## Opera, quae quidem extant, omnia . . . nuncque primum & Graece Latine in lucem edita . . . adiecta quoque sunt Eutocii Ascalonitae in eosdem Archimedis libros commentaria item Graece & Latine, nunquam antea excusa.

Basle: Joannes Hervagius, 1544.

First edition of one of the key scientific books of the Renaissance, representing a decisive step forward in the history of mathematics, containing the first printings of the majority of the surviving works of the greatest mathematician, physicist and engineer of antiquity. This is a fascinating copy with numerous contemporary annotations by a well-informed reader, both in the margins and in the text itself (in a minuscule neat hand that does not obscure the original text). In addition, there is a full-page manuscript entitled ‘Cristiani Hugenii / Alia demonstratio propositionis 18 / Archimedis de spiralibus / ad paginam 111,’ and including a large geometrical diagram, which provides Huygens' alternative proof to a proposition about Archimedes’ spiral demonstrated on the facing page of text. This suggests that the annotator was probably a member of Huygens’ circle. This book constitutes “the first printing of the original Greek text of seven Archimedean mathematical texts, accompanied by Jacopo de Cremona’s Latin translation from a manuscript corrected by Regiomontanus, and the commentaries (in both Greek and Latin) of the sixth-century mathematician Eutocius of Ascalon” (Norman). “Archimedes – together with Newton and Gauss – is generally regarded as one of the greatest mathematicians the world has ever known, and if his influence had not been overshadowed at first by Aristotle, Euclid and Plato, the progress of modern mathematics might have been much faster. As it was, his influence began to take full effect only after this first printed edition which enabled Descartes, Galileo, and Newton in particular to build on what he had begun” (PMM). The seven treatises included in the present work are: *On the Sphere & Cylinder*; *On the Measurement of the Circle*; *On Conoids & Spheroids*; *On Spirals*; *On the Equilibrium of Planes* (and *Centres of Gravity*); *The Arenarius*, or *Sand-Reckoner*; and *On the Quadrature of the Parabola*. “Publication of this *editio princeps* inspired a multiplication of texts on Archimedes and his methods, which exerted a strong influence on the development of mathematics during the sixteenth and seventeenth centuries. One of the important effects of that influence can be seen in Kepler’s *Astronomia nova *(1609), in which Archimedes’ so-called ‘exhaustion procedure’ was applied to the measurement of time elapsed between any two points if Mars’s orbit” (Norman). “Apart from one small tract published in 1503 and an imperfect edition by Tartaglia in 1543, [this] is the first complete edition of Archimedes’ works” (PMM). This volume also includes for the first time the description of the heliocentric system of Aristarchus, who had conceived this theory centuries before Copernicus.

“The principal results in *On the Sphere and Cylinder* (in two books) are that the surface area *S* of any sphere of radius *r* is four times that of its greatest circle (in modern notation, *S = *4*πr*^{2}) and that the volume *V* of a sphere is two-thirds that of the cylinder in which it is inscribed (leading immediately to the formula for the volume, *V = *4/3*πr*^{3}). Archimedes was proud enough of the latter discovery to leave instructions for his tomb to be marked with a sphere inscribed in a cylinder. Marcus Tullius Cicero (106–43 BC) found the tomb, overgrown with vegetation, a century and a half after Archimedes’ death.

“*Measurement of the Circle* is a fragment of a longer work in which π, the ratio of the circumference to the diameter of a circle, is shown to lie between the limits of 3 10/71 and 3 1/7. Archimedes’ approach to determining π, which consists of inscribing and circumscribing regular polygons with a large number of sides, was followed by everyone until the development of infinite series expansions in India during the 15th century and in Europe during the 17th century. That work also contains accurate approximations (expressed as ratios of integers) to the square roots of 3 and several large numbers.

“*On Conoids and Spheroids* deals with determining the volumes of the segments of solids formed by the revolution of a conic section (circle, ellipse, parabola, or hyperbola) about its axis. In modern terms, those are problems of integration.

“*On Spirals* develops many properties of tangents to, and areas associated with, the spiral of Archimedes—i.e., the locus of a point moving with uniform speed along a straight line that itself is rotating with uniform speed about a fixed point. It was one of only a few curves beyond the straight line and the conic sections known in antiquity.

