# John Wallis

English mathematician

John Wallis (November 23, 1616 – October 28, 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. He is credited with introducing the symbol ∞ to represent the concept of infinity. He similarly used 1/∞ for an infinitesimal. He was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of mathematics.

Portrait of John Wallis
by Sir Godfrey Kneller, 1st Baronet

## Quotes

• [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.
• 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.
• Autobiography (1696), Notes and Records of the Royal Society of London (1970) Vol. 25, p. 27.

### Arithmetica Infinitorum (1656)

Translation via Jacqueline A. Stedal, The Arithmetic of Infinitesimals: John Wallis 1656 (2004) unless otherwise indicated
• You may find this work (if I judge rightly) quite new. For I see no reason why I should not proclaim it; nor do I believe that others will take it wrongly. ...it teaches all by a new method, introduced by me for the first time into geometry, and with such clarity that in these more abstruse problems no-one (as far as I know) has used...
• This method of mine takes its beginnings where Cavalieri ends his Method of indivisibles. ...for as his was the Geometry of indivisibles, so I have chosen to call my method the Arithmetic of infinitesimals.
• Around 1650 I came across the mathematical writings of Torricelli, where among other things, 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.

### Treatise of Algebra (1685)

Source
• [W]hereas Nature, in propriety of Speech, doth not admit more than Three (Local) Dimensions, (Length, Breadth and Thickness, in Lines, Surfaces and Solids;) it may justly seem improper to talk of a Solid (of three Dimensions) drawn into a Fourth, Fifth, Sixth, or further Dimension.
A Line drawn into a Line, shall make a Plane or Surface; this drawn into a Line, shall make a Solid. But if this Solid be drawn into a Line, or this Plane into a Plane, what shall it make? A Plano-plane? This is a Monster in Nature, and less possible than a Chimera or a Centaure. For Length, Breadth and Thickness, take up the whole of Space. Nor can our Fansie imagine how there should be a Fourth Local Dimension beyond these Three.

### A Discourse of Combinations, Alterations, and Aliquot Parts (1685)

As published in Jakob Bernoulli, John Wallis, The Doctrine of Permutations and Combinations (1795)
• Let as many Numbers, as you please, be proposed to be Combined: Suppose Five, which we will call a b c d e. Put, in so many Lines, Numbers, in duple proportion, beginning with 1. The Sum (31) is the Number of Sumptions, or Elections; wherein, one or more of them, may several ways be taken. Hence subduct (5) the Number of the Numbers proposed; because each of them may once be taken singly. And the Remainder (26) shews how many ways they may be taken in Combination; (namely, Two or more at once.) And, consequently, how many Products may be had by the Multiplication of any two or more of them so taken. But the same Sum (31) without such Subduction, shews how many Aliquot Parts there are in the greatest of those Products, (that is, in the Number made by the continual Multiplication of all the Numbers proposed,) a b c d e. For every one of those Sumptions, are Aliquot Parts of a b c d e, except the last, (which is the whole,) and instead thereof, 1 is also an Aliquot Part; which makes the number of Aliquot Parts, the same with the Number of Sumptions. Only here is to be understood, (which the Rule should have intimated;) that, all the Numbers proposed, are to be Prime Numbers, and each distinct from the other. For if any of them be Compound Numbers, or any Two of them be the same, the Rule for Aliquot Parts will not hold.
