Thomas Harriot on continuous compounding

4 BSHM Bulletin<strong>Harriot</strong> supposes that the interest is added n times per annum, at the equivalentrate of 1/nb, for seven years. In that situation, one pound invested for seven years willyield (1 þ 1/nb) 7n pounds. If n is allowed to become arbitrarily large, the result iscontinuous compounding. <strong>Harriot</strong> first applied the binomial formula to the casesn ¼ 1, 2, 3, writing down the expansions ofb 2 ð1 þ 1=bÞ 7 , b 2 ð1 þ 1=2bÞ 14 , b 2 ð1 þ 1=3bÞ 21 :He then considered the expansion of b 2 (1 þ 1/nb) 7n for general n, putting b ¼ 10 sothat the annual rate was 10% (the maximum legal rate at that time) and themultiplier b 2 conveniently represented 100 pounds. For example, the fourth term isDownloaded by [Norman Biggs] at 11:48 24 January 2013ð7n 2Þð7n 1Þð7nÞn 3 : 6bWhen b ¼ 10 and n is large this is approximately 7 3 /60 ¼ 343/60, as shown in themanuscript. The later terms become small very quickly, due to the presence in thedenominator of powers of 10 and factorials. Thus <strong>Harriot</strong> correctly inferred thatthe yield does not increase unboundedly. He calculated that 100 pounds invested at10% for seven years, with continuous compounding, will yield 201 pounds 7 shillingsand 6 pence, plus .06205 pence. Remarkably, this answer is very close to the truevalue, which is 201 pounds 7 shillings and 6 pence, plus .06458...pence. The phrase‘not 7/100’ in <strong>Harriot</strong>’s manuscript suggests that he had estimated the sum of theterms that he ignored, and was confident that the true value was very close to his.We now know that the true value is given by a simple and elegant formula, whichcan be implemented by pressing a few keys on an electronic calculator, but thismethod was not available to <strong>Harriot</strong>. The object of this article is to describe how therelevant mathematics was developed.The hyperbolic logarithmThe arithmetical algorithms for multiplication and division are significantly moreintricate than those for addition and subtraction. In the early seventeenth century thetime was ripe for an invention which, in modern terms, we describe on the followinglines. For every number s assign a corresponding number L(s), in such a way thatmultiplying s and t corresponds to adding L(s) and L(t):LðstÞ ¼LðsÞþLðtÞ:If tables of s and L(s) are available, then st can be found by first adding L(s) and L(t),and then finding the number st that corresponds to the sum.The first person to devise a workable system of this kind was a Scottishnobleman, John Napier. His system was published in 1614. It was quite complicated,and was designed specifically for calculations in trigonometry. But to Napier goesthe credit for the original invention, and for the word logarithm. (For the avoidanceof doubt it must be stressed that Napier did not invent the so-called naturallogarithm, although this claim can be found in some older books on the history ofthe subject.)A few years later Henry Briggs, professor at Gresham College in London, tookup Napier’s idea. When the two men met they discussed the best way to set up asystem that would facilitate the making of the necessary tables. Since multiplying by1 has no effect, this should be represented by adding 0: in other words, the logarithm

6 BSHM BulletinThe inverse problemAlthough Newton’s 1671 manuscript was not published at the time, parts of itgradually entered the public domain. In 1676, in the first of his famous letters toLeibniz/Oldenbourg (Turnbull 1960), Newton stated his binomial theorem forfractional exponents in the form of an infinite series for the expression (P þ PQ) m/n .He asserted that, taking the first term to be A ¼ P m/n , the successive terms B, C, D, ...can be calculated recursively by the ruleDownloaded by [Norman Biggs] at 11:48 24 January 2013B ¼ m n AQ, C ¼ m n2nm 2nBQ, D ¼ CQ, ...:3nIf the parameters are chosen suitably, the terms get small very quickly. That wasessentially <strong>Harriot</strong>’s method, although in his case the exponent was an integer andthe series was finite (but its length was unbounded).Newton thought it necessary to explain the meaning of the exponent m/n at somelength, by relating it to nth roots and mth powers, where n and m are integers. Thisadds weight to the suggestion that the notion of ‘raising a number to the power y’(and the corresponding functional relationship) was not fully understood at thattime. However, Newton had made another discovery that was to be crucial in settingup the theory of exponents.One of Newton’s favourite methods for solving an equation was to write one ofthe variables in the form of a power series in another variable, and then ‘equatecoefficients’. This will work (sometimes) even when the original relationship is itself apower series, such as the series for the hyperbolic logarithm of 1 þ x obtained byMercator and Newton himself:y ¼ xx 22 þ x33x 44 þ:In this case x ¼ 0 when y ¼ 0, so we can try x ¼ ay þ by 2 þ cy 3 þ dy 4 þ.Substituting for x on the right-hand side, and equating coefficients of powers of y,we obtain a set of equations for the coefficients a, b, c, d, .... These can be solved,giving a ¼ 1, b ¼ 1 2 , c ¼ 1 6 , d ¼ 124, and so on. It follows that 1 þ x, the antilogarithm ofy is equal to1 þ y þ y22 þ y36 þ y424 þ ,where the coefficient of y r is 1/(1.2.3...r). In this way Newton obtained what we nowcall the exponential series.Newton’s technique was published in John Wallis’s Treatise of Algebra (1685).Wallis gave many examples and, in particular, in Example VII of Chapter XCV(p 343) he showed how ‘a Logarithm being given, we may find the Number to whichit belongs’. He chose points B ¼ (0, a) and D ¼ (0, x) and stated, without proof, aninfinite series for the length BD ¼ x a in terms of the area z of the region betweenBD and the hyperbola xy ¼ ab:BD ¼ z b þ zz2abb þz36aab 3 þz424a 3 b 4 þ:Putting a ¼ b ¼ 1 we obtain the exponential series.

