## Elementorum geometricorum lib. XV. Cum expositione Theonis in priores XIII a Bartholomaeo Veneto latinitate donata, Campani in omnes, et Hypsiclis Alexandrini in duos postremos. His adiecta sunt phaenomena, catoptrica et optica, deinde protheoria Marini et data, postremum vero, opusculum de levi et ponderoso, hactenus non visum, eiusdem autoris.

Basel: Johann Herwagen, August 1537.

First edition of the first complete assembly of Euclidean texts, a very attractive copy bound and annotated at Wittenburg within a few years of publication. This first Basel edition contains both Campanus of Novara’s mediaeval and Bartolomeo Zamberti’s Renaissance versions of the *Elements*, and a new introduction by Philipp Melanchthon (present here but often removed or mutilated by the censor), together with all the other major Euclidean texts: the *Phaenomena*, *Optica*, *Catoptrica*, *Data*, and the *Opusculum de levi et ponderoso*, the last of which is printed here for the first time. Herwagen had previously published the first Greek text of the *Elements*, in 1533. This is a typographically handsome and textually significant edition of the “the oldest mathematical textbook in the world still in common use today … The *Elements* is a compilation of all earlier Greek mathematical knowledge since Pythagoras, organized into a consistent system so that each theorem follows logically from its predecessor; and in this simplicity lies the secret of its success … The *Elements *remained the common school textbook of geometry for hundreds of years” (PMM on the first Latin edition). This book “has exercised an influence upon the human mind greater than that of any other work except the Bible” (DSB). This is a particularly interesting copy of the first Basel edition in its original state and with numerous contemporary annotations by an informed reader. This Basel edition was reprinted in 1546 and 1558.

*Provenance*: inscription to title ‘Petrus Dornhoffer emptus est liber iste Witenbergae duobus florenis 1545’. Dornhoffer, from Linz, enrolled on 16 June 1539 at the University of Freiburg, and then on 4 April 1545 at the University of Wittenburg where, he tells us, he acquired the book for 2 fl. His neat and careful marginalia, in Latin and Greek, appear in the preface and the first three books of the *Elements*. On the front pastedown he has made notes on geometry and provided a tabular summary of the *Elements*.

*Binding*: executed in 1545, when Dornhoffer purchased the volume. The rolls forming two rectangular compartments on each cover, depicting ‘Fides’, ‘Charitas’ and ‘Spes’, carry the initials ‘A F’, identified by Haebler as Andreas Franckow of Wittenberg, who became a master binder in 1534 (Haebler I, p. 110). Another copy of the present work, accompanied by the 1533 Greek edition, also bound by Franckow in 1545, was sold at Sotheby’s, London, 7 November 2017, lot 109, £52,500.

“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).

Euclid’s *Phaenomena* [pp. 483-504] is a textbook of what the Greeks called *sphaeric*, intended for use by students of astronomy. It was included in the collection of astronomical works which Pappus calls *The Treasury of Astronomy*, alternatively known as *The Little Astronomy*, in contrast with Ptolemy’s *Syniaxis*, or *Great Astronomy*. It consists of a preface and sixteen propositions. The preface gives reasons for believing that the universe is a sphere and includes some definitions of technical terms. Euclid in this work is the first writer to use “horizon” absolutely—Autolycus had written of the “horizon (i.e., bounding) circle”—and he introduces the term “meridian circle.” The propositions set out the geometry of the rotation of the celestial sphere and prove that stars situated in certain positions will rise or set at certain times.

Proclus attributes to Euclid a book entitled *Specularia* (or *Catoptrica*) [pp. 504-515], that is, on mirrors. The work which bears that name in the editions of Euclid is certainly not by him but is a later compilation, and Proclus is generally regarded as having made a mistake. If the later compilation is the work of Theon, as may well be the case, it would have been quite easy for Proclus to have assigned it to Euclid inadvertently.

