Three Ways to Compute Pi.

To accompany this presentation, which will take place on 10/10/10 in room 514 of the Jefferson Library from 1330 to 1500, we will discuss the following books. The first seven items listed below deal with π. The others are classics and will be shown for their own interest.

Here is the text of what I will say about Machin's Formula for Computing Pi. Power Point for the presentation.


Œuvres d'Archimède. Traduites littéralmement, avec un commentaire par F. Peyrard, Paris 1807.
SPEC: WPT QA31 A693 1807.   WPT abbreviates "West Point Treasure".

This is a "Thayer Book," one of the thousand books that Thayer purchased in France in 1815-1817. At this time, books were typically sold in quires and the purchaser had them bound to specification. Thayer's favorite book-dealer was a man named Kilian. Note that the front cover is stamped "U. S. / Military Academy / West-Point" (with a hyphen). Thayer purchased a special stamp to do this embossing. This is a typical early nineteenth-century French binding. The marbled endpapers are also common. The book has been repaired. The portrait of Archimedes facing the title page is nice, but certainly not authentic.

Of special interest is the diagram for the the first proposition of The Measurement of the Circle, pp. 116-122.This diagram appears on the frontispiece of Gregorius a Sancto Vincentio 1647, which we shall view shortly.

Also of interest is The Quadrature of the Parabola, pp. 318-347. The diagrams show that he is slicing up the parabolic segment and then hanging the pieces on a balance (p. 332). His argument for the sum of a specific geometric series is on pp. 343-344.

Gregorius a Sancto Vincentio,
P. Gregorii a Sto Vincento Opvs geometricvm qvadratvrae circvli et sectionvm coni, decem libris comprehensum,
Antverpiae, Apvd Ioannem et Iacobvm Mevrsios, 1647. 
QA444 .S155 1647 

This large volume (over 1250 pages) was written in the 1620s but his Jesuit superiors refused to let him publish it then. It contains the first presentation of the summation of infinite geometric series, a method of trisecting angles using infinite series, and the result Saint Vincent considered his most important: a method for squaring the circle. Alas, this result was incorrect, as Huygens first pointed out in 1651. Although this error destroyed his reputation, the work contains much of value which influenced Leibniz, Wallace, and Wren. The most important result for the calculus is a surprising connection between the natural logarithm and the rectangular hyperbola, namely the idea that we use today to define the logarithm (about p. 594). 

The frontispiece of the Opus geometricum is the most magnificent allegory in all of mathematical publishing. In the foreground, Archimedes is drawing the diagram for his proof of the area of a circle. Cowering attentively behind him is Euclid, who is looking on in awe. The character anachronistically wearing swim goggles has not been identified. Wading in the estuary is Neptune, whose banner carries the slogan "Plus ultra," there is more beyond this ancient geometry, yet the ancients are prevented from getting there by the Pillars of Hercules. But Gregorius has discovered this new land of mathematics---at least, this frontispiece claims so. In the background the sunbeam carries the words "Mutat quadrata rotundis" (the square is changed into a circle) which are illustrated by the putto holding the square frame which focuses the sunbeam into a circle on the ground. Note that the putti are tracing it out with a compass, and that the circle is correctly drawn in perspective as an ellipse. Here is more detail about the frontispiece.

This volume was once owned by René François de Sluse (1622-1685), who developed a method for finding tangents to algebraic curves just before Newton (1642-1727) discovered his own. The volume also contains notes which, I conjecture, were written by Sluse. The numerous slips of paper in the volume and the many marginal notes could be by Sluse. Someone has shaded the diagrams on p. 528 and 535.

Jones, William
, 1675-1749
Synopsis palmariorum matheseos: or, a New Introduction to the Mathematics.
SPEC QA35 .J6 1706

On p. 243, Jones gives James Gregory's series for the arctangent, introduces the symbol π for the ratio of the circumference to the diameter of a circle, and mentions John Machin. Then on p. 263, Jones gives Machin's series (not formula) and states that Machin used it to compute π "True to above a 100 Places." This book is the first to use π in our modern sense; the word "periphery" is used in this context, so it explains the choice of the letter π.

Maseres, Francis
, 1731-1824
Scriptores logarithmici; or a Collection of Several Curious Tracts on the Nature and Construction of Logarithms, vol. 3.
SPEC QA59 .M4 vol.3

It was serendipity that led me to this volume which contains "A most easy and expeditious method of squaring the circle," pp.  155-164, which is an exposition of Machin's work. The diagram and discussion on p. 161 goes a long way to explaining the procedure, but does not explain where the 1/5 comes from in Machin's Formula, which is not even given explicitly here. Looking at his very clever way of doing the computation of π shows that the computation was not as hard as one might originally think; see pp. 162-164. These volumes of Maseres are a real treasure trove and would make a good firstie project. 

