Search Results for Wallis

Biographies

  1. Wallis biography
    • John Wallis .
    • John Wallis's father was the Reverend John Wallis who had become a minister in Ashford in 1602.
    • The Reverend Wallis married Joanna Chapman, who was his second wife, in 1612 and John was the third of their five children.
    • Writing in his autobiography, Wallis comments [Notes and Records Roy.
    • He also studied logic at this school but mathematics was not considered important in the best schools of the time, so Wallis did not come in contact with that topic at school.
    • It was during the 1631 Christmas holidays that Wallis first came in contact with mathematics when his brother taught him the rules of arithmetic.
    • Wallis found that mathematics [Notes and Records Roy.
    • In 1637 Wallis received his BA and continued his studies receiving his Master's Degree in 1640.
    • It was during this time that the first of two events which shaped Wallis's future took place:- .
    • one evening at supper, a letter in cipher was brought in, relating to the capture of Chichester on 27 December 1642, which Wallis in two hours succeeded in deciphering.
    • This was the time of the Civil War between the Royalists and Parliamentarians and Wallis used his skills in cryptography in decoding Royalist messages for the Parliamentarians.
    • In this same year his mother died and this left Wallis as a man of independent means since he inherited a major estate in Kent.
    • In 1644 Wallis became secretary to the clergy at Westminster and through this he was given a fellowship at Queen's College, Cambridge.
    • Wallis wrote:- .
    • In this passage we have modernised Wallis's English a little to make it more easily understood.
    • We talked above about two events which shaped Wallis's future, the first being cryptography.
    • Wallis wrote a book Treatise of Angular Sections which remained unpublished for forty years.
    • He also discovered methods of solving equations of degree four which were similar to those which Harriot had found but Wallis claimed that he made the discoveries himself, not being aware of Harriot's contributions until later.
    • Cromwell held Wallis in high regard, not just for his political views but also for his scholarship.
    • Wallis held the Savilian Chair for over 50 years until his death and, even if he was appointed for the wrong reasons, he most certainly deserved to hold the chair.
    • This was not the only position which Wallis would hold at Oxford.
    • Certainly Wallis's opponents believed that he became keeper of the University archives because of his support for Cromwell.
    • Even if this were the case, as with the Savilian Chair, Wallis carried out his duties extremely well and fully deserved the post.
    • Although Wallis was a Parliamentarian he certainly spoke out against the execution of Charles I and, in 1648, had signed a document opposing the execution.
    • This was done in good faith for although Wallis used his undoubted political skills to gain what wanted at times, there was never any suggestion that he was anything other than an honest man.
    • Wallis, however, gained by signing the petition against the King's execution for, in 1660 when the monarchy was restored and Charles II came to the throne, Wallis had his appointment in the Savilian Chair confirmed by the King.
    • Charles II went even further for he appointed Wallis as a royal chaplain and, in 1661, nominated him as a member of a committee set up to revise the prayer book.
    • Wallis contributed substantially to the origins of calculus and was the most influential English mathematician before Newton.
    • Wallis's most famous work was Arithmetica infinitorum which he published in 1656.
    • In this work Wallis established the formula .
    • Wallis discovered this result when he was attempting to compute the integral of (1 - x2)½ from 0 to 1 and hence to find the area of a circle of unit radius.
    • About the beginning of my mathematical studies, as soon as the works of our celebrated countryman, Dr Wallis, fell into my hands, by considering the Series, by the Intercalation of which, he exhibits the Area of the Circle and the Hyperbola..
    • In his Tract on Conic Sections (1655) Wallis described the curves that are obtained as cross sections by cutting a cone with a plane as properties of algebraic coordinates:- .
    • Wallis developed methods in the style of Descartes analytical treatment and he was the first English mathematician to use these new techniques.
    • This work is also famed for the first use of the symbol ∞ which was chosen by Wallis to represent a curve which one could traced out infinitely many times.
    • Wallis was also an important early historian of mathematics and in his Treatise on Algebra he gives a wealth of valuable historical material.
    • In Treatise on Algebra Wallis accepts negative roots and complex roots.
    • One highly controversial section in this work is one in which Wallis claims that Descartes' knowledge of algebra was gained directly from Harriot.
    • Wallis received criticism for these claims immediately the book was published, but the subject is still of interest to historians of mathematics today.
    • The claims made by Wallis on this topic have never been shown false to everyone's complete satisfaction.
    • Wallis made other contributions to the history of mathematics by restoring some ancient Greek texts such as Ptolemy's Harmonics, Aristarchus's On the magnitudes and distances of the sun and moon and Archimedes' Sand-reckoner.
    • Wallis became involved in a bitter dispute with Hobbes, who although a fine scholar, was far below Wallis's class as a mathematician.
    • Wallis's book Arithmetica infinitorum with his methods was in press at the time and he refuted Hobbes claims.
    • of Wallis with the pamphlet Six lessons to the Professors of Mathematics at the Institute of Sir Henry Savile.
    • Wallis replied with the pamphlet Due Correction for Mr Hobbes, or School Discipline for not saying his Lessons Aright to which Hobbes wrote the pamphlet The Marks of the Absurd Geometry, Rural Language etc.
    • of Doctor Wallis.
    • Wallis replied:- .
    • One aspect of Wallis's mathematical skills has not yet been mentioned, namely his great ability to do mental calculations.
    • It was a feat which was rightly considered remarkable, and Oldenburg, the Secretary of the Royal Society, sent a colleague to investigate how Wallis did it.
    • Hearne, writing of Wallis in 1885, describes him a follows:- .
    • Honours awarded to John Wallis .
    • http://www-history.mcs.st-andrews.ac.uk/Biographies/Wallis.html .