Volume 28 (2013) 7In the 1680s Jakob (James) Bernouilli also studied the theory of infinite series.His derivation of the exponential series was not based on inverting the logarithmicseries, but arose from the mathematics of continuous compounding—the methodfirst used by <strong>Harriot</strong>. It is acknowledged (Hofmann 2008) that in the winter of1690–91 Bernouilli obtained the exponential series from the binomial series ‘in a boldbut formally unsatisfactory manner’. His papers were collected in the De seriebusinfinitis, published posthumously in 1713. In the discussion of continuouscompounding (p 303) the exponential series appears in the formDownloaded by [Norman Biggs] at 11:48 24 January 20131 þ x s þ x21:2:s 2 þx31:2:3:s 3 þ:At this point Bernouilli does not identify the expression as the inverse of a hyperboliclogarithm, although elsewhere in his papers he had observed that the exponentialseries can be regarded as the inverse of the logarithmic series.Another derivation of the exponential series from the binomial series, but arisingfrom a different problem, is due to Abraham de Moivre. In Problem V of hisDoctrine of chances (1718), de Moivre set out to find the ratio x/q, where q can bearbitrarily large and x is related to q by the equation1 þ 1 x¼ 2:qHis method was to expand the left-hand side by ‘Sir Isaac Newton’s theorem’,obtaining1 þ x xðx 1Þ xðx 1Þðx 2Þþq 1:2:q 2 þ1:2:3:q 3 þ:When x/q is fixed, but q is allowed to be large, he obtained the series1 þ x q þ x22q 2 þ x36q 3 þ:He remarked that the sum of this series is ‘the number whose hyperbolic logarithm isx/q’. Since the sum must be 2 in this case, the answer is that x/q is the hyperboliclogarithm of 2. The calculation shows that de Moivre knew that he needed Newton’sversion of the binomial theorem, since the exponent x might not be an integer. Also,he understood the relationship between the exponential series and the hyperboliclogarithm.Continuous compounding according to John ArbuthnotIt is clear from the evidence given above that, in the early years of the eighteenthcentury, mathematicians were familiar with the exponential series1 þ x þ x21:2 þ x31:2:3 þ x41:2:3:4 þ:However, its sum was still being described by the cumbersome phrase ‘the numberwhose hyperbolic logarithm is x’. An instance of this practice, specifically referring tocontinuous compounding, occurs in a little-known book written by a slightly betterknownmathematician, John Arbuthnot (1727).

