The Arithmetic of Infinitesimals or a New Method of Inquiring into the Quadrature of Curves, and other more difficult mathematical problems

  • John Wallis
Chapter
Part of the Sources and Studies in the History of Mathematics and Physical Sciences book series (SHMP)

Abstract

If there is proposed a series,1 of quantities in arithmetic proportion (or as the natural sequence of numbers)2 continually increasing, beginning from a point or 0 (that is, nought, or nothing),3 thus as 0, 1, 2, 3, 4, etc., let it be proposed to inquire what is the ratio of the sum of all of them, to the sum of the same number of terms equal to the greatest.

Keywords

Infinite Series Fourth Power Cube Root Equal Height Fourth Root 
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References

  1. 1.
    Wallis used the Latin word series in two ways: (1) to denote a list of terms denned according to some rule; this meaning has been translated as ‘sequence’, and: (2) to denote a (finite or infinite) collection of such terms, usually (but not necessarily) summed; this meaning has been translated as ‘series’, even though it does not correspond exactly to the modern mathematical understanding of the word (see Glossary).Google Scholar
  2. 2.
    Quantities in arithmetic proportion (or arithmetic proportionals) increase or decrease by regular addition of a fixed quantity, thus: a, a + d, a + 2d, a + 3d,…. The sequence of natural numbers 0, 1, 2, 3,… is the simplest example.Google Scholar
  3. 3.
    By allowing his sequences to begin ‘from a point or 0’, Wallis was implying that the quantities can be taken either from geometry (magnitudes) or from arithmetic (numbers).Google Scholar
  4. 4.
    Inductione, (by induction) is not to be understood here in the modern formal sense of mathematical induction. Wallis used ‘by induction’ here and throughout simply to mean that a well established pattern could reasonably be assumed to continue.Google Scholar
  5. 5.
    Wallis’s reasoning seems to break down immediately at this point, because if his series contains an infinite number of terms increasing indefinitely it can have no greatest term. What he is really thinking of, however, though he does not yet make it clear, is a series with a finite greatest term l, arrived at by m steps of size d, thus 0, d, 2d, 3d,…, md = l. When m is finite it is clear that the sum of terms is 1/2 (m + 1)l, or, to (m + 1)l as 1 to 2. Wallis allowed the number of steps m to become infinitely large, by making d arbitrarily small, indeed infinitesimally small, but in such a way that md remains always equal to l and is therefore finite. In that case, Wallis argued (‘by induction’) that the same ratio of 1 to 2 would still hold.Google Scholar
  6. 6.
    Triangulum enim constat quasi ex infinitis redis parallelis was the phrase to which Thomas Hobbes later objected so strongly (‘“as it were” is no phrase of a geometrician’); Hobbes 1656, 46.Google Scholar
  7. 7.
    A parabola is a curve whose equation in modern notation, in its simplest form, is y n = kx. For the common (or simple) parabola n = 2, while for a cubical, biquadratic or supersolid parabola, n = 3, 4 or 5, respectively. By parabola Wallis always meant the simple parabola; the others he described as paraboloeides, translated as ‘higher parabolas’. Wallis distinguished also between Tight and inclined parabolas (cut from right or inclined cones): in a right conic the ordinates are at right angles to the diameter. An erect parabolic conoid is the solid formed by rotation of a right parabola around its axis o symmetry (its diameter). A parabolic pyramid is a pyramid with polygonal cross-sections parallel to the base and parabolic cross-sections through the vertex. In Proposition 4 the solid is based on the simple parabola y 2 = kx, so that if x 1 , x 2,… are arithmetically proportional then so are y1 2, y2 2.Google Scholar
  8. 8.
    The words ‘(Quam spuriam dicimus)’, ‘which we call spurious’ were added when the Arithmetica infinitorum was reprinted in 1695.Google Scholar
  9. 9.
    In 1695 Wallis added a note at this point to explain that by spiral he meant not the Archimedean spiral itself, but the sum of arcs of similar sectors, inscribed inside the Archimedean spiral; this he called the spurious spiral. The result stated in Proposition 5 does not hold for the true Archimedean spiral. The first revolution of the Archimedean spiral is equal in length to a half parabola whose base is the greatest radius of the spiral and whose axis is half the circumference of the coterminous circle. This result was discovered by Roberval and published in Mersenne’s Cogitata physico-mathematica in 1644 (Book II, De hydraulico, 129), but Wallis read it there only in 1656 and added a hasty Scholium or Comment after Proposition 13 to explain his own results. Wallis failed to understand that the true spiral is generated from a uniform motion along the radius, and an accelerated motion along a steadily increasing circumference (hence the analogy with the parabola which is similarly generated by uniform motion in one direction and accelerated motion in another) and his failure rendered Propositions 5 to 15 somewhat meaningless. Hobbes, who had discussed the problem with Roberval and understood the correct argument, immediately pointed out Wallis’s error, but Wallis persisted in it even in his reply to Hobbes in his Elenchus of 1656. For further discussion of this problem see Jesseph 1999, 117–125.Google Scholar