• Ch.I Of the variety of Elections, or Choice, in taking or leaving One or more, out of a certain Number of things proposed.
• Suppose we a certain Number of things exposed, different each from other, as a, b, c, d, e, &c. The question is, how many ways the order of these may be varied? as, for instance, how many changes may be Rung upon a certain Number of Bells; or, how many ways (by way of Anagram) a certain Number of (different) Letters may be differently ordered?
1) If the thing exposed be but One, as a, it is certain, that the order can be but one. That is 1.
2) If Two be exposed, as a, b, it is also manifest, that they may be taken in a double order, as ab, ba, and no more. That is 1 x 2 = 2.
3) If Three be exposed; as a, b, c: Then, beginning with a, the other two b, c, may (by art. 2,) be disposed according to Two different orders, as bc, cb; whence arise Two Changes (or varieties of order) beginning with a as abc, acb: And, in like manner it may be shewed, that there be as many beginning with b; because the other two, a, c, may be so varied, as bac, bca. And again as many beginning with c as cab, cba. And therefore, in all, Three times Two. That is 1 x 2, x 3 = 6.
4) If Four be exposed as a, b, c, d; Then, beginning with a, the other Three may (by art. preceeding) be disposed six several ways. And (by the same reason) as many beginning with b, and as many beginning with c, and as many beginning with d. And therefore, in all, Four times six, or 24. That is, the Number answering to the case next foregoing, so many times taken as is the Number of things here exposed. That is 1 x 2 x 3, x 4 = 6 x 4 = 24.
5) And in like manner it may be shewed, that this Number 24 Multiplied by 5, that is 120 = 24 x 5 = 1 x 2 x 3 x 4 x 5, is the number of alternations (or changes of order) of Five things exposed. (Or, the Number of Changes on Five Bells.) For each of these five being put in the first place, the other four will (by art. preceeding) admit of 24 varieties, that is, in all, five times 24. And in like manner, this Number 120 Multiplied by 6, shews the Number of Alternations of 6 things exposed; and so onward, by continual Multiplication by the conse quent Numbers 7, 8, 9, &c.
6) That is, how many so ever of Numbers, in their natural Consecution, beginning from 1, being continually Multiplied, give us the Number of Alternations (or Change of order) of which so many things are capable as is the last of the Numbers so Multiplied. As for instance, the Number of Changes in Ringing Five Bells, is 1 x 2 x 3 x 4 x 5 = 120. In Six Bells, 1 x 2 x 3 x 4 x 5 x 6 = 120 x 6 = 720. In Seven Bells, 720 x 7 = 5040. In Eight Bells, 5040 x 8 = 40320, And so onward, as far as we please.
• Ch.II Of Alternations, or the different Change of Order, in any Number of Things proposed.