8 BSHM BulletinDownloaded by [Norman Biggs] at 11:48 24 January 2013Arbuthnot was born in Scotland and moved to London in 1691. Here hetranslated a book by Huygens on games of chance and, incidentally, introduced theword probability into the language of mathematics. He was also qualified inmedicine, and was appointed physician to Queen Anne in 1705. He became a Fellowof the Royal Society, and was a member of the commission appointed to investigatethe Newton–Leibniz controversy. His book Tables of ancient coins, weights andmeasures is mainly concerned with Greek and Roman weights and measures, asubject that had previously engaged several other scholars. The book was highlyregarded by some contemporaries, but its central contention, concerning the mass ofthe Roman denarius, is now discredited in numismatic circles.For all its failings, Arbuthnot’s book is a mine of information on the writings ofclassical authors. For example, ‘Chapter 22, Of the interest of money’, begins with alist of references to the various rates of interest charged in Roman times. Then, witha sudden change of key, Arbuthnot writes:A monthly interest is higher than an annual one of the same rate, because itoperates by compound interest. This suggests to me the following Problem. Therate per annum being given, to find the greatest Sum which is to be made of onePound, supposing the interest payable every indivisible moment of time.His argument proceeds as follows. If the interval of compounding is t, we require(1 þ rt) 1/t which ‘by Newton’s Theorem’ is1 þ r þ 1 t 1 3t þ2 r2 2t2þ r 3 þ 1 6t þ 11t2 6t 3r 4 þ:624Supposing that t denotes ‘an indivisible Moment of time, and therefore it is equal tonothing’, he obtains1 þ r þ r2 2 þ r3 6 þ r424 þ:He works out the answer for 10,000,000 pounds invested for one year at 6%,obtaining 10,618,365.4 pounds. Finally, he remarks that:the Problem is likewise solv’d by a Table of Logarithms, as follows. Multiply rinto .43429448 ..., viz. the Reciprocal of the Hyperbolick Logarithm of 10; andthe product will be the Logarithm of the number requir’d, which will be found bythe Common Tables.Essentially, Arbuthnot had identified the sum of his series as ‘the numberwhose hyperbolic logarithm is r’. But he could not identify it as e r , because thesymbol e was still unknown. It is thought that e was first used by LeonhardEuler in an unpublished letter written in 1728, the year after Arbuthnot’s book waspublished.In 1731 Euler referred to it as ‘the number whose hyperbolic logarithm is 1’, andit first occurs in print in the first volume of his Mechanica (1736). By introducingthis notation, Euler dispelled the mists that had formed over many decades. It wasnow clear that the sum of the exponential series with parameter x (the numberwhose hyperbolic logarithm is x) is the xth power of the sum of the series when x ¼ 1(the number e). In other words, the exponential function is the inverse of the

Volume 28 (2013) 9logarithmic function. <strong>Harriot</strong>’s result on continuous compounding is an instance ofthe rule that, for all x,1 þ x nnapproaches e x as n becomes large. This is such a neat result that Euler chose to definee x in this way.AcknowledgementsDownloaded by [Norman Biggs] at 11:48 24 January 2013I am very grateful to Jacqueline Stedall for deploying her extensive knowledge ofseventeenth century mathematics (and <strong>Harriot</strong> in particular) in order to comment onan early version of this article. The transcript of <strong>Harriot</strong>’s manuscript is published bypermission of the British Library Board.BibliographyArbuthnot, J, Tables of ancient coins, weights and measures, London, 1727.Beery, J, ‘Formulating figurate numbers’, BHSM Bulletin, 24 (2009), 78–91.Beery, J and Stedall, J, <strong>Thomas</strong> <strong>Harriot</strong>’s doctrine of triangular numbers: the MagisteriaMagna, European Mathematical Society, 2009.Bernouilli, Jakob, Ars Conjectandi ...De Seriebus Infinitis, Basle, 1713.de Moivre, Abraham, The doctrine of chances, London, 1718.de Morgan, Augustus, Arithmetical books, London, 1847.de Saint-Vincent, Gre´goire, Opus geometricum quadraturae ciculi et sectionum coni, Antwerp,1647.Euler, Leonhard, Mechanica, St Petersburg, 1736.Evans, A B (ed), Francesco Balducci Pegolotti: La Pratica della mercatura, Medieval Academyof America, 1936.Franci, R and Toti Rigatelli, L, ‘Fourteenth-century Italian algebra’, in Cynthia Hay (ed),Mathematics from manuscript to print 1300–1600, Clarendon Press, 1988.Hofmann, J E, ‘Jakob (Jacques) Bernouilli’, Complete dictionary of scientific biography,Encyclopedia.com, 2008.Lewin, G, ‘Compound interest in the seventeenth century’, Journal of the Institute of Actuaries,108 (1981), 423–442.Malcolm, N and Stedall, J, John Pell (1611–1685) and his correspondence with Sir CharlesCavendish, Oxford University Press, 2005.Mercator, Nicolaus, Logarithmotechnia, London, 1668.Plofker, K, ‘Mathematics in India’, in Victor Katz (ed), The mathematics of Egypt,Mesopotamia, China, India, and Islam, Princeton University Press, 2007.Stevin, Simon, Tafelen van Interest, Antwerp, 1582.Stevin, Simon, De Thiende, Leiden, 1585, also published in French as La disme, inLa pratiqued’arithmetique, Leiden, 1585.Stedall, J, ‘Symbolism, combinations, and visual imagery in the work of <strong>Thomas</strong> <strong>Harriot</strong>’,Historia mathematica, 34 (2007), 380–401.Turnbull, H W, The mathematical correspondence of Isaac Newton, Vol. II, CambridgeUniversity Press, 1960.Wallis, J, A treatise of algebra, both historical and practical ..., London, 1685.Whiteside, D T (ed), Mathematical papers of Isaac Newton, Vol. III, Cambridge UniversityPress, 1969.