Rare Books at West Point

The following 25 books were chosen for a show and tell to Professor Shell-Gellasch's The History of Mathematics from a West Point Perspective course and Professor Rickey's Analysis class on April 3-4, 2003.  A few of the books below were chosen for their rarity, beauty, and mathematical significance, several more were chosen because they have been discussed in class, and many of them were also chosen because of the impact they have had on your education. 

You can access the West Point library on the web by
            http://www-internal.library.usma.edu/search/
Not every book in the collection is in this catalog. For a fuller list of the mathematical works, see A Station Favorable to the Pursuits of Science: Primary Materials in the History of Mathematics at the United States Military Academy, by Joe Albree, David C. Arney, and V. Frederick Rickey (who are not pictured on the back cover of the book). 

The books we shall look at are listed here in chronological order.

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 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.

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.

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 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.

Euclid,  Eukleidou ta sozomena = Euclidis quœ supersunt omnia / ex recensione Davidis Gregorii, M.D., Astronomiœ Professoris Saviliani, & R.S.S,  Oxoniœ : E Theatro Sheldoniano, 1703.  QA31 .E86 1703 

This work, which is edited by David Gregory, is the first collected works of Euclid. The text is in two columns, Latin at the foredge and Greek adjacent to the gutter. The reason for choosing this work is its wonderful frontispiece. A very similar frontispiece appeared in a 1710 edition of Apollonius.

Newton, Isaac, Sir, 1642-1727,  Sir Isaac Newton's Two treatises: Of the quadrature of curves, and Analysis by equations of an infinite number of terms, explained: containing the treatises themselves, translated into English, with a large commentary: in which the demonstrations are supplied where wanting, the doctrine illustrated, and the whole accommodated to the capacities of beginners, for whom it is chiefly designed. By John Stewart.  London, Printed by J. Bettenham, at the expence of the Society for the Encouragement of Learning; and sold by J. Nourse, 1745.  QA35 .N57 1745

Bernoulli, Jean, 1667-1748.  Johannis Bernoulli ... opera omnia, tam antea sparsim edita, quam hactenus inedita .. , Lausannæ & Genevæ, sumptibus M. M. Bousquet & sociorum, 1742. 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.

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.

Thayer binding. Not in the catalog. 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 curves were 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. Here you will find his summation of the squares of the reciprocals of the integers. This is Eulers "pre-calculus" book --- he only uses algebraic methods, no infinitesimal ones --- The differential and integral calculus were treated in 2 + 3 additional volumes.

Hutton, Charles, 1737-1823,  A mathematical and philosophical dictionary: containing an explanation of the terms, and an account of the several subjects, comprized under the heads mathematics, astronomy, and philosophy both natural and experimental: with an historical account of the rise, progress, and present state of these sciences: also memoirs of the lives and writings of the most eminent authors, both ancient and modern, who by their discoveries or improvements have contributed to the advancement of them ... With many cuts and copper-plates, London, Printed by J. Davis, for J. Johnson; and G. G. and J. Robinson, 1796-95.  Thayer: Q121 .H9 1795  2 volumes.

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.)

It is in this work that Lagrange introduces the  f '(x)  notation for derivatives.

Journal de l'Ecole polytechnique. Paris : École polytechnique, 1798- . ANNEX-PERSHING CNTR. LIB. HAS vol. 4(1799).

Note: My memory is that we have a long run of this; in fact I am sure of this for I looked at volume 9 a few years ago.  I would like to look at just a couple of volumes, say 1, 9 and 15, but it does not matter much which ones.

Agnesi, Maria Gaetana, 1718-1799. Analytical institutions : in four books / originally written in Italian by Donna Maria Gaetana Agnesi ; translated into English by John Colson ; now first printed, from the translator's manuscript, under the inspection of the John Hellins. London : Printed by Wilks and Taylor, 1801.  QA35 .A2713 1801  2 volumes.

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 four cases. This is a wonderful example of how abstraction makes things easier. Of course the "witch of Agnesi" is here. That sad term results from a mistranslation by Colson.

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.

Gauss, Carl Friedrich, 1777-1855,  Recherches arithmétiques, par M. Ch.- Fr.-Gauss, traduites par A.-C.-M. Poullet-Delisle,  Paris, Courcier, 1807.  QA241 .G28 

The Disquisitiones arithmeticae (1801) is justly one of the most famous and influential books in number theory. This French translation is quite rare. The book begins with the definition of congruence and treats its basic properties. The last chapter shows that if  n  is a power of two times one or more distinct Fermat primes, then one can construct --- with Euclidean tools --- a regular n-gon. The discovery of the 17-gon is what induced the young Gauss to choose mathematics over philology. Gauss did not --- although this is often got wrong --- show that certain regular polygons are non-constructible. That is due to Wantzel in 1837.

Garnier, Jean Guillaume, 1766-1840, Geometrie analytique ou application de l’algebra a la geometrie, Paris: Courcier, 1813. 

Thayer binding.  Not in the catalog.

Fourier, Jean Baptiste Joseph, Théorie analytique de la chaleur. Paris : F. Didot, 1822.  QC254 .F55 1822 

For you students of analysis, this work was seminal for much of nineteenth century mathematics. In this work on the conduction of heat, Fourier introduces his series.

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.

Poisson, Simeon-Denis. Théorie mathématique de la chaleur, Paris : Bachelier, 1835. QC254 .P7

Monge, Gaspard, 1746-1818, Géométrie descriptive, Paris, Bachelier, 1827. 5. éd., augm. d'une théorie des ombres et de la perspective, extraite des papiers de l'auteur, par M. Brisson .  QA501 .M74 1827 

Bourdon, Louis Pierre Marie, 1779-1854,  Elements of algebra tr. from the French of M. Bourdon, for the use of the cadets of the U. S. Military Academy. By Lieut. Edward C. Ross,  New York: E. B. Clayton, 1831.  QA154 .B77 1831 

Edward Ross taught mathematics at USMA as a first class cadet in 1820-1821, and stayed on till 1833, serving as "principal assistant professor" under Charles Davies. Later, 1840-1848, he taught at Kenyon College in Ohio, where Rutherford B. Hayes was one of his students. He ended his career at CCNY. This book was used as a text at USMA from 1832 to 1839, being replaced by another "translation" by Davies --- and it was used until 1900! It would be an interesting project to compare the different editions and translations of Bourdon.

Olivier, Théodore, 1793-1853,  Cours de géométrie descriptive,  Paris, Carilian-Goeury et V. Dalmont, 1852-53,  2. éd.  QA501 .O48 1852  atlas and text.

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

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. The last time I asked about it, it was not located.

Grant, Ulysses S., 1822-1885, USMA 1843.  A drawing in descriptive geometry.

He uses U. H. in his signature and the drawing is also signed by Professor Church. This was out last fall. Sorry, but I don't know how to describe these things more carefully.

Church, Albert Ensign, 1807-1878, USMA 1828,  Elements of the differential and integral calculus. Arranged by Albert E. Church,  New York, Wiley and Putnam, 1842.  QA303 .C558 

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.

Smith, Charles, 1844-1916.  An elementary treatise on conic sections. London ; New York : Macmillan, 1906. Textbooks: QA485 .S62 1906.

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.

Prepared by V. Frederick Rickey, 31 March 2003.