  2. Hobbes biography
    • The new method of indivisibles, as put forward by Cavalieri, was accepted by Hobbes but he rejected Wallis's version as given in Arithmetica infinitorum.
    • Jesseph writes of Hobbes' attempt to square the circle [Squaring the circle : The war between Hobbes and Wallis (Chicago, 1999).',5)">5]:- .
    • This was a phrase that Wallis would pour scorn on when he attacked Hobbes' ideas.
    • Wallis attacked the whole of Hobbes' mathematical work of De Corpore and a vigorous argument between the two arose which lasted for 25 years.
    • To Hobbes mathematics was geometry and only geometry, and Wallis's Algebra he described as:- .
    • Hobbes responded to the attack by Wallis and others of De Corpore by publishing Six Lessons to the Professors of Mathematics in the University of Oxford in 1656.
    • Wallis replied with telling mathematical arguments, but also with unfair charges of disloyalty.
    • Hobbes could win arguments when his morality was attacked, but when it came to mathematics Wallis had a clear upper hand understanding mathematics far more deeply than Hobbes.
    • Hobbes writes about himself in the third person (see for example [Squaring the circle : The war between Hobbes and Wallis (Chicago, 1999).',5)">5]):- .

  3. Brouncker biography
    • It is doubtful whether Brouncker learned more than arithmetic at Oxford, for Wallis, giving the status of mathematics at this time, wrote:- .
    • He had to keep out of the limelight to avoid paying for his Royalist views so he worked away corresponding with Wallis and solving some difficult mathematical problems which we look at below.
    • Through Wallis, and others with whom he was corresponding, he became involved with a group of scientists who met in Gresham College London.
    • This result, written up in around ten pages, was added by Wallis to his treatise Arithmetica Infinitorum and probably first discovered by Brouncker in 1654.
    • Wallis told Huygens of this result and Huygens expressed strong doubts that it was true.
    • It appeared in a paper published by Brouncker in the Philosophical Transactions of the Royal Society of 1668 but he clearly states that this result is the one referred to by Wallis in 1665.
    • In 1659 Brouncker's improvement of Neile's computation of the arc length of the semicubical parabola ay2 = x3 appeared in Wallis's work De Cycloide et de Corporibus inde Genitis.

  4. Kruskal William biography
    • Allen Wallis, the founding Chairman of the University of Chicago's Statistics Department, offered Kruskal the position of instructor at Chicago with the agreement that he would become an assistant professor once he was awarded his doctorate.
    • At Chicago he began a joint research project with Wallis, while Scheffe had left Columbia University and Howard Levene took over as Kruskal's advisor.
    • In 1952 Kruskal published two important papers both of which came out of his work with Wallis.
    • The paper A nonparametric test for the several sample problem was authored by Kruskal alone while the second Use of ranks in one-criterion analysis of variance was a joint publication with Wallis.
    • It was in two parts, the first being on robustness along the lines he had been working on at Columbia, the second part on the work he had undertaken at Chicago with Wallis.
    • [The Kruskal-Wallis] test is found today under that name as part of every major statistical computation system.
    • He appointed Allen Wallis to head the Commission and appointed Kruskal and others such as Tukey as members.

