University Of Cambridge Faculty

"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."

"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."

"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."

"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."

"However, it is not unlikely that the Arabs, who received from the Indians the numeral figures (which the Greeks knew not), did from them also receive the use of them, and many profound speculations concerning them, which neither Latins nor Greeks know, till that now of late we have learned them from thence. From the Indians also they might learn their algebra, rather than from Diophantus."

"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."

"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."

"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."

"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."

"I saw, there was little or no Help to bee exspected from others; but that if I should have further Occasions of that Kind, I must trust to my owne Industry, and such Observations as the present Case should afford. And indeed the Nature of the Thing is scarce capable of any other Directions; every new Cipher allmost being contrived in a new Way, which doth not admit any constant Method for the finding of it out: But hee that will do any Thing in it, must first furnish himself with Patience and Sagacity, as well as hee may, and then Consilium in arenâ capere, and make the best Conjectures hee can, till hee shall happen upon something that hee may conclude for Truth."

"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."

"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."

"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."

"I was... informed, that Baptista Porta, and one or two more, had written somewhat of that Subject, upon this Information I was willing to see whether I might from any of them find any Directions, that might help mee, if I should afterwards have the like Occasion: But I found very little in any of them for my Purpose. Their Businesse being for the most Part, onely to shew how to write in Cipher, (which was not my Work,) and that Things so written were beyond the Skill of Men to decipher. Onely in Baptista Porta (who alone if I mistake not, hath written any Thing to Purpose about deciphering, and was it seemes famous in his Time for his Abilities that Way;) I found that there were some general Directions, such as were obvious from the Nature of the Thing, and which I had before of myself taken Notice of, and made use of so far as the Nature of an intricate Cipher would permit. But the Truth of it is, there are scarce any of his Rules, which the present Way of Cipher (which is now much improved, beyond what, it seemes, it was in his Days) doth not in a Manner wholly elude..."

"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."

"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."

"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."

"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."

"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."

"Taylor's method... has no recommendation from its neatness and perspicuity, but is justly censured by John Bernoulli for its obscure conciseness."

"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."

"The methods of the Bernoullis and of Taylor, were held, at the time of their invention, to be most complete and exact. Several imperfections, however, belong to them. They do not apply to problems involving three or more properties; nor do they extend to cases involving differentials of a higher order than the first: for instance, they will not solve the problem, in which a curve is required, that with its radius of curvature and evolute shall contain the least area. Secondly, they do not extend to cases, in which the analytical expression contains, besides x, y, and their differentials, integral expressions; for instance, they will not solve the second case proposed in James Bernoulli's Programma if the Isoperimetrical condition be excluded; for then the arc s, an integral, since it =\int \!dx \sqrt(1+\frac{dy^2}{dx^2}), is not given. Thirdly, they do not extend to cases, in which the differential function, expressing the maximum should depend on a quantity, not given except under the form of a differential equation, and that not integrable; for instance, they will not solve the case of the curve of the quickest descent, in a resisting medium, the descending body being solicited by any forces whatever."

"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."

"The Authors who write near the beginnings of science, are, in general the most instructive: they take the reader more along with them, shew him the real difficulties, and, which is a main point, teach him the subject, the way by which they themselves learned it."

"About the year 1645 while, I lived in London (at a time, when, by our Civil Wars, Academical Studies were much interrupted in both our Universities:) beside the Conversation of divers eminent Divines, as to matters Theological; I had the opportunity of being acquainted with divers worthy Persons, inquisitive into Natural Philosophy, and other parts of Humane Learning; And particularly of what hath been called the New Philosophy or Experimental Philosophy. We did by agreement, divers of us, meet weekly in London on a certain day, to treat and discourse of such affairs. ...Some of which were then but New Discoveries, and others not so generally known and imbraced, as now they are, with other things appertaining to what hath been called The New Philosophy; which, from the times of Galileo at Florence, and Sr. Francis Bacon (Lord Verulam) in England, hath been much cultivated in Italy, France, Germany, and other Parts abroad, as well as with us in England. About the year 1648, 1649, some of our company being removed to Oxford (first Dr. Wilkins, then I, and soon after Dr. Goddard) our company divided. Those in London continued to meet there as before... Those meetings in London continued, and (after the King's Return in 1660) were increased with the accession of divers worthy and Honorable Persons; and were afterwards incorporated by the name of the Royal Society, &c. and so continue to this day."

"To history we shall adhere no farther, than is sufficient to preserve an unbroken series of methods gradually becoming more exact and extensive; the series beginning with the first rude, though perfectly just, method of James Bernoulli, and ending with Lagrange's exquisite and refined Calculus of Variations."

"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."

"These Exponents they call Logarithms, which are Artificial Numbers, so answering to the Natural Numbers, as that the addition and Subtraction of these, answers to the Multiplication and Division of the Natural Numbers. By this means, (the Tables being once made) the Work of Multiplication and Division is performed by Addition and Subtraction; and consequently that of Squaring and Cubing, by Duplication and Triplication; and that of Extracting the Square and Cubic Root, by Bisection and Trisection; and the like in the higher Powers."

"Although I am not aware of having omitted any thing that is requisite to the full explanation of the subject, yet I cannot flatter myself that it will be thoroughly understood from this Work alone. For, in general it may be laid down as true, that no doctrine, of novelty and intricacy, can be completely taught by a single Treatise. It seems to be indispensably necessary for the student, that the subject should be put under several points of view: that if not apprehended under one, it may be under another."