“*On the Equilibrium of Planes* (or *Centres of Gravity of Planes*; in two books) is mainly concerned with establishing the centres of gravity of various rectilinear plane figures and segments of the parabola and the paraboloid. The first book purports to establish the ‘law of the lever’ (magnitudes balance at distances from the fulcrum in inverse ratio to their weights), and it is mainly on the basis of that treatise that Archimedes has been called the founder of theoretical mechanics. Much of that book, however, is undoubtedly not authentic, consisting as it does of inept later additions or reworkings, and it seems likely that the basic principle of the law of the lever and—possibly—the concept of the centre of gravity were established on a mathematical basis by scholars earlier than Archimedes. His contribution was rather to extend those concepts to conic sections.

“*Quadrature of the Parabola* demonstrates, first by ‘mechanical’ means and then by conventional geometric methods, that the area of any segment of a parabola is 4/3 of the area of the triangle having the same base and height as that segment.

“*The Sand-Reckoner* is a small treatise that is a jeu d’esprit written for the layman—it is addressed to Gelon, son of Hieron [see below]—that nevertheless contains some profoundly original mathematics. Its object is to remedy the inadequacies of the Greek numerical notation system by showing how to express a huge number—the number of grains of sand that it would take to fill the whole of the universe. What Archimedes does, in effect, is to create a place-value system of notation, with a base of 100,000,000. (That was apparently a completely original idea, since he had no knowledge of the contemporary Babylonian place-value system with base 60.) The work is also of interest because it gives the most detailed surviving description of the heliocentric system of Aristarchus of Samos (c. 310–230 BC) and because it contains an account of an ingenious procedure that Archimedes used to determine the Sun’s apparent diameter by observation with an instrument …

“Archimedes’ mathematical proofs and presentation exhibit great boldness and originality of thought on the one hand and extreme rigour on the other, meeting the highest standards of contemporary geometry. While he arrived at the formulas for the surface area and volume of a sphere by ‘mechanical’ reasoning involving infinitesimals, in his actual proofs of the results in *Sphere and Cylinder* he uses only the rigorous methods of successive finite approximation that had been invented by Eudoxus of Cnidus in the 4th century BC. These methods, of which Archimedes was a master, are the standard procedure in all his works on higher geometry that deal with proving results about areas and volumes. Their mathematical rigour stands in strong contrast to the ‘proofs’ of the first practitioners of integral calculus in the 17th century, when infinitesimals were reintroduced into mathematics. Yet Archimedes’ results are no less impressive than theirs. The same freedom from conventional ways of thinking is apparent in the arithmetical field in *Sand-Reckoner*, which shows a deep understanding of the nature of the numerical system” (Britannica).

Although Eutocius (480-540) was not an original thinker, his commentaries contain much historical information which might otherwise have been lost. It is to Eutocius that we owe the Archimedean solution of a cubic by means of intersecting conics, referred to in *On the Sphere & Cylinder* (Book II.4) but not otherwise extant except through his commentary. Eutocius also records the solution of the original problem of II.4 by Diocles (c. 240 – c. 180 BC), avoiding the use of the cubic, and the solution by Dionysodorus (c. 250 – c. 190 BC) of the auxiliary cubic. It is thought that Eutocius did not know of the four remaining works, *On Conoids & Spheroids*, *On Spirals*; *The Sand-Reckoner*, and *On the Quadrature of the Parabola*.

