Paris: Melchior Mondiere, 1625.
Editio princeps of this important text by Euclid, his only work in pure geometry, other than the Elements, to have survived in Greek. It is here accompanied by a commentary, or rather an introduction, by Marinus of Naples (5th century AD), the pupil and biographer of Proclus. The work is of particular interest given contemporary developments in French geometry — Descartes, Mersenne, Fermat, etc., to whose circle the translator Claude Hardy belonged..
Very rare editio princeps of this important text by Euclid, his only work in pure geometry, other than the Elements, to have survived in Greek. It is here accompanied by a commentary, or rather an introduction, by Marinus of Naples (5th century AD), the pupil and biographer of Proclus. Although the importance of the first printing of any Euclidean text goes without saying, the work is of particular interest given contemporary developments in French geometry – Descartes, Mersenne, Fermat, etc., to whose circle the translator Claude Hardy belonged. Euclid’s Data opens with the passive perfect participle ‘δεδομενα,’ which means ‘given’; its Latin form ‘data’ remained in the title in modern times. “The Data … 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. Rectilinear 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” (DSB). The Data is more concerned with ‘problems’ than with ‘theorems’. In theorems, the goal is to show the truth of a claim; in problems, it is to perform a task after being given certain objects (e.g., given a straight line segment, construct an equilateral triangle on it). The more you are ‘given’, the easier is the fulfillment of the task. The Data provides a mechanism for extending what one has been given by the terms of the problem. This is a tool for the solution of problems. “A clue to the purpose of the Data is given by its inclusion in what Pappus calls the Treasury of Analysis [or Collection] … The Data is a collection of hints on analysis” (DSB). The relative obscurity of the Data, compared to the Elements, might be explained by the evolution of Greek mathematics, which in its early stages focused on the solution of special problems, but later concentrated more on the systematic arrangement of theorems. This is a very rare book. OCLC lists copies at Chicago, Harvard, New York Public, Stanford, and Wisconsin in US. ABPC/RBH lists only three other copies.
There are “two characteristic features of a classical analysis of a [geometrical] problem: it proceeded by means of a concept ‘given,’ and it was performed with respect to a figure in which the required elements were supposed to be drawn already. The latter was indicated by such phrases as ‘factum jam sit,’ ‘Let it be done.’ Which served as a standard reminder that the subsequent argument was an analysis. The at-first-sight contradictory approach, namely to assume a problem solved in order to find its solution, was seen as the essential feature of analytical reasoning. In the supposed figure some elements were given at the outset; some were directly constructible from those originally given, and some required more steps. The analysis used a kind of shorthand, codified largely in Euclid’s Data, for finding the constructible (‘given’) elements in the figure. The geometer used that shorthand as it were to plot a path from the primary given elements to the elements he ultimately wanted to construct.
“In the Data Euclid distinguished between three modes of being given: given in magnitude, given in position, and given in kind. Geometrical entities (line segments, angles, rectilinear figures) were ‘given in magnitude’ if, as Euclid phrased it: ‘we can assign equals to them.’ The third mode applied to rectilinear figures (triangles, polygons); such a figure could be ‘given in kind’, which meant that its angles and the ratios of its sides were given, but not its size. Thus if a figure was given in kind, this meant that another figure similar to it could be placed anywhere in the plane. For ratios there was only one mode of being given: a ratio was given if a ratio equal to it could be obtained, which meant in effect that two magnitudes could be produced whose ratio was equal to the given ratio.
“Euclid’s Data contained some 100 problems of the form ‘If A is given in mode α, then B is given in mode β.’ In all these propositions, the consequent (B) was constructible by straight lines and circles from the antecedent (A). These propositions, then, provided the steps by which the geometer planned the route from the given elements to the required ones. Once that route was planned, the construction could be written out along the same path.
“Once a route from the given to the required elements was planned in the analysis by means of the Data, the geometer could then proceed to the synthesis, i.e., the actual construction, by writing out the separate construction steps along that route … Nevertheless, the locus classicus on analysis, Pappus’s explanation at the beginning of Collection VII, stressed that the arguments in analysis and synthesis were each other’s reverse … It appears, however, that the actual practice of analysis was we find it in classical and early modern sources calls for a more nuanced view. Beyond the fact that analysis started with the assumption of the required, its direction of argument was not so definite … In the case of analysis on the basis of the Data, the obvious order was the same as that of the synthesis, namely from the given to the required” (Bos, Redefining Geometrical Exactness (2012), pp. 100-102).
The most interesting propositions are a group of four which are exercises in geometrical algebra corresponding to Elements 11.28, 29. Proposition 58 reads: “If a given area be applied to a given straight line so as to be deficient by a figure given in form, the breadths of the deficiency are given;” Proposition 84, which depends upon it, reads: “If two straight lines contain a given area in a given angle, and if one of them is greater than the other by a given quantity, then each of them is given.” These propositions are together equivalent to asserting the existence of the solution of a certain quadratic equation. Propositions 59 and 85 give the corresponding theorems for the excess, and are again equivalent to a quadratic equation.
Claude Hardy (1598?-1678) was a lawyer by profession, but took part in the weekly meetings of Roberval, Mersenne, and the other French geometricians in the Académie Mersenne, and was a friend of Claude Mydorge, who introduced him to Descartes. In his Examen of 1630, and again in his Refutation of 1638, Hardy exposed the fallacy of Paul Yvon’s solution to the problem of the duplication of the cube, a problem which attracted the attention of several seventeenth century writers, including Viéte, Descartes, Fermat, and Newton. Hardy also engaged in the dispute between Fermat and Descartes over the former’s method of maxima and minima; Hardy, together with Desargues and Mydorge, supported Descartes, while Fermat found two zealous defenders in Roberval and Pascal. “Hardy owed his greatest fame, however, to his knowledge of Arabic and other exotic languages, and in particular, to his edition of Euclid’s Data (1625), the editio princeps of the Greek text, together with a Latin translation” (DSB, under Hardy).
Hardy “edited the works here from manuscripts provided by Nicolas Rigault from the Bibliothèque royale. Menge (in volume 6 of Euclid's Opera omnia, edited by Heiberg & Menge, 1896) believed these manuscripts to be Par. 2366, 1981 and 2347 (all sixteenth-century), but also observed a number of readings pointing to a Vatican manuscript. Menge himself established the text of the Data from the famous Vat. Gr. 190 (P), ibid. 204 (Vat.), a manuscript in Bologna (b), and two fourteenth-century manuscripts, one in Paris (Par. Gr. 2448, z) and one in Florence (a, from Demetrius Cydones)” (Macclesfield catalogue).
DSB IV.524; Brunet 11.1081; Graesse II, p. 511; Hoffmann II, p. 167; Riccardi, Bib. Euclidea 1625; Steck VIII.10.
4to (223 x 178 mm), pp 8, 181, [3:errata], text in Latin and Greek in parallel columns, printer’s device on title, woodcut initials and headpieces, woodcut diagrams in text, printed marginal notes. Contemporary limp vellum. A very fine and completely unrestored copy.