The *Perspectiva *(or *Optica*) [pp. 516-536], which is also attributed to Euclid by Proclus, is also attested by Pappus, who includes it in the *Little Astronomy*. An elementary treatise in perspective, it was the first Greek work on the subject and remained the only one until Ptolemy wrote in the middle of the second century. It starts with definitions, some of them really postulates, the first of which assumes, in the Platonic tradition, that vision is caused by rays proceeding from the eye to the object. It is implied that the rays are straight. The second states that the figure contained by the rays is a cone which has its vertex in the eye and its base at the extremities of the object seen. Definition 4 makes the fundamental assumption that “Things seen under a greater angle appear greater, and those under a lesser angle less, while things seen under equal angles appear equal.” From proposition 6 it is easy to deduce that parallel lines appear to meet. There are groups of propositions relating to the appearances of spheres, cones, and cylinders. Propositions 37 and 38 prove that if a straight line moves so that it always appears to be the same size, the locus of its extremities is a circle with the eye at the center or on the circumference. The book contains fifty-eight propositions of similar character. It was written before the *Phaenomena*, for it is cited in the preface of that work.

The *Data* (pp. 537-585) is the only work by Euclid on pure geometry, other than the *Elements*, to have survived in the original Greek. It “is closely connected with books I-VI of the *Elements*. It is concerned with the different senses in which things are said to be given. Thus areas, straight lines, angles, and ratios are said to be “given in magnitude” when we can make others equal to them. Rectilineal figures are “given in species” or “given in form” when their angles and the ratio of their sides are given. Points, lines, and angles are “given in position” when they always occupy the same place, and so on. After the definitions there follow ninety-four propositions, in which the object is to prove that if certain elements of a figure are given, other elements are also given in one of the defined senses …

“No work by Euclid on mechanics is extant in Greek, nor is he credited with any mechanical works by ancient writers. According to Arabic sources, however, he wrote a *Book on the Heavy and the Light*, and when Hervagius was about to publish his 1537 edition there was brought to him a mutilated fragment, *De levi et ponderoso*, which he included as one of Euclid’s works [pp. 585-586]. In 1900 Curtze published this side by side with a *Liber Euclidis de gravi et levi et de comparatione corporum ad invicem* which he had found in a Dresden manuscript. It is clearly the same work expressed in rather different language and, as Duhem observed, it is the most precise exposition that we possess of the Aristotelian dynamics of freely moving bodies. Duhem himself found in Paris a manuscript fragment of the same work, and in 1952 Moody and Clagett published a text [*The Medieval Science of Weights*], with English translation, based chiefly on a manuscript in the Bodleian Library at Oxford … as Clagett points out, “The only dynamics that had been formulated at all, in the time in which Euclid lived, was the dynamics of Aristotle.” In Clagett’s judgment, “No solid evidence has been presented sufficient to determine the question of authenticity one way or the other” (DSB).

“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).

The first printed edition of Euclid, in 1482, was a Latin translation by Campanus (c, 1220-96), based principally on the 12th-century translation from the Arabic by Adelard of Bath. In 1505, Bartolomeo Zamberti (1473-1543) published the first Latin translation from a Greek manuscript, claiming that he had restored the text from the corruptions, as he saw them, introduced by Campanus. The present Basel edition reproduces both the Campanus and Zamberti versions *in toto*: each theorem and proof first occurs *ex Campano* and is immediately followed by its mate and proof *Theon ex Zamberto*. The *additiones* due to Campanus appear in place but are appropriately set off and indicated as such. “Herwagen had migrated from Strasburg about 1528, when he acquired the citizenship of Basle, and married Gertrude, widow of the learned Basel printer John Froben and the daughter of the scholar and patron of letters Wolfgang Lachner. He was the first printer to inset Euclid’s diagrams in the text . . . In August, 1537, he published a Latin version of the *Elements*, followed by other works attributed to Euclid. It is in roman type and contains three pages of introduction by Philip Melanchthon addressed ‘studiosis adolescentibus’. From many copies this introduction has been removed by the clerical censor who has added his stamp. As there does not appear to be anything objectionable in the introduction itself, this action of the censor must have arisen from hostility to the writer of it” (Thomas-Stanford).

