John Wallis

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For other people named John Wallis, see John Wallis (disambiguation). John Wallis_sentence_0

John Wallis_table_infobox_0

John WallisJohn Wallis_header_cell_0_0_0
BornJohn Wallis_header_cell_0_1_0 3 December O.S. 23 November] 1616

Ashford, Kent, EnglandJohn Wallis_cell_0_1_1

DiedJohn Wallis_header_cell_0_2_0 8 November 1703(1703-11-08) (aged 86) O.S. 28 October 1703]

Oxford, Oxfordshire, EnglandJohn Wallis_cell_0_2_1

NationalityJohn Wallis_header_cell_0_3_0 EnglishJohn Wallis_cell_0_3_1
EducationJohn Wallis_header_cell_0_4_0 Felsted School, Emmanuel College, CambridgeJohn Wallis_cell_0_4_1
Known forJohn Wallis_header_cell_0_5_0 Wallis product

Inventing the symbol ∞ Extending Cavalieri's quadrature formula Coining the term "momentum"John Wallis_cell_0_5_1

FieldsJohn Wallis_header_cell_0_6_0 MathematicsJohn Wallis_cell_0_6_1
InstitutionsJohn Wallis_header_cell_0_7_0 John Wallis_cell_0_7_1
Academic advisorsJohn Wallis_header_cell_0_8_0 William OughtredJohn Wallis_cell_0_8_1
Notable studentsJohn Wallis_header_cell_0_9_0 William BrounckerJohn Wallis_cell_0_9_1

John Wallis (/ˈwɒlɪs/; Latin: Wallisius; 3 December O.S. John Wallis_sentence_1

23 November] 1616 – 8 November O.S. John Wallis_sentence_2

28 October] 1703) was an English clergyman and mathematician who is given partial credit for the development of infinitesimal calculus. John Wallis_sentence_3

Between 1643 and 1689 he served as chief cryptographer for Parliament and, later, the royal court. John Wallis_sentence_4

He is credited with introducing the symbol ∞ to represent the concept of infinity. John Wallis_sentence_5

He similarly used 1/∞ for an infinitesimal. John Wallis_sentence_6

John Wallis was a contemporary of Newton and one of the greatest intellectuals of the early renaissance of mathematics. John Wallis_sentence_7

Life John Wallis_section_0

John Wallis was born in Ashford, Kent. John Wallis_sentence_8

He was the third of five children of Reverend John Wallis and Joanna Chapman. John Wallis_sentence_9

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. John Wallis_sentence_10

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). John Wallis_sentence_11

At the school in Felsted, Wallis learned how to speak and write Latin. John Wallis_sentence_12

By this time, he also was proficient in French, Greek, and Hebrew. John Wallis_sentence_13

As it was intended he should be a doctor, he was sent in 1632 to Emmanuel College, Cambridge. John Wallis_sentence_14

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. John Wallis_sentence_15

His interests, however, centred on mathematics. John Wallis_sentence_16

He received his Bachelor of Arts degree in 1637 and a Master's in 1640, afterwards entering the priesthood. John Wallis_sentence_17

From 1643 to 1649, he served as a nonvoting scribe at the Westminster Assembly. John Wallis_sentence_18

He was elected to a fellowship at Queens' College, Cambridge in 1644, from which he had to resign following his marriage. John Wallis_sentence_19

Throughout this time, Wallis had been close to the Parliamentarian party, perhaps as a result of his exposure to Holbeach at Felsted School. John Wallis_sentence_20

He rendered them great practical assistance in deciphering Royalist dispatches. John Wallis_sentence_21

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. John Wallis_sentence_22

Most ciphers were ad hoc methods relying on a secret algorithm, as opposed to systems based on a variable key. John Wallis_sentence_23

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. John Wallis_sentence_24

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. John Wallis_sentence_25

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. John Wallis_sentence_26

He was finally able to indulge his mathematical interests, mastering William Oughtred's Clavis Mathematicae in a few weeks in 1647. John Wallis_sentence_27

He soon began to write his own treatises, dealing with a wide range of topics, which he continued for the rest of his life. John Wallis_sentence_28

Wallis wrote the first survey about mathematical concepts in England where he discussed the Hindu-Arabic system. John Wallis_sentence_29

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. John Wallis_sentence_30

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 8 November O.S. John Wallis_sentence_31

28 October] 1703. John Wallis_sentence_32

In 1650, Wallis was ordained as a minister. John Wallis_sentence_33

After, he spent two years with Sir Richard Darley and Lady Vere as a private chaplain. John Wallis_sentence_34

In 1661, he was one of twelve Presbyterian representatives at the Savoy Conference. John Wallis_sentence_35

