## Elementa geometriae. [Translated from the Arabic by Adelard of Bath (c. 1080-c. 1152). Edited by Giovanni Campano da Novara (1220-96).]

Venice: Erhard Ratdolt, 25th May 1482.

First edition of the “oldest mathematical textbook still in common use today” (PMM), This book “has exercised an influence upon the human mind greater than that of any other work except the Bible” (*DSB*). Euclid’s *Elements* is the only work of classical antiquity to have remained continuously in print, and to be used continuously as a textbook from the pre-Christian era to the 20th century. It is the foundation work not only for geometry but also for number theory. Euclid’s *Elements of Geometry* is a compilation of early Greek mathematical knowledge, synthesized and systematically presented by Euclid in ca. 300 BC. Books I-IV are devoted to plane geometry, Book V deals with the theory of proportions, and Book VI with the similarity of plane figures. Books VII-IX are on number theory, Book X on commensurability and incommensurability, Books XI-XII explore three dimensional geometric objects, and Book XIII deals with the construction of the five regular solids. The text is the standard late-medieval recension of Campanus of Novara, based principally on the 12th-century translation from the Arabic by Adelard of Bath. In fact, Adelard left three Latin versions of Euclid. Campanus’s text is a free reworking of earlier Latin translations, mainly Adelard’s second version (an abbreviated paraphrase), with additional proofs that make it “the most adequate Arabic-Latin Euclid of all … With an eye to making the *Elements *as self-contained as possible, he devoted considerable care to the elucidation and discussion of what he felt to be obscure and debatable points” (DSB). This text was printed more than a dozen times in the late-15th and 16th century. The “decisive influence of Euclid’s geometrical conception of mathematics is reflected in two of the supreme works in the history of thought, Newton’s *Principia* and Kant’s *Kritik der reinen Vernunft*” (DSB). Ratdolt’s edition is one of the most beautifully printed of early scientific books, and is the first dated book with diagrams (Stillwell). His method of printing diagrams in the margins to illustrate a mathematical text became a model for much subsequent scientific publishing. The method used to is still a matter of scholarly debate: although traditionally described as woodcuts, it is probable that printer’s “rules” were used, i.e., thin strips of metal, type high, which were bent and cut and adjusted and set into a substance that would hold them (and pieces of type) in place.

Born ca. 300 BC in Alexandria, Egypt, “Euclid compiled his *Elements* from a number of works of earlier men. Among these are Hippocrates of Chios (flourished *c.* 440 BC), not to be confused with the physician Hippocrates of Cos (*c.* 460–375 BC). The latest compiler before Euclid was Theudius, whose textbook was used in the Academy and was probably the one used by Aristotle (384–322 BC). The older elements were at once superseded by Euclid’s and then forgotten. For his subject matter Euclid doubtless drew upon all his predecessors, but it is clear that the whole design of his work was his own …

“Euclid understood that building a logical and rigorous geometry depends on the foundation—a foundation that Euclid began in Book I with 23 definitions (such as “a point is that which has no part” and “a line is a length without breadth”), five unproved assumptions that Euclid called postulates (now known as axioms), and five further unproved assumptions that he called common notions. Book I then proves elementary theorems about triangles and parallelograms and ends with the Pythagorean theorem …

“The subject of Book II has been called geometric algebra because it states algebraic identities as theorems about equivalent geometric figures. Book II contains a construction of “the section,” the division of a line into two parts such that the ratio of the larger to the smaller segment is equal to the ratio of the original line to the larger segment. (This division was renamed the golden section in the Renaissance after artists and architects rediscovered its pleasing proportions.) Book II also generalizes the Pythagorean theorem to arbitrary triangles, a result that is equivalent to the law of cosines. Book III deals with properties of circles and Book IV with the construction of regular polygons, in particular the pentagon.

“Book V shifts from plane geometry to expound a general theory of ratios and proportions that is attributed by Proclus (along with Book XII) to Eudoxus of Cnidus (*c.* 395/390–342/337 BC). While Book V can be read independently of the rest of the *Elements*, its solution to the problem of incommensurables (irrational numbers) is essential to later books. In addition, it formed the foundation for a geometric theory of numbers until an analytic theory developed in the late 19th century. Book VI applies this theory of ratios to plane geometry, mainly triangles and parallelograms, culminating in the “application of areas,” a procedure for solving quadratic problems by geometric means.

