## Cylindricorum et annularium libri IV: item De circulorum volutione per planum, dissertatio physiomath[i]ca.

Antwerp: Jacob van Meurs, 1651.

First edition, very rare, and a fine copy. “Tacquet’s most important mathematical work, *Cylindricorum et annularium,* contained a number of original theorems on cylinders and rings. Its main importance, however, lay in its concern with questions of method. Tacquet rejected all notions [originating with Cavalieri] that solids are composed of planes, planes of lines, and so on, except as heuristic devices for finding solutions. The approach he adopted was that of Luca Valerio and Gregorius [of Saint-Vincent], an essentially Archimedean method” (DSB). “Tacquet’s criticisms must have been effective, because indivisibles became homogeneous magnitudes as a result of innovations introduced during the course of the seventeenth century” (Rossini, p. 465). The historian of mathematics Henri Bosmans “states that it was Tacquet’s decisive influence, followed by Pascal’s large-scale implementation, which allowed what is sometimes referred to as the passage from indivisibles to infinitesimals” (Julien, p. 187). “In this work the ideas that the tangent and the area under a curve were inverse to each other appeared. It arises from the way that Tacquet thought of curves generated by moving points, but not actually comprising of points. Of course this idea is an early form of what would become clear when the calculus was invented, namely that the derivative and integral were inverse to each other. This book had a considerable effect on Pascal and was important in setting the scene for the invention of the calculus” (MacTutor). Bosmans has pointed out the debt owed to Tacquet’s *Cylindricorum et annularium* by Pascal in his preparation of the *Lettres de Dettonville* (1659). Indeed, in the *Lettre à Carcavy *Pascal writes that Tacquet’s book is full of “learned geometry”, praises how Tacquet “handles the indivisibles with all desirable rigour”, and refers specifically to Tacquet for a result on the approximation of a surface by inscribed and circumscribed polyhedra (see Descotes). “The change in Pascal to a clear point of view with respect to infinitesimals may well have come from Pascal’s reading of Tacquet’s *Cylindricorum et annularium*, in which the author denied the validity of concluding anything about the ratio of surfaces from the ratio of their indivisibles, or lines” (Boyer, p. 151). This is a rare work, particularly in commerce. ABPC/RBH list only the Turner copy from the University of Keele, offered by a prominent London dealer in 2002, and subsequently sold at Reiss in 2005 (quite a poor copy, in modern binding, with some paper repairs and a partially removed library stamp). Ours is a fine, untouched copy in contemporary binding.

*Provenance*: Contemporary ownership inscription on title of “Conde da Torre”, possibly the Portuguese nobleman João de Mascarenhas (1633-1681), second Count of Torre and first Marquis of Fronteira.

“The Jesuit mathematician André Tacquet (1612-60) was, by the standards of his time, a man of the world. Although he may never have left his native Flanders, his network of correspondents spanned Europe’s religious divide, reaching to Italy and France but also to Protestant Holland and England. Only months before his death he entertained the Dutch polymath Christiaan Huygens, who had travelled to Antwerp with the express purpose of meeting Tacquet, by then regarded as one of the brightest mathematical stars ever to come out of the Society of Jesus … It was his mathematical excellence that transcended 17^{th}-century prejudices. In England, Henry Oldenburg, secretary of the Royal Society of London and no friend of the Jesuits, spent so much time describing Tacquet’s *Opera mathematica* at the Society’s meeting in January 1669 that he felt compelled to apologise to the fellows for abusing their patience. But it was, he insisted, ‘one of the best books ever written on mathematics’” (Alexander, p. 118).

“In 1651 André Tacquet, the urbane Fleming whose work was celebrated by Catholics and Protestants alike, published his*Cylindricorum et annularium libri IV* (‘Four books on cylinders and rings’), a work dedicated to the study of geometrical features of these figures and their applications. Befitting a Jesuit publication, the frontispiece shows two angels, bathed in divine light, holding up a ring enclosing the book’s title; on the ground below them a band of cherubs is busy putting the theory into practice. The implication is clear: divine mathematics, universal and perfectly rational, orders and arranges the physical world to the best possible effect. It is a fetching visual depiction of the Jesuit view of the role and nature of mathematics.

