Exercitationes geometricae sex. I. De priori methodo indiuisibilium. II. De posteriori methodo indiuisibilium. III. In Paulum Guldinum è Societate Iesu dicta indiuisibilia oppugnantem. IV. De vsu eorumdem ind. in potestatibus cossicis. V. De vsu dictorum ind. in vnif. diffor. grauibus. VI. De quibusdam propositionibus miscellaneis, quarum synopsim versa pagina ostendit.

Bologna: Giacomo Monti, 1647.

First edition, rare, of Cavalieri’s second work on integration methods, a sequel to and elaboration of his Geometria indivisibilibus (1635), one of the most important forerunners of the integral calculus. Cavalieri’s ‘method of indivisibles’, considers an area as made up of an indefinite number of equidistant parallel line segments (‘omnes lineae’), and a solid as made up of an indefinite number of parallel plane areas. It states that, if two planar figures are contained between a pair of parallel lines, and if the lengths of the two segments cut by them from any line parallel to the including lines are always in a given ratio, then the areas of the two planar pieces are also in this ratio (there is an analogous principle for the determination of volumes). Cavalieri’s principle provided a simple and effective alternative to the Archimedean method of exhaustion, and was used by Kepler, Galileo, Cavalieri’s pupil Torricelli, Wallis, Pascal, and others. The present work contains numerous applications of the method (see below), as well as replies to those who criticised the earlier work. In his copy of the present work, held by the Senate House Library, University of London, Augustus de Morgan wrote in 1852, “This work must not be confounded with the Geometria of Cavalieri published in 1635, and reprinted after his death. But of the two, this is the work which most established his claim to have ushered in the dawn of the integral calculus”. The Exercitationes also contains an important analysis of the focal properties of spherical surfaces and lenses. ABPC/RBH lists only three copies since Honeyman.

“For Cavalieri a surface consists of an indefinite number of equidistant parallel straight lines and a solid of a set of equidistant parallel planes. These constitute the line and surface indivisibles respectively. For plane figures (or solids) a regula, that is, a line (or plane) drawn through the vertex, is the starting point. The regula moves parallel to itself until it comes into coincidence with a second line (or plane) termed the base or tangens opposita. The intercepts (lines or plane sections) of the regula with the original plane (or solid) figure are the elements or indivisibles making up the totality of lines (or planes) of the figure.

“In the techniques developed by Cavalieri the indivisibles of two or more configurations are associated together in the form of ratios and from these ratios the relations between the areas (or volumes) of the figures themselves are derived. In moving from a relation between the sums of the indivisibles and thus to a relation between the spaces the infinite is employed, but purely in an auxiliary role.

“Cavalieri’s defence of indivisible methods was based primarily on the idea of a device or artificium which works rather than on any definite or dogmatic views as to the nature of indivisibles and the spaces which they occupy. Nevertheless the classic problem of the nature of the continuum imposed itself upon him and, from the outset, he felt himself obliged to try to meet some of the arguments which he felt might be directed against his methods. He admits that some might well doubt the possibility of comparing an indefinite number of lines, or planes (indefinitae numero lineae, vel plana). When such lines (or planes) are compared, he says, it is not the numbers of such lines which are considered but the spaces which they occupy. Since each space is enclosed it is bounded and one can add to it or take away from it without knowing the actual number of lines or planes. Whether indeed the continuum consists of indivisibles or of something else neither the space nor the continuum is directly measurable. The totalities, or sums, of the indivisibles making up such spaces are, however, always comparable. To establish a relation between the areas of plane figures, or the volumes of solid bodies, it is therefore sufficient to compare the sums of the lines, or planes, developed by any regula.

“The foundations for Cavalieri's indivisible techniques rest upon two distinct and complementary approaches which he designates by the terms collective and distributive respectively. In the first, the collective sums, Σ l1 and Σ l2, of the line (or surface) indivisibles for two figures P1 and P2 are obtained separately and then used to establish the ratio of the areas (or volumes) of the figures themselves. If, for example, Σ l1 / Σ l2 = α/β, then P1/P2 = α/β. This approach was given the most extensive application by Cavalieri and he exploited it with skill and ingenuity to obtain a fascinating collection of new results which he exhibited in the Exercitationes [Geometriae Sex, 1647]. The distributive theory, on the other hand, was developed primarily in order to meet the philosophic objections which Cavalieri felt might be raised against the comparison of indefinite numbers of lines and planes. Fundamental here is Cavalieri's Theorem: the spaces (areas or volumes) of two enclosed figures (plane or solid) are equal provided that any system of parallel lines (or planes) cuts off equal intercepts on each. In brief, if for every pair of corresponding intercepts l1 and l2, l1 = l2, then P1 = P2. An immediate extension follows: if l1/ l2 = α/β, then P1/P2 = α/β. Cavalieri himself only made use of this method in a limited number of cases where α/β is constant for all such pairs of intercepts. This technique, however, in the hands of Gregoire de Saint-Vincent and others in the seventeenth century, became a valuable means of integration by geometric transformation. Ultimately, whichever of the two methods was applied, Cavalieri was prepared to concede that absolute rigour required in each case an Archimedean proof with completion by reductio ad absurdum” (Baron, The Origins of the Infinitesimal Calculus, 1969, pp. 124-6).

“The concept of indivisibles does sometimes show up fleetingly in the history of human thought: for example, in a passage by the eleventh-century Hebrew philosopher and mathematician Abraham bar Hiyya (Savasorda); in occasional speculations — more philosophical than mathematical — by the medieval Scholastics; in a passage by Leonardo da Vinci; in Kepler’s Nova stereometria doliorum ... [and in] Galileo ...