  10. 10.
    By figura, or ‘figure’, Wallis always meant a plane figure, enclosed by lines and having area. In particular, a circle is a plane figure with area, while the line bounding it is the circumference. Google Scholar
  11. 11.
    In duplicata ratione, literally ‘in duplicate ratio’ or ‘in twice the ratio’. In the Classical geometrical context the ‘ratio’ (or power) associated with quantities in arithmetic proportion is 1, the ‘ratio’ associated with their squares is 2 and with their cubes 3. It is not a great step from ‘ratio’ in this sense to ‘index’, but Wallis did not make that move formally until Proposition 64. ‘In duplicate ratio’ is translated here and elsewhere by the more familiar phrase ‘as the square of.Google Scholar
  12. 12.
    A ‘segment’ in Propositions 11 to 18 is to be understood as a portion of length.Google Scholar
  13. 13.
    Ducta in, or, ‘drawn into’. The outcome, or ‘product’, of such a construction is an area delineated by a rectangle or square. As the mathematical paradigm shifted from geometry to arithmetic, ducta in came to have the meaning of ‘multiplied by’, and the ‘product’ was the result of the multiplication. The geometrical word ‘square’ is still used for the product of two equal quantities, and Wallis also used ‘rectangle’ for the product of two unequal quantities (see Proposition 120).Google Scholar
  14. 14.
    This Comment was added after Wallis had discovered the rectification of the true Archimedean spiral in Mersenne’s Cogitata, in 1656, when most of the Arithmetica infinitorum was already printed; see note 9.Google Scholar
  15. 15.
    One such learned man was Wallis himself, see note 9. Wallis took up the same theme again at much greater length in the Comment following Proposition 182.Google Scholar
  16. 16.
    For a parabola with equation y 2 = kx, the length of the diameter, or intercepted diameter, at a given point is given by the x-coordinate, while the length of the ordinate is given by the y-coordinate.Google Scholar
  17. 17.
    The latus rectum of a conic is the total length of the ordinates passing through the focus. For a parabola with equation y 2 = kx (therefore with focus at (k, 0)) the latus rectum is 2k. Google Scholar
  18. 18.
    Note here that the altitude, or height, of a parallelogram is the distance along the diameter of the parabola.Google Scholar
  19. 19.
    De dimensione parabolae solidique hyperbolici, Torricelli 1644, 95–111; 101.Google Scholar
  20. 20.
    Wallis has mistakenly written ‘diameter’ here.Google Scholar
  21. 21.
    See Proposition 45.Google Scholar
  22. 22.
    Fractionis denominators, sive consequente rationis, literally ‘the denominator of the fraction, or the consequent term of the ratio’.Google Scholar
  23. 23… ut tandem quolibet assignabili minor evadat, (ut patet;) si in infinitum procedatur, prorsus evaniturus est. Google Scholar
  24. 24.
    In subduplicata ratione, literally ‘in half ratio’; see also note to Proposition 10.Google Scholar
  25. 25.
    In triplicata ratione, literally ‘in triplicate ratio’; see also note to Propositions 10 and 23.Google Scholar
  26. 26.
    The Proposition referred to here is actually 26.Google Scholar
  27. 27.
    Aequalem quam proxime, literally ‘very nearly equal’, but Wallis uses the phrase quam proxime here and elsewhere in a rather stronger sense, to mean ‘as close as one wishes’ (see also, for example, the Comment to Proposition 190).Google Scholar
  28. 28.
    The (right) sine of an arc is half the length of the chord connecting its ends. For an arc subtending an angle 2θ at the centre of a circle of radius, its ‘sine’ is therefore r sin θ. The length of the arc itself is . The versed sine is the distance between the centre of the arc and the chord connecting its ends, that is, r(1 - cosθ).Google Scholar
  29. 29.
    An infinite series should be understood in the sense of Proposition 2, that is, an increasing series with a finite greatest term reached by an infinite number of infinitesimally small steps.Google Scholar
  30. 30.
    Note Wallis’s clear distinction between geometric descriptions: laterals (or sides), squares, cubes, biquadrates, and arithmetic descriptions in terms of powers. Google Scholar
  31. 31.
    At this point in the 1695 edition Wallis inserted a section headed Monitum with further comments on spirals; see Wallis 1693–99, I, 385–387.Google Scholar
  32. 32.
    Wallis or his printers gave the wrong diagram in this Proposition: he needed a right parabola as in the Comment to Proposition 38, and as given here.Google Scholar