Maseres, Francis, 1731-1824
Dissertation on the Use of the Negative Sign in Algebra.
SPEC QA211 .M37

Maseres' believes that this is the first work after Jones 1706 that contains Machin's Series. Maseres gives Machin's proof of Machin's formula and then launches into a computation of π, but instead of computing the 100 places that Machin did, he only computes about 20. But this illustrates all of the main ideas. It appears that the computation, while it does require accuracy and perseverance, is not as hard as only might imagine.

Here is a volume that needs repair. You can see how crude the first attempt was. The library is in need of a good bookbinder, so if you know one, speak up.

Hutton, Charles,
A Treatise on Mensuration, both in Theory and Practice.

Finally we have an explanation of the mysterious 1/5 that occurs in Machin's Formula.

Euler, Leonhard,
Introduction a l'analyse infinitesimale, par Leonard Euler; tranduitee du Latin en Francais, avec des notes & eclaircissements, par J. B. Labey. 1796, 1797, 2 volumes. QA35 .E9 1796

This book has a "Thayer binding," being one of the books he published in Paris in 1815-1817. This is one of Euler's most famous works. The Latin original, Introductio in analysin infinitorum, was published in 1748. The contents of Euler's seven (yes 7) volumes on the calculus are much closer to what we teach today than are the original works of Newton and Leibniz or the rigorous work of Cauchy and Weierstrass. In Euler's calculus the fundamental objects of study are functions (see the table of contents, p. xiij); this does not seem innovative but earlier the concept of a curve was fundamental. Here the trigonometric functions on the unit circle were disseminated to the mathematical community. The logarithmic and exponential functions are treated as inverse functions (Chapter 8, §126, p.92 of the French). Here you will find his summation of the squares of the reciprocals of the integers. This is Euler's "pre-calculus" book – he only uses algebraic methods, no infinitesimal ones. The differential and integral calculus were treated in 2 + 3 additional volumes. Euler's formula is on p. 102, §138. In §142 there is a Machin type identity; earlier, in E74, Euler reduces the search for Machin type identities to the solution of a Diophantine Equations.

The second volume is devoted entirely to analytic geometry and to the classification of curves. Note the special way that Theyer had the plates bound. They are pasted onto blank pages so that they fold entirely out of the volume.

, Roger  (1682-1716)
Harmonia mensurarum, sive analysis & synthesis per rationum & angulorum mensuras promotæ: accedunt alia opuscula mathematica, per Rogerum Cotesium. Edidit et auxit Robertus Smith, SPEC  QA35 .C67

Cotes discovered that the derivative of the sine is the cosine. See p. 3 of the second pagination.

Euler, Leonhard,
Institutionum calculi integralis,
Petropoli : Impensis Academiae Imperialis Scientiarum, 1768-1770. 3 vols: QA308 .E88 1768.

Although Euler has discussed the trigonometric functions in several earlier works, we see here why he really needs them: to solve first order linear differential equations with constant coefficients. See volume 2, §1132, p. 392.

It is very likely that Ferdinand Hassler sold this book and other Latin titles to the Academy in 1825.

Bernoulli, Jakob
Ars conjectandi.
QA3 .B485 1713

This is a seminal work on probability; yet it is the appendix on infinite series that is most interesting to us. The Bernoulli numbers first appear on p. 97, where formulas for sums of powers of integers are given (up to the 10th power). The term Bernoulli's numbers was used for the first time by Abraham De Moivre (1667-1754) and also by Leonhard Euler (1707-1783) in 1755 (Institutiones calculi differentialis, Part 2, section 155. E212). The first part of the work has been translated into English in 2006 by Edith Sylla, who, incidentally, is the great-great-granddaughter of Jared Mansfield, the first professor of mathematics at West Point.

Bernoulli, Jean
, 1667-1748.  
Johannis Bernoulli ... opera omnia, tam antea sparsim edita, quam hactenus inedita
.. , Lausannæ & Genevæ, sumptibus M. M. Bousquet & sociorum.
QA3 .B52 vol. 3

These four volumes constitute the collected works of Johann Bernoulli. The third volume contains his lectures on the integral calculus. These were given in Paris in 1691-1692 to L'Hospital and a footnote on p. 387 states that his lectures on the differential calculus were published by L'Hospital in the first calculus book, 1696. What he does not say is that L'Hospital hired him to do mathematics for him.

We will take special note of volume 3 which contains Bernoulli's lectures on the integral calculus and its footnote dealing with L'Hospital's text.  

Lagrange, Joseph Louis, 1736-1813.