  5. Pascal biography
    • Although Pascal was not the first to study the Pascal triangle, his work on the topic in Treatise on the Arithmetical Triangle was the most important on this topic and, through the work of Wallis, Pascal's work on the binomial coefficients was to lead Newton to his discovery of the general binomial theorem for fractional and negative powers.
    • Pascal published a challenge offering two prizes for solutions to these problems to Wren, Laloubere, Leibniz, Huygens, Wallis, Fermat and several other mathematicians.
    • Wallis and Laloubere entered the competition but Laloubere's solution was wrong and Wallis was also not successful.

  6. Newton biography
    • Newton also studied Wallis's Algebra and it appears that his first original mathematical work came from his study of this text.
    • He read Wallis's method for finding a square of equal area to a parabola and a hyperbola which used indivisibles.
    • Newton made notes on Wallis's treatment of series but also devised his own proofs of the theorems writing:- .
    • Thus Wallis doth it, but it may be done thus ..

  7. Halley biography
    • Halley was appointed Savilian professor of geometry at Oxford in 1704 following the death of Wallis.
    • Dr Wallis is dead - Mr Halley expects his place - who now talks, swears and drinks brandy like a sea captain.
    • Of those of our English nation he spoke in particular of Sir Henry Savile; but his greatest encomiums were upon Dr Wallis and Mr Newton ..

  8. Wren biography
    • Newton, never one to give excessive praise to others, states in the Principia that he ranks Wren together with Wallis and Huygens as the leading mathematicians of the day.
    • Wren's mathematical work now exists, if at all, in detached fragments rescued from oblivion, some in print, and a little more in bare outline in the published work of contemporaries, especially Wallis.
    • Work on the logarithmic spiral, which had been rectified by Wallis in the late 1650s, led Wren to note that it was possible to consider an area preserving transformation which would transform a cone into a solid logarithmic spiral.

  9. Wilkins biography
    • Wallis writes that the beginnings of that Society were:- .
    • about the year 1645 (if not sooner) when Dr Wilkins (then Chaplain to the Prince Elector Palatine in London) [and a number of other men including Wallis himself] met weekly on a certain day and hour, under a certain penalty, and a weekly contribution for the charge of experiments, with certain rules agreed amongst us, to treat and discourse of such affairs..
    • Returning to the account given by Wallis of the founding of the Royal Society:- .

  10. Descartes biography
    • Wallis in Algebra (1685) strongly argues that the ideas of La Geometrie were copied from Harriot.
    • Wallis writes:- .
    • There seems little to justify Wallis's claim, which was probably made partly through patriotism but also through his just desires to give Harriot more credit for his work.

  11. Hevelius Johannes biography
    • Hevelius also corresponded with most of the leading astronomers including Wallis, Flamsteed and Halley in England, and Gassendi and Boulliau in France.
    • Wallis, who took the opportunity to have a go at Hooke with whom he had fallen out, reviewed Hevelius's letter in the Philosophical Transactions of the Royal Society of 1685.
    • I hear Dr Wallis has taken up the cudgels and vindicated Hevelius against Hooke.

  12. Huygens biography
    • He was greatly impressed with Wallis and the other English scientists whom he met and, from this time on, he was to continue his contacts with this group.
    • Wallis and Wren also answered this question.

  13. Guarini biography
    • (Aristotelis loca mathematicis, Bolgna, 1615), and were espoused by John Wallis, the best English mathematician of the day, when he stated that beyond the equalities of quantity remained the similitude between qualities, i.
    • Guarini referred to Wallis among the few authorities for his mathematics - both were involved in summation to infinity.

  14. Mengoli biography
    • His books were nevertheless widely distributed in the seventeenth century, and were known to Collins, Wallis, and Leibniz; they were then almost forgotten, so that Mengoli's work has been studied again only recently.
    • Both of these were influenced by his contribution, in the case of Leibniz the influence was direct as he read Mengoli's work while in the case of Newton he knew of it indirectly through studying Wallis.

  15. Hooke biography
    • He began to study at Oxford at a particularly significant time for Thomas Willis, Seth Ward, Robert Boyle, John Wilkins, John Wallis, Christopher Wren and William Petty were among those who regularly met as the "Oxford branch" of the "invisible college" or the "philosophical college" which had been set up in 1648-49 when some of the scientists meeting in London moved to Oxford.
    • And particularly that of the oval figure of the Earth which was read by me to this Society about 27 years since upon the occasion of the carrying the pendulum clocks to sea and at two other times since, though I have had the ill fortune not to be heard, and I conceive there are some present that may very well remember and do know that Mr Newton did not send up that addition to his book till some weeks after I had read and showed the experiments and demonstration thereof in this place and had answered the reproachful letter of Dr Wallis from Oxford.

