“John Wallis”的意思、由来-开放百科全书

John Wallis was born in Ashford, Kent,he was the third of five children of Reverend John Wallis and Joanna Chapman. He was initially educated at a school in Ashford but moved to James Movat's school in Tenterden in 1625 following an outbreak of plague. Wallis was first exposed to mathematics in 1631, at Felsted School (then known as Martin Holbeach's school in Felsted); he enjoyed maths, but his study was erratic, since "mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical" (Scriba 1970). At the school in Felsted, Wallis learned how to speak and write Latin. By this time, he also was proficient in other languages such French, Greek, and Hebrew.^[6] As it was intended he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge.^[7] While there, he kept an act on the doctrine of the circulation of the blood; that was said to have been the first occasion in Europe on which this theory was publicly maintained in a disputation. His interests, however, centred on mathematics. He received his Bachelor of Arts degree in 1637 and a Master's in 1640, afterwards entering the priesthood. From 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. He was elected to a fellowship at Queens' College, Cambridge in 1644, from which he had to resign following his marriage.

Throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. He rendered them great practical assistance in deciphering Royalist dispatches. The quality of cryptography at that time was mixed; despite the individual successes of mathematicians such as François Viète, the principles underlying cipher design and analysis were very poorly understood. Most ciphers were ad hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. Wallis realised that the latter were far more secure – even describing them as "unbreakable", though he was not confident enough in this assertion to encourage revealing cryptographic algorithms. He was also concerned about the use of ciphers by foreign powers, refusing, for example, Gottfried Leibniz's request of 1697 to teach Hanoverian students about cryptography.^[8]

Returning to London – he had been made chaplain at St Gabriel Fenchurch in 1643 – Wallis joined the group of scientists that was later to evolve into the Royal Society. He was finally able to indulge his mathematical interests, mastering William Oughtred's Clavis Mathematicae in a few weeks in 1647. He soon began to write his own treatises, dealing with a wide range of topics, which he continued for the rest of his life. Wallis wrote the first survey about mathematical concepts in England where he discussed the Hindu-Arabic system.^[9]

Wallis joined the moderate Presbyterians in signing the remonstrance against the execution of Charles I, by which he incurred the lasting hostility of the Independents. In spite of their opposition he was appointed in 1649 to the Savilian Chair of Geometry at Oxford University, where he lived until his death on 28 October 1703 (O.S.). In 1650, Wallis was ordained as a minister. After, he spent two years with Sir Richard Darley and Lady Vere as a private chaplain. In 1661, he was one of twelve Presbyterian representatives at the Savoy Conference.

Besides his mathematical works he wrote on theology, logic, English grammar and philosophy, and he was involved in devising a system for teaching a deaf boy to speak at Littlecote House.^[10] William Holder had earlier taught a deaf man, Alexander Popham, to speak "plainly and distinctly, and with a good and graceful tone".^[11] Wallis later claimed credit for this, leading Holder to accuse Wallis of "rifling his Neighbours, and adorning himself with their spoyls".^[12]

Wallis' appointment as Savilian Professor of Geometry at the Oxford University

Contributions to mathematics

Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. In his Opera Mathematica I (1695) he introduced the term "continued fraction".

Wallis rejected as absurd 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 argument that negative numbers are greater than infinity involves the quotient

and considering what happens as x approaches and then crosses the point x = 0 from the positive side.) Despite this he is generally credited as the originator of the idea of the number line, in which numbers are represented geometrically in a line with the negative numbers represented by lengths opposite in direction to lengths of positive numbers.^[14]

Analytical geometry

In 1655, Wallis published a treatise on conic sections in which they were defined analytically. This was the earliest book in which these curves are considered and defined as curves of the second degree. It helped to remove some of the perceived difficulty and obscurity of René Descartes' work on analytic geometry.

In the Treatise on the Conic Sections Wallis popularised the symbol ∞ for infinity. He wrote, “I suppose any plane (following the Geometry of Indivisibles of Cavalieri) to be made up of an infinite number of parallel lines, or as I would prefer, of an infinite number of parallelograms of the same altitude; (let the altitude of each one of these be an infinitely small part 1/∞ of the whole altitude, and let the symbol ∞ denote Infinity) and the altitude of all to make up the altitude of the figure.”^[15]

Integral calculus

Arithmetica Infinitorum, the most important of Wallis's works, was published in 1656. In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extended, but some ideas were open to criticism. He began, after a short tract on conic sections, by developing the standard notation for powers, extending them from positive integers to rational numbers:

Leaving the numerous algebraic applications of this discovery, he next proceeded to find, by integration, the area enclosed between the curve y = x^m, the axis of x, and any ordinate x = h, and he proved that the ratio of this area to that of the parallelogram on the same base and of the same height is 1/(m + 1), extending Cavalieri's quadrature formula. He apparently assumed that the same result would be true also for the curve y = ax^m, where a is any constant, and m any number positive or negative, but he discussed only the case of the parabola in which m = 2 and the hyperbola in which m = −1. In the latter case, his interpretation of the result is incorrect. He then showed that similar results may be written down for any curve of the form

and hence that, if the ordinate y of a curve can be expanded in powers of x, its area can be determined: thus he says that if the equation of the curve is y = x⁰ + x¹ + x² + ..., its area would be x + x²/2 + x³/3 + ... He then applied this to the quadrature of the curves y = (x − x²)⁰, y = (x − x²)¹, y = (x − x²)², etc., taken between the limits x = 0 and x = 1. He shows that the areas are, respectively, 1, 1/6, 1/30, 1/140, etc. He next considered curves of the form y = x^1/m and established the theorem that the area bounded by this curve and the lines x = 0 and x = 1 is equal to the area of the rectangle on the same base and of the same altitude as m : m + 1. This is equivalent to computing

He illustrated this by the parabola, in which case m = 2. He stated, but did not prove, the corresponding result for a curve of the form y = x^p/q.

Wallis showed considerable ingenuity in reducing the equations of curves to the forms given above, but, as he was unacquainted with the binomial theorem, he could not effect the quadrature of the circle, whose equation is

, since he was unable to expand this in powers of x. He laid down, however, the principle of interpolation. Thus, as the ordinate of the circle

is the geometrical mean of the ordinates of the curves

and

, it might be supposed that, as an approximation, the area of the semicircle

which is

might be taken as the geometrical mean of the values of

that is, 1 and

; this is equivalent to taking

or 3.26... as the value of π. But, Wallis argued, we have in fact a series

... and therefore the term interpolated between 1 and

ought to be chosen so as to obey the law of this series{{Clarify|date=July 2010}}. This, by an elaborate method that is not described here in detail, leads to a value for the interpolated term which is equivalent to taking

In this work also the formation and properties of continued fractions are discussed, the subject having been brought into prominence by Brouncker's use of these fractions.

A few years later, in 1659, Wallis published a tract containing the solution of the problems on the cycloid which had been proposed by Blaise Pascal. In this he incidentally explained how the principles laid down in his Arithmetica Infinitorum could be used for the rectification of algebraic curves and gave a solution of the problem to rectify (i.e., find the length of) the semicubical parabola x³ = ay², which had been discovered in 1657 by his pupil William Neile. Since all attempts to rectify the ellipse and hyperbola had been (necessarily) ineffectual, it had been supposed that no curves could be rectified, as indeed Descartes had definitely asserted to be the case. The logarithmic spiral had been rectified by Evangelista Torricelli and was the first curved line (other than the circle) whose length was determined, but the extension by Neile and Wallis to an algebraic curve was novel. The cycloid was the next curve rectified; this was done by Christopher Wren in 1658.

Early in 1658 a similar discovery, independent of that of Neile, was made by van Heuraët, and this was published by van Schooten in his edition of Descartes's Geometria in 1659. Van Heuraët's method is as follows. He supposes the curve to be referred to rectangular axes; if this be so, and if (x, y) be the coordinates of any point on it, and n be the length of the normal{{Clarify|date=July 2010}}, and if another point whose coordinates are (x, η) be taken such that η : h = n : y, where h is a constant; then, if ds be the element of the length of the required curve, we have by similar triangles ds : dx = n : y. Therefore, h ds = η dx. Hence, if the area of the locus of the point (x, η) can be found, the first curve can be rectified. In this way van Heuraët effected the rectification of the curve y³ = ax² but added that the rectification of the parabola y² = ax is impossible. since it requires the quadrature of the hyperbola. The solutions given by Neile and Wallis are somewhat similar to that given by van Heuraët, though no general rule is enunciated, and the analysis is clumsy. A third method was suggested by Fermat in 1660, but it is inelegant and laborious.

Collision of bodies

The theory of the collision of bodies was propounded by the Royal Society in 1668 for the consideration of mathematicians. Wallis, Christopher Wren, and Christian Huygens sent correct and similar solutions, all depending on what is now called the conservation of momentum; but, while Wren and Huygens confined their theory to perfectly elastic bodies (elastic collision), Wallis considered also imperfectly elastic bodies (inelastic collision). This was followed in 1669 by a work on statics (centres of gravity), and in 1670 by one on dynamics: these provide a convenient synopsis of what was then known on the subject.

