American University Rare Book Room

Tartaglia, Niccolo (died 1557), 1568.
Qvesiti et inventioni diverse de Nicolo Tartaglia, di novo restampati con vna gionta al sesto libro, nella quale si mostra duoi modi di redur una città inespugnabile. La divisione et continentia di tvtta l’opra nel seguente foglio si trouara notata. Con privilegio.

• First edition was 1554. An important early work in the science of gunnery. The phrase "point blank" was introduced here, referring to his gunner's quadrant. A partial English translation was published in 1788 as Three bookes of colloquies concerning the arte of shooting : in great and small peeces [sic] of artillerie.

Euclid (fl. ca. 300 BC), 1570.
The elements of geometrie of the most auncient philosopher Euclide of Megara faithfully (now first) translated into the Englishe toung by H. Billingsley, citizen of London; whereunto are annexed certaine scholies, annotations, and inventions, of the best mathematiciens, both of time past and in this our age; with a very fruitfull praeface made by M.I. Dee, specifying the chief mathematicall scieces, what they are, and whereunto commodious; where, also are disclosed certaine new secrets mathematicall and mechanicall, untill these our daies, greatly missed. London : Imprinted by Iohn Daye, 1570. QA31.E8 B6 1570

• The first edition of Euclid to appear in English is a magnificent volume (or two, as it is bound at AU). Most impressive is book XI with it's fold up diagrams, one of which is pictured in Katz's A History of Mathematics, p. 365 (you can make a nice overhead from it).

William Oughtred (1575-1660), 1647.
The key of the mathematicks new forged and filed: together with a Treatise of the resolution of all kinds of affected aequations in numbers. With the rule of compound usury; and demonstration of the rule of false position. And a most easie art of delineating all manner of plaine sun-dyalls. Geometrically taught, by Will. Oughtred.  [There are no copies of this work listed in WorldCat (or OCLC the Online Computer Library Caltagoue), yet there are two copies at American University. Only one has the portrait of Oughtred.]QA33 .O96 1647

• This is the first English edition of Oughtred's Clavis mathematicae (1637), a work which influenced the young Newton.

Rene Descartes (1596-1650), 1649.
Philosophiae naturalis principia mathematica. Lugduni Batavorum : Ex Officina Ioannis Maire, 1649. QA551 .D44 1649

• Bound together here with one of the most important philosophical works of Descartes is the first Latin translation of his Geometrie, the second appendix of his Discourse de la Methode (1637). This work contains commentary by Florimond de Beaune (1601-1652) and Frans van Schooten (1615-1660). It is interesting to compare the bulk of the commentaries in this work with those in the second Latin edition, but you will need to go to the Naval Observatory Library to see a copy.

Diophantus of Alexandria (third century, AD), 1670.
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. AU: LIB Special Collections Rare: QA33.D24 B1 1670 .

• This work contains the first printed statement of Fermat's Last Theorem.  Fermat read the 1620 edition of Diophantus and wrote his famous comments in the margins of that work, now doubt lost. This edition was edited by Fermat's son Samuel. Every middle school teacher should discuss numeris multangulis in their classes. The solution of FLT by Andrew Wiles has been memorialized by Tom Lehrer in his song "That's Mathematics."

Melder, Christiiani
Evclidis elementorvm sex priores libri recogniti opera Christiani Melder, Ludg. Batav. Amst.: Apud Danielem, Abrahamum & Adrianum `a Gaesbeeck, 1673. In Michalowicz collection.

• We shall look at similarities in the frontispieces in this work and the following two. For details, see "Why have a frontispiece? Examples from the Michalowicz Collection at American University," by Fasanelli and Rickey.

Tacquet, Andrea (1612-1660)
Elementa Euclidea geometriæ, planæac solidæ; et selecta ex Archimede theoremata, quibus accedit trigonometria, auctore Andrea Tacquet, Soc. Jesu Sacerdote, & Matheseos Profes­sore. Cum notis, et additamentis Gulielmi Whiston. A.M. matheseos Professoris Lucasiani. Postrema Editio. Cui in aliena manu juventutis accessit ab aliena manu brevis de sectionibus conicis tractatus. Neapoli: Benedicti Gessari, 1744. In Michalowicz collection.

Chambers, Ephriam (1680?-1740)
Cyclopædia: or, an universal dictionary of arts and sciences. . . . By E. Chambers, F.R.S. With the supplement and modern improvements, incorporated in one alphabet. By Abraham Rees, . . . In four volumes. . . . . London: printed for J. F. and C. Rivington, A. Hamilton, T. Payne and Son, W. Owen, B. White and Son [and 24 others in London], 1786-88. Volume 1 only is in the Michalowicz collection.

Euler, Leonhard (1707-1783), 1748.
Introductio in analysin infinitorum.  Provenance: donated by Bern Dibner; signed by Michaëlis Angeli Giacomelli; stamped with: Libreria Antiquaria S. Bocca, Font. Borohesse 27-Roma.  QA33 .S87 1748b .