  16. Collins biography
    • He corresponded with Barrow, David Gregory, James Gregory, Newton, Wallis, Borelli, Huygens, Leibniz, Tschirnhaus and Sluze.
    • Collins published books by Barrow and Wallis and left a collection of 2000 books and an uncounted number of manuscripts.

  17. Boyle biography
    • At Oxford he joined a group of forward looking scientists, including John Wilkins, John Wallis who was the Savilian Professor of Geometry, Seth Ward who was the Savilian Professor of Astronomy, and Christopher Wren who would succeed Ward as Savilian Professor of Astronomy in 1661.
    • Wallis found someone whom he considered particularly suitable to be Boyle's wife and wrote to him saying:- .

  18. Cocker biography
    • Certainly he did not remain long at his new address since he is mentioned in letters of Collins to Wallis in 1666 and 1667 and by that time he was living in Northampton.
    • As Wallis writes in [Ann.

  19. Heuraet biography
    • He seemed successful in this but then Huygens wrote to Wallis in June 1659 making a claim which historians believe is simply untrue.
    • Wallis, who championed Neile's cause, strongly disagreed.

  20. Bernoulli Jacob biography
    • Jacob Bernoulli also studied the work of Wallis and Barrow and through these he became interested in infinitesimal geometry.

  21. Jones biography
    • Wallis writes in [Dictionary of National Biography (Oxford, 2004).',2)">2]:- .

  22. Boulliau biography
    • Then in 1682 he published Opus novum ad arithmeticam infinitorum which he claimed clarified the Arithmetica infinitorum of Wallis.

  23. Sluze biography
    • He corresponded with many mathematicians in England, France and other European countries, for example he was in regular contact with Blaise Pascal, Christiaan Huygens, and John Wallis.
    • Oldenburg transmitted mathematical details to de Sluze based on the information given him by Wallis, Brouncker and Wren concerning their latest research [Henry Oldenburg: Shaping the Royal Society (Oxford University Press, Oxford, 2002).',3)">3]:- .

  24. Kutta biography
    • He wrote a paper on Wallis's 1659 work on integration and the length of an ellipse.

  25. De Witt biography
    • Let us end by quoting the praise that Huygens gave to de Witt in a letter written to Wallis 6 June 1659:- .

  26. Neile biography
    • Neile's work on this appeared in Wallis's De Cycloide in 1659.

  27. Le Paige biography
    • He published Sluze's correspondence with Pascal, Huygens, Oldenburg and Wallis.

  28. Dechales biography
    • As Moritz Cantor points out in [Voresungen uber Geschichte der Mathematik III (Leipzig, 1913), 4-6, 15-19.',3)">3] Dechales rarely mentions the work of Mydorge, Desargues, Pascal, Fermat, Descartes, or Wallis [Dictionary of Scientific Biography (New York 1970-1990).',1)">1]:- .

  29. Beltrami biography
    • He compared Saccheri's results with those of Borelli, Wallis, Clavius and the non-euclidean geometry of Lobachevsky and Bolyai.

  30. Varignon biography
    • He gave a new unified treatment of particular cases which had already been studied by Wallis, Huygens, Leibniz and Newton.

  31. Donaldson biography
    • After returning to Oxford he was appointed Wallis Professor of Mathematics in 1985, before moving to Imperial College, London in 1999.

  32. Bougainville biography
    • Continuing he landed on Tahiti which he found had been discovered eight months earlier in 1767 by the Englishman Samuel Wallis.

  33. Gregory David biography
    • David Gregory certainly supported Newton strongly in the Newton - Leibniz controversy arguing, as did Gregory's friend Wallis, that Leibniz had learnt of the calculus through a letter from Collins.

  34. Fermat biography
    • The second of the two problems, namely to find all solutions of Nx2 + 1 = y2 for N not a square, was however solved by Wallis and Brouncker and they developed continued fractions in their solution.

  35. Euler biography
    • They include works by Varignon, Descartes, Newton, Galileo, van Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis.

  36. Montroll biography
    • A A Maradudin, K E Shuler and R F Wallis write in a tribute to Montroll by the University of California:- .

  37. Tschirnhaus biography
    • He also met John Collins in London and John Wallis in Oxford.
    • He showed Collins and Wallis his methods for solving equations, but these turned out to be special cases of known results.

  38. Ward Seth biography
    • At that time Oxford was the home of many illustrious men of science, among whom may be mentioned John Wilkins, the Warden of Wadham; Robert Boyle; Thomas Willis; Jonathan Coddard; and John Wallis.

  39. Oughtred biography
    • He had many pupils but the most famous were John Wallis, Christopher Wren and Richard Delamain.

