The best material model of a cat is another, or preferably the same, cat.
Arturo Rosenblueth and Norbert Wiener, Philosophy of Science, 1945.
During the first half of the twentieth century, David Eugene Smith (1860-1944) was a moving force in the world of mathematics education. As the chairman of the mathematics education department at Columbia University’s Teachers College, Smith led the way in teaching reforms attuned to the Progressive Education Movement. He firmly believed that the teaching of mathematics should be closely associated with the history of the subject. As an historian of mathematics, he wrote and lectured widely on the subject and also collected historical mathematical materials: texts, documents and artifacts. Smith befriended the wealthy New York book publisher and bibliophile, George Arthur Plimpton (1855-1936). While Plimpton was an avid collector of materials from the liberal arts that comprised “the tools of learning” for western civilization, under Smith’s influence Plimpton greatly enriched his collection with mathematical manuscripts and many early Renaissance texts on arithmetic. When Plimpton died in 1936, he bequeathed his collection to Columbia University. Similarly, beginning in 1931, David Eugene Smith began donating his extensive collection of mathematical memorabilia: historical texts; correspondence; portraits of famous mathematicians; signatures and concrete artifacts to the Columbia University Library.
Today, these two collections exist as rich resources for understanding the development of matlhematics and the lives and work of many of the persons responsible for its advance. These archives are available to researchers through the the Rare Book and Manuscript Collection at Columbia University. The Mathematical Association of America, in cooperation with the Columbia University Libraries, is pleased to display a selection of items, Mathematical Treasures, from these two separate collections. The editors of Convergence would like to particularly thank Dr. Michael Ryan, Director of Rare Books and Manuscripts, and Jennifer Lee, Librarian for Public Service and Programs, for their assistance in making this display possible.
We present here a very few items from these two collections. More will be posted over the next several weeks. Each item is posted in this article at the standard web resolution of 72dpi. If you right-click on the name of the item, found in the first sentence of the item description, you can download the item as a tif file at a resolution of approximately 200dpi as well. (Since these files are in the range of 3-5mb, the downloading may take some time.) That version should be suitable for most purposes in a classroom setting. If you want a version in even higher resolution, please contact the editors. (Note that if you just click on the name of the item, you will get the tif file on your screen, again after a long wait, but you may not be able to save it to your computer.)
Plimpton 322, an Old Babylonian tablet from Larsa, has four columns of numbers, two of which, most experts believe, contain, in each of the fifteen rows, two of the three numbers in a Pythagorean triple. This tablet was first analyzed by Otto Neugebauer and Abraham Sachs in their 1945 book, Mathematical Cuneiform Texts (New Haven, American Oriental Society). There have been numerous discussions of this tablet since that time. In particular, two articles, "Sherlock Holmes in Babylon," (1980) by R. Creighton Buck, and "Words and Pictures: New Light on Plimpton 322," (2002) by Eleanor Robson are included in Marlow Anderson, Victor Katz, & Robin Wilson, eds., Sherlock Holmes in Babylon and Other Tales of Mathematical History (Washington: Mathematical Association of America, 2004), pp. 5-26. Further references to the literature are included in those two articles. More recently, Jöran Friberg, in A Remarkable Collection of Babylonian Mathematics Texts (New York: Springer, 2007) (pp. 433-452) has challenged the interpretation of the numbers on the tablet as parts of Pythagorean triples.
Emilie du Chatelet portrait
This portrait of Émilie du Châtelet is from David Eugene Smith's extensive collection of portraits, but is not easily found elsewhere. Du Châtelet (1706-1749) for many years was the one person in France with whom Voltaire could easily discuss philosophical and scientific issues, and they co-authored a book dealing with Newton's philosophy. She is most famous for her translation of Newton's Principia into French, the only such translation for many years thereafter. For a detailed biography and access to other portraits of her, click here.
Charles Dodgson portrait
This photograph of Charles Dodgson (Lewis Carroll) (1832-1898), from the David Eugene Smith collection, was taken by Dodgson himself. Although Dodgson is best known for his Alice series of books, he wrote several mathematics texts and stated and proved a very general theorem specifying the nature of the set of solutions to an arbitrary system of linear equations. For more biographical information, click here.
Ars Magna Title Page
This is the title page from the Ars Magna, by Gerolamo Cardano, published in 1545. The page contains an engraving of a portrait of the author. This is the text in which the algebraic solution of cubic equations was first printed. Although the initial discovery of the solution of one type of cubic equation was due to Scipione del Ferro, and solution methods for at least three types were worked out by Niccolò Tartaglia, it was Cardano, along with his student Lodovico Ferrari, who worked out the details for thirteen cases and then published them in this book. The book also contains Ferrari's basic method of solution for quartic equations, as well as much else.
The title page may be translated as follows:
Book one of The Great Art, or the Rules of Algebra, by Gerolamo Cardano, most distinguished mathematician, philosopher, and physician, which is the tenth in order of the whole work on arithmetic, which is titled the perfect work.
You have in this book, diligent reader, the rules of algebra (in Italian, called rules of the coss), so abounding in new discoveries and demonstrations by the author, more than seventy, that earlier works now amount to little (or, in the vernacular, are washed out). It disentanbles the knots not only where one term is equal to another or two to one, but also where two are equal to two or three to one. It is a pleasure, therefore, to publishe this book separately so that, this most abstruse and unsurpassed treasury of all of arithmetic being brought to light, and as, in a theater exposed to the sight of all, its readers may be encouraged and will all the more readily embrace and with less aversion study thoroughly the remaining books of this perfect work, which will be published volume by volume.