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. John Wallis_sentence_36

William Holder had earlier taught a deaf man, Alexander Popham, to speak "plainly and distinctly, and with a good and graceful tone". John Wallis_sentence_37

Wallis later claimed credit for this, leading Holder to accuse Wallis of "rifling his Neighbours, and adorning himself with their spoyls". John Wallis_sentence_38

Wallis' appointment as Savilian Professor of Geometry at the Oxford University John Wallis_section_1

The Parliamentary visitation of Oxford that began in 1647 removed many senior academics from their positions, including (in November 1648) the Savilian Professors of Geometry and Astronomy. John Wallis_sentence_39

In 1649 Wallis was appointed as Savilian Professor of Geometry. John Wallis_sentence_40

Wallis seems to have been chosen largely on political grounds (as perhaps had been his Royalist predecessor Peter Turner, who despite his appointment to two professorships never published any mathematical works); while Wallis was perhaps the nation's leading cryptographer and was part of an informal group of scientists that would later become the Royal Society, he had no particular reputation as a mathematician. John Wallis_sentence_41

Nonetheless, Wallis' appointment proved richly justified by his subsequent work during the 54 years he served as Savilian Professor. John Wallis_sentence_42

Contributions to mathematics John Wallis_section_2

Wallis made significant contributions to trigonometry, calculus, geometry, and the analysis of infinite series. John Wallis_sentence_43

In his Opera Mathematica I (1695) he introduced the term "continued fraction". John Wallis_sentence_44

Analytic geometry John Wallis_section_3

In 1655, Wallis published a treatise on conic sections in which they were defined analytically. John Wallis_sentence_45

This was the earliest book in which these curves are considered and defined as curves of the second degree. John Wallis_sentence_46

It helped to remove some of the perceived difficulty and obscurity of René Descartes' work on analytic geometry. John Wallis_sentence_47

In the Treatise on the Conic Sections Wallis popularised the symbol ∞ for infinity. John Wallis_sentence_48

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." John Wallis_sentence_49

Integral calculus John Wallis_section_4

Arithmetica Infinitorum, the most important of Wallis's works, was published in 1656. John Wallis_sentence_50

In this treatise the methods of analysis of Descartes and Cavalieri were systematised and extended, but some ideas were open to criticism. John Wallis_sentence_51

He began, after a short tract on conic sections, by developing the standard notation for powers, extending them from positive integers to rational numbers: John Wallis_sentence_52

Leaving the numerous algebraic applications of this discovery, he next proceeded to find, by integration, the area enclosed between the curve y = x, 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. John Wallis_sentence_53

He apparently assumed that the same result would be true also for the curve y = ax, 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. John Wallis_sentence_54

In the latter case, his interpretation of the result is incorrect. John Wallis_sentence_55

He then showed that similar results may be written down for any curve of the form John Wallis_sentence_56

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 = x + x + x + ..., its area would be x + x/2 + x/3 + ... John Wallis_sentence_57

He then applied this to the quadrature of the curves y = (x − x), y = (x − x), y = (x − x), etc., taken between the limits x = 0 and x = 1. John Wallis_sentence_58

He shows that the areas are, respectively, 1, 1/6, 1/30, 1/140, etc. John Wallis_sentence_59

He next considered curves of the form y = x 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. John Wallis_sentence_60

This is equivalent to computing John Wallis_sentence_61

He illustrated this by the parabola, in which case m = 2. John Wallis_sentence_62

He stated, but did not prove, the corresponding result for a curve of the form y = x. John Wallis_sentence_63

(which is now known as the Wallis product). John Wallis_sentence_64

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. John Wallis_sentence_65

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. John Wallis_sentence_66

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 x = ay, which had been discovered in 1657 by his pupil William Neile. John Wallis_sentence_67

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. John Wallis_sentence_68

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. John Wallis_sentence_69

The cycloid was the next curve rectified; this was done by Christopher Wren in 1658. John Wallis_sentence_70

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. John Wallis_sentence_71

Van Heuraët's method is as follows. John Wallis_sentence_72

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, 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. John Wallis_sentence_73

Therefore, h ds = η dx. John Wallis_sentence_74

Hence, if the area of the locus of the point (x, η) can be found, the first curve can be rectified. John Wallis_sentence_75

In this way van Heuraët effected the rectification of the curve y = ax but added that the rectification of the parabola y = ax is impossible. John Wallis_sentence_76

since it requires the quadrature of the hyperbola. John Wallis_sentence_77

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. John Wallis_sentence_78

A third method was suggested by Fermat in 1660, but it is inelegant and laborious. John Wallis_sentence_79