“The *Cylindricorum et annularium* is Tacquet’s most celebrated work, the one that established his reputation as one of Europe’s most original and creative mathematicians. As it turned out, it may have been a bit too ‘original and creative’ for his superiors: when Tacquet sent a copy of the book the newly appointed superior general, Goswin Nickel, the general’s response was surprisingly cool. After thanking the mathematician and congratulating him on the book, Nickel added that it would be better if Tacquet applied his impressive gifts to producing textbooks of elementary geometry for use by students at the Society’s colleges, rather than original works aimed at a select audience of professional mathematicians … Tacquet, a good soldier in the Army of Christ, obeyed. From then on he published no more original work, but concentrating instead on producing textbooks, some of which are of such quality that they became standards in the field for over a century …

“In his critique, Tacquet is respectful, even deferential, toward his rivals. He refers to Cavalieri as ‘a noble geometer’ and insists that he ‘does not wish to detract from the deserved glory’ of Cavalieri’s ‘most beautiful invention’ [*Geometria indivisibilibus*, 1635]. Tacquet knew of what he spoke, because he was himself deeply familiar with the work of Cavalieri and Torricelli and was no less capable than they of using their method to arrive at new results. But once he gets beyond his congenial style and mathematical mastery, it becomes clear that Tacquet’s opposition to the infinitely small is … unyielding … ‘I cannot consider the method of proof by indivisibles as either legitimate or geometrical,’ he states flatly at the opening of his discussion of indivisibles. ‘It proceeds from lines to surfaces, from surfaces to solids, and applies to the surface the quality or proportion obtained from the lines, and transfers what was obtained from the surfaces to the solid.’ ‘By this method,’ he concludes, ‘nothing can be proven by anyone’” (*ibid*., pp. 161-2).

“André (or Andreas) Tacquet resembles his contemporary, Torricelli, in the generality of his adoption from his predecessors of varied infinitesimal methods. In his *Cylindricorum et annularium libri IV* he gave, for example, four demonstrations of the proposition that the volume of a sphere is equal to that of a cylindrical wedge whose base is half a great circle of the sphere, and whose altitude is equal to the circumference of the sphere. This theorem had been given by a number of mathematicians since Kepler, as well as by Archimedes in the *Method*, probably not then extant. Tacquet, however, after proving the theorem in two ways by the use of inscribed and circumscribed figures, gave two further demonstrations by indivisibles, based on the equality of triangles and circular sections. Torricelli had himself been satisfied with the rigor of proofs by means of indivisibles, although he supplied alternative demonstrations for the benefit of others. Tacquet, on the other hand, said that he did not consider that the method of Cavalieri was to be admitted as either legitimate or geometrical. He maintained that the cylindrical wedge could not, in all strictness, be considered as made up of triangles; nor could the sphere be regarded as composed of circles … A geometrical magnitude, he asserted, is made up only of *homogenea*, that is, parts of like dimension – a solid of small solids, and area of small areas, and a line of small lines – and not of *heterogenea*, or parts of a lower dimension, as Cavalieri had maintained. He therefore felt that a proposed magnitude is exhausted (a word he undoubtedly acquired from Gregory of St. Vincent) by inscribing *homogenea* within them ‘as in the manner of the ancients’” (Boyer, pp. 139-140).

Tacquet gave a famous example where Cavalieri’s method led to incorrect results. On pp. 23-24, he considers a right-angled triangle with one horizontal and one vertical side. Rotating this triangle around the vertical side generates a cone. Each plane section of the cone parallel to the base determines a circle, and the circumference of each of these circles bears the same ratio to its radius (namely, 2π, to use our notation). Since the surface of the cone is made up of all these circular cross-sections, and the triangle is made up of all the radii, Cavalieri’s method would imply that the same ratio is also that between the surface area of the cone and the area of the triangle. But this is not the case.

Archimedes had used a double *reductio ad absurdum* style of proof to find areas and volumes, and this argument continued to be used until the publication of Cavalieri’s work. To show that the area of a given region is equal to *A*, Archimedes showed that, for any number *B* smaller than *A*, an inscribed figure could be constructed whose area is greater than *B* (the inscribed area was usually composed of rectangles or triangles so that its area could easily be determined). This shows that the area of the given region cannot be smaller than *A*. A similar argument with circumscribed figures shows that the area cannot be larger than *A*. This technique is usually referred to as the ‘method of exhaustion.’

