## [Opera Geometrica]. De sphaera et solidis sphaeralibus libri duo in quibus Archimedis doctrina de sphaera et cylindro denuo componitur, latiùs promouetur. [De motu gravium naturaliter descendentium et proiectorum libri duo – De dimensione parabolae solidique hyperbolici problemata duo].

Florence: Amadoro Massa & Lorenzo de Landis, 1644.

First edition of the only book by Torricelli published in his lifetime, and an outstanding copy, uncut in original boards. The *Opera geometrica* contains his most important works on mathematics and physics. It diffused and considerably advanced the new geometry of indivisibles begun by Cavalieri, and (in *De motu gravium*) continued the study of the parabolic motion of projectiles begun by Galileo. “As Torricelli acquired increasing familiarity with the method of indivisibles, he reached the point of surpassing the master – as Cavalieri himself said” (DSB). “Torricelli far outdid his master Cavalieri in the flexibility and perspicuity of his use of the method of indivisibles in making new discoveries. One of the novel results which pleased him greatly was the determination, in 1641 [and included in the present work, p. 115 of *De dimensione parabolae*], that the volume of an infinitely long solid, obtained by revolving about its asymptote a portion of the equilateral hyperbola, was finite … Torricelli’s proof is interesting in that it makes use of the idea of cylindrical indivisibles, whereas those of Cavalieri had invariably been plane” (Boyer, pp. 125-6). This is “a gem of the mathematical literature of the time” (DSB). In the *Opera geometrica*, Torricelli “worked out the consequences of laws of falling (also in parabolic ballistics) formulated by Galileo, and verified them by a number of ingenious experiments. In a particular section [pp. 200-201] Galileo’s laws are applied to liquids flowing from apertures in vessels, and what is known as Torricelli’s law [that the efflux velocity of a jet of liquid spurting from a small hole at the bottom of a vessel is equal to that which a single drop of the liquid would have if it could fall freely in a vacuum from the level of the top of the liquid] was stated and experimentally proved” (*Biographical Dictionary of Scientists*). On the basis of this result, “Ernst Mach proclaimed Torricelli the founder of hydrodynamics” (DSB). The final section of the *Opera *also contains Torricelli’s study of the cycloid, the curve traced out by a point on the circumference of a circle rolling along a straight line; this was the first printed work on the cycloid. “By the Method of Indivisibles he demonstrated its area to be triple that of the revolving circle, and published his solution. The same quadrature had been effected a few years earlier by Roberval in France, but his solution was not known to the Italians” (Cajori, p. 172) – and Roberval’s proof was not published until 1693. In *De motu gravium*, Torricelli also develops an original method of drawing tangents to curves. “Amongst all those who contributed to the development of infinitesimal processes before Newton and Leibniz, Torricelli exhibited most clearly the link between the two operations now known as differentiation and integration” (Baron, p. 185). The only other uncut copy in original boards listed by ABPC/RBH is the Norman copy (Sotheby’s, June 15, 1998, lot 822, $19,550).

“Mathematical research occupied Torricelli’s entire life. During his youth he had studied the classics of Greek geometry, which dealt with infinitesimal questions by the method of progressive elimination. But since the beginning of the seventeenth century the classical method had often been replaced by more intuitive processes; the first examples were given by Kepler, who in determining areas and volumes abandoned Archimedean methods in favor of more expeditious processes differing from problem to problem and hence difficult to imitate. After many years of meditation, Cavalieri, in his geometry of indivisibles (1635), drew attention to an organic process, toward which Roberval, Fermat, and Descartes had been moving almost in the same year; the coincidence shows that the time was ripe for new geometrical approaches.

“The new geometry considered every plane figure as being formed by an infinity of chords intercepted within the figure by a system of parallel straight lines; every chord was then considered as a rectangle of infinitesimal thickness—the indivisible, according to the term introduced by Galileo. From the assumed or verified relations between the indivisibles it was possible to deduce the relations between the totalities through Cavalieri’s principle, which may be stated as follows: Given two plane figures comprised between parallel straight lines, if all the straight lines parallel thereto determine in the two figures segments having a constant relation, then the areas of the two figures also have the same relation. The principle is easily extended to solid figures. In essence Cavalieri’s geometry, the first step toward infinitesimal calculus, replaced the potential mathematical infinity and infinitesimal of the Greek geometricians with the present infinity and infinitesimal.

