EUCLID of Alexandria. Das sibend, acht vnd neünt Buch, des hochberühmbten Mathematici Euclidis Megarensis.
Augsburg: Valentine Ottmar, 10 April 1555.

Very rare first appearance of any part of Euclid’s work in German, and one of the earliest vernacular editions (preceded only by Italian translations). This is a fine copy with a distinguished provenance.

Although this work contains only three of Euclid’s ‘books’ 7-9, it contains essentially all of his results relating to arithmetic and number theory, and forms a coherent collection. The six earlier books, first published in German translation in 1562, treat the geometry of lines and circles, while the later books deal mainly with three-dimensional geometry. Book 7 deals strictly with elementary number theory: divisibility, prime numbers, and the Euclidean algorithm for finding the greatest common divisor and least common multiple. Here Euclid also establishes the fundamental theorem of arithmetic, that (in modern terms) every positive integer can be written as a product of primes in an essentially unique way. Book 8 deals with proportions in number theory and geometric sequences. Book 9 applies the results of the preceding two books. Here is Euclid’s great proof that there are infinitely many prime numbers. He also gives the sum of a geometric series, and the construction of even perfect numbers (i.e. numbers that are equal to the sum of their divisors).

The editor of the present work, Johann Scheubel (1494-1570), was professor of mathematics at the University of Tübingen. Scheubel had earlier produced a Latin edition of Books I-VI (Basel, 1550); an arithmetic text, De numeris et diversis rationibus seu regulis computationum opusculum (Leipzig, 1545), in which he found binomial coefficients by means of a Pascal triangle a century before Pascal made that device famous; and an algebra text, Algebrae compendiosa facilisque descriptio (Paris, 1552).

Provenance: 1. Jakob Fugger (1542-98), member of the famous German family of merchant princes. Then the richest family in Europe, the Fuggers were generous patrons of the arts and learning and philanthropists, notably at Augsburg, their principal residence. Their fortune was largely built on a virtual monopoly in the mining and trading of silver, copper, and mercury. 2. Constantin Caratheodory (1873-1950) was a distinguished mathematician who made significant contributions to the theory of functions, the calculus of variations and measure theory. He held a number of academic posts, eventually succeeding Felix Klein at Göttingen.

Very rare: No copies in the major collections of Macclesfield, Honeyman, Horblit, or De Vitry. The last collection included just under 200 editions of Euclid. We can locate no auction records from the past 50 years. OCLC: Brown, Columbia, British Library, National Library of Wales.

Thomas-Stanford, C. Early editions of Euclid’s Elements, XVI (p. 54); Steck, M. Bib. Euclideana, III.54 (p. 65); Grässe, II, p. 513; Riccardi, P. Bib. Euclidea, p. 19, no. 1555. Not in Schweiger, STC German, or Adams.

4to: 203 x 145 mm. Pp. [12, last two blank], CCXXXIIII, [6, last two blank], woodcut diagrams in text. Contemporary German calf gilt, title of work and super ex-libris of Jakob Fugger; ‘iac f’ lettered in gilt on upper cover, edges gilt and gauffered, some light browning, lacking two pairs of ties, spine damaged at head and foot, corners slightly worn, remains of blue and yellow silk ties. Bookplate of Constantin Caratheodory on front paste-down. A fine and unsophisticated copy.