**Mathematical Treasures in the West Point Library**

This show-and-tell session in the Jefferson Library, which is specifically designed for cadets in MA100, is scheduled for Friday 5 November 2010 during the third and fourth hours and again during Dean's hour.

**1703
Euclid**,

This book has a stunning frontispiece that relates that mathematics is a civilized endeavor done by individuals. For more information about the frontispiece click here.

**1807
Archimedes**

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 first
proposition of *The Measurement of the Circle*, pp. 116-122. He is
finding a "formula" for the area of a circle.

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.

**Euler, Leonhard,** 1707-1783, *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

Thayer binding. 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 work of Newton and Leibniz or the rigorous work of Cauchy and Weierstrass. In Euler's calculus the fundamental objects of study are functions; this does not seem innovative but earlier the concept of a curve was fundamental. Here the trigonometricfunctionson the unit circle were disseminated to the mathematical community. Euler introduces the unit circle and also the symbol π. 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.

**1706
Jones, William**, 1675-1749

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. However, this book was not "discovered" until 1895; before that Euler was given credit for introducing the symbol π

**1825
Lacroix, Silvestre François**,
1765-1843

**1826
Hassler, Ferdinand Rudolph**,
1770-1843, Elements of analytic trigonometry:
plane and spherical, New York: The author,
1826.
QA531 .H35 1826

One Special Collections' copy contains tipped in copy of the author's copyright papers for this work as well as letter dated 1807 regarding Hassler's acceptance of the post as Professor of Mathematics US Military Academy.

**1836
Davies, Charles, **1798-1876,
USMA 1815, Elements of the differential and
integral calculus, New York : Wiley & Long,
1836. QA303
.D249 1836

The Preface begins ominously: "The Differential and Integral Calculus is justly considered the most difficult branch of the pure Mathematics. .. . . it cannot be mastered without patient and severe study." His derivation of the derivative of the sine function is similar to what we do today (pp. 66-68). Then when the derivative formulas are collected together, a student has drawn two hands pointing to the formulas, and added the comment "To be remembered." This is good advice. On pages 77-78, the proof of Machin's formula is sketched and then it is used to approximate π. A student has corrected several misprints.

**1890
Legendre, Adrien Marie**, 1752-1833.

Cover incised "Grant, U.S. West Point N `02." There is a poem on the endcover.

* Staff Records*, June 1905.

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).

1817-1818 **
Delafield, Richard, 1798-1873, USMA 1818**,

Drawings in Descriptive Geometry.

This volume of drawings was a new acquisition in 1990. The volume is about 24'' wide and 12'' high and about half an inch thick. It bears a paper label indicating that it is "The first book of geometry used in the United States," a claim which is almost certainly false, for Euclid would have been used previously at, e.g., Harvard. The front endpaper carries a comment closer to the truth: "This book was the first book ever gotten up in Descriptive Geometry in US. Prof. Crozet used no textbook, instructing his pupils by lecture only." There is an inventory of these drawings by Cadet Richard C. Bell, USMA `93. These should be matched up with Crozet's book on descriptive geometry. Delafield never taught at the academy; this is further evidence that these are cadet drawings; some of them are dated 1818 (e.g., a drawing of "The ionic order" is dated march 31st 1818).

**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.

**1906
Smith, Charles**, 1844-1916.

Various editions of this book were used at West Point from 1899 to 1919. This copy was used by William Cooper Foote, USMA 1913. The lessons covered in 1909-1910 are written on the front endpapers. Note the many handwritten notes and "mimeographed" interpolations. This text was much maligned by cadets. The 1914 Howitzer, p. 18, has a sketch of a cadet holding his copy of Smith and being carried off to the Insane Asylum for Hopeless Cases. In this copy, Foote has underlined some phrases in the preface such as "very easy" and "as simple a manner as possible for the benefit of beginners," and then he has added his own "NOTE: This book is the best known example of biting sarcasm and bitter irony in the world."

**1617
Copernicus, Nicholaus ** (1473-1543)

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.

**1543
Copernicus, Nicholas**

De revolvtionibvs orbium cœlestium

QB41 .C76 1543

When I was looking at the list of books damaged or destroyed in the fire of 1838, 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 the third edition, which was indeed damaged. But the first edition was in good shape. This is the book that gives us the word "revolution" in its current political sense. To learn more about the history of this book, readThe Book Nobody Read(2004) by Owne Gingerich. Since people are always interested in the value of books, we should note that a first edition ofDe revolutionibuswas sold at Christie's recently.

The famous diagram of the solar system, with the sun at the center is on page 9 verso. The title of the book ends with the sentence: "Igitur eme, lege, fruere," buy this book, read this book, enjoy this book.

Prepared October 2010. If you have questions, contact Professor V. Frederick Rickey.