“After overcoming his initial mistrust of the new method, Torricelli used it as a heuristic instrument for the discovery of new propositions, which he then demonstrated by the classical methods. The promiscuous use of the two methods—that of indivisibles for discovery and the Archimedean process for demonstration—is very frequent in the *Opera geometrica*. The first part of *De sphaera et solidis sphaeralibus*, compiled around 1641, studies figures arising through rotation of a regular polygon inscribed in or circumscribed about a circle around one of its axes of symmetry … he classifies such rotation solids into six kinds, studies their properties, and presents some new propositions and new metrical relations for the round bodies of elementary geometry. The second section of the volume deals with the motion of projectiles …

In the third section, apart from giving twenty demonstrations of Archimedes’ theorem on squaring the parabola, Torricelli shows that the area comprised between the cycloid and its base is equal to three times the area of the generating circle. As an appendix to this part of the work there is a study of the volume generated by a plane area animated by a helicoid motion round an axis of its plane, with the demonstration that it equals the volume generated by the area in a complete rotation round the same axis. Torricelli applies this elegant theorem to various problems and in particular to the surface of a screw with a square thread, which he shows to be equal to a convenient part of a paraboloid with one pitch.

“[Torricelli] extended [Cavalieri’s] theory by using curved indivisibles, based on the following fundamental concept: In order to allow comparison of two plane figures, the first is cut by a system of curves and the second by a system of parallel straight lines; if each curved indivisible of the first is equal to the corresponding indivisible of the second, the two figures are equal in area. The simplest example is given by comparison of a circle divided into infinitesimal concentric rings with a triangle (having the rectified circumference as base and the radius as height) divided into infinitesimal strips parallel to the base. From the equality of the rings to the corresponding strips it is concluded that the area of the circle is equal to the area of the triangle.

“The principle is also extended to solid figures. Torricelli gave the most brilliant application of it in 1641 by proving a new theorem, a gem of the mathematical literature of the time. The theorem, published in *Opera geometrica*, is as follows: [take a point *x = a* (> 0) on the hyperbola *xy* = 1, and rotate the part of the hyperbola with 0 < *x* ≤ *a* and *y *> 0 around the y-axis]. Although such area is infinite in size, the solid it generates by rotating round the asymptote [*y*-axis], although unlimited in extent, nevertheless has a finite volume, calculated by Torricelli as *π/a*. Torricelli’s proof, greatly admired by Cavalieri and imitated by Fermat, consists in supposing the solid generated by rotation to be composed of an infinite number of cylindrical surfaces of axis *x*, all having an equal lateral area, all placed in biunivocal correspondence with the sections of a suitable cylinder, and all equal to the surfaces of that cylinder: the principle of curved indivisibles allows the conclusion that the volume of this cylinder is equal to the volume of the solid generated by rotation of the section of the hyperbola considered … Still using curved indivisibles, Torricelli found, among other things, the volume of the solid limited by two plane surfaces and by any lateral surface, in particular the volume of barrels. In 1643 the results were communicated to Fermat, Descartes, and Roberval, who found them very elegant and correct …

“Torricelli made other important contributions to mathematics during his studies of mechanics. In *De motu gravium* he continued the study of the parabolic motion of projectiles, begun by Galileo, and observed that if the acceleratory force were to cease at any point of the trajectory, the projectile would move in the direction of the tangent to the trajectory. He made use of this observation, earning Galileo’s congratulations, to draw the tangent at a point of the Archimedean spiral, or the cycloid, considering the curves as described by a point endowed with two simultaneous motions. In unpublished notes the question is thoroughly studied in more general treatment. A point is considered that is endowed with two simultaneous motions, one uniform and the other varying, directed along two straight lines perpendicular to each other. After constructing the curve for distance as a function of time, Torricelli shows that the tangent at any point of the curve forms with the time axis an angle the tangent of which measures the speed of that moving object at the point. In substance this recognizes the inverse character of the operations of integration and differentiation, which from the fundamental theorem of the calculus, published in 1670 by Isaac Barrow, who among his predecessors mentioned Galileo, Cavalieri, and Torricelli. But not even Barrow understood the importance of the theorem, which was first demonstrated by Newton …