  33. 33.
    Here the squares are to be understood as geometrical objects, since Wallis goes on to compare them with similar planes. Google Scholar
  34. 35.
    This is the first time in the text that Wallis uses area as an absolute quantity (rather than expressing it as a ratio).Google Scholar
  35. 36.
    Although this is the first time Wallis has formally defined the concept of a fractional index, he has already used the idea implicitly; see for example the Comment to Proposition 45 where the fractions associated with square roots and cube roots are taken to be 1/2 and 1/3 (or, in classical language, subduplicate and subtriplicate ratios).Google Scholar
  36. 37.
    Si Parallelogrami rectae omnes, in rectas Triangulis respective ducantur,…’. As pointed out in the note to Proposition 11, the verb ducere (in) was used to describe the construction of a perpendicular, the ‘product’ being the square or rectangle so defined. Such a construction gives the geometrical equivalent of multiplication in arithmetic.Google Scholar
  37. 38.
    A literal translation of ‘…inter rectas sic multiplicatas’; Wallis is now blurring the classical distinction between geometry (which deals with lines) and arithmetic (which deals with numbers).Google Scholar
  38. 39.
    Here the planes, or products, are represented by a single line, or ordinate, of the same magnitude.Google Scholar
  39. 40.
    The lines DE are those in the lower left part of the diagram. The lines on the upper right illustrate Proposition 77, which follows.Google Scholar
  40. 41.
    Wallis used composition and resolution for inverse processes such as addition and subtraction; or multiplication and division; or raising to powers and taking roots. Other writers, however, used the terms as equivalents of synthesis and analysis. Google Scholar
  41. 42.
    Si Pyramis ad Triangulum respective applicetur,…’. The verb applicare (ad), literally ‘to lay to’, was used for the geometrical construction of setting an area against a line (or a solid against an area), the geometrical equivalent of division in arithmetic (see also the notes to Propositions 11 and 75).Google Scholar
  42. 43.
    The third proportional of two (ordered) quantities x and y is y 2 /x (since x : y = y : y 2 /x).Google Scholar
  43. 44.
    Wallis used Oughtred’s notation dAq · DAq :: · for ratio, throughout Propositions 99 and 100.Google Scholar
  44. 45.
    Torricelli 1644, 115–116.Google Scholar
  45. 46.
    Wallis has mistakenly written ‘divides’.Google Scholar
  46. 47.
    Wallis is now discussing solids of revolution.Google Scholar
  47. 48.
    Wallis wants to approach R by an infinite number of small steps, which would suggest that A = ∞, so there is a serious contradiction here. The problem arises from Wallis’s concept of an area as a sum of lines: for him the area of a rectangle with base R is equivalent to R taken infinitely many times. But if the altitude of the rectangle is A, its area is AR, leading Wallis to state that A is equivalent to ‘infinitely many times’, or ‘the number of all the terms’.Google Scholar
  48. 49.
    Inversae, here translated as ‘reversed’, means ‘decreasing instead of increasing’.Google Scholar
  49. 50.
    Since Wallis is speaking of multiplication of series, the Latin rectangulorum would here more naturally be translated as ‘of the products’. I have kept the more literal translation ‘of the rectangles’, because Wallis goes on to compare these ‘rectangles’ with squares or other geometrical figures.Google Scholar
  50. 52.
    Rectangulorum, or ‘products’, as in Proposition 120.Google Scholar
  51. 53.
    Here Wallis needs what he has needed all along, the binomial theorem for fractional indices.Google Scholar
  52. 54.
    Here for the first time Wallis used some of the algebraic formulae that he developed in On conic sections. Google Scholar
  53. 55.
    That is, the limitation that both the transverse diameter and the maximum intercepted diameter must be equal to the latus rectum. Google Scholar
  54. 56.
    This is virtually the last of Wallis’s geometric examples; from now on his investigations are based almost entirely on arithmetic.Google Scholar
  55. 57.
    In seipsam inverse positam, that is ‘taken backwards’ or ‘reversed’. Bear in mind that Wallis’s series, though it has an infinite number of terms, has a finite greatest term, and so the terms can be taken in either direction.Google Scholar
  56. 58.
    In seipsam directe positam-, Wallis uses ‘directly’ here to mean ‘forward’. Earlier (in Propositions 99, 102, 103, 104, 106) he spoke of direct and reciprocal proportion; the two uses of ‘direct’ are linked in that the powers, or indices, of direct series go forwards (1,2,3,…) whereas the indices of reciprocal or inverse series go backwards (-1,-2,-3,…).Google Scholar
  57. 60.
    Wallis has mistakenly written ‘first powers’.Google Scholar
  58. 61.