“Books VII–IX contain elements of number theory, where number (*arithmos*) means positive integers greater than 1. Beginning with 22 new definitions—such as unity, even, odd, and prime—these books develop various properties of the positive integers. For instance, Book VII describes a method, *antanaresis* (now known as the Euclidean algorithm), for finding the greatest common divisor of two or more numbers; Book VIII examines numbers in continued proportions, now known as geometric sequences (such as *ax*, *ax*^{2}, *ax*^{3}, *ax*^{4}, …); and Book IX proves that there are an infinite number of primes.

“According to Proclus, Books X and XIII incorporate the work of the Pythagorean Thaetetus (*c.* 417–369 BC). Book X, which comprises roughly one-fourth of the *Elements*, seems disproportionate to the importance of its classification of incommensurable lines and areas (although study of this book would inspire Johannes Kepler [1571–1630] in his search for a cosmological model).

“Books XI–XIII examine three-dimensional figures, in Greek *stereometria*. Book XI concerns the intersections of planes, lines, and parallelepipeds (solids with parallel parallelograms as opposite faces). Book XII applies Eudoxus’s method of exhaustion to prove that the areas of circles are to one another as the squares of their diameters and that the volumes of spheres are to one another as the cubes of their diameters. Book XIII culminates with the construction of the five regular Platonic solids (pyramid, cube, octahedron, dodecahedron, icosahedron) in a given sphere” (*Britannica*).

“The significance of Euclid’s *Elements* in the history of thought is twofold. In the first place, it introduced into mathematical reasoning new standards of rigor which remained throughout the subsequent history of Greek mathematics and, after a period of logical slackness following the revival of mathematics, have been equaled again only in the past two centuries. In the second place, it marked a decisive step in the geometrization of mathematics … It was Euclid in his *Elements*, possibly under the influence of that philosopher who inscribed over the doors of the Academy ‘God is for ever doing geometry,’ who ensured that the geometrical form of proof should dominate mathematics. This decisive influence of Euclid’s geometrical conception of mathematics is reflected in two of the supreme works in the history of thought, Newton’s *Principia* and Kant’s *Kritik der reinen Vernunft*. Newton’s work is cast in the form of geometrical proofs that Euclid had made the rule even though Newton had discovered the calculus, which would have served him better and made him more easily understood by subsequent generations; and Kant’s belief in the universal validity of Euclidean geometry led him to a transcendental aesthetic which governs all his speculations on knowledge and perception. It was only toward the end of the nineteenth century that the spell of Euclidean geometry began to weaken and that a desire for the ‘arithmetization of mathematics’ began to manifest itself; and only in the second quarter of the twentieth century, with the development of quantum mechanics, have we seen a return in the physical sciences to a neo-Pythagorean view of number as the secret of all things. Euclid’s reign has been a long one; and although he may have been deposed from sole authority, he is still a power in the land” (*DSB*).

This copy is in the second state, as found in almost all copies printed on paper. The second state has leaves a1-a9 set differently from the first state: the heading on a1v is in two lines rather than three and is set in the same type as the text rather than heading type; the three-sided woodcut border and woodcut initial “P” are added to a2r; the headline in red on a2r beings “Preclarissimus liber elementorum;” and headlines do not begin until a10r. “The two outer pages of sheet c1 also differ, having evidently been reprinted owing to errors in the text and the diagram … of the 12th proposition of the 4th book” (B.M.C.). Copies of the first state include those printed on vellum in the British Library and the Bibliothèque Nationale.

BMC V, 285 (IB. 20513); Dibner, *Heralds of Science* 100; Essling 282; Goff E-113; Grolier/Horblit 27; GW 9428; Klebs 383.1; Norman 729; PMM 25; Redgrave 26; Sander 2605; Stillwell *Science* 163; Thomas-Stanford 1a; Goff E-113.

Chancery folio (311 x 212mm). 138 leaves (dedication leaf supplied from another copy). A tall copy, with none of the diagrams cropped or shaved. Incipit printed in red, woodcut three-quarter border [Redgrave border 3], woodcut initials, and more than 500 woodcut and type-rule geometric diagrams (dedication leaf with worming to text and some green paint offset, lower margin of leaf a2 restored; small portions of leaves c8, f1, and f2 renewed; occasional light marginal dampstaining, some spots and stains). Modern blind-tooled morocco custom full morocco box.

Item #6135

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Price:
$375,000.00
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