In the *Cylindricorum et annularium* Tacquet gives two proofs of most of his results on rings and cylinders, the first using a modified form of the exhaustion technique, the second using indivisibles; the precision and rigour of the traditional method is repeatedly stressed. But Tacquet introduces a number of innovations in the use of the method of exhaustion.

“Tacquet’s book has two theorems dealing with exhaustion, which are the foundation for nearly all other theorems in the book. The first proposition of the first book is reminiscent of Valerio’s theorem [*De Centro Gravitatis Solidorum Libri Tres*, 1604]:

Let *A *and *B* be two magnitudes, either areas or volumes, and let the ratio of *E* to *F *be given. If one can consecutively inscribe into *A* and *B* a sequence of magnitudes that relate to one another as *E* to *F*, and if these magnitudes exhaust *A* and *B* (i.e., they differ from these by an arbitrarily small amount), then the magnitude *A* will relate to the magnitude *B* as *E* to *F*.

“Tacquet’s general theorem has the advantage that he does not have to repeat a double *reductio ad absurdum* with each proof …

“The first proposition of the second book introduces another exhaustion method:

If a sequence of magnitudes *A _{i,n}* and

*B*can be inscribed in magnitudes

_{i,n}*A*and

*B*, and if likewise a sequence of magnitudes

*A*and

_{c,n}*B*can be circumscribed about magnitudes

_{c,n}*A*and

*B*, and if moreover

*A*and

_{i,n}*A*exhaust

_{c,n}*A*and for the corresponding magnitudes we have

*A*and

_{i,n }/ B_{i,n}= E/F*A*, then

_{c,n }/ B_{c,n}= E/F*A/B = E/F.*

“The simplification lies in the fact that it is no longer necessary to exhaust the inscribed and circumscribed magnitudes for each of the magnitudes *A* and *B*, as it suffices to calculate the ratio for one or the other …

“An important new concept is found in Tacquet’s definitions of surfaces and solids. He defines the cylinder, for instance, as a solid that is generated by the movement of a circle in such a way that one of the points of the circle segment moves along a straight line. The axis of this cylinder is the straight line joining the centre of two of the generated circles. Despite this definition he does not accept that the cylinder is composed of circles” (*ibid*., pp. 214-5).

An important example of Tacquet’s ‘kinetic’ method of generating curves and surfaces is contained in the second part of the *Cylindricorum et annularium*, entitled ‘Dissertatio physico-mathematica de circulorum volutionibus,’ in which Tacquet studies the cycloid, a curve traced out by a point on the circumference of a circle as it is rolled along a straight line. This curve was to be the focus of Blaise Pascal’s work on indivisibles, published in the *Lettres de A. Dettonville* (1659).

“Although Pascal was undoubtedly attracted by the power of indivisible methods, he was impressed by the careful geometrical approach of Grégoire de Saint-Vincent and swayed by the vigorous criticism of Cavalierian indivisibles launched by André Tacquet in his work *Cylindricorum et annularium*. Pascal was accordingly impelled to examine carefully the basis for the use of indivisibles in geometry” (Baron, pp. 199-200).

“Blaise Pascal in a sense represents the highest development of the method of infinitesimals carried out under the tradition of classical geometry … Pascal was not a professional geometer, and as a result his geometrical work was accomplished in two periods which were separated by an interval of mathematical inactivity (from 1654 to 1658) during which he devoted his interests to theology. These two periods, moreover, are characterised by somewhat different views as to the nature of infinitesimals … In this connection he enunciated, in the *Potestatum numericarum summa* of 1654, the theorem on the integral of *x ^{n}* … The essential point in Pascal’s demonstration is the omission of terms of lower dimension … The geometrical intuition of indivisibles of lower dimension was carried over into arithmetic to justify the neglect of certain terms of lower degree …

“In the later period of his mathematical activity, his view appears to be modified. In connection with problems such as those in his *Traité des sinus du quart de cercle* of 1659 [contained in the *Lettres de Dettonville*], … he used the language of infinitesimals in speaking of the sum of all the ordinates; but he added that one need not fear to do this, inasmuch as what is really meant is the sum of arbitrarily small rectangles” (Boyer, pp. 147-151).