“In *De motu gravium* Torricelli seeks to demonstrate Galileo’s principle regarding equal velocities of free fall of weights along inclined planes of equal height. He bases his demonstration on another principle, now called Torricelli’s principle but known to Galileo, according to which a rigid system of a number of bodies can move spontaneously on the earth’s surface only if its center of gravity descends. After applying the principle to movement through chords of a circle and parabola, Torricelli turns to the motion of projectiles and, generalizing Galileo’s doctrine, considers launching at any oblique angle—whereas Galileo had considered horizontal launching only. He demonstrates in general from Galileo’s incidental observation that if at any point of the trajectory a projectile is relaunched in the opposite direction at a speed equal to that which it had at such point, the projectile will follow the same trajectory in the reverse direction. The proposition is equivalent to saying that dynamic phenomena are reversible—that the time of Galileo’s mechanics is ordered but without direction. Among the many theorems of external ballistics, Torricelli shows that the parabolas corresponding to a given initial speed and to different inclinations are all tangents to the same parabola (known as the safety parabola or Torricelli’s parabola, the first example of an envelope curve of a family of curves).

“The treatise concludes with five numerical tables. The first four are trigonometric tables giving the values of sin 2α, sin^{2}α, ½tan α, and sin α, respectively, for every degree between 0° and 90°; with these tables, when the initial speed and angle of fire are known, all the other elements characteristic of the trajectory can be calculated. The fifth table gives the angle of inclination, when the distance to which the projectile is to be launched and the maximum range of the weapon are known. In the final analysis these are firing tables, the practical value of which is emphasized by the description of their use in Italian, easier than Latin for artillerymen to understand. Italian is also the language used for the concluding description of a new square that made it easier for gunners to calculate elevation of the weapon.

“The treatise also refers to the movement of water in a paragraph so important that Ernst Mach proclaimed Torricelli the founder of hydrodynamics. Torricelli’s aim was to determine the efflux velocity of a jet of liquid spurting from a small orifice in the bottom of a receptacle. Through experiment he had noted that if the liquid was made to spurt upward, the jet reached a height less than the level of the liquid in the receptacle. He supposed, therefore, that if all the resistances to motion were nil, the jet would reach the level of the liquid. From this hypothesis, equivalent to a conservation principle, he deduced the theorem that bears his name: The velocity of the jet at the point of efflux is equal to that which a single drop of the liquid would have if it could fall freely in a vacuum from the level of the top of the liquid at the orifice of efflux. Torricelli also showed that if the hole is made in a wall of the receptacle, the jet of fluid will be parabolic in form; he then ended the paragraph with interesting observations on the breaking of the fluid stream into drops and on the effects of air resistance. Torricelli’s skill in hydraulics was so well known to his contemporaries that he was approached for advice on freeing the Val di Chiana from stagnant waters, and he suggested the method of reclamation by filling.

Torricelli (1608-47) “attended the mathematics and philosophy courses of the Jesuit school at Faenza, showing such outstanding aptitude that his uncle was persuaded to send him to Rome for further education at the school run by Benedetto Castelli, a member of his order who was a mathematician and hydraulic engineer, and a former pupil of Galileo’s. Castelli took a great liking to the youth, realized his exceptional genius, and engaged him as his secretary … In 1641 Torricelli was again in Rome; he had asked Castelli and other mathematicians for their opinions of a treatise on motion that amplified the doctrine on the motion of projectiles that Galileo had expounded in the third day of the *Discorsi e dimostrazioni matematiche intorno a due nuove scienze* . . . (Leiden, 1638). Castelli considered the work excellent; told Galileo about it; and in April 1641, on his way from Rome to Venice through Pisa and Florence, after appointing Torricelli to give lectures in his absence, submitted the manuscript to Galileo, proposing that the latter should accept Torricelli as assistant in drawing up the two ‘days’ he was thinking of adding to the *Discorsi*. Galileo agreed and invited Torricelli to join him at Arcetri. But Castelli’s delay in returning to Rome and the death of Torricelli’s mother, who had moved to Rome with her other children, compelled Torricelli to postpone his arrival at Arcