De Centro Gravitatis Solidorum Libri Tres. Rome: B[artolomeo] Bonfadino, 1604. [Bound with]: Quadratura Parabolae per simplex falsum. Et altera quàm secunda Archimedis expeditior ad Martium Columnam. Rome: L[epido] Facio, 1606.

Rome: Bonfadino; Facio, 1604; 1606.

First editions of these two rare and fundamental works on determining centres of gravity – these are fascinating copies with numerous contemporary annotations in an untouched contemporary binding. In these books Valerio transformed the Archimedean ‘method of exhaustion’ into a series of important theorems “so that for a whole class of convex curves and solids it was no longer necessary to establish results by special methods” (Baron, p. 101). Valerio was thus able to determine for the first time the centres of gravity of a number of ‘Archimedean solids,’ notably the hemispheroid and hyperboloid “which is undoubtedly one of the most difficult results in classical-Renaissance mathematics” (Napolitani & Saito, p. 108). These works were also important for initiating the transition from classical Greek geometry to modern mathematics (see below) – for example, according to Divizia, De CentroGravitatis anticipated the concept of limit and the integral calculus. Valerio strongly influenced Galileo, through his correspondence and these two books, and he was singled out for praise in the Discorsi, where he is described as ‘our greatest geometer, the New Archimedes of our age’. “Valerio introduced important changes and novelties into the mathematics of his time – especially at the methodological level. He opened up a new road which was to be followed by several others: notably, Cavalieri, with his theory of indivisibles, and, above all, Descartes with his geometry of curves. Mathematics, as a result, was totally transformed, and its language and methods have since undergone a revolution” (ibid., p. 74). “Valerio was the first to break with the Greek model of the mathematical object: he introduces, and uses, classes of figures defined by one or more properties, to prove theorems that are more general and productive (in particular Theorem II-32); he invented [sic] the method of exhaustion, codifying in a series of general theorems, the classical technique to establish quantitative relationships between geometric figures … The way to these methods had been opened by Valerio, and Cavalieri proceeded further ahead: but then little remained to be done. The introduction of ars analytica into geometry was to break this wall; it was not so long before Descartes opened a practically infinite new world for mathematicians, by identifying the curves with their equations. It is difficult, however, to imagine Cavalieri and Descartes without Valerio, to imagine their radical innovations without the methodological novelties of De centro gravitatis solidorum … the impact of Valerio’s work provoked a crisis in the rigidity of the classical paradigm, and helped to create the conceptual context that was to lead to the birth of modern mathematics” (ibid., pp. 120-3). OCLC lists six copies of De Centro Gravitatis and four of Quadratura Parabolae in US.

Provenance: Signature on title ‘Ignatis Braccii’ (?), possibly Ignatius Braccius, author of Phoenicis effigies in numismatis, et gemma, quae in Museo Gualdino asseruantur (1637). Full page of manuscript remarks in a contemporary hand on front free endpaper discussing the work of Archimedes, Galileo and Valerio on centres of gravity, and numerous marginal annotations in the same hand throughout.

“The rediscovery of Greek mathematics, especially of Greek geometry, in the sixteenth century lies at the basis of many of the conceptual revolutions of the following century … The period of rediscovery may be considered to have come to an end in 1575, the year in which both Maurolico and Commandino died. This date marks the beginning of the assimilation phase, an activity which soon became increasingly important. The assimilation of Archimedes involved the completion and the revision of his work; it implied a methodological reflection on the whole subject of geometry of measure and its relationship to mechanics …

“Luca Valerio was one of the most important figures in this process of assimilation of ancient geometry. He was born in Naples in 1553, and entered the Collegio Romano of the Jesuits at the age of 17. There he attended the academy of mathematics directed by Christopher Clavius; when he left the Society in 1580, he was already an established mathematician … However, it was only after several years that he succeeded in finding a position worthy of his talents: in 1600 he obtained the chair of mathematics at the University ‘La Sapienza’ in Rome, thanks to the protection of Pope Clement VIII Aldobrandini. And thus, at last, he was able to devote himself to research. His work, Three Books on the Center of Gravity of Solids (De Centro Gravitatis Solidorum libri tres) was published in Rome in 1604, and won him a lasting reputation, which continued long after his death in 1618. Valerio’s reflections on the classics regularly follow the same pattern: he takes a demonstrative technique already available (which can often easily be identified with a technique used by Archimedes) for a particular case, and then he transforms the specific properties of that particular case which makes the technique valid, into the definition of a particular class of figures (so that that same technique is applicable to all the figures of this newly defined class). This was a completely original approach, which led to new vistas of mathematical research …

