**Rare Book Display for the NSF Workshop on Improving
College Mathematics Teaching Through Faculty Development. **

2010 June 15. Come to the third floor of the Jefferson Library for a show-and-tell about some of the treasures in the collection.

**1570
**

After a book on proportion (pp. 1-171) there is an edition of
the famous *Ars magna* (originally 1445; pp. 1-163 in the second
pagination). The third section of this volume is *De regula aliza *(for a
discussion of this book, see Hutton's *Dictionary*, I, 71).

Chapter II lists the 22 cases of quadratic and cubic equations. The Wilmer translation expresses these in modern symbolism. The later introduction of negative numbers and zero makes this much simpler, thus showing the value of abstraction. Rule II in Chaptere XXXVII is supported by this example: "Divide 10 into two parts the product of which is 30 or 40. It is clear that this case is impossible. Nevertheless we will work thus". The solutions are 5 ± √-15. It is interesting to see the notation that Cardano uses.

**1647
Gregorius a Sancto Vincentio, **
1585-1667, 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.

The frontispiece of the

Opus geometricumis 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 ismore beyondthis 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.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.

**1649
Descartes, René,** 1596-1650,

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 just over 100 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.

**1703
Euclid**,

For information about the frontispiece click here.

**1802
Mansfield, Jared, **1759-1830,
Essays, mathematical
and physical : containing new theories and illustrations of some very important
and difficult subjects of the sciences. Never before published,
New-Haven : Printed by W. W. Morse, [1802].
QA7 .M28

Contents: Of negative quantities in algebra.--Goniometrical properties.--Nautical astronomy.--Of the longitude.--Orbicular motion.--Investigation of the loci.--Fluxionary analysis.--Theory of gunnery.--Theory of the moon.--Appendix: New tables for computing the latitude and longitude at sea, by means of double altitudes and lunar distances.

Copy p. [1] has signature of: "Samuel S. Smith March 13th 1817"; Smith was an 1818 USMA graduate and a mathematics teacher at USMA 1818-1828.

**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 Hasslers' acceptance of 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.

**1850
Davies, Charles**, 1798-1876, USMA 1815

**1842
Church, Albert Ensign**, 1807-1878, USMA 1828

**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."

**
1819
Pascal, Blaise, 1623-1662 , Œuvres
de Blaise Pascal. Paris, Lefèvre, 1819. SPECIAL
COLL B1900 .A2 1819 vol. 1, 2, and 3. **

**1809**

**Gauss, Carl Friedrich**,
1777-1855*Theoria motus corporum coelestium in sectionibus
conicis solem ambientium*. Hamburgi : sumtibus F. Perthes et I.H. Besser,
1809. SPECIAL COLL QB355 .G25

**1857**

**Gauss, Carl Friedrich**,
1777-1855*Theory of the motion of the heavenly bodies moving about the
sun in conic sections: a translation of Gauss's "Theoria motus." With an
appendix*.

Hutton, Charles

**1890
Legendre, Adrien Marie, **1752-1833,
Elements
of geometry and trigonometry : from the works of A. M. Legendre / adapted to the
course of mathematical instruction in the United States by Charles Davies ;
edited by J. Howard Van Amringe, New York : American Book Co. c1890.
QA529 .L43 1890

Note: Would like to have the copy with class list facing back cover. U. S. Grant III and Douglas McArthur are listed on these class lists.

**1801
Agnesi, Maria Gaetana**, 1718-1799.

The Italian original of this work was published the same year as Euler's Introductio but there is little comparison between the books. She wrote it to educate her younger brothers and it was printed at home. The section on equations of straight lines is interesting as she does not have negative numbers and so there are six cases (vol. 1, pp. 90-95). This is a wonderful example of how abstraction makes things easier. Of course the "witch of Agnesi" is here (vol. 1, p. 222; vol. 2, p. 79). That sad term results from a mistranslation by Colson.

**1738
Maupertuis**, (1698-1759)

QB283 .M413 1738

English translation of *La figure de la terre* (1738). Newton argued in
his *Princpia* (1687) that the Earth is an oblate spheroid, but this was
disputed by Cassini. Using data from surveys in Lapland and Peru, Maupertuis
computed the curvature of the Earth in those locations in two different ways and
thus found the constants in the equation of the ellipsoid representing the
Earth. This is explained in Part II, Chapter IX, pp. 164-167 and Figure 17.

**1713
Bernoulli, Jakob** (1654-1705)

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, section155. 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.

**1730
**

QA302 .L613 1730

This is a double translation of the very first (1696) calculus book, L'Hospital'sAnalysis of the Infinitely Small for the Study of Curved Lines;The French is translated into English and the Leibnizian notation into Newtonian. L'Hospital's famous rule is on p. 191.

**1796-1797
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 more 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 trigonometric functions on the unit circle were disseminated to the mathematical community (Chapter 8, §126, p. 92 of the French). The logarithmic and exponential functions are treated as inverse functions. 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.

The preface begins with a comment that is still true today: "Often I have considered the fact that most of the difficulties which block the progress of students trying to learn analysis stem from this: that although they understand little of ordinary algebra, still they attempt this more subtle art."

**1768-1770
Euler, Leonhard,** 1707-1783,
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 §1132, p. 392.

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

**1797
Lagrange, Joseph Louis, 1736-1813.**

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.

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.

Prepared by V. Frederick Rickey, 9 June 2010.