# Quinto Libro degli Elementi d’Euclide ovvero Scienza Universale delle Proporzioni Spiegata colla Dottrina del Galileo.

4to. [2] ff., (8), 149 pp, (3), 153-230 pp., 231-232 ff., 233-284 pp., 2 folding engraved plates.
bound with

**[VIVIANI, Vincenzo]**. *Enodatio Problematum universis geometris Propositorum […] Praemissis, horum occasione, Tentamentis Variis ad Solutionem illustris veterum Problematis De Anguli Trisectione*. Florence, Gugliantini, 1677. [2] ff, (6), 63 pp., 4 folding engraved plates. With presentation inscription to verso of half-title of first work signed by Viviani. Bound in contemporary calf with spine in six compartments. A wonderfully fresh copy, light toning and foxing to one or two leaves, otherwise excellent.

First complete edition (second; first 1674, see below) of this important Galileianum, **an assembly of previously unpublished writings by Galileo**, together with texts by Torricelli and Viviani himself, **inscribed by Viviani** to an unknown (scored) receipient.

Vincenzo Viviani resided with Galileo at Arectri from October 1639 as his pupil, amanuensis, and assistant, and together with Torricelli, spent the last months of Galileo’s life with him recording the master’s final meditations on the relationship between mathematics and physics. The first chapter of this work, a dialogue entitled “Quinto Libro degli Elementi d’Euclide... spiegata colla Dottrina del Galileo” was dictated by Galileo to Torricelli in November 1641. (Galileo died January 9, 1642.) Though on his deathbed, it was to be the beginning of still another book continuing the discussion between his three old interlocutors from the *Two New Sciences*.

In this dialogue, printed here and edited from a manuscript given to Viviani by Cardinal De’ Medici, Galileo reflects upon two definitions found in Euclid’s *Elements*, “same ratio” and “compound ratio,” which were “the two most important keys taken from antiquity in creating Galileo’s mathematical physics, so that his exposition of them as the last act of his scientific career reflected his earliest scientific steps at Pisa and Padua.

Like the Leaning Tower affair, this dialogue linked his last days with his first; Galileo had come full circle” (Drake, p. 421). As a young man, Galileo was profoundly influenced by the Elements, especially Books Five and Six which contained the Eudoxian theory of proportion. “The importance of the Eudoxian proportion theory to Galileo’s science cannot be exaggerated. Until the application of algebra to the general solution of geometrical (as well as arithmetical) problems, not achieved until after Galileo’s work was completed, rigorous connection of mathematics with physical events was possible only through some theory of proportionality.... Eudoxian theory establishing proportionality between continuous magnitudes was essential to any great advance over medieval physics” (Drake, p. 4).

Viviani “with the rigor and prolixity of the ancients…devoted an appendix to geometric problems, among which was one on the trisection of an angle, solved by the use of the cylindrical spiral or of a cycloid; another was the problem of duplicating the cube, solved by means of conics or of the cubic xy2 =k” (*DSB*).

After Galileo’s death, the Church prohibited Viviani from publishing a complete edition of his works and pursuing the evolution of his mathematical ideas. This prohibition must have been extremely difficult on someone who carried the seeds for the next generation in Italy. He turned instead to the study of the geometry of the ancients. (He published an Italian version of Euclid’s *Elements* in 1690.) But he is, perhaps, best remembered by scholars of Galileo for his association with his teacher. Though a work about Euclid’s Fifth Book, *Quinto Libro* is very much an occasion to approach Galileo from the very sources of his thinking by someone who was both intimate with him and capable of assessing his ideas.

The first edition of this work appeared in 1674, and consisted of 149 numbered pages, followed by 3 unnumbered pages of addenda and privilege; the work was reissued in 1676, virtually doubled in size, in part answering queries posed by a student in Leiden in 1675; this much augmented edition also adds two engraved plates. Generally, this augmented edition has the date 1674 on the title, but the date 1676 in the preface and on the verso of the final leaf; in the present copy, the title reads 1674, but the preface is dated 1674, and 1676 is the latest date found on the final page. It seems clear that one can distinguish between a first edition of 1674 containing 150-odd pages without illustrations, and another of 284-odd pages, the latter with illustrations. Our London colleague W.P. Watson (Cat. 11.54) describes a copy in a contemporary binding whose contents are identical to this one, with the exception that it apparently had the date 1674 on the last leaf; he calls this “first edition, unrecorded issue”. In our opinion, the number of minor variations between title-page, dedication and colophon suggest to us that any copy with the additional leaves possibly dates from 1676, and that the presence of earlier dates in dedication or colophon reflects the availability of these leaves rather than genuine “issues”.

Viviani’s *De Anguli Trisectione* represents the mathematician’s notable attempt to tackle one of the most famous problems in mathematics, here developing a method based on the equilateral hyperbola or the conchoid (Ball, *A Short History of Mathematics*, p. 316).

* Cinti, 151; Riccardi II.626.2; Stillman Drake,*Galileo at Work*.

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Price:
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