“The objects of classical Greek geometry were always particular objects, given by a more or less axiomatized construction procedure … And the aim of geometry was to study these objects, and determine their properties … An immediate consequence of this notion of mathematical objects is that for the Greeks, general objects could not exist. Nothing like our ‘curves’ existed in classical geometry. Various curved lines existed, of course – the circle, the conic sections, the conchoid, the cissoid, the spiral, the quadratrix – but no single conceptual operational category existed that included all of them … If no general object existed, then general methods could not exist, either … It is usually stated that Archimedes, and the Greeks in general, used the ‘method of exhaustion’. Strictly speaking, however, this method did not exist in Greek geometry, at least as a codified method to compare areas and volumes. True, a whole series of theorems concerning areas and volumes by Euclid and Archimedes is based on a common technique: in order to prove that two figures are equal, they demonstrated by means of a double reductioad absurdum that one of the two can neither be greater nor smaller than the other. This double reduction is generally obtained by using ‘known’ figures that approximate the given figure … The ‘method of exhaustion’, then, is a convenient historiographic label that makes it possible to identify a posteriori certain procedures having indubitable similarity, but nothing more.

“The problem Valerio solved in his De Centro Gravitatis Solidorum was that of determining the center of gravity of all the solids the Renaissance inherited from Euclidean and Archimedean geometry: sphere, cone, pyramid, prism, cylinder, polyhedra, paraboloid, hyperboloid, ellipsoid, and their parts. For brevity, we shall call them ‘Archimedean solids’. This seems to be a fairly restricted problem, but it had not yet been solved, and indeed was one of the problems most widely studied by mathematicians of the late sixteenth century, partly because it constituted a necessary component of the completion of Archimedes’ works, who had only dealt with the centers of gravity of plane figures in those works known at that time. [In The Method, rediscovered in 1906, Archimedes investigated the centres of gravity of various solids.] In 1565 Commandino published Liber de Centro Gravitatis Solidorum, which, however, was far from being satisfactory. Several proofs were flawed, and above all, the most thorny questions had not even been dealt with: the determination of the centers of gravity of segments of the sphere and the ellipsoid, and of the center of gravity of the hyperboloid. Many other scholars dedicated themselves to this task, including Francesco Maurolico, Simon Stevin, the young Galileo, and Valerio’s teacher, Clavius. However, their results had either not been published, or were not generally known. And none of them had ever arrived at the determination of the centers of gravity of the hemisphere or the hyperboloid.

“One element common to all the sixteenth-century attempts that are known to us is their substantial adherence to the ancient ‘classical’ approach: the centers of gravity of the various solids were dealt with case by case, introducing ad hoc techniques for each solid studied. Valerio followed a different method. Instead of dealing with the problem of the determination of the centers of gravity and quadrature case by case, he constructed an enormous edifice of theorems valid for a whole class of figures (those he called circa axim and circa diametrum: roughly speaking, the former is the solid figure generated by rotation around an axis, the latter the plane figure with an axis of symmetry) … Valerio was thus taking a significant step ahead in departing from classical mathematics. [For example,] the determination of the center of gravity of a hyperboloid (as Valerio proudly observes, ‘attempted by nobody before’, ‘antea tentata nemini’) is reduced to the application of a general theorem regarding the centers of gravity of figures circa axim and circa diametrum. Valerio was fully aware of this breakthrough in methodology, which had a considerable influence on the subsequent development of mathematics, in particular on Cavalieri.

