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Learning by ReadingOriginal Mathematics

Ranjan Roy

In spite of the large number of good textbooksavailable today, our undergraduate and graduatestudents can greatly benefit by regarding thecorpus of received original mathematical works asaccessible, fascinating, relevant reading materialwith direct bearing on their coursework. Whenmathematicians such as Landen, Daniel Bernoulli, orNewton discussed summation of series, differenceequations, or areas under curves, they wrote withthe passion and enthusiasm that accompanies thediscovery of new insights and relationships. Byreading these original works, our students maymore clearly see mathematics as dynamic andevolving. They can better understand theoremsand formulas as arising out of real and significantmathematical questions, with connections to otherproblems. Moreover, many older works presenttheir results in simple form, without the elaboratetechniques developed later. Also, the results inolder works may be incomplete or lacking inrigor; this apparent deficiency gives students anopportunity to provide that rigor, or to understandthe need for further refinement.

Reflecting a sense of excitement and thenew prospects revealed by his discovery of in-finite series, in 1670–71 Newton wrote in hisDe Methodis [5]

I am amazed that it has occurred to no one (ifyou except N. Mercator with his quadratureof the hyperbola) to fit the doctrine recentlyestablished for decimal numbers in similarfashion to variables, especially since the wayis then open to more striking consequences.For since this doctrine in species has the

Ranjan Roy is professor of mathematics at Beloit College.His email address is royr@beloit.edu.

same relationship to Algebra that the doctrinein decimal numbers has to common Arith-metic, its operations of Addition, Subtraction,Multiplication, Division and Root-extractionmay easily be learnt from the latter’s pro-vided the reader be skilled in each, bothArithmetic and Algebra, and appreciate thecorrespondence between decimal numbersand algebraic terms continued to infinity … .

Using Newton’s analogy between arithmetic andalgebra, just as 1/3 or

√3 could be written as

infinite decimals, 1/(1+ x2) and√

1+ x2 could beexpanded in infinite series. As one of the “morestriking consequences”, he applied the method ofsuccessive approximation for solving an algebraicequation f (y) = 0, gleaned from his study of Viète,to solve a polynomial equation f (x, y) = 0, butusing infinite series. Newton showed that sucha solution had to be of the form y = xαg(x),with α rational and g(x) a power series. Hethereby obtained the first statement of the implicitfunction theorem. The factor α was determined bythe method called Newton’s polygon; interestingly,in his De Analysi of 1669, Newton had not yet takenthis factor into account, though a simple exampleshows the need for it. Against the backdrop ofDescartes’s geometry, Newton’s results appear in ameaningful context, motivated by his effort to findthe area under a curve defined by f (x, y) = 0, aproblem he solved by integrating y = xαg(x) termby term.

Students may be concerned that Newton did notconsider convergence questions, and this itself pro-vides ample food for thought. Indeed, the algebraicgeometer Abhyankar [1] wrote, “…Newton’s proof,being algorithmic, applies equally well to power

October 2011 Notices of the AMS 1285

series, whether they converge or not. Moreover,and that is the main point, Newton’s algorithmicproof leads to numerous other existence theorems,while Puiseux’s existential proof does not do so.”In his Adventures of a Mathematician, Ulam recallsBanach’s remark, “Good mathematicians see analo-gies between theorems or theories, the very bestones see analogies between analogies.” In Newton,one finds analogy being employed by a master.

In the work of Daniel Bernoulli, students mayperceive the dynamic development of mathematicalideas, as well as the intellectual growth and joyof discovery within Bernoulli himself. In 1724Daniel Bernoulli wrote in his Exercitationes thatthere was no formula for the general term ofthe sequence 1, 3, 4, 7, 11, 18, … , a sequencesatisfying the second-order difference equationan + an−1 = an+1. His cousin Niklaus I Bernoulliinformed him that, in fact, the general term couldbe expressed as [

(1+√

5)/2]n + [

(1−√

5)/2]n.

On rethinking, Daniel Bernoulli discovered themethod for solving linear difference equationsusually presented in textbooks. He explained ina 1728 paper [2] that if x1, x2, . . . , xm were thesolutions of the algebraic equation

∑mk=0 βkxk = 0,

then the general solution of the difference equationm∑k=0

βkan+k = 0 would be an =m∑k=1

Akxnk

where A1, A2, . . . , Am were arbitrary constants de-termined by the m initial values. In 1717 deMoivre had published the method of generatingfunctions, or, in his words, recurrent series, tosolve difference equations. Bernoulli’s novel ideawas the construction of the general solution as alinear combination of special solutions, a methodso original that it took a decade to extend it tolinear differential equations. In fact, Euler wrote toJohann Bernoulli in September 1739 that he wasastonished to discover that the solution of a lineardifferential equation with constant coefficients wasdetermined by an algebraic equation.