“In contrast to Euclid’s *Elements*, the writings of Archimedes were not widely known in antiquity. Survival of their texts was due to interest in Archimedes’ writings at the Byzantine capital of Constantinople from the sixth through the tenth centuries. “It is true that before that time individual works of Archimedes were obviously studied at Alexandria, since Archimedes was often quoted by three eminent mathematicians of Alexandria: Hero, Pappus, and Theon. But it is with the activity of Eutocius of Ascalon, who was born toward the end of the fifth century and studied at Alexandria, that the textual history of a collected edition of Archimedes properly begins. Eutocius composed commentaries on three of Archimedes’ works: *On the Sphere and the Cylinder*, *On the Measurement of the Circle*, and *On the Equilibrium of Planes*. These were no doubt the most popular of Archimedes’ works at that time … The works of Archimedes and the commentaries of Eutocius were studied and taught by Isidore of Miletus (442-537) and Anthemius of Tralles (474-534), Justinian’s architects of Hagia Sophia in Constantinople. It was apparently Isidore who was responsible for the first collected edition of at least the three works commented on by Eutocius as well as the commentaries. Later Byzantine authors seem gradually to have added other works to this first collected edition until the ninth century when the educational reformer Leon of Thessalonica produced the compilation represented by Greek manuscript A (adopting the designation used by the editor, J. L. Heiberg [*Opera omnia, cum commentariis Eutocii*, 3 vols., Leipzig, 1880-1]). Manuscript A contained all of the Greek works now known excepting *On Floating Bodies*, *On the Method*, *Stomachion*, and *The Cattle Problem*. This was one of the two manuscripts available to William of Moerbeke (1215-86) when he made his Latin translations in 1269. It was the source, directly or indirectly, of all of the Renaissance copies of Archimedes. A second Byzantine manuscript, designated as B, included only the mechanical works: *On the Equilibrium of Planes*, *On the Quadrature of the Parabola* and *On Floating Bodies* (and possibly *On Spirals*). It too was available to Moerbeke, but it disappears after an early fourteenth-century reference. Finally we can mention a third Byzantine manuscript, C, a palimpsest whose Archimedean parts are in a hand of the tenth century. It was not available to the Latin West in the Middle Ages, or indeed in modern times until its identification by Heiberg in 1906 at Constantinople (where it had been brought from Jerusalem).

“In the fifteenth century, knowledge of Archimedes in Europe began to expand. A new Latin translation was made by James of Cremona (1400-56) in about 1450 by order of Pope Nicholas V. Since this translation was made exclusively from manuscript A, the translation failed to include *On Floating Bodies*, but it did include the two treatises in A omitted by Moerbeke, namely *The Sand Reckoner* and Eutocius' *Commentary on the* *Measurement of the Circle*. It appears that this new translation was made with an eye on Moerbeke’s translation. . . . There are at least nine extant manuscripts of this translation, one of which was corrected by Regiomontanus and brought to Germany about 1468 … Greek manuscript A itself was copied a number of times. Cardinal Bessarion had one copy prepared between 1449 and 1468 (MS E). Another (MS D) was made from A when it was in the possession of the well-known humanist George [Giorgio] Valla (1447-99). The fate of A and its various copies has been traced skilfully by J. L. Heiberg in his edition of Archimedes' *Opera*. The last known use of manuscript A occurred in 1544, after which time it seems to have disappeared.

“The first printed Archimedean materials were in fact merely Latin excerpts that appeared in George Valla’s *De expetendis et fugiendis rebus opus* (Venice, 1501) and were based on his reading of manuscript A. But the earliest actual printed texts of Archimedes were the Moerbeke translations of *On the Measurement of the Circle* and *On the Quadrature of the Parabola* (*Teragonismus, id est circuli quadratura* etc.) published from the Madrid manuscript by L[uca] Gaurico (Venice, 1503). In 1543 also at Venice N[iccolo] Tartaglia republished the same two translations directly from Gaurico’s work, and in addition, from the same Madrid manuscript, the Moerbeke translations of *On the Equilbrium of Planes* and Book I of *On Floating Bodes* (leaving the erroneous impression that he had made these translations from a Greek manuscript, which he had not since he merely repeated the texts of the Madrid manuscript, with virtually all their errors) … The key event, however, in the further spread of Archimedes was the aforementioned *editio princeps* of the Greek text with the accompanying Latin translation of James of Cremona at Basel in 1544” (Marshall Clagett in DSB).

For this *editio princeps* the editor Thomas Gechauff, called Venatorius (d. 1551), was able to use the above-mentioned manuscript of James of Cremona’s Latin translation corrected by Regiomontanus, which included the commentaries of Eutocius. For the Greek text Gechauff used a manuscript which had been acquired in Rome by humanist Willibald Pirckheimer (1470-1530), and is preserved today in Nuremberg City Library. Gechauf, Nuremberg scholar and theologian, was born about 1490 and was a pupil of Johannes Schöner (1477-1547) and a friend of Pirckheimer. He wrote in both Latin and German, published an edition of Aristophanes ‘Plutus’ (1531), and his name is found in some works in conjunction with that of Andreas Osiander (1498-1552), who famously added the preface to Copernicus.