  40. Leibniz biography
    • Leibniz demanded a retraction saying that he had never heard of the calculus of fluxions until he had read the works of Wallis.

History Topics

  1. Pell's equation
    • Several mathematicians participated in Fermat's challenge, in particular Frenicle de Bessy, Brouncker and Wallis.
    • There followed an exchange of letters between these mathematicians during 1657-58 which Wallis published in Commercium epistolicum in 1658.
    • In Commercium epistolicum Wallis gave two methods of proving Brahmagupta's lemma which are both essentially equivalent to the argument we gave at the beginning of this article based on the result .
    • Wallis published Treatise on Algebra in 1685 and Chapter 98 of that work is devoted to giving methods to solve Pell's equation based on the exchange of letters he had published in Commercium epistolicum in 1658.
    • However, in his algebra text Wallis put all the methods into a standard form.
    • Wallis, describing Brouncker's method, had made that claim, as had Fermat when commenting on the solutions proposed to his challenge.
    • He was, of course, aware of the work of Brouncker on Pell's equation as presented by Wallis, but he was totally unaware of the contributions of the Indian mathematicians.
    • The other major contribution of Euler was in naming the equation "Pell's equation" and it is generally believed that he gave it that name because he confused Brouncker and Pell, thinking that the major contributions which Wallis had reported on as due to Brouncker were in fact the work of Pell.

  2. Trigonometric functions
    • The notation Si.2 was used by Cavalieri, s co arc by Oughtred and S by Wallis.
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    • Cavalieri used Ta and Ta.2, Oughtred used t arc and t co arc while Wallis used T and t.
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    • Cavalieri used Se and Se.2, Oughtred used se arc and sec co arc while Wallis used s and σ.

  3. Mental arithmetic
    • First we mention John Wallis whose calculating powers are described in [Mathematical Recreations and Essays (London, 1940).',2)" onmouseover="window.status='Click to see reference';return true">2]:- .
    • [Wallis] occupied himself in finding (mentally) the integral part of the square root of 3 cross 1040; and several hours afterwards wrote down the result from memory.
    • However, in one respect Wallis is very different from others we describe in that he was 53 years old when he performed the above feats.

  4. Real numbers 2
    • If we move forward almost exactly 100 years to the publication of A treatise of Algebra by Wallis in 1684 we find that he accepts, without any great enthusiasm, the use of Stevin's decimals.
    • However, Wallis understood that there were proportions which did not fall within this definition of number, such as those associated with the area and circumference of a circle:- .
    • For Wallis there were a variety of ways that one might achieve this approximation, so coming as close as one pleased.

  5. Pell's equation references
    • A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
    • A A Antropov, Wallis' method of "approximations" as applied to the solution of the equation xÛ - nyÛ = 1 in integers (Russian), Istor.-Mat.

  6. Pell's equation references
    • A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
    • A A Antropov, Wallis' method of "approximations" as applied to the solution of the equation xÛ - nyÛ = 1 in integers (Russian), Istor.-Mat.

  7. Elliptic functions

  8. Longitude2
    • In 1662 the group from Gresham College, which included John Wilkins, John Wallis and Robert Hooke, and other groups of scientists, became the Royal Society of London for the Promotion of Natural Knowledge.
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  9. Pi history

  10. Infinity
    • The symboln ∞ n which we use for infinity today, was first used by John Wallis who used it in De sectionibus conicis in 1655 and again in Arithmetica infinitorum in 1656.

  11. Infinity references
    • S Probst, Infinity and creation : the origin of the controversy between Thomas Hobbes and the Savilian professors Seth Ward and John Wallis, British J.

  12. Infinity references
    • S Probst, Infinity and creation : the origin of the controversy between Thomas Hobbes and the Savilian professors Seth Ward and John Wallis, British J.

  13. Non-Euclidean geometry
    • One such 'proof' was given by Wallis in 1663 when he thought he had deduced the fifth postulate, but he had actually shown it to be equivalent to:- .
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Famous Curves

  1. Cycloid
    • Wallis and Lalouere entered but Lalouere's solution was wrong and Wallis was also not successful.

  2. Neiles
    • Wallis published the method in 1659 giving Neile the credit.
    • He was a pupil of Wallis and showed great promise.

  3. Cissoid
    • Huygens and Wallis found, in 1658, that the area between the curve and its asymptote was 3πa2.

Societies etc

  1. London Royal Society
    • The first group of such men included Robert Moray, Robert Boyle, John Wilkins, John Wallis, John Evelyn, Christopher Wren and William Petty.
    • We are particularly lucky to have a description of the beginnings of the Society from John Wallis (see for example [7]):- .

