John Wallis and Transcendence: Measuring the Circle

John Wallis and Transcendence: measuring the circle

January 25, 2010

MATH 160 (seminar-course) Tuesdays 2-4 p.m. Sci Center Room 530

Barry Mazur and Mark Reeder

What does it mean to know a length, an area, a volume? We do it by relating lengths to other lengths, areas to areas, volumes to volumes. Archimedes had his “mechanical” method and modern mathematics has the Calculus. John Wallis grappled with an approach that lies somewhere in between Archimedes and the Calculus. We will read John Wallis’s treatise The arithmetic of inﬁnitesimals to learn what this is1.

Archimedes2 related the area of a circle to that of a certain triangle built on its circumference. He related the area of parabolic segments to the area of appropriate triangles, and the volumes of spheres to that of cones and cylinders. Lurking behind many of these wonderful theorems is π (which providentially cancels out in all of the ratios that occur in these theorems). How can we study this (as it turns out, transcendental) number π? Wallis shows how the circle forces us to expand the idea of Number to include such things as π, and he gives new ways of representing such numbers. (His own Proposition 191 was quite a shocker to him—and remains a startling thing to us: it is great fun trying to get into his way of thinking, and watch it emerge.) It is also constructive, and Wallis describes (sometimes tortuously) his path to the result. In Wallis’s exposition one can glimpse, as well, an implied rebuke to the ancients, who–in contrast to him— were fond of hiding their ideas; this is made explicit by Wallis in one of several interesting comments he makes. Our main source will be the recent English translation:

AI = The Arithmetic Of Inﬁnitesimals: John Wallis 1656 by J. A. Stedall (Springer)

We will also be reading some relevant other texts (all in English) taken from translations of Euclid’s Elements and various works of Archimedes, and Jacob Bernoulli’s Art of Conjecturing. For some of the Archimedes material, the web-page http://my.harvard.edu/icb/icb.do?keyword=k53966 to the seminar-course History of Science 206r (Spring 2008-2009): Archimedes and the Archimedean Tradition might be useful.

1The enrollment of the class is limited to 15 students, and undergraduates may enroll only with permission of the instructors. The prerequisite is a working knowledge of the Calculus. 2in Proposition 1 of his “Measurement of the Circle”

1 1 Possible Dates

1. January 26: Introduction—The ancients:

(a) Euclid: Elements XII.1, XII.2 (b) Archimedes: Measurement of the Circle

Readings:

(a) (Overview) pages 1-3 of Stedall’s “The Discovery of Wonders” and pages xi-xvi of the Introduction to AI. (b) (Plane ﬁgures compared to squares of lengths) From The Thirteen books of Euclid’s Elements (edited by Sir Thomas Heath) Dover Publications, Volume 3; pages: 363-378. (c) (The Measurement of the Circle) The Works of Archimedes, Sir Thomas L. Heath, ed., Dover, 1953; pp. 91-93. (d) (The handout) “On the measurement of the circle.” (e) (The Quadrature of the Parabola) The Works of Archimedes, Sir Thomas L. Heath, ed., Dover, 1953, pp. 233-235, 244-252.

2. February 2:

(a) Continuation of discussion of MC and QP. (b) Some of Archimedes’ explicit results regarding rational approximation of π. (c) Some modern expressions for π.

Readings: Read the items listed for Feb. 9 discussion; see below.

3. February 9: Introduction to Sums of powers 1k + 2k + ... + nk with Geometric applications.

(a) Propositions 1-4 k = 1. pp 13-16 (b) Propositions 19-23 k = 2, with discussion. pp 26-28 (c) Leonardo of Pisa “The Book of Squares.” Propositions 10, 11; pp. 39-48 in Fibonacci, The Book of Squares (annotated and translated by L.E. Sigler) Academic Press.

Readings: Read the items listed for Feb. 16 discussion; see below. Also: pp. 205-217 in Ja- cob Bernoulli’s The Art of Conjecturing (translated by E.D. Sylla) Johns Hopkins University Press.

4. February 16: Continuation of sums of powers 1k + 2k + ... + nk

(a) Propositions 39-42, 44-50 k ≥ 3. pp 39-47

5. February 23: Comparison with Bernoulli and Euler

(a) J. Bernoulli’s The Art of Conjecturing. (b) Euler expressing power sums in terms of Bernoulli numbers.

6. March 2: First interpolation— Fractional and negative exponents

(a) Props 53-57 fractional exponents. pp 48-51

2 (b) Props 64-71 recursion and geometric applications. pp 56-60 (c) Props 73-107 sums of negative powers, harmonic series, regions of inﬁnite extent but ﬁnite volume. pp 61-81

7. March 9: Continuation of Propositions 73-107

(a) Compare props 102, 103 with zeta function. (b) Compare with Or`esme’sproof of divergence of harmonic series (c) Diatribe Against the Ancients in comment after Prop 107. pp 81-82

8. March 23: Wallisian “Integration” of (1 − x1/p)q for integer q.

(a) Props 108-132, ﬁrst version of The Table. pp 82-102

9. March 30: Extending The Table to fractional q.

(a) Propositions 133-169, second version of table. pp103-122, 126-130 (b) Digression: application to Cyclotomy. Extended comment pp 122-126

10. April 6: Figurate numbers:

(a) (triangular numbers etc) which appear in The Table. (b) Propositions 169-182. pp 129-139 (c) Recursive method for sums of powers. First part of comment after prop 182 pp 139-141.

11. April 13: Completion of the table

(a) Prop 183-189. pp 150-157. (b) Stuck! Invokes Proteus [(Compare with Kepler/Mars 3 : comment after prop 189 p.158. (c) Suspicion that box is non-algebraic: Transformational Comment after prop 190 p.161.

12. April 20: Wallis’ product formula for π.

(a) Appears to give up, oﬀers for posterity some inequalities extracted from The Table, and suddenly arrives at product formula for box, including error estimate. prop 191. pp 164-167 (b) “The same another way”: Brounckner continued fractions. pp 164 and elsewhere

13. April 27: Summing up

(a) Summarizing The Table via interpolation curves: prop 192-194 (b) letter to Oughtred p.9. (c) What Newton learned from Wallis (from Newton’s notes).

3“While thus triumphing over Mars, and preparing for him, as for one already vanquished, tabular prisons and equated excentric fetters, it is buzzed here and there that the victory is vain, and that the war is raging anew as violently as before. For the enemy left at home a despised captive has burst all the chains of the equatons, and broken forth from the prisons of the tables.” Astronomia nova, chapter 51