Daniel Bernoulli applied his discovery to findthe numerically largest solution of an algebraicequation. From the general solution, he easilydetermined the numerically largest value to beapproximately an+1/an, for n sufficiently large.Bernoulli commented that his result, even if notuseful, was beautiful. But, in fact, he applied it toobtain the zeros of some Laguerre polynomialsarising in his work on the frequencies of oscillationof hanging chains. At the end of his 1728 paper,he solved the difference equation

an+1 − 2 cosθan + an−1 = 0.

Surprisingly, this gave him a new derivation of deMoivre’s 1707 formula

2 cos(nθ) = (cosθ + i sinθ)n + (cosθ − i sinθ)n ,

from which de Moivre [4], in a few lines, hadobtained the important and useful formula

x2n − 2anxn cos(nθ)+ a2n

=n−1∏k=0

(x2 − 2ax cos

(θ + 2kπ

n

)+ a2

).

It would be a fascinating experience for a studentto work through the derivation of this factorizationformula from a difference equation.

Landen’s efforts to sum∑∞n=1 1/n2 produced a

brilliant and essentially correct idea. But, sincecomplex analysis had not been developed, hissolution was incomplete, with many loose ends.Students may study Landen’s insights with greatbenefit, as they attempt to fill in the gaps and renderhis work more coherent; they may observe thatroutine calculations can yield results of tremendoussignificance.

In 1729 Euler discussed the dilogarithm function

Li2(x) = −∫ x

0

log(1− t)t

dt =∞∑n=1

xn

n2,

where the last equality would hold for |x| ≤ 1;though he found some interesting results, hewas unable to evaluate Li2(1) from the integralrepresentation. In a paper of 1760 in the Philo-sophical Transactions, Landen [3] showed that thisevaluation required the use of log(−1). Unawareof Euler’s work on logarithms, Landen set out todetermine log(−1) by first differentiating x = sinzto obtain dz/

√−1 = dx/

√x2 − 1. Integrating this,

he gotz√−1= log

(x+√x2 − 1√−1

).

He set z = π/2 and x = 1 so that log√−1 =

−π/(2√−1). Since a square root must have two

values, he concluded

(1) log (−1) = ± π√−1.

To evaluate Li2(1), he started with the calculation

x−1 + x−2

2+ x

−3

3+ · · ·

(2)

= log1

1− 1/x= log x+ log

11− x −

π√−1,

where he chose the negative sign from equation(1). Next, Landen divided (2) by x to obtain(3)

− Li2(1/x) = −π√−1

log x+ 12(log x)2 + Li2(x)+C.

To find C, he first took x = 1 to get C =−2

∑∞n=1 1/n2 and to evaluate this series, he

set x = −1 in (3). But this time, he pragmati-cally chose the positive sign in (1) and arrivedat

∞∑n=1

1n2= π

2

6.

1286 Notices of the AMS Volume 58, Number 9

As his computations indicate, formula (3) iscorrect only for x ≥ 1. Landen proceeded to divide(3) by x and then integrate; he repeated the processindefinitely. The resulting formulas gave him thevalues of

∞∑n=1

1n2k and

∞∑n=1

(−1)n−1

(2n− 1)2k+1.

Observe that the series Li2(x) and Li2(1/x) convergewhen x = eiθ , with 0 < θ ≤ 2π . When this valueof x is substituted in (3), the resulting formulais incorrect; choosing the plus sign in (3) makesthe formula valid, yielding the Fourier seriesexpansion of the second Bernoulli polynomial!Clearly, a student of complex analysis could learna great deal by delving into Landen’s remarkablesummations.

An ever-increasing number of old mathematicspapers and books are easily available online andin print, making it very convenient to use suchmaterial in class discussions, assignments, ormath club activities. For example, Newton’s basicanalogy may be presented in a calculus lecture, andNewton’s polygon assigned as a project. The wealthof old mathematics applicable in our courses isastonishing, including both circumscribed andopen-ended problems, and this treasure trove is awonderful teaching resource.

References[1] S. S. Abhyankar, Historical ramblings in algebraic

geometry and related algebra, Amer. Math. Monthly83 (1976), 417.

[2] D. Bernoulli, Observationes de Seriebus Recurren-tibus, Comm. Acad. Sci. Imper. Petrop. 3 (1728),85–100. Reprinted in Werke, vol. 2.

[3] John Landen, A new method of computing the sumsof certain series, Phil. Trans. 51, part 2 (1760), 553–565.

[4] A. de Moivre, Miscellanea Analytica, London, 1730,1–2.

[5] I. Newton, Mathematical Papers of Isaac Newton, vol.3 (D. T. Whiteside, ed.), Cambridge University Press,Cambridge, 1969, 33–35.

October 2011 Notices of the AMS 1287

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