Collision of bodies John Wallis_section_5

The theory of the collision of bodies was propounded by the Royal Society in 1668 for the consideration of mathematicians. John Wallis_sentence_80

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). John Wallis_sentence_81

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. John Wallis_sentence_82

Algebra John Wallis_section_6

In 1685 Wallis published Algebra, preceded by a historical account of the development of the subject, which contains a great deal of valuable information. John Wallis_sentence_83

The second edition, issued in 1693 and forming the second volume of his Opera, was considerably enlarged. John Wallis_sentence_84

This algebra is noteworthy as containing the first systematic use of formulae. John Wallis_sentence_85

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. John Wallis_sentence_86

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 John Wallis_sentence_87

John Wallis_description_list_0

  • s = vt,John Wallis_item_0_0

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 John Wallis_sentence_88

John Wallis_description_list_1

  • s1 : s2 = v1t1 : v2t2.John Wallis_item_1_1

Geometry John Wallis_section_7

He is usually credited with the proof of the Pythagorean theorem using similar triangles. John Wallis_sentence_89

However, Thabit Ibn Qurra (AD 901), an Arab mathematician, had produced a generalisation of the Pythagorean theorem applicable to all triangles six centuries earlier. John Wallis_sentence_90

It is a reasonable conjecture that Wallis was aware of Thabit's work. John Wallis_sentence_91

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 1298 on the parallel postulate. John Wallis_sentence_92

The book was based on his father's thoughts and presented one of the earliest arguments for a non-Euclidean hypothesis equivalent to the parallel postulate. John Wallis_sentence_93

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. John Wallis_sentence_94

He found that Euclid's fifth postulate is equivalent to the one currently named "Wallis postulate" after him. John Wallis_sentence_95

This postulate states that "On a given finite straight line it is always possible to construct a triangle similar to a given triangle". John Wallis_sentence_96

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. John Wallis_sentence_97

Unlike other authors, he realised that the unbounded growth of a triangle was not guaranteed by the four first postulates. John Wallis_sentence_98

Calculator John Wallis_section_8

Another aspect of Wallis's mathematical skills was his ability to do mental calculations. John Wallis_sentence_99

He slept badly and often did mental calculations as he lay awake in his bed. John Wallis_sentence_100

One night he calculated in his head the square root of a number with 53 digits. John Wallis_sentence_101

In the morning he dictated the 27-digit square root of the number, still entirely from memory. John Wallis_sentence_102

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. John Wallis_sentence_103

It was considered important enough to merit discussion in the Philosophical Transactions of the Royal Society of 1685. John Wallis_sentence_104

Educational Background John Wallis_section_9

John Wallis_unordered_list_2

  • Cambridge, M.A., Oxford, D.D.John Wallis_item_2_2
  • Grammar School at Tenterden, Kent, 1625–31.John Wallis_item_2_3
  • School of Martin Holbeach at Felsted, Essex, 1631–2.John Wallis_item_2_4
  • Cambridge University, Emmanuel College, 1632–40; B.A., 1637; M.A., 1640.John Wallis_item_2_5
  • D.D. at Oxford in 1654John Wallis_item_2_6

Musical theory John Wallis_section_10

Wallis translated into Latin works of Ptolemy and Bryennius, and Porphyrius's commentary on Ptolemy. John Wallis_sentence_105

He also published three letters to Henry Oldenburg concerning tuning. John Wallis_sentence_106

He approved of equal temperament, which was being used in England's organs. John Wallis_sentence_107

Other works John Wallis_section_11

His Institutio logicae, published in 1687, was very popular. John Wallis_sentence_108

The Grammatica linguae Anglicanae was a work on English grammar, that remained in print well into the eighteenth century. John Wallis_sentence_109

He also published on theology. John Wallis_sentence_110

Family John Wallis_section_12

On 14 March 1645 he married Susanna Glynde (c. 1600 – 16 March 1687). John Wallis_sentence_111

They had three children: John Wallis_sentence_112

John Wallis_ordered_list_3

  1. Anne Blencoe (4 June 1656 – 5 April 1718), married Sir John Blencowe (30 November 1642 – 6 May 1726) in 1675, with issueJohn Wallis_item_3_7
  2. John Wallis (26 December 1650 – 14 March 1717), MP for Wallingford 1690–1695, married Elizabeth Harris (d. 1693) on 1 February 1682, with issue: one son and two daughtersJohn Wallis_item_3_8
  3. Elizabeth Wallis (1658–1703), married William Benson (1649–1691) of Towcester, died with no issueJohn Wallis_item_3_9

See also John Wallis_section_13

John Wallis_unordered_list_4

Credits to the contents of this page go to the authors of the corresponding Wikipedia page: Wallis.