### Of Logarithms, Their Invention and Use (1685)

Chapter 12 of Treatise of Algebra, as published in Mathematical Tables (1717) Contrived after a most Comprehensive Method: viz. Henry Hopkey, A Table of Logarithms, from 1 to 101000. To which is added (upon the same Page) The Differences and Proportional Parts, whereby the Logarithm of any Number under 10,000,000 may be easily found. Tables of Natural Sines, Tangents, and Secants, with their Logaritms, and Logarithmick Differences to every Minute of the Quadrant. Tables of Natural Versed Sines, and their Logarithms, to every Minute of the Quadrant. With their Construction and Use. By Mr. Briggs, Dr. Wallis, Mr. Halley, Mr. Abr. Sharp, Savilian Professors of Geometry, the University of Oxford.
• Logarithms was first of all Invented (without any Example of any before him, that I know of) by John Neper... And soon after by himself (with the assistance of Henry Briggs...) reduced to a better form, and perfected. The invention was greedily embraced (and deservedly) by Learned Men. ...in a short time, it became generally known, and greedily embraced in all Parts, as of unspeakable Advantage; especially for Ease and Expedition in Trigonometrical Calculations.
• 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.

### Dr. Wallis's Account of some Passages of his own Life (1696)

E Coll. MSS. Smithianis penes Editorem, Vol. 22 p. 38. as contained in Ch.XII of Peter Langtoft's Chronicle, (as illustrated and improved by Robert of Brunne) from the Death of Cadwalader to the end of K. Edward the First's Reign. Transcrib'd and now first publish'd, from a MS. in the Inner-Temple Library by Thomas Hearne, M.A. (1725)
• It was always my affectation even from a child, in all pieces of Learning or Knowledge, not merely to learn by rote, which is soon forgotten, but to know the grounds or reasons of what I learn; to inform my Judgement, as well as furnish my Memory; and thereby, make a better Impression on both.
• This suiting my humor so well; I did thenceforth prosecute it, (at School and in the University) not as a formal Study, but as a pleasing Diversion, at spare hours; as books of Arithmetick or others Mathematical fel occasionally in my way. For I had none to direct me, what books to read, or what to seek, or in what Method to proceed. For Mathematicks, (at that time, with us) were scarce looked upon as Academical Studies, but rather Mechanical; as the business of Traders, Merchants, Seamen, Carpenters, Surveyors of Lands, or the like; and perhaps some Almanack-makers in London.
• I made no Scruple of diverting (from the common Road of Studies then in fashion) to any part of Useful Learning. Presuming, that Knowledge is no Burthen; and, if of any part thereof I should afterwards have no occasion to make use, it would at least do me no hurt; And what of it l might or might not have occasion for, I could not then foresee.
• As to Divinity, (on which I had an eye from the first,) l had the happiness of a strict and Religious Education, all along from a Child: Whereby I was not only preserved from vicious Courses, and acquainted with Religious Exercises; but was early instructed in the Principles of Religion, and Catachetical Divinity, and the frequent Reading of Scripture, and other good Books, and diligent attendance on Sermons. (And whatever other Studies I followed, I was careful not to neglect this.) And became timely acquainted with Systematick and Polemick Theology. And had the repute of a good Proficient therein.
• In Hilary Term 1636, 7. I took the Degree of Batchelor of Arts; and in 1640, the Degree of Master of Arts, and then left Emanuel College; and the same year I entered into Holy Orders, ordained by Bishop Curle, then Bishop of Winchester. I then lived a Chaplain for about a year, in the house of Sr. Richard Darley, (an antient worthy Knight,) at Buttercramb in Yorkshire, and then, for two years more, with the Lady Vere, (the Widdow of the Lord Horatio Vere,) partly in London, and partly at Castlc-Hedingham in Essex, the antient seat of the Earls of Oxford.
• 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.
• 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.
• On March 4. 1644, 5. I married Susanna daughter of John and Rachel Glyde of Northjam in Sussex; born there about the end of January 1621, 2. and baptised Feb. 3 following. By whom I have (beside other children who died young) a Son and two Daughters now surviving; John born Dec. 26 1650. Anne born June 4. 1656. and Elizabeth born Sept. 23 1658. ...My Wife died at Oxford Mar. 17. 1686, 7. after we had been married more than 42 years.
• 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.
• 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.
• 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.
• 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.

### An Essay on the Art of Decyphering (1737)

In which is inserted a discourse of Dr. Wallis. Now first publish'd from his original manuscript in the publick library at Oxford
• 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.
• 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.
• 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.
• 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...
• 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.