Bosmans sees Pascal’s *Potestatum numericarum summa* as containing two mutually incompatible ideas about indivisibles. On the one hand, Pascal sometimes regards indivisibles as rigorously null quantities, as had Cavalieri. On the other hand, he sometimes regards indivisibles as simply quantities that are negligible in comparison to other quantities. “Bosmans then strongly underlines the difference with the clarity of the *Lettres de Dettonville*, where Pascal expresses himself with impeccable rigour, substituting for the strict indivisibles of Cavalieri homogeneous quantities whosesums differ from that to be measured by less than any given quantity. He then finds the reason for this progress in the reading that Pascal would have made between 1654 and 1658 of the book published by Tacquet in 1651” (Descotes, pp. 1-2, our translation).

In the last section of the *Lettre à Carcavy*, Pascal refers specifically to Tacquet in his discussion of the problem of finding the area of a surface obtained by rotating a curve around a vertical axis. When an infinitesimal section of the curve is rotated, one obtains a circular band; these bands together make up the whole surface. “This is properly what, according to Dettonville, Tacquet has demonstrated: ‘The sum of these semi-circumferences of the surface of the semi-solid makes up this very surface (as others have demonstrated, among them Tacquet)’. We find in fact in the *Cylindricorum et annularium*, Book II, 1st part, a Proposition VI which corresponds to Pascal’s words. Its object is to prove that, if we consider in a great-circle BICQ on a sphere with diameter BC, if we inscribe and circumscribe regular polygons on the semi-circle BIC, and if we rotate these polygons around the diameter BC, they inscribe and circumscribe in the sphere with solids whose surfaces differ from that of the sphere by a quantity which can be made as small as one wishes. We see how it accords with Pascal’s thought: the inscribed and circumscribed segments generate bands during the rotation which, at the limit, can be said to compose the curved surface. In accordance with his principles, Father Tacquet demonstrates it in the manner of the Ancients” (*ibid*., p. 4).

“Tacquet was the son of Pierre Tacquet, a merchant, and Agnes Wandelen of Nuremberg. His father apparently died while the boy was still young but left the family with some means. Tacquet received an excellent education in the Jesuit collège of his native town, and a contemporary report describes him as a gifted if somewhat delicate child. In 1629 he entered the Jesuit order as a novice and spent the first two years in Malines and the next four in Louvain, where he studied logic, physics, and mathematics. His mathematics teacher was William Boelmans, a student of and secretary to Gregorius Saint Vincent. After his preliminary training Tacquet taught in various Jesuit *collège**s* for five years, notably Greek and poetry at Bruges from 1637 to 1639. From 1640 to 1644 he studied theology in Louvain and in 1644-45 he taught mathematics there. He took his vows on 1 November 1646 and subsequently taught mathematics in the collèges of Louvain (1649-55) and Antwerp (1645-49, 1655-60)” (DSB).

A second edition of the *Cylindricorum et annularium* was published in 1659, with the addition of a fifth book devoted to an unrelated subject, the paradox of ‘Aristotle’s wheel’. The perceived unsuitability of the *Cylindricorum at annularium* for the Jesuit colleges may explain why it was not added to all copies of the *Opera mathematica* (1669) (it was not present in the Macclesfield copy, for example).

De Backer-Sommervogel VII, 1806, 3; Poggendorff II, 1064. Alexander, *Infinitesimal*, 2014. Baron, *The Origins of the Infinitesimal Calculus*, 1969. Boyer, *The History of the Calculus and its Conceptual Development*, 1949. Descotes, ‘Documents relatifs aux *lettres de A. Dettonville *(I). Pascal et le Père Tacquet,’ *Courrier du Centre international Blaise Pascal* 14 (1992), pp. 1-13. Julien (ed.), *Seventeenth-Century Indivisibles Revisited*, 2015. Malet, *From indivisibles to infinitesimals*, 1996. Meskens, *Between Tradition and Innovation*: *Gregorio a San Vicente and the Flemish Jesuit Mathematics School*, 2021. Rossini, ‘Giordano Bruno and Bonaventura Cavalieri’s theories of indivisibles: a case of shared knowledge,’ *Intellectual History Review *28 (2018), pp. 461-476.

Small 4to (217 x 162 mm), pp. [xx], 284, [4], with 18 folding engraved plates (the first 9 bound preceding title, the remainder at end, as in the instructions to binder on p. [286]), full-page engraving on title, second work with special half-title (occasional light browning and spotting). Contemporary yapped limp vellum with manuscript title on spine, remains of ties (a few stains and light rubbing).

Item #5427

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Price:
$15,000.00
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