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.

Cardano, Girolamo, 1501-1576
Hieronymi Cardani ... Opus nouum de proportionibus numerorum, motuum, ponderum, sonorum, aliarvmque rerum mensurandarum, non solum geometrico more stabilitum, sed etiam uarijs experimentis & obseruationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, & in v libros digestum. Praeterea Artis magnæ, sive de regulis algebraicis, liber vnus, abstrusissimus & inexhaustus plane totius arithmeticæ thesaurus, ab authore recens multis in locis recognitus & auctus. Item. De aliza regula liber, hoc est, algebraicæ logisticæ suæ, numeros recondita numerandi subtilitate, secundum geometricas quantitates inquirentis, necessaria coronis, nunc demum in lucem edita. Basileæ: Ex Officina Henricpetrina, 1570.   EARLY EURO IMP-SPEC   QA33 .C27 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.

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.

, Eukleidou ta sōzomena = Euclidis quæ supersunt omnia / ex recensione Davidis Gregorii, M.D., Astronomiæ Professoris Saviliani, & R.S.S., Oxoniæ : E Theatro Sheldoniano, 1703. SPECIAL COLL.  QA31 .E86 1703.

For information about the frontispiece click here.

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.

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.

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.

Davies, Charles
, 1798-1876, USMA 1815 The logic and utility of mathematics, with the best methods of instruction explained and illustrated. By Charles Davies, LL.D.  New York, A.S. Barnes & co.; Cincinnati, H.W. Derby & company, 1850.   SPECIAL COLL  QA9 .D24

The work ends with notes on what the mathematical curriculum should include. It mirrors what has been taught at West Point during the second quarter of the nineteenth century: Arithmetic, algebra, geometry, plane and spherical trigonometry, surveying and leveling, descriptive geometry, shades, shadows, and perspective, analytic geometry and the differential and integral calculus (pp. 341-351). This curriculum matches the series of books that Davies wrote (see an advertisement at the end).


Church, Albert Ensign
, 1807-1878, USMA 1828, Elements of the differential and integral calculus. New York, Wiley and Putnam, 1842   SPECIAL COLL   QA303 .C558 and TEXTBOOKS-SPEC   QA303 .C558 

Church uses a variant of Machin's formula to compute π. There is no discussion of convergence. In the discussion of max-min problems, he rightly points out that the hard part is finding an equation to relate the variables. He correctly comments that "No general rule can well be given by which this expression can be found (p. 91). In contrast with modern books, he gives only a dozen examples.

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. 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."

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



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



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. Boston, Little, Brown and Co., 1857. Davis, Charles Henry, 1807-1877, Translator. SPECIAL COLL  QB355 .G27


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.   WP THAYER-SPEC Q121 .H9 1795  vol. 1 and 2.


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.

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 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.

,  (1698-1759)
The figure of the earth : determined from observations made by order of the French King, at the Polar circle / by Messrs de Maupertuis, Camus, Clairaut, Le Monnier, the Abbé Outhier and Mr. Celsius. London : Printed for T. Cox ... [and 3 others], 1738.
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.

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.

L'Hospital, marquis de (Guillaume François Antoine)  (1661-1704)
The method of fluxions both direct and inverse: the former being a translation from the celebrated Marquis De L'Hospital's Analyse des infinements petits / and the latter supply'd by the translator, E. Stone.
 QA302 .L613 1730

This is a double translation of the very first (1696) calculus book, L'Hospital's Analysis 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.

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."

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.

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.

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.