that algebra would enable an 'absolute contemplation' of the conic sections by directly expressing... 'primary and essential affections' from which secondary affections could be 'deduced by calculation'. ...Both the use of symbols to express essences and their manipulation to obtain easy results characterized Wallis..."

"Of the Oxford mathematician John Wallis... Sorbière wrote that his appearance inclined one to laughter and that he suffered from bad breath that was "noxious in conversation." Wallis' only hope, according to Sorbière, was to be purified by the "Air of the court of London." For the Society's nemesis Thomas Hobbes, however, who was also Wallis's personal enemy, Sorbière had only praise."

"It is not unknown to those who know any Thing of publike Affairs, of how great Concernment it is, especially in civill Commotions, for those who are to manage such Transactions, to be furnished with continuall Intelligence from their Correspondents, yet so as to conceal their Councells and Resolutions from the adverse Party. And to this Purpose, in all Ages, much Care and lndustry hath been still used, how in Matters of Consequence, to convey Intelligence safely and secretly to those with whom they hold Correspondence, so as not to bee intercepted by the Enemy, or if intercepted, at least not discovered. And as this is no where of more Concernment, so no where more difficult, than in civill Wars, where the intermingling of opposite Parties makes it difficult, if not impossible, to distinguish Friends and Foes."

"Upon this Occasion many Methods have been invented of secret Writing, or Writing in Cipher, a Thing heretofore scarce known to any but the Secretaries of Princes, or others of like Condition; but of late Years, during our Commotions and civill Wars in England, grown very common and familiar, so that now there is scarce a Person of Quality, but is more or lesse acquainted with it, and doth as there is Occasion, make use of it."

"Thus in Compliance with your repeated desires, I have given you a short account of divers passages of my life, 'till I have now come to more than fourscore years of age. How well I have acquitted my self in each, is for others rather to say, than for Your friend and servant John Wallis. Oxford January 29. 1696, 7."

"If any ask, with what Confidence I durst adventure upon a Task so unusuall, as interpreting of Letters committed to Cipher; I shall only give this plain Account thereof."

"I made it my business to examine things to the bottom; and reduce effects to their first principles and original causes. Thereby the better to understand the true ground of what hath been delivered to us from the Antients, and to make further improvements of it. What proficiency I made therein, I leave to the Judgement of those who have thought it worth their while to peruse what I have published therein from time to time; and the favorable opinion of those skilled therein, at home and abroad."

"In the year 1660 being importuned by some friends of his, I undertook so to teach Mr. Daniel Whalley of Northampton, who had been Deaf and Dumb from a Child. I began the work in 1661, and in little more than a year's time, I had taught him to pronounce distinctly any words, so as I directed him... and in good measure to understand a Language and express his own mind in writing; And he had in that time read over to me distinctly (the whole or greatest part of) the English Bible; and did pretty well understand (at least) the Historical part of it. In the year 1662 I did the like for Mr. Alexander Popham... I have since that time (upon the same account) taught divers Persons (and some of them very considerable) to speak plain and distinctly, who did before hesitate and stutter very much; and others, to pronounce such words or letters, as before they thought impossible for them to do: by teaching them how to rectify such mistakes in the formation, as by some natural impediment, or acquired Custome, they had been subject to."

"Partly out of my owne Curiosity, partly to satisfy the Gentleman's Importunity that did request it, I resolved to try what I could do in it: And having projected the best Methods I could think of for the effecting it, I found yet so hard a Task, that I did divers Times give it over as desperate: Yet, after some Intermissions, resuming it againe, I did at last overcome the Difficulty; but with so much Paines and Expense of Time as I am not willing to mention; though yet I did not repent of that Labour, when I had discovered thereby, that it was a Businesse, which though with much Difficulty, was yet capable to bee effected."

"It hath been my Lot to live in a time, wherein have been many and great Changes and Alterations. It hath been my endeavour all along, to act by moderate Principles, between the Extremities on either hand, in a moderate compliance with the Powers in being, in those places, where it hath been my Lot to live, without the fierce and violent animosities usual in such Cases, against all, that did not act just as I did, knowing that there were many worthy Persons engaged on either side. And willing whatever side was upmost, to promote (as I was able) any good design for the true Interest of Religion, of Learning, and the publick good; and ready so to do good Offices, as there was Opportunity; And, if things could not be just, as I could wish, to make the best of what is: And hereby, (thro' God's gracious Providence) have been able to live easy, and useful, though not Great."

"About the beginning of our Civil Wars, in the year 1642, a Chaplain of Sr. Will. Waller's (one evening as we were sitting down to Supper at the Lady Vere's in London, with whom I then dwelt,) shewed me an intercepted Letter written in Cipher. He shewed it me as a Curiosity (and it was indeed the first thing I had ever seen written in Cipher.) And asked me between jeast and earnest, whether I could make any thing of it. And he was surprised when I said (upon the first view) perhaps I might, if it proved no more but a new Alphabet. It was about ten a clock when we rose from Supper. I then withdrew to my chamber to consider of it. And by the number of different Characters therein, (not above 22 or 23:) I judged that it could not be more than a new Alphabet, and in about 2 hours time (before I went to bed) I had deciphered it; and I sent a Copy of it (so deciphered) the next morning to him from whom I had it. And this was my first attempt at Deciphering."

"Being encouraged by... success, beyond expectation; I afterwards ventured on many others and scarce missed of any, that I undertook, for many years, during our civil Wars, and afterwards. But of late years, the French Methods of Cipher are grown so intricate beyond what it was wont to be, that I have failed of many; tho' I have master'd divers of them. Of such deciphered Letters, there be copies of divers remaining in the Archives of the Bodleyan Library in Oxford; and many more in my own Custody, and with the Secretaries of State."

"The Occasion of that Assembly was this; The Parliament which then was, (or the prevailing part of them,) were ingaged in a War with the King. ...The Issue of which War, proved very different from what was said to be at first intended. As is usual in such cases; the power of the sword frequently passing from hand to hand and those who begin a War, not being able to foresee where it wil end."