• 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.
• Amir R. Alexander, Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (2014)
• 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:
${\displaystyle {\frac {-1}{\quad 1}}={\frac {\quad 1}{-1}}}$
...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 "number line,"... 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.
• 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.
• John Davys, An Essay on the Art of Decyphering (1737)
• 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.
• John Davys, An Essay on the Art of Decyphering (1737)
• 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.
• Pierre de Fermat, Letter to Kenelm Digby (August 15, 1657) translation in Jacqueline A. Stedall, A Discourse Concering Algebra: English Algebra to 1685 (2002) as quoted by Niccolò Guicciardini, Isaac Newton on Mathematical Certainty and Method (2009)
• [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.
• Pierre de Fermat, Letter to Kenelm Digby (June, 1658) in Wallis, Commercium Epistolicum translated in Wallis, A Treatise of Algebra (1685) & as quoted by Niccolò Guicciardini, Isaac Newton on Mathematical Certainty and Method (2009)
• 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.
• Fritz Gesztesy, Spectral Theory and Mathematical Physics in Proceedngs of Symposia in Pure Mathematics (2007) Vol.76, Part 2
• 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
${\displaystyle {\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdot \cdots }$ .
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
${\displaystyle {\frac {2}{\pi }}={\frac {1\cdot 3}{2\cdot 2}}\cdot {\frac {3\cdot 5}{4\cdot 4}}\cdot {\frac {5\cdot 7}{6\cdot 6}}\cdot \cdots {\frac {(2n-1)\cdot (2n+1)}{2n\cdot 2n}}\cdot \cdots }$ .
• Fritz Gesztesy, Spectral Theory and Mathematical Physics in Proceedngs of Symposia in Pure Mathematics (2007) Vol.76, Part 2
• 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.
• Niccolò Guicciardini, Isaac Newton on Mathematical Certainty and Method (2009)
• 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.
• Herbert Edwin Hawkes, William Arthur Luby, Frank Charles Touton, First Course in Algebra (1917)
• 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.
• Herbert Edwin Hawkes, William Arthur Luby, Frank Charles Touton, First Course in Algebra (1917)
• 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.
• Thomas Hobbes, "Six Lessons to the Professors of the Mathematics, One of Geometry, the other of Astronomy, in the Chairs set up by the Noble and Learned Sir Henry Savil, in the University of Oxford" (1656) in The English Works of Thomas Hobbes of Malmesbury, Vol.7 (1845) ed. Sir William Molesworth, Bart.
• 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.
• Thomas Hobbes, "Six Lessons to the Professors of the Mathematics, One of Geometry, the other of Astronomy, in the Chairs set up by the Noble and Learned Sir Henry Savil, in the University of Oxford" (1656) in The English Works of Thomas Hobbes of Malmesbury, Vol.7 (1845) ed. Sir William Molesworth, Bart.
• 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"...
• Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) p. 214.
• 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.
• Morris Kline, Mathematical Thought from Ancient to Modern Times (1972)
• [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 conic sections [(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.
• John Stillwell, Mathematics and Its History (1989, 2002) 2nd edition, p. 115.
• 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.
• Thomas Taylor, Preface, The Elements of the True Arithmetic of Infinites. In Which All the Propositions in the Arithmetic of Infinites Invented by Dr. Wallis, Relative to the Summation of Infinite Series, and, also, the Principle of the Doctrine of Fluxions are Demonstrated to be False; and the Nature of Infinitesimals is Unfolded (1809) p. v.