“However, this was not the only important methodological innovation of Valerio. A second decisive step was his invention of the method of exhaustion … we believe that Valerio should be credited with the first systematization of this method dealing with comparison of areas and volumes, codified in the first three theorems of De Centro Gravitatis, and not just assimilation of a more or less standardized approach or know-how in Greek mathematics” (Napolitani & Saito, pp. 67-72). “It was Valerio who achieved this step: he raised a series of proof techniques to the level of a method, which, from Grégoire de Saint Vincent on, was to be called the method of exhaustion” (ibid., p. 93). “Valerio makes a wide use of this principle in De Centro Gravitatis: the most famous case is the determination of the ratio between the sphere and the circumscribed cylinder (De Centro Gravitatis, II-12), praised by Galileo in the ‘First Day’ of his Two New Sciences (Discorsi Intorno a Due Nuove Scienze) published in 1638 and mentioned in almost all subsequent literature” (ibid., p. 92).

Valerio himself regarded these first three theorems of Book 2 as being particularly important: “here you will find many things necessary for this research, which should find their place of their own right in geometry. Especially the first three propositions of the second book: you will understand that by means of these a large – and extremely difficult – part of geometry has been set on a royal road by straightforward and general demonstration” (De Centro Gravitatis, p. 2 – translation from Napolitani & Saito, p. 71). “Modern studies … have always tended to see, in these propositions, an anticipation not only of the concept of limits, but even of the theorem that the limit of a ratio is equal to the ratio of the limits” (p. 86). “One of the most interesting lemmas in the books [of De Centro Gravitatis] says in effect that if lim x = a and lim y = b, and if x/y = c = constant, then a/b = lim x/lim y = lim x/y = c, which is basically the same as lemma IV of book I of Newton’s Principia and as Cavalieri’s principle” (DSB).

Perhaps the crucial result in the whole work is Theorem II-32, “in which Valerio proves that if two figures decreasing around the same axis or diameter (one of which may well be a solid circa axim, and the other a plane figure circa diametrum, as Valerio underlines), always have proportional sections when they are cut by any two planes parallel to the base, then they have the same center of gravity” (Napolitani & Saito, p. 95). Theorem II-32 will prove to be an extremely powerful instrument in the hands of Valerio: it enables him to determine the centers of gravity of ‘all’ the solids: ‘many of these in two ways, some in three ways’ (De Centro Gravitatis, p. 2). Here we mean by ‘all the solids’ what we call ‘Archimedean solids,’ in particular conoids and spheroids and their segments … For Renaissance geometry, the center of gravity of the parabolic conoid was a sort of extrema Thule: nobody, not even Maurolico – probably the greatest mathematician of the XVI century – had proceeded beyond it. Furthermore, the result was already known from Archimedes’ Floating Bodies, and this work had guided Commandino in his unsuccessful attempt to devise a convincing proof. Valerio goes well beyond these Pillars of Hercules: he easily succeeds in determining centers of gravity which were wholly unknown in the mathematics of his period: in particular, that of the hemispheroid and – above all – that of the hyperbolic conoid (hyperboloid)” (Napolitani & Saito, pp. 107-8).

In 1587-88 Galileo had written his own work on the centre of gravity of the paraboloid, Theoremata de centro gravitatis solidorum. Valerio met Galileo in Pisa in 1590 and influenced him to renew his studies on centres of gravity. “Early in 1609 Galileo sent his demonstration that a parabolic line through the corners of a rectangle divided its area in the ratio of one-third to two-thirds to a friend in Rome for delivery to Luca Valerio, whose book on centers of gravity and quadrature of the parabola (published in 1604) he greatly admired. Galileo had forgotten his meeting with Valerio at Pisa nearly twenty years before, of which the Roman mathematician reminded him in reply, praising Galileo’s demonstration. The correspondence thus opened resulted in Galileo’s sending to Valerio for criticism two principles upon which he intended to establish his treatise on motion, now greatly expanded, in June 1609” (Drake, p. 136).