  2. London Royal Society
    • The first group of such men included Robert Moray, Robert Boyle, John Wilkins, John Wallis, John Evelyn, Christopher Wren and William Petty.
    • We are particularly lucky to have a description of the beginnings of the Society from John Wallis (see for example [7]):- .

  3. History of the Royal Society
    • The first group of such men included Robert Boyle, John Wilkins, John Wallis, John Evelyn, Robert Hooke, Christopher Wren and William Petty.
    • We are particularly lucky to have a description of the beginnings of the Society from John Wallis:- .

  4. Aubrey's Brief Lives

  5. Wilks Award of the ASS
    • 1980 W Allen Wallis .

  6. Savilian Chairs
    • 1649 John Wallis .

  7. Fellow of the Royal Society
    • John Wallis 1663 .

References

  1. References for Wallis
    • References for John Wallis .
    • J F Scott, The Mathematical Work of John Wallis (1616-1703) (London, 1938, New York, 1981).
    • C J Scriba, Studien zur Mathematik des John Wallis (1616-1703) (Wiesbaden, 1966).
    • A A Antropov, Two methods for the solution of Pell's equation in the work of J Wallis (Russian), Istor.
    • A A Antropov, Wallis' method of 'approximations' as applied to the solution of the equation xÛ - nyÛ = 1 in integers (Russian), Istor.-Mat.
    • M H Bektasova, Wallis' 'Algebra' (Russian), in Collection of questions on mathematics and mechanics (Russian) 8 (Alma-Ata, 1976), 3-17, 226.
    • L I Cerkalova, Composite ratios in Wallis (Russian), Jaroslav.
    • D Dennis and J Confrey, The creation of continuous exponents : a study of the methods and epistemology of John Wallis, in Research in collegiate mathematics education II (Providence, RI, 1996), 33-60.
    • J Dutka, Wallis's product, Brouncker's continued fraction, and Leibniz's series, Arch.
    • D H Fowler, An approximation technique, and its use by Wallis and Taylor, Arch.
    • I A Golovinskii, Interpolation of sequences in the work of Wallis and Euler (Russian), in History and methodology of the natural sciences XX (Russian) (Moscow, 1978), 62-68.
    • K Hara, Pascal et Wallis au sujet de la cycloide, Ann.
    • K Hill, Neither ancient nor modern : Wallis and Barrow on the composition of continua.
    • K Hill, Neither ancient nor modern : Wallis and Barrow on the composition of continua.
    • J E Hofmann, Leibniz und Wallis, Studia Leibnitiana 5 (1973), 245-281.
    • F D Kramar, The origins of vector algebra in Wallis' mechanics (Russian), Voprosy Istor.
    • Wallis (Russian), Istor.-Mat.
    • F D Kramar, Questions of the foundations of analysis in the works of Wallis and Newton (Russian), Trudy Sem.
    • L Maieru, John Wallis : a reading of the polemics between Peletier and Clavius concerning the angle of contact (Italian), in Conference on the History of Mathematics (Rende, 1991), 315-364.
    • A Malet, Barrow, Wallis, and the remaking of seventeenth century indivisibles, Centaurus 39 (1) (1997), 67-92.
    • T Murata, Un traite heuristique japonais contemporain de Wallis et de Newton, Comment.
    • T Murata, Wallis' 'Arithmetica infinitorum' and Takebe's 'Tetsujutsu sankei' : what underlies their similarities and dissimilarities?, Historia Sci.
    • T F Nikonova, The first attempt to compile a history of algebra by the English mathematician John Wallis (Russian), Moskov.
    • S Probst, Infinity and creation : the origin of the controversy between Thomas Hobbes and the Savilian professors Seth Ward and John Wallis, British J.
    • H Pycior, Mathematics and philosophy : Wallis, Hobbes, Barrow and Berkeley, Journal of the History of Ideas 48 (1987), 265-287.
    • C J Scriba, The autobiograhy of John Wallis, F.R.S., Notes and Records Roy.
    • C J Scriba, A tentative index of the correspondence of John Wallis, F.R.S., Notes and Records Roy.
    • C J Scriba, Wallis und Harriot, Centaurus 10 (1965), 248-257.
    • J F Scott, The Reverend John Wallis, F.R.S., Notes and Records Royal Society of London 15 (1960-61), 57-68.
    • T A Tokareva, The 'Treatise of algebra both historical and practical' of John Wallis (Russian), Istor.-Mat.
    • G U Yule, John Wallis 1616-1703 , Notes and Records Royal Society of London 2 (1939).
    • http://www-history.mcs.st-andrews.ac.uk/References/Wallis.html .