In this famous introduction, “Melanchton began and ended his praises of the art [mathematics] with the famous sign over the door of Plato’s Academy: ‘let no one enter who is untrained in geometry.’ Melanchton suggested two interpretations of Plato’s sign. First, the philosopher may have intended mathematics to be a prerequisite for the study of the other arts, particularly philosophy. For although geometry had its uses for practical men who made buildings or pots, ultimately it was the philosopher who really needed it. In a passage that would resonate with later readers, Melanchton claimed that the natural philosophy and physics of Aristotle could not be understood without a grounding in mathematics. Moving beyond the natural world, Melanchton noted that mathematics drew men out of their concern for worldly things to a consideration of the heavens: at first, quite literally through the measurement of the stars, but eventually ‘it [carries] the aspiring souls back to their homeland and into consort with celestial beings – even to the vision of God …

“Melanchton went on to consider a second interpretation of Plato’s sign which, again, emphasized the effect of the art on the practitioner rather than its utility in the world. Ethical behaviour was, as Plato showed, a kind of geometrical harmony; thus mathematics and a knowledge of geometrical harmony in itself was necessary for the achievement of virtue. Underscoring this lofty, intellectual understanding of mathematics, Melanchton went on to say that he was not addressing those who have no care for the liberal arts, nor those who aimed only for mercantile gain … He hoped that anyone who opened the book in which his preface appeared would be reminded by his opening words – ‘let no one enter who is untrained in geometry’ – that they should always aim for the most sublime uses of this art, and not linger in calculation and measurement.

“It was in this context that Melanchton then turned to historical anecdote, and related the story of the wreck of Aristippus, the follower of Socrates and founder of the Cyrenaic school:

‘When Aristippus lost everything he owned in a shipwreck, he nevertheless reached the shore of Rhodes in safety along with a few companions. The story goes that, while walking along the beach, he noticed some geometrical figures in elaborate constructions. Although the sea had stripped them of all their provisions and thrown them up onto some unknown land, once Aristippus saw these figures he bid his companions to be of good heart, saying that he had seen the footprints of men, and was glad for himself and the others because they had not been washed up on some barbarous shore; and he assured them that humanity towards shipwrecked strangers would not be wanting in men who cultivated the study of these arts. How I wish that those footprints of men which Aristippus marvelled at on the shore were more frequent in our schools. For these arts have lain deserted and neglected for many centuries now’ …

From this striking story, Melanchton retained the notion that mathematics was a mental possession, but otherwise quite recast the intention of the anecdote. The pursuit of geometry, he argued, revealed a people who had achieved a high level of *moral* mastery, along with intellectual culture. They would be just, and would know how to show humanity to the lost – a conceit that Melanchton brilliantly turned around to a criticism of the state of the contemporary university.

“This is the extent of the historical engagement in Melanchton’s preface. Nevertheless, this frequently printed encomium deserves mention because of its influence on later writers on the history of mathematics. The strong Platonism would remain a feature of such histories. His retelling of the Aristippus story would be picked up by several later writers, and would eventually become virtually symbolic of Greek mathematics” (Goulding, *Defending Hypatia. **Ramus, Savile, and the Renaissance Rediscovery of Mathematical History* (2010), pp. 14-16).

Adams E974; BM STC, German Books p. 288; Houzeau-Lancaster 832; Honeyman 977; Sotheran 1200 (“The first complete edition of Euclid’s works”); Steck, *Bibliographia Euclideana* III.33; Thomas-Stanford 9; VD16 E4154. See PMM 25 for the first Latin edition (1482).

Folio (310 x 205 mm), pp. [7], [1, blank], 587, [1, colophon]; printer’s device to title and last page, woodcut initials (a few coloured in red) and head-pieces, woodcut diagrams within text throughout; a few small worm holes to first few leaves and to gutter margin at very end, very occasional light marginal damp stains, slight wear to fore-edge of first few leaves from metal catch; a very good copy in contemporary German pigskin over wooden boards, bevelled edges, sides decorated in blind with stamps and rolls to a panel design dated 1545, four raised bands to spine, brass catches and clasps; some slight rubbing and wear to extremities, some staining to lower cover; near contemporary annotations in red and brown ink to front pastedown and title and some marginalia to first three books of the Elements.

Item #4692

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Price:
$12,500.00
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