“In December 1612, having completed his Sunspot Letters, Galileo considered having the Linceans print his work on centers of gravity of paraboloids, probably in the form of a letter to Luca Valerio. Ensuing correspondence shows that he sent this to Cesi, who in discussing the matter with Valerio found that he was then revising his own book on a similar subject, published in 1603-4” (ibid., p. 202). “Galileo repeatedly stated that he had given up the idea of publishing his early work, Theoremata, because ‘some time later, he ran across the book of Luca Valerio, a prince of geometers, and saw that this resolved the entire subject without omitting anything; hence he went no further, though his own advances were made along quite a different road from that taken by Valerio’ (English translation by Stillman Drake in Galileo, Two New Sciences (1974), p. 260). It may be conjectured that one of the things he read in the book of ‘a prince of geometers’ which convinced him to abandon his research in this field was the determination of the center of gravity of the hyperboloid: Valerio’s result in fact refuted what Galileo thought he had demonstrated” (Napolitani & Saito, p. 108, n. 1). ”In the end Valerio did not reprint his book, and Galileo eventually placed his investigations of centers of gravity in an appendix to his own last book in 1638” (Drake, p. 202).

In his sequel to the main work, Quadratura Parabolae, Valerio used the results on centres of gravity of solids in De Centro Gravitatis to establish Archimedes’ formula for the area of a segment of a parabola – ‘The area of any segment of a parabola is four-thirds that of the triangle which has the same base as the segment and equal height’ – which Archimedes had established in his treatise The Quadrature of the Parabola. The crucial result was Valerio’s proof in De Centro Gravitatis that the centre of gravity of a hemisphere was at the point on its axis 3/8 of the distance from the centre of the sphere to its circumference.

“In his 1606 Quadrature of the Parabola through a Simple Error [Valerio] gave an ingenious proof which used centers of gravity to re-derive Archimedes’ area formula for the parabola. Much of Valerio’s 1606 work consisted of showing [what] his determination of the center of gravity of the hemisphere implied [about] the center of gravity of a semiparabola … Th[e] result is equivalent to the integral computation that shows that the x-coordinate of the center of gravity of the semiparabola bounded by the y-axis, the curve y = x2, and the line y = a2, is x = 3a/8, but Valerio’s lengthy exhaustion proof is entirely geometrical and shows in essence that since the area π(a2x2) of any slice of a hemisphere of radius a perpendicular to the x-axis is proportional to the corresponding slice a2x2 of the semiparabola, the two figures will balance in equilibrium when suspended from the same point on a line parallel to the x-axis [and hence have the same centre of gravity – this is an application of Theorem II-32 in De Centro Gravitatis]” (Leahy, p. 268). See Leahy (pp. 269-270) for Valerio’s deduction of Archimedes’ area formula for the parabola from this result, which makes use of Archimedes’ Law of the Lever: ‘Two magnitudes balance at distances reciprocally proportional to the magnitudes’.

De Centro Gravitatis was dedicated to Pope Clement VIII, who had been a pupil of Valerio in Rome; he also thanked Clement VIII’s nephew Cardinal Pietro Aldobrandini for his support. Valerio “had to hasten the publication of the book in 1604 to satisfy his protectors, Pope Clement VIII and his nephew, Cardinal Pietro Aldobrandini: he even states that he wrote the whole of the third book in only one month, October 1603. The existence of a previous version of the work consisting of only two books is confirmed by Valerio himself, though no trace of this version survives now” (Napolitani & Saito, p. 73).

Riccardi I 570 1 & 2 (“raro e pregiato”); Carli Favaro 10, 24; Roberts & Trent 332-3. Baron, The Origins of the Infinitesimal Calculus, 1969. DSB XIII, 560-1. Divizia, Remarks on ’De centro gravitatis solidorum’ of Luca Valerio’, Physis 25 (1983), pp. 227-249. Drake, Galileo at Work, 1978. Leahy, ‘The method of Archimedes in the seventeenth century,’ American Mathematical Monthly 125 (2018), pp. 267–272.Napolitani & Saito, ‘Royal Road or Labyrinth? Luca Valerio’s De Centro Gravitatis Solidorum and the beginnings of modern mathematics,’ Bollettino di Storia delle Scienze Matematiche 24 (2004), pp. 67-124.

Small 4to, pp. [viii], 80, 92, 94 [2, blank]; 20, 23, [1, blank], with printer’s device on first title and numerous woodcut diagrams in the text. Numerous marginal annotations in a contemporary hand. Contemporary vellum. A fine copy.

Item #4695

Price: $25,000.00

See all items by