  2. References for Barrow
    • K Hill, Neither ancient nor modern : Wallis and Barrow on the composition of continua.
    • K Hill, Neither ancient nor modern : Wallis and Barrow on the composition of continua.
    • A Malet, Barrow, Wallis, and the remaking of seventeenth century indivisibles, Centaurus 39 (1) (1997), 67-92.
    • H Pycior, Mathematics and philosophy : Wallis, Hobbes, Barrow and Berkeley, Journal of the History of Ideas 48 (1987), 265-287.

  3. References for Hobbes
    • D M Jesseph, Squaring the circle : The war between Hobbes and Wallis (Chicago, 1999).
    • F Cajori, Controversies between Wallis, Hobbes, and Barrow, Math.
    • S Probst, Infinity and creation : the origin of the controversy between Thomas Hobbes and the Savilian professors Seth Ward and John Wallis, British J.
    • H Pycior, Mathematics and philosophy : Wallis, Hobbes, Barrow and Berkeley, Journal of the History of Ideas 48 (1987), 265-287.

  4. References for Pell
    • P J Wallis, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    • P J Wallis, An early mathematical manifesto - John Pell's 'Idea of Mathematics', Durham Research Review 18 (1967), 139-148.

  5. References for Brouncker
    • J Dutka, Wallis's product, Brouncker's continued fraction, and Leibniz's series, Arch.

  6. References for Stirling
    • P J Wallis, Biography in Dictionary of Scientific Biography (New York 1970-1990).

  7. References for Green
    • P J Wallis, Biography in Dictionary of Scientific Biography (New York 1970-1990).

  8. References for Newton
    • F D Kramar, Questions of the foundations of analysis in the works of Wallis and Newton (Russian), Trudy Sem.

  9. References for Cocker
    • R Wallis, Edward Cocker (1632?-1676) and his arithmetick: De Morgan demolished, Ann.

  10. References for Szekeres
    • J R Giles and J S Wallis, George Szekeres.

  11. References for Simpson
    • P J Wallis, Biography in Dictionary of Scientific Biography (New York 1970-1990).

  12. References for Ward Seth
    • Mathematics: John Wallis and Seth Ward; Newton, in The Cambridge History of English and American Literature VIII (XV.

  13. References for Harriot
    • C J Scriba, Wallis und Harriot, Centaurus 10 (1965), 248-257.

  14. References for Taylor
    • D H Fowler, An approximation technique, and its use by Wallis and Taylor, Arch.

  15. References for Jones
    • Biography by Ruth Wallis, in Dictionary of National Biography (Oxford, 2004).

  16. References for Berkeley
    • H Pycior, Mathematics and philosophy : Wallis, Hobbes, Barrow and Berkeley, Journal of the History of Ideas 48 (1987), 265-287.

  17. References for Pascal
    • K Hara, Pascal et Wallis au sujet de la cycloide, Ann.

  18. References for Clavius
    • L Maieru, John Wallis : a reading of the polemics between Peletier and Clavius concerning the angle of contact (Italian), in Conference on the History of Mathematics (Rende, 1991), 315-364.

  19. References for Leibniz
    • J E Hofmann, Leibniz und Wallis, Studia Leibnitiana 5 (1973), 245-281.

Additional material

  1. Mathematicians and Music 2.2
    • Returning to the beginning of the Harmonic period let us consider the musical writings which were issued in the seventeenth century by such mathematicians as Kepler, Wallis, Mersenne, Desargues, Descartes and Christian Huygens.
    • Markedly contrasted to Kepler in abilities and habits of thought was John Wallis, the notably able Savilian professor at Oxford University, where a brilliant mathematical school was developed under his direction.
    • The first of these is a Greek and Latin edition of Ptolemy's Harmony, and Porphyry's third century commentary on the same, with an extensive appendix by Wallis on ancient and modern music.
    • Among other writings of Wallis on acoustics and music may be mentioned four memoirs published in the Philosophical Transactions, and bearing the following titles: "On the trembling of consonant strings," "On the division of the monochord, or section of the musical canon," "On the imperfections of an organ," and "On the strange effects of music in former times." .
    • The Franciscan friar Marin Mersenne, Wallis's senior by nearly 30 years, is known to the general run of mathematicians through the numbers with which his name is associated and which arise in discussion of perfect numbers.