“In Cavalieri we come to a rational systematization of the method of indivisibles, a method that not only is deemed useful in the search for new results but also, contrary to what Archimedes assumed, is regarded as valid, when appropriately modified, for purposes of demonstrating theorems.

“At this point a primary question arises: What significance did Cavalieri attribute to his indivisibles? This mathematician, while perfectly familiar with the subtle philosophical questions connected with the problem of the possibility of constituting continuous magnitudes by indivisibles, seeks to establish a method independent of the subject's hypotheses, which would be valid whatever the concept formed in this regard ... It must be further pointed out, according to L. Lombardo Radice, that the Cavalieri view of the indivisibles has given us a deeper conception of the sets: it is not necessary that the elements of the set be assigned or assignable; rather it suffices that a precise criterion exists for determining whether or not an element belongs to the set” (DSB).

The Geometria was extensively read and its contents hotly debated. The most serious attack came from Paul Guldin in his Centrobaryca (1635-41). He accused Cavalieri of plagiarising Kepler’s Nova Stereometria Doliorum (1615) and Sover’s Curvi ac Recti Proportio (1630). More seriously, Guldin “criticized Cavalieri’s use of indivisibles in his Geometria indivisibilibus (1635), asserting not only that the method had been taken from Kepler but also that since the number of indivisibles was infinite, they could not be compared with one another. Furthermore, he pointed out a number of fallacies to which the method of indivisibles appeared to lead. In 1647, after the death of Guldin, Cavalieri published Exercitationes geometricae sex, in which he defended himself against the first charge by pointing out that his method differed from that of Kepler in that it made use only of indivisibles, and against the second by observing that the two infinities of elements to be compared are of the same kind.” (DSB). Cavalieri’s attempts to resolve the paradoxes raised by Guldin led to a somewhat clearer formulation of his method, which he presents in exercitatio I and II, and it was this formulation that came to be used by later 17th century mathematicians.

The Exercitationes contains a wealth of new applications of the method of indivisibles. In exercitatio IV Cavalieri presents a generalization of the method of indivisibles which enabled him to deal with algebraic curves of degrees higher than two. Using this he obtained (in modern terms) the integration of xn for n = 1,2,3,… This had been given in the case n = 2 in the Geometria (but in that case the result was known to Archimedes). Exercitatio V contains the first general proof of ‘Pappus’ theorem’ relating the volume of a solid of revolution to the area of the surface being rotated. This theorem had been proved in some special cases in Guldin’s Centrobaryca. Cavalieri not only provided a general proof in the Exercitationes, but also accused Guldin of plagiarism, as the result appears in certain editions of Pappus’ works (although it is now thought to be a later interpolation). Exercitatio VI treats miscellaneous topics, including the description of a hydraulic pump which Cavalieri designed for the monastery of S. Maria della Mascarella in Bologna, where he held the honorary position of prior.

The Exercitationes also contains an important contribution to optics. “Cavalieri took the first important step beyond Kepler in analysing the focal properties of spherical surfaces and lenses in Exercitationes geometricae sex. He begins the section “On the foci of lenses” [pp. 458-95, Props. VII-XIX of exercitatio VI] by observing that whereas conic sections have precise foci, spherical lenses, as Kepler showed, possess a focus only to a very close approximation. He then succeeded in deriving a general rule for the focal point for all varieties of lenses, by assuming, as Kepler did, thin lenses and the small-angle approximation, but without now mentioning the newly discovered sine law of refraction. Using these approximations, he found for the focus f of a lens with index of refraction 3/2, (ρ1 ± ρ2)/ρ1 = 2ρ2/f, where ρ1,2 represent the radii of the two surfaces and the plus or minus sign is taken depending on whether the surfaces face in the opposite or the same direction, respectively. Cavalieri derived these results by means of single paraxial rays – not pencils – very much like a modern elementary textbook” (Alan Shapiro in Before Newton: The Life and Times of Isaac Barrow, M. Feingold (ed.), pp. 127-8).

Cavalieri (1598-1647) was a Jesuit and professor of mathematics at Bologna. He considered himself a disciple of Galileo, whom he had met through his teacher Benedetto Castelli. It was Galileo who urged Cavalieri to look into problems of the calculus, and who praised him by stating that ‘few, if any, since Archimedes, have delved as far and as deep into the science of geometry’. Galileo included Cavalieri’s theory in a discussion of the theory of matter in the First Day and in the discussion of accelerated motion in the Third Day of his Discorsi e dimostrazioni matematiche (1638). In the last months of his life, Cavalieri became too ill to attend to the publication of the Exercitationes, and it was his friend Stefano degli Angeli (1623-97) who made the final edits and saw the book through the press.

Brunet I, 1697: 'ouvrage très-recommandable'; Honeyman 649; Macclesfield 505; Riccardi I, 329 (‘una delle più preziose opere del nostro autore’); Sotheran I, 734: ‘containing the earliest demonstration of the theorems of Pappus, and the first determination of focal distances of glass lenses’. See K. Andersen, ‘Cavalieri’s method of indivisibles,’ Arch. Hist. Exact Sci. 31 291-367 (springerlink.com/content/t353114221102655/fulltext.pdf).

4to (230 x 162mm), pp. [12], 543, [1], with woodcut title vignette, woodcut initials and head- and tail-pieces, and numerous woodcut geometrical diagrams in text (some light spotting throughout). Eighteenth-century calf with gilt lettering and impressions on the spine (neatly restored).

Item #5536

Price: $15,000.00