    Wallis is referring here to the second and later editions of William Oughtred’s Clavis mathematicae, those published from 1647 onwards; in the first (1631) edition, Oughtred’s note on figurate numbers appears at Chapter 18, note 16. Wallis was involved in correcting the third Latin edition of the Clavis for publication at Oxford in 1652. See Stedall 2002, 55–87.Google Scholar
  59. 62.
    Note the mixture of geometry and algebra in this theorem. Triangles and sides are geometrical concepts, while an arithmetic mean can be constructed either geometrically or algebraically. The final step in Wallis’s argument, however, the construction of new triangular numbers from given sides is purely algebraic: there is no physical meaning to a triangular number based on a side of ½ or 1 ½ or 2 ½ points.Google Scholar
  60. 63.
    Serierum characteribus, literally ‘ from the properties of the sequences’. Since for Wallis the properties of each sequence had now come to be defined by an algebraic formula, character is translated from here onwards by ‘formula’.Google Scholar
  61. 64.
    This formula was included only in the edition reprinted in the Opera mathematica in 1695; it is included here for completeness.Google Scholar
  62. 65.
    These formulae were first written down symbolically, almost exactly as Wallis has them here, by Thomas Harriot about fifty years earlier, except that Harriot used nn etc. where Wallis later wrote l 2 etc.; see British Library Add MS 6782, ff. 108, reproduced in Lohne 1979, 294. Harriot also discovered the same method of generating the numbers by successive multiplication. The formulae and the method of generating them were also known to Fermat who, however, expressed the results verbally: ‘The last side multiplied by the next greater makes twice the triangle. The last side multiplied by the triangle of the next greater side makes the three times the pyramid. The last side multiplied by the pyramid of the next greater side makes four times the triangulo-triangle. And so on by the same progression ad infinitum’; Fermat to Roberval, 4 November 1636, Fermat 1891–1912, II, 84–85, see also Mahoney 1974, 230.Google Scholar
  63. 66.
    From his formulae for figurate numbers Wallis is about to derive further results on sums of powers. It seems that Fermat was in possession of the same facts but worked the other way round: from sums of powers to formulae for figurate numbers: see Mahoney 1974, 229–233.Google Scholar
  64. 67.
    When the Arithmetica infinitorum was reprinted in 1695 Wallis added further (lengthy) calculations for the sums of sequences of fifth and sixth powers; see Wallis 1695, I, 449–452.Google Scholar
  65. 68.
    The ‘altitude’ of each space is the length of the segment along the diameter of the parabola.Google Scholar
  66. 69.
    The ‘width’ of each parallelogram (actually a rectangle) is the length of the ordinate that bounds it.Google Scholar
  67. 70.
    Wallis mistakenly has Proposition 178 here.Google Scholar
  68. 71.
    Here Wallis takes A to represent any term of the first sequence, thus ∞, 1, ½,….Google Scholar
  69. 72.
    Here Wallis uses A in two distinct ways: (i) to represent in a general way the first terms of the even sequences (see note 5), but also (ii) to denote in particular the first term of the second sequence.Google Scholar
  70. 73.
    The sequence of first terms.Google Scholar
  71. 75.
    Resolution is here used as the opposite of composition, thus of subtraction as opposed to addition, division as opposed to multiplication, or extraction of roots as opposed to composition of powers.Google Scholar
  72. 76.
    Non metitur, literally ‘by which it is not measured’.Google Scholar
  73. 77.
    Wallis’s argument here is that β/a is the smaller of the two quantities a/α and β/a (because of the decreasing ratio), and is therefore less than the square root of their product. Wallis does not make himself entirely clear, and Christiaan Huygens was puzzled by this part of the argument, and failed to understand why Wallis went on to take a square root; see Huygens to Wallis, [11]/21 July 1656, Beeley and Scriba 2003, 189–192.Google Scholar
  74. 79.
    The first fraction, beginning with zero, oscillates between zero and infinity, but multiplied by the next fraction, beginning with 2, it is supposed to make 1.Google Scholar
  75. 81.
    Wallis has mistakenly written ‘greater’ (major) here.Google Scholar
  76. 82.
    Propositus, or proposed, or given.Google Scholar
  77. 83.
    Quaesitus, or sought out.Google Scholar
  78. 84.
    Descartes’ definition of a geometric curve is not that it should be smooth and regular, but that it can be described by a single equation in the coordinates. Wallis has failed to find such an equation for the odd curves, but goes on to argue (see below) that such formulae or equations must exist.Google Scholar

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© Springer Science+Business Media New York 2004

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  • John Wallis

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