This is one of Euler's most famous works. The contents of Euler's seven (yes 7) volumes on the calculus are much closer to what we teach today than are the original works of Newton and Leibniz or the rigorous work of Cauchy and Weierstrass. In Euler's calculus the fundamental objects of study are functions (see the table of contents, p. xiij); 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. The logarithmic and exponential functions are treated as inverse functions (Chapter 8, §126). 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 in §138. In §142 there is a Machin type identity for computing π; earlier. The second volume is devoted entirely to analytic geometry and to the classification of curves.

 Euler, Leonhard (1707-1783), 1818.
An Introduction to the Elements of Algebra, Designed for the Use of those Who are Acquainted Only with the First Principles of Arithmetic. Selected from the Algebra of Euler [by John Farrar].  QA152 .F243 1818 .

This work appeared first in Russian, then in German. AU holds 7 English editions, so this would be a good example to study the changes in editions.

Newton, Isaac (1642-1727), 1769.
Universal arithmetick: or, A treatise of arithmetical composition and resolution. Written in Latin by Sir Isaac Newton. Translated by the late Mr. Ralphson; and rev. and cor. by Mr. Cunn. To which is added, a treatise upon the measures of ratios, by James Maguire, A.M. The whole illustrated and explained, in a series of notes, by the Rev. Theaker Wilder, London, Printed for W. Johnston, 1769. AU: LIB Special Collections Martin: QA35 .N564 .

• Newton's most widely read work.

Maria Gaetana Agnesi (1718-1799), 1801
Analytical institutions in four books: originally written in Italian / By Donna Maria Gaetana Agnesi ... Tr. into English by the late Rev. John Colson ... Now first printed, from the translator’s manuscript, under the inspection of the Rev. John Hellins, London : Printed by Taylor and Wilks, 1801. AU: LIB Special Collections: QA35 .A2.

• A wonderful work. It should be reprinted. For details see my "Agnesi vs. Euler: Out with the old, in with the new."

L'Hospital, Guillaume (1661-1704), 1730.
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, London: Printed for William Innys, 1730.  AU: LIB Special Collections Martin:  QA302 .S877 1730

• This is an English translation of the first calculus book. It is a double translation: From French to English and from Leibnizian notation to Newtonian.

Playfair, John (1748-181), 1814
Elements of geometry: containing the first six books of Euclid, with a supplement on the properties of the circle, the intersections of planes, and the geometry of solids / by John Playfair, 2d American ed. with improvements.  Boston, Printed by T. B. Wait and Sons for F. Nichols, 1814.  AU: LIB Special Collections Martin:  QA451 .P6 1814

• A famous geometry book that appeared in many editions.

J.-L. (Jean-Louis) Boucharlat (d. 1848), 1828.
An elementary treatise on the differential and integral calculus. Cambridge : W. P. Grant, 1828. Translation of Elemens de calcul differentiel et de calcul integral. QA303 .B752 1828

• Chosen as an example of the many calculus texts in the Martin Collection at AU. This one is important in that it was used as a textbook at the United States Military Academy at West Point. Being a translation, it shows the French influence on American mathematics early in the nineteenth-century.

Euclid, edited by Byrne, Oliver (1810-1880), 1847.
The first six books of the elements of Euclid, in which coloured diagrams and symbols are used instead of letters for the greater ease of learners, London, W. Pickering, 1847. QA451 .B9 1847 .

• The entire text is on the web. The AU library has a dozen books by Byrne. This one is expensive as the publisher has become famous for excellent printing. But it has recently been reprinted and so you can buy your own copy.

Davies, Charles (1798-1876), 1847.
Arithmetic: designed for academies and schools, uniting the inductive reasoning of the French with the practical methods of the English system, with full illustrations of the method of cancellation. Special Collections Martin. QA103 .D25 1847

• Charles Davies was department head of mathematics at the U.S. Military Academy at West Point from 1823 to 1837. He became America's most prolific author of mathematics textbooks in the mid-nineteenth century.

Carroll, Lewis, (1832-1898), 1879. 
Euclid and his modern rivals, London, Macmillan and Co., 1879. QA28 .D6

• This work has been reprinted by Dover. Carroll's monogram is on the half-title.

Rufus Fuller, 1893
A double discovery. The square of the circle. Boston, MA, Printed for the author, 1893. QA467 .F96 1893

• This is an example of crank literature. The first clue is that the frontispiece is a picture of the author. The second is that the work is "Printed for the author." The third is that Diagram 16 claims that pi equals 3949/27889 exactly. Serious mathematicians don't do such things. Nonetheless, the book does have value as a curiosity. Students should be shown things like this as a simple exercise in sorting the wheat from the chaff in the vast mathematical literature. Individuals interested in this topic should consult Mathematical Cranks (1994) and The Trisectors (1994), both by Underwood Dudley.