Manuscripts A and B are now lost. However, after disappearing into a European private collection in the early twentieth century, the third key record of Archimedes’ texts discussed above, the tenth century Byzantine manuscript C, known as the Archimedes Palimpsest, re-appeared at a Christie’s auction in New York on October 28, 1998, where it was purchased by a private collector in the United States. Since then it has been made widely available to scholars, and has been the subject of much research. It contains the only extant manuscript of Archimedes’ *Method Concerning Mechanical Theorems*, which describes how he used a ‘mechanical’ method to arrive at some of his key discoveries, including the area of a parabolic segment and the surface area and volume of a sphere. The technique consists of dividing each of two figures into an infinite but equal number of infinitesimally thin strips, then ‘weighing’ each corresponding pair of these strips against each other on a notional balance to obtain the ratio of the two original figures. Archimedes emphasizes that this procedure, though useful as a heuristic method, does not constitute a rigorous proof. Nevertheless, his method is a clear precursor of Cavalieri’s method of indivisibles (1635), and of the integral calculus of Newton and Leibniz.

Archimedes (c. 287 – 212/211 BC) “probably spent some time in Egypt early in his career, but he resided for most of his life in Syracuse, the principal Greek city-state in Sicily, where he was on intimate terms with its king, Hieron II. Archimedes published his works in the form of correspondence with the principal mathematicians of his time, including the Alexandrian scholars Conon of Samos and Eratosthenes of Cyrene. He played an important role in the defence of Syracuse against the siege laid by the Romans in 213 BC by constructing war machines so effective that they long delayed the capture of the city. When Syracuse eventually fell to the Roman general Marcus Claudius Marcellus in the autumn of 212 or spring of 211 BC, Archimedes was killed in the sack of the city.

“Far more details survive about the life of Archimedes than about any other ancient scientist, but they are largely anecdotal, reflecting the impression that his mechanical genius made on the popular imagination. Thus, he is credited with inventing the Archimedes screw, and he is supposed to have made two ‘spheres’ that Marcellus took back to Rome—one a star globe and the other a device (the details of which are uncertain) for mechanically representing the motions of the Sun, the Moon, and the planets. The story that he determined the proportion of gold and silver in a wreath made for Hieron by weighing it in water is probably true, but the version that has him leaping from the bath in which he supposedly got the idea and running naked through the streets shouting ‘Heur ka!’ (“I have found it!”) is popular embellishment. Equally apocryphal are the stories that he used a huge array of mirrors to burn the Roman ships besieging Syracuse; that he said, ‘Give me a place to stand and I will move the Earth’; and that a Roman soldier killed him because he refused to leave his mathematical diagrams—although all are popular reflections of his real interest in catoptrics (the branch of optics dealing with the reflection of light from mirrors, plane or curved), mechanics, and pure mathematics” (Britannica).

Active in the early 6th century, Eutocius apparently was a pupil of the Neo-Platonist Ammonius Saccas (175-242), and perhaps a colleague of Anthemius of Tralles. If so, he was trained as a Neo-Platonist philosopher. In this tradition, it was customary to pay attention to the mathematical sciences and even to write some commentaries on them, but Eutocius is the only Neo-Platonist we know to concentrate uniquely on mathematical commentary. In addition to his commentaries on Archimedes, he also wrote an important commentary on the first four books of the *Conics* of Apollonius (c. 262 BC – c. 190 BC).

PMM 72; Adams A1531; Dibner 137; Grolier/Horblit 5; Hoffman I, 228; Macclesfield 179 & 180; Norman 61.

Folio (310 x 205 mm), pp. [8], 1-139, [1], [8], [1], 2-163, [1], [4], 1-65, [1], 1-68, [1, colophon]. Numerous woodcut diagrams and initials, text in Greek and Latin. Seventeenth-century (Dutch?) calf with gilt-stamped armorial on covers (rebacked retaining the original endpapers), red speckled edges.

Item #4500

**
Price:
$75,000.00
**