### A History of the Study of Mathematics at Cambridge (1889)

W. W. Rouse Ball, source
• 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 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 ${\displaystyle x^{-n}}$  stood for the reciprocal of ${\displaystyle x^{n}}$  and that ${\displaystyle x^{\frac {p}{q}}}$  stood for the ${\displaystyle q^{th}}$  root of ${\displaystyle x^{p}}$ . He next proceeded to find by the method of indivisibles the area enclosed between the curve ${\displaystyle y=x^{m}}$ , the axis of ${\displaystyle x}$ , and any ordinate ${\displaystyle x=h}$ ; and he proved that this was to the parallelogram on the same base and of the same altitude in the ratio ${\displaystyle 1:m+1}$ . He apparently assumed that the same result would also be true for the curve ${\displaystyle y=ax^{m}}$ , where ${\displaystyle a}$  is any constant. In this result ${\displaystyle m}$  may be any number positive or negative, and he considered in particular the case of the parabola in which ${\displaystyle m=2}$ , and that of the hyperbola in which ${\displaystyle m=-1}$ : 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 ${\displaystyle y=\sum {ax^{m}}}$ ; 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 ${\displaystyle y=x^{0}+x^{1}+x^{2}+...}$  its area would be ${\displaystyle y=x+{\frac {1}{2}}x^{2}+{\frac {1}{3}}x^{3}+...}$  He then applied this to the quadrature of the curves ${\displaystyle y=(1-x^{2})^{0}}$ , ${\displaystyle y=(1-x^{2})^{1}}$ , ${\displaystyle y=(1-x^{2})^{2}}$ , ${\displaystyle y=(1-x^{2})^{3}}$ , &c. taken between the limits ${\displaystyle x=0}$  and ${\displaystyle x=1}$ : and shewed that the areas are respectively
${\displaystyle 1,\quad {\frac {2}{3}},\quad {\frac {8}{15}},\quad {\frac {16}{35}},\quad \&c}$ .
• He next considered curves of the form ${\displaystyle y=x^{\frac {1}{m}}}$  and established the theorem that the area bounded by the curve, the axis of ${\displaystyle x}$ , and the ordinate ${\displaystyle x=1}$  is to the area of the rectangle on the same base and of the same altitude as ${\displaystyle m:m+1}$ . This is equivalent to finding the value of ${\displaystyle \int _{0}^{1}x^{\frac {1}{m}}dx}$ . He illustrated this by the parabola in which ${\displaystyle m=2}$ . He stated but did not prove the corresponding result for a curve of the form ${\displaystyle y=x^{\frac {p}{q}}}$ .
• As he was unacquainted with the binomial theorem he could not effect the quadrature of the circle, whose equation is ${\displaystyle y=(1-x^{2})^{\frac {1}{2}}}$ , since he was unable to expand this in powers of ${\displaystyle x}$ . 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 ${\displaystyle y=(1-x^{2})^{0}}$  and ${\displaystyle y=(1-x^{2})^{1}}$ , so as an approximation its area might be taken as the geometrical mean between ${\displaystyle 1}$  and ${\displaystyle {\frac {2}{3}}}$ . This is equivalent to taking ${\displaystyle 4{\sqrt {\frac {2}{3}}}}$  or 3.26... as the value of ${\displaystyle \pi }$ . But, he continued, we have in fact a series ${\displaystyle 1,{\frac {2}{3}},{\frac {8}{15}},{\frac {16}{35}},...}$  and thus the term interpolated between ${\displaystyle 1}$  and ${\displaystyle {\frac {2}{3}}}$  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
${\displaystyle \pi =2{\frac {2\cdot 2\cdot 4\cdot 4\cdot 6\cdot 6\cdot 8\cdot 8...}{1\cdot 3\cdot 3\cdot 5\cdot 5\cdot 7\cdot 7\cdot 9...}}}$
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.
• In 1659 Wallis published a tract on cycloids in which incidentally he explained how the principles laid down in his Arithmetica infinitorum could be applied to the rectification of algebraic curves: and in the following year one of his pupils, by name William Neil, applied the rule to rectify the semicubical parabola ${\displaystyle x^{3}=ay^{2}}$ . 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.
• 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.
• 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.
• In 1686 Wallis published an Algebra, preceded by a historical account of the development of the subject which contains a great deal of valuable information... This algebra is noteworthy as containing the first systematic use of formulae.
• A particle moving with a uniform velocity would be denoted by Wallis by the formula s = vt, ...while previous writers would have denoted the same relation by stating what is equivalent to the proposition s1 : s2 = v1t1 : v2t2 (see e.g. Newton's Principia, bk. I. sect. I., lemma 10 or 11).
• Wallis rejected as absurd and inconceivable the now usual idea of a negative number as being less than nothing, but accepted the view that it is something greater than infinity.
• The writings of Wallis published between 1655 and 1665 revealed and explained to all students the principles of those new methods which distinguish modern from classical mathematics. His reputation has been somewhat overshadowed by that of Newton, but his work was absolutely first class in quality. Under his influence a brilliant mathematical school arose at Oxford. In particular I may mention Wren, Hooke, and Halley as among the most eminent of his pupils. But the movement was shortlived, and there were no successors of equal ability to take up their work.