  2. Charles Bossut on Leibniz and Newton
    • Leibniz, justly feeling himself hurt by this priority of invention ascribed to Newton, and the consequence maliciously insinuated, answered with great moderation, that Facio no doubt spoke solely on his own authority; that he could not believe it was with Newton's approbation; that he would not enter into any dispute with that celebrated man, for whom he had the profoundest veneration, as he had shown on all occasions; that, when they when they had both coincided in some geometrical inventions, Newton himself had declared in his Principia that neither had borrowed anything from the other; that, when he published his differential calculus in 1884, he had been master of it about eight years; that about the same time, it was true, Newton had informed him, but without any explanation, of his knowing how to draw tangents by a general method, which was not impeded by irrational quantities; but that he could not judge whether this method were the differential calculus since Huygens, who at that time was unacquainted with this calculus, equally affirmed himself to be in possession of a method which had the same advantages; that the work of an English writer, in which the calculus was explained in a positive manner was the preface to Wallis's Algebra, not published till 1693; that, relying on all these circumstances, he appealed entirely to the testimony and candour of Newton, etc.
    • Newton, gifted by nature with superior intellect, was born at a time when Harriot, Wren, Wallis, Barrow, and others, had already rendered the mathematical sciences flourishing in England, enjoyed likewise the advantage of receiving lessons from Barrow in his early youth at Cambridge.
    • While they agree that the evolution of radicals into series is a considerable step made by Newton, they immediately perceive, without the assistance of any subsequent and conjectural light, that the methods of Fermat, Wallis, and Barrow, might have been employed to find the results concerning quadratures which Newton contents himself with enunciating; since, after the evolution of radicals, if there be any, nothing more is necessary but to sum up the monomial quantities.
    • His reasons are in substance, first, the Commercium epistolicum exhibits no vestige of Newton's having employed dotted letters to denote fluxions in the writings alleged; secondly, in the Principia, where the author had so frequently occasion for employing this calculus and giving it's algorithm, he has not done it; he proceeds everywhere by means of lines and figures without any determinate analysis, and simply in the manner of Huygens, Roberval, Cavalleri, etc,: thirdy, the dotted letters first began to appear in the third volume of Wallis's Works, several years after the differential calculus was everywhere known; fourthly, the true method of differencing differences, or of taking the fluxions of fluxions, was unknown to Newton, since even in his treatise on quadratures, not published till 1704, the rule he gives at the end for determining the fluxions of all orders by considering these fluxions as the terms of the power of a binomial formed of a variable quantity, and it's first fluxion, and treating the first fluxion as constant, is false except simply for the term which answers to the first fluxion: fifthly, at the same period of 1704 Newton was not versed in the integral calculus of differential equations which Leibniz and the two Bernoullis had already carried so far; otherwise he would not have failed to treat this part of the analysis of infinities, the most difficult, and at least as worthy of being promulgated and carried to perfection as the quadratures on which he enlarged so much.

  3. The Tercentenary of the birth of James Gregory
    • Huygens and Wallis attacked his arguments remorselessly, but Gregory never admitted defeat: he gave way, now here now there, and anticipating the fighting spirit of his later kinsman, Rob Roy, he inevitably reappeared undaunted, to renew the attack from some fresh quarter.

  4. Gibson History 2 - Mathematics in the schools
    • The low value thus placed on mathematics is not, however, peculiar to Scotland; John Wallis's Account of some Passages in his Own Life (quoted in Adamson's Short History of Education, p.

  5. Florian Cajori on William Oughtred
    • It was during the second half of the seventeenth century that Sir Isaac Newton, surrounded by a group of great men - Wallis, Hooke, Barrow, Halley, Cotes - carried on his epoch-making researches in mathematics, astronomy, and physics.

Quotations

  1. A quotation by Wallis
    • A quotation by John Wallis .
    • http://www-history.mcs.st-andrews.ac.uk/Quotations/Wallis.html .

  2. A quotation by Lie
    • Descartes, Cavalieri, Fermat and Wallis.

Chronology

  1. Mathematical Chronology
    • Wallis publishes Arithmetica infinitorum which uses interpolation methods to evaluate integrals.
    • Wallis publishes his Mechanica (Mechanics) which is a detailed mathematical study of mechanics.
    • Wallis publishes De Algebra Tractatus (Treatise of Algebra) which contains the first published account of Newton's binomial theorem.

  2. Chronology for 1650 to 1675
    • Wallis publishes Arithmetica infinitorum which uses interpolation methods to evaluate integrals.
    • Wallis publishes his Mechanica (Mechanics) which is a detailed mathematical study of mechanics.

  3. Chronology for 1675 to 1700
    • Wallis publishes De Algebra Tractatus (Treatise of Algebra) which contains the first published account of Newton's binomial theorem.

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JOC/BS August 2001