Théorie des fonctions analytiques, contenant les principes du calcul différentiel, dégagés de toute considération d'infiniment petits ou d'évanouissans, de limites ou de fluxions, et réduits à l'analyse algébrique des quantités finies,
Paris, Impr. de la République, prairial an v [1797]. Thayer collection: QA300 .L2 1797. Reprinted in Journal de l'École polytechnique, 9. cahier, t. III (2 p. l., viii, 276 p.)

The title of this work sets the theme: to develop calculus without infinitesimals, limits or fluxions; that is to say, using only the tools of algebra.  It is in this work that Lagrange introduces the  f '(x)  notation for derivatives, p. 14. He uses series to find derivatives; see, e.g., the derivation of the derivative of sine and cosine, p. 27-28. This volume was donated to the academy to honor Williston Fish, USMA 1881. It has a typical early nineteenth century binding and was once owned by Michael Chasles, an excellent geometer.

Staff Records, June 1905.

These are the records of the Academic Board, a committee consisting of the Heads of the Academic Departments.

We shall look for George Patton's name. He was 109/125 in French (p. 197). "Cadet Patton was of doubtful proficiency in Mathematics (conic sections) " (pp. 198, 201). "It was then moved that it be recommended to the War Department that Cadet Paton be turned back to join the incoming 4th class. Carried. Ayes 11, Noes 1, Absent 1." (p. 202). Finally, on p. 216 we find the "Fourth Class arranged according to General Merit." Patton was deficient in mathematics and had scores of 44.22 out of 50 in English, 57.05 out of 75 in French, 39.03 out of 40 in Drill and Regulations, and 46.51 in Conduct. Combining these he was "Deficient" in "General Merit" (p. 216).

Grant, Ulysses S., 1822-1885, USMA 1843.   Two drawings in descriptive geometry.

He uses U. H. in his signature and the drawings are also signed by Professor Church.

Copernicus, Nicholaus        (1473-1543)
Nicolai Copernici Torinensis. Astronomia instavrata, libris sex comprehensa, qui De revolutionibus orbium cælestium inscribuntur / Nunc demum post 75 ab obitu authoris annum integritate suæ restituta, notisque illustrata, opera & studio d. Nicolai Mvlerii.
EARLY EURO IMP-SPEC    QB41 .C76 1617.

When I was looking at the list of books damaged or destroyed in the fire of 1837, my heart sank when I saw "Copernici insturata" for I knew that Copernicus published only one book in his lifetime. I immediately asked to see the copy and I was first given this copy of the third edition, which was indeed damaged. But our copy of the first edition was in good shape. Neither of these works is in the 1822 library catalog, but both are in the 1830, so it is almost certain that they were sold to the Academy by Ferdinand Hassler who taught mathematics at West Point from 1807 to 1809. The diagram of the solar system is on p. 21.

Copernicus, Nicholas

De revolvtionibvs orbium cœlestium
QB41 .C76 1543

This is the book that gives us the word "revolution" in its current political sense. To learn more about the history of this book, read The Book Nobody Read (2004) by Owen Gingerich. Since people are always interested in the value of books, we should note that a first edition of De revolutionibus was sold at Christie's in 2008.

The title of the book ends with the sentence: "Igitur eme, lege, fruere," buy this book, read this book, enjoy this book. In the dedication, anticipating that his work will have detractors, Copernicus writes "Mathematica mathematicis scribuntur" ‒ mathematics is written for mathematicians (folio iv verso). The famous diagram of the solar system, with the sun at the center is on folio 9 verso.

The original manuscript, first edition, and partial English translation are all on line.

Descartes, René,
1596-1650,  Geometria  à Renato Des Cartes, anno 1637 Gallicè edita ; nunc autem cum notis Florimondi de Beavne, in curiâ Blœsensi consiliarii regii, in linguam Latinam versa, & commentariis illustrata ; operâ atque studio Francisci à Schooten,  Lugduni Batavorum : Ex Officinâ Ioannis Maire, 1649.  QA33 .D43 1649 

This is the first Latin edition of the appendix on geometry of Descartes's Discours de la méthode (1637). It is sad that this is the first edition. The second, 1659-1660, is more important for it influenced both Newton and Liebniz. The value of this work is the commentaries and new treatises on analytic geometry. The 1637 original was only 118 pages, this is almost 350, the second edition is nearly 1000 page. Perhaps the most important result in this work was von Heurat's rectification of the semi-cubical parabola, for it led Newton to the Fundamental Theorem of the Calculus. As often happens in early books, the printers make mistakes. In this volume the second and third pages of the index should be before page 1, but they follow page 8. Note the neat diagram for one of Descartes' ovals on p. 61. As to notation, this is the oldest book that one can read with ease (only the equal's sign is unusual).

 Prepared by V. Frederick Rickey, September 2010.