(Translation adopted from that of T. Richard Witmer, The Great Art (Cambridge: MIT Press, 1968.)
This is a page from al-Khwarimi's algebra text, Kitab al-jabr wa l-muqabala, written in about 825, the first extant algebra text, by Muhammad ibn Musa al-Khwarizmi. This copy itself is undated, however. It corresponds to page 15 in the translation by Frederic Rosen: The Algebra of Muhammed ben Musa (London: Oriental Translation Fund, 1831), which is also available in a reprinting in the series on Islamic Mathematics and Astronomy, from the Institute for the History of Arabic-Islamic Science at the Johann Wolfgang Goethe University, Frankfurt am Main. On this page is al-Khwarizmi's proof of the rule for solving a quadratic equation of the form "squares plus roots equal numbers" (x2 + bx = c). The central square in the diagram represents the square on the unknown. The four rectangles on the four sides of the square each have width b/4. Thus the area of the central square plus the four rectangles is c. The square is then completed by adding the four corner squares, each of side b/4. Thus, the area of the large square is, in modern notation, x2 + bx + b2/4 = (x + b/2)2, and this is in turn equal to c + b2 /4. The solution to the equation is then evident.
Lilavati of Bhaskara
This is a page from a manuscript of the Lilavati of Bhaskara II (1114-1185). This manuscript dates from 1650. The rule for the problem problem illustrated here is in verse 151, while the problem itself is in verse 152:
These verses and much else from the Lilavati may be found in Kim Plofker, "Mathematics in India", in Victor Katz, ed., The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook (Princeton: Princeton University Press, 2007), pp. 385-514.
De Divina Proportione, by Luca Pacioli
This is the Tree of Proportions and Proportionality from the De Divina Proportione of Luca Pacioli, published in 1509. (Convergence has published Leonardo's woodcuts, taken from this book.) Some of the terms in Pacioli's tree are familiar today; some are taken originally from the study of proportions by Nicomachus in his Arithmetic; but the meaning of some other terms are not generally known. It would be a worthwhile project to go through the tree and decipher the meanings of all of the terms in it.
Robert Recorde's Pathway to Knowledge
This is the first page of Pathway to Knowledge by Robert Recorde (1510-1558). The book is essentially a version of Euclid's Elements, with commentary by Recorde to make it easier to understand. On this page are Euclid's first three postulates, that one can draw a line between any two points, that one can extend a straight line, and that one can draw a circle with a given center and a given radius.
The Treviso Arithmetic
This is a page from the Treviso Arithmetic (1478), the earliest known example of a printed book on arithmetic. The work has no title, and no author's name is found anywhere in the book. It was printed in Treviso, a town about 26 km northwest of Venice. In many book catalogues, it is called the Arte del Abbaco (Art of Calculation), because it demonstrates not only how to use the Hindu-Arabic decimal place-value system, but how to solve numerous practical problems, mostly using the rule of three. The problem that begins in the middle of the left-hand page of this illustration is the following:
The first calculation for this problem, toward the bottom of the page, converts both amounts of money into grossi, given that there are 24 grossi in a ducat. The next instruction is to multiply each partner's amount by the length of time it was invested; so on the next page, Sebastiano's amount is multiplied by 24 and Jacomo's by 18.
A complete translation and analysis of the Treviso Arithmetic can be found in Frank Swetz, Capitalism and Arithmetic: The New Math of the 15th Century (La Salle, IL: Open Court, 1987). For further information on the Treviso Arithmetic, click here.
On Triangles, by Regiomontanus
This is the title page of On Triangles by Regiomontanus (Johannes Müller) (1436-1476). Although the work was written in 1464, it was not published until 1533. The page can be translated as follows:
A complete translation and analysis of On Triangles is available in Baranabas Hughes, trans., Regiomontanus on Triangles (Madison: University of Wisconsin Press, 1967).
The George Arthur Plimpton Collection
The Plimpton library was formally presented to Columbia University in 1936 shortly before the donor's death. The collection of more than sixteen thousand volumes was assembled by George Arthur Plimpton who served as a board member of the textbook publisher Ginn & Company, to show the development of "our tools of learning." He stated his notable purpose in the preface to his The Education of Shakespeare as "the privilege to get together the manuscripts and books which are more or less responsible for our present civilization, because they are the books from which the youth of many centuries have received their education." In general, the Plimpton Library may be described as an assemblage of notable treatises on the liberal arts, particularly grammar, rhetoric, arithmetic, algebra, geometry, geography, astronomy and handwriting. Represented in the Library are the forms of knowledge from the most rudimentary, the hornbook, to the most sublime heights reached in the writings of Aristotle, Donatus, Cicero, Boethius, Euclid, Ptolemy, Pliny and Petrus Lombardus. It is hardly surprising that one of the earliest items in the collection may be the most remarkable, a cuneiform clay tablet on which is written in Old Babylonian (1900-1600BCE) script a mathematical listing of Pythagorean triples.
The David Eugene Smith Collection
When Smith began giving his collection to the Columbia University Libraries in 1931, it included 12,000 printed books on the history of mathematics, ranging from the 15th through the 20th century. It also included 35 boxes of historical documents relating to mathematics; 140 boxes of his own professional papers; 350 volumes of western European manuscripts dating from the 15th to the early 20th century; 670 volumes of Oriental (primarily Arabic and Persian) manuscripts dating from the 8th to the early 20th century; 88 volumes of Chinese manuscripts; 363 volumes of Japanese