### "Squaring the Circle" A History of the Problem (1913)

E. W. Hobson, source, taken from Ch.3 "The Second Period."
• The foundations of the new analysis were laid in the second half of the seventeenth century when Newton... and Leibnitz... founded the Differential and Integral Calculus, the ground having been to some extent prepared by the labours of Huyghens, Fermat, Wallis, and others. By this great invention of Newton and Leibnitz, and with the help of the brothers James Bernoulli... and John Bernoulli... the ideas and methods of the Mathematicians underwent a radical transformation which naturally had a profound effect upon our problem. The first effect of the new analysis was to replace the old geometrical or semi-geometrical methods of calculating ${\displaystyle \pi }$  by others in which analytical expressions formed according to definite laws were used, and which could be employed for the calculation of ${\displaystyle \pi }$  to any assigned degree of approximation.
• The first result of this kind was due to John Wallis... undergraduate at Emmanuel College, Fellow of Queen's College, and afterwards Savilian Professor of Geometry at Oxford. He was the first to formulate the modern arithmetic theory of limits, the fundamental importance of which, however, has only during the last half century received its due recognition; it is now regarded as lying at the very foundation of analysis. Wallis gave in his Arithmetica Infinitorum the expression
${\displaystyle {\frac {\pi }{2}}={\frac {2}{1}}\cdot {\frac {2}{3}}\cdot {\frac {4}{3}}\cdot {\frac {4}{5}}\cdot {\frac {6}{5}}\cdot {\frac {6}{7}}\cdot {\frac {8}{7}}\cdot {\frac {8}{9}}\cdots }$
for ${\displaystyle \pi }$  as an infinite product, and he shewed that the approximation obtained at stopping at any fraction in the expression on the right is in defect or in excess of the value ${\displaystyle {\frac {\pi }{2}}}$  according as the fraction is proper or improper. This expression was obtained by an ingenious method depending on the expression for ${\displaystyle {\frac {\pi }{8}}}$  the area of a semi-circle of diameter 1 as the definite integral ${\displaystyle \int \limits _{0}^{1}{\sqrt {x-x^{2}}}dx}$ . The expression has the advantage over that of Vieta that the operations required are all rational ones.
• Lord Brounckner... communicated without proof to Wallis the [continued fraction] expression
${\displaystyle {\frac {4}{\pi }}=1+{\frac {1}{2+}}{\frac {9}{2+}}{\frac {25}{2+}}{\frac {49}{2+}}\cdots }$ ,
a proof of which was given by Wallis in his Arithmetica Infinitorum. It was afterwards shewn by Euler that Wallis' formula could be obtained from the development of the sine and cosine in infinite products, and that Brounckner's expression is a particular case of much more general theorems.

### History of Mathematics (1923)

David Eugene Smith, Vol.1
• Of the contemporaries of Newton one of the most prominent was John Wallis. ...Wallis was a voluminous writer, and not only are his writings erudite, but they show a genius in mathematics... He was one of the first to recognize the significance of the generalization of exponents to include negative and fractional as well as positive and integral numbers. He recognized also the importance of Cavalieri's method of indivisibles, and employed it in the quadrature of such curves as y=xn, y=x1/n, and y=x0 + x1 + x2 +... He failed in his attempts at the approximate quadrature of the circle by means of series because he was not in possession of the general form of the binomial theorem. He reached the result, however, by another method. He also obtained the equivalent of ${\displaystyle ds=\!dx{\sqrt {1+({\frac {dy}{dx}})^{2}}}}$  for the length of an element of a curve, thus connecting the problem of rectification with that of quadrature.
• In 1673 he wrote his great work De Algebra Tractatus; Historicus & Practicus, of which an English edition appeared in 1685. In this there is seen the first serious attempt in England to write on the history of mathematics, and the result shows a wide range of reading of classical literature of the science. This work is also noteworthy because it contains the first of an effort to represent the imaginary number graphically by the method now used. The effort stopped short of success but was an ingenious beginning.
• Wallis was in sympathy with Greek mathematics and astronomy, editing parts of the works of Archimedes, Eutocius, Ptolemy, and Aristarchus; but at the same time he recognized the fact that the analytic method was to replace the synthetic, as when he defined a conic as a curve of the second degree instead of as a section of a cone, and treated it by the aid of coordinates.
• His writings include works on mechanics, sound, astronomy, the tides, the laws of motion, the Torricellian tube, botany, physiology, music, the calendar (in opposition to the Gregorian reform), geology, and the compass,—a range too wide to allow of the greatest success in any of the lines of his activity. He was also an ingenious cryptologist and assisted the government in deciphering diplomatic messages.
• Among his interesting discoveries was the relation
${\displaystyle {\frac {4}{\pi }}={\frac {3}{2}}\cdot {\frac {3}{4}}\cdot {\frac {5}{4}}\cdot {\frac {5}{6}}\cdot {\frac {7}{6}}\cdot {\frac {7}{8}}\cdots }$
one of the early values of π involving infinite products.
• Footnote: see his Opera Mathematica, I, 441