"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."

"Mathematicks were not, at the time, looked upon as Accademical Learning, but the business of Traders, Merchants, Seamen, Carpenters, land-measurers, or the like; or perhaps some Almanak-makers in London. And of more than 200 at that time in our College, I do not know of any two that had more of Mathematicks than myself, which was but very little; having never made it my serious studie (otherwise than as a pleasant diversion) till some little time before I was designed for a Professor in it."

"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."

"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."

"‘Yet some few of such investigations we have in the five first propositions of Euclid’s thirteenth book … seems to be the work of Theo, […] rather than of Euclid himself.’"

"I came across the mathematical writings of Torricelli... which... I read in... 1651... where... he expounds the geometry of indivisibles of Cavalieri. ...His method, as taught by Torricelli... was indeed all the more welcome to me because I do not know that anything of that kind was observed in the thinking of almost any mathematician I had previously met; for what holds for most... concerning the circle... usually had by polygons with an infinite number of sides, and... the circumference by... an infinite number of infinitely short lines... could.., it seemed to me, with... changes, be... adjusted to other problems; and... by that means examine... Euclid, Appolonius and especially... Archimedes. ...I began to think ...whether this might bring ...light to the quadrature of the circle."

"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."

"[Mathematics were] scarce looked upon as Academical studies but rather Mechanical... And among more than two hundred students (at that time) in our college, I do not know of any two (perhaps not any) who had more of Mathematicks than I, (if so much) which was then but little; and but very few, in that whole university. For the study of Mathematicks was at that time more cultivated in London than in the universities."

"On a novel plan, I have combined the historical progress with the scientific developement of the subject; and endeavoured to lay down and inculcate the principles of the Calculus, whilst I traced its gradual and successive improvements. ...there is little doubt, the student's curiosity and attention will be more excited and sustained, when he finds history blended with science, and the demonstration of formulae accompanied with the object and the causes of their invention, than by a mere analytical exposition of the principles of the subject. He will have an opportunity of observing how a calculus, from simple beginnings, by easy steps, and seemingly the slightest improvements, is advanced to perfection; his curiosity too, may be stimulated to an examination of the works of the contemporaries of Newton; works once read and celebrated: yet the writings of the Bernoullis are not antiquated from loss of beauty, nor deserve neglect..."

"There is another point... and that is the method of demonstration by geometrical figures. In the first solution of Isoperimetrical problems, the Bernoullis use diagrams and their properties. Euler, in his early essays, does the same; then, as he improves the calculus he gets rid of constructions. In his Treatise [footnote: Methodus inveniendi, &c.], he introduces geometrical figures, but almost entirely, for the purpose of illustration: and finally, in the tenth volume of the Novi Comm. Petrop. as Lagrange had done in the Miscellanea Taurinensea, he expounds the calculus, in its most refined state, entirely without the aid of diagrams and their properties. A similar history will belong to every other method of calculation, that has been advanced to any degree of perfection."

"In 1810 a work was published in Cambridge under the following title—A Treatise on Isoperimetrical Problems and the Calculus of Variations. By Robert Woodhouse... This work details the history of the Calculus of Variations from its origin until the close of the eighteenth century, and has obtained a high reputation for accuracy and clearness. During the present century some of the most eminent mathematicians have endeavored to enlarge the boundaries of the subject, and it seemed probable that a survey of what had been accomplished would not be destitute of interest and value. Accordingly the present work has been undertaken... As the early history of the Calculus of Variations had been already so ably written, it was unnecessary to go over it again; but it seemed convenient to commence with a short account of the two works of Lagrange and a work of Lacroix..."

"I will not say that a witness shall not be asked to what may tend to disparage him: that would prevent an investigation into the character of the witness, which may often be of importance to ascertain. I think those questions only should not be allowed to be asked which have a direct and immediate effect to disgrace or disparage the witness."

"The act of a single Judge, unless adopted by the Court to which he belongs, is of no validity. As the Courts do not sit in vacation, many things are done by the Judges individually; but their acts, when recognised, become the acts of the Court."

"It is true that Courts of equity, in administering justice, sometimes go further than the Courts of law."

"Amendments ought not to be made, except in cases where the alteration is of such a nature as that no one can be misled by it."

"Inconvenience arising from the operation of an Act of Parliament can be no ground of argument in a Court of law."

"No man can come into a British Court of justice to seek the assistance of the law who founds his claim upon a contravention of the British laws."

"In matters where a company is not restrained by Parliament they have a right to make reasonable regulations; but it will always be a question whether their regulations are reasonable or not."

"When Courts adopt a fiction they must necessarily support it."

"Intellectually he was a giant; I have hardly ever known anyone who gave me such an impression of the power and grasp of his mind. Like Mr. Gladstone, he would sometimes, when he was talking to me in my room, get interested in his subject, and, with great emphasis, and in an unnecessarily loud voice, deliver a speech which, if it had been taken down, would have been an appreciable addition to the sum of human thought. In Council he rarely spoke, but when he did so, always with the same thunderous voice and commanding air, he invariably convinced everybody and carried his point. His knowledge, his sagacity, his insight were wonderful, and they were by no means confined to questions of law."