Algebra

In 1685 Wallis published Algebra, preceded by a historical account of the development of the subject, which contains a great deal of valuable information. The second edition, issued in 1693 and forming the second volume of his Opera, was considerably enlarged. This algebra is noteworthy as containing the first systematic use of formulae. A given magnitude is here represented by the numerical ratio which it bears to the unit of the same kind of magnitude: thus, when Wallis wants to compare two lengths he regards each as containing so many units of length. This perhaps will be made clearer by noting that the relation between the space described in any time by a particle moving with a uniform velocity is denoted by Wallis by the formula

where s is the number representing the ratio of the space described to the unit of length; while the previous writers would have denoted the same relation by stating what is equivalent to the proposition

Geometry

He is usually credited with the proof of the Pythagorean theorem using similar triangles. However, Thabit Ibn Qurra (AD 901), an Arab mathematician, had produced a generalisation of the Pythagorean theorem applicable to all triangles six centuries earlier. It is a reasonable conjecture that Wallis was aware of Thabit's work.^[16]

Wallis was also inspired by the works of Islamic mathematician Sadr al-Tusi, the son of Nasir al-Din al-Tusi, particularly by al-Tusi's book written in AD 1298 on the parallel postulate. The book was based on his father's thoughts which presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. After reading this, Wallis then wrote about his ideas as he developed his own thoughts about the postulate, trying to prove it also with similar triangles.^[17]

He found that Euclid's fifth postulate is equivalent to the one currently named "Wallis postulate" after him. This postulate states that "On a given finite straight line it is always possible to construct a triangle similar to a given triangle". This result was encompassed in a trend trying to deduce Euclid's fifth from the other four postulates which today is known to be impossible. Unlike other authors, he realised that the unbounded growth of a triangle was not guaranteed by the four first postulates.^[18]

Calculator

Another aspect of Wallis's mathematical skills was his ability to do mental calculations. He slept badly and often did mental calculations as he lay awake in his bed. One night he calculated in his head the square root of a number with 53 digits. In the morning he dictated the 27-digit square root of the number, still entirely from memory. It was a feat that was considered remarkable, and Henry Oldenburg, the Secretary of the Royal Society, sent a colleague to investigate how Wallis did it. It was considered important enough to merit discussion in the Philosophical Transactions of the Royal Society of 1685.^[19]^[20]

Educational Background

Contributions with Newton

John Wallis was also considered an editor of Isaac Newton’s mathematical work. Wallis was always open to the idea of making algebraic methods in print form, however Isaac Newton was not. Newton believed that algebra was not “worthy” enough to be published in any kind of form. Wallis eventually convinced Newton to the idea, and Newton agreed to publish some of his algebraic methods in Wallis’s published books “Algebra”, and “Latin Opera”. Working with Newton, Wallis learned a lot from it which helped him come up with more advanced ideas later in the future.^[21] Newton and Wallis shared very distinct opinions in the midst of their works, Newton believed that math could reveal things about the natural world. He observed the works of Newton and there were times when plagiarism was an obstacle in their works because they both had very similar instances in their ideals.^[9]

Musical theory

Wallis translated into Latin works of Ptolemy, Bryennius, and Porphyrius's commentary on Ptolemy. He also published three letters to Henry Oldenburg concerning tuning. He approved of equal temperament that was being used in England's organs.^[22]

Other works

His Institutio logicae, published in 1687, was very popular.^[23] The Grammatica linguae Anglicanae was a work on English grammar, that remained in print well into the eighteenth century. He also published on theology.^[23]

Family

On 14 March 1645 he married Susanna Glynde (16?? – 16 March 1687), They had three children:

See also

Footnotes

1. ^Joseph Frederick Scott, The mathematical work of John Wallis (1616-1703), Taylor and Francis, 1938, p. 109.
2. ^Random House Dictionary.
3. ^Old Style dates: 23 November 1616 – 28 October 1703.
4. ^{{cite journal|author=Smith, David Eugene|authorlink=David Eugene Smith|title=John Wallis As a Cryptographer|journal=Bulletin of the American Mathematical Society|year=1917|volume=24|issue=2|pages=82–96|mr=1560009|doi=10.1090/s0002-9904-1917-03015-7}}
5. ^{{Cite journal|last=Kearns|first=D. A.|date=1958|title=John Wallis and complex numbers|journal=The Mathematics Teacher|volume=51|issue=5|pages=373–374|jstor=27955680}}
6. ^{{Cite journal|last=Yule|first=G. Udny|date=1939|title=John Wallis, D.D., F.R.S.|journal=Notes and Records of the Royal Society of London|volume=2|issue=1|pages=74–82|jstor=3087253}}
7. ^{{acad|id=WLS632J|name=Wallys, John}}
8. ^{{citation|first=David|last=Kahn|title=The Codebreakers: The Story of Secret Writing|year=1967|place=New York|publisher=Macmillan|lccn=63016109|page=169}}
9. ^¹4
10. ^{{cite news|title=Find could end 350-year science dispute|url=http://news.bbc.co.uk/1/hi/health/7511446.stm|accessdate=5 May 2018|publisher=BBC|date=26 July 2008}}
11. ^W. Holder, W. (1668). "Of an Experiment, Concerning Deafness". Philosophical Transactions of the Royal Society 3, pp. 665–668.
12. ^Holder, Philosophical Transactions of the Royal Society, supplement, 10.
13. ^{{cite news| url= https://www.maths.ox.ac.uk/system/files/media/John%20Wallis.pdf | accessdate=2 January 2019}}
14. ^{{cite book|author=Martínez, Alberto A.|title=Negative Math: How Mathematical Rules Can Be Positively Bent|url=https://books.google.com/books?id=8HSodlYby9MC&pg=PA22|accessdate=9 June 2013|year=2006|publisher=Princeton University Press|isbn=978-0-691-12309-7|page=22}}
15. ^Scott, J.F. 1981. ‘’The Mathematical Work of John Wallis, D.D., F.R.S. (1616–1703)’’. Chelsea Publishing Co. New York, NY. p. 18.
16. ^{{cite book|first=G.G.|last=Joseph|title=The Crest of the Peacock: Non-European Roots of Mathematics|edition=2|publisher=Penguin|year=2000|isbn=978-0-14-027778-4|page=337}}
17. ^The Mathematics of Egypt, Mesopotamia, China, India, and Islam:A Sourcebook Victor J. Katz Princeton University Press
18. ^{{citation|first=David M.|last=Burton|title=The History of Mathematics / An Introduction|edition=7th|year=2011|publisher=McGraw-Hill|isbn=978-0-07-338315-6|page=566}}
19. ^Dr. Wallis (1685) "Two extracts of the Journal of the Phil. Soc. of Oxford; one containing a paper, communicated March 31, 1685, by the Reverend Dr. Wallis, president of that society, concerning the strength of memory when applied with due attention; … ", Philosophical Transactions of the Royal Society of London, 15 : 1269-1271. Available on-line at: Royal Society of London{{dead link|date=March 2018 |bot=InternetArchiveBot |fix-attempted=yes }}
20. ^{{citation|title=The Common Scientist of the Seventeenth Century: A Study of the Dublin Philosophical Society, 1683–1708|volume=15|series=Routledge Library Editions: History & Philosophy of Science|first=K. Theodore|last=Hoppen|publisher=Routledge|year=2013|isbn=9781135028541|url=https://books.google.com/books?id=PHK7ErKBiCEC&pg=PA157|page=157}}
21. ^{{Cite journal|last=Guicciardini|first=Niccolò|date=2012|title=JOHN WALLIS AS EDITOR OF NEWTON'S MATHEMATICAL WORK|journal=Notes and Records of the Royal Society of London|volume=66|issue=1|pages=3–17|jstor=41723282|doi=10.1098/rsnr.2011.0051}}
22. ^David Damschoder and David Russell Williams, Music Theory from Zarlino to Schenker: A Bibliography and Guide (Stytvesant, NY: Pendragon Press, 1990), p. 374.
23. ^¹²{{Cite EB1911 |wstitle=Wallis, John |volume=28 |page=284–285}}
24. ^Joan Thirsk, ‘Blencowe , Anne, Lady Blencowe (1656–1718)’, Oxford Dictionary of National Biography, Oxford University Press, Oct 2005; online edn, Jan 2007 accessed 16 Nov 2016
25. ^WALLIS, John (1650-1717), of Soundness, Nettlebed, Oxon. | History of Parliament Online
26. ^Early Modern Letters Online : Person

References

Stedall, Jacqueline A. (2001). Of Our Own Nation: John Wallis's Account of Mathematical Learning in Medieval England. Historia Mathematica /, 28, 73.

Life