Library Tour for the Center for Lifetime Study
The following items have been especially chosen to fit into the themes that we have dealt with in this course.
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 of Horace, "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.
Diophantus of Alexandria, Diophanti Alexandrini Arithmeticorum libri sex, et De numeris multangulis liber unus ; cum commentariis C. G. Bacheti v. c. & obseruationibus d. P. de Fermat ... Accessit Doctrinae analyticae inuentum nouum, collectum ex varijs eiusdem d. de Fermat epistolis, Tolosae: B. Bosc, 1670. EARLY EURO IMP-SPEC QA31 .D55 1670.
It is in this work that Fermat's Last Theorem appears for the first time.
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.
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. WP THAYER-SPEC 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 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.
In Chapter VIII, p. 92-93, when Euler comments that the circumference of a circle cannot be expressed exactly as a rational number, i.e., that pi is irrational, there is a footnote that reads "Cette proposition a été démontrée par Lambert, Mémoires de Berlin; main on en trouvera une démonstration plus simple dan les Élémens de Géométrie du C. Legenddre, qui ont paru depuis peu." The simple demonstratiion cited here is by Fourier. After giving a decimal approximation of pi, the translation reads "Pour abéger j'écrirai π au lieu de ce nombre". This is a faithful translation of Euler's Latin; he says that he is introducing the symbol. This is clear evidence that this was done independently of William Jones in his Synopsis palmariorum matheseos (1706), p. 243.
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.
Archimedes. Oeuvres d'Archimède, traduites littéralement, avec un commentaire, par F. Peyrard; Suivies d'un mémoire du traducteur, sur un nouveau miroir ardent, et d'un autre memoire de M. Delambre, sur l'arithmétique des Greces. Paris: Chez François Buisson, 1807.
This frontispiece is an engraving of a bust of Archimedes. This volume has a "Thayer binding" and was one of the books that Sylvanus Thayer purchased in France when he was there, 1815-1817.
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.
Euler, Leonhard, 1707-1783, Elements of algebra by Leonard Euler; translated from the French, with the notes of M. Bernoulli, &c. and the additions of M. de La Grange; to which is prefixed a memoir of the life and character of Euler by Francis Horner, London : Printed for Longman, Orme, 1822. SPEC: QA154 .E85 1822
Euler wrote this work, which begins with the elements of algebra and proceeds through quite advanced work on Diophantine equations, in order to train an amenuensis.
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.
Grant, Ulysses S., 1822-1885, USMA 1843. Two drawings 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.
Davies, Charles, 1798-1876, USMA 1815, Elements of algebra : including Strums' theorem / translated from the French of M. Bourdon ; adapted to the course of mathematical instruction in the United States by Charles Davies, New York : A.S. Barnes & Co., 1846, c1844. QA154 .D25 1846
NOTE: Would prefer to have the copy signed by Cas S. Lee
Church, Albert Ensign,1807-1878, USMA 1828, Elements of analytical geometry. New-York, G. P. Putnam, 1851. TEXTBOOKS-SPEC QA551 .C56 1851.
Lewis Carroll (pseud), 1832-1898, An elementary treatise on determinants : with their application to simultaneous linear equations and algebraical geometry by Charles L. Dodgson, London: Macmillan, 1867. SPEC QA191 .C3 1867.
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.
Charlotte Agnas Scott, 1858-1931, An introductory account of certain modern ideas and methods in plane analytical geometry, London and New York, Macmillan, 1894. SPEC QA552 .S42
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.
Omar Khayyam, Rubáiyát of Omar Khayyám, New York : T.Y. Crowell, [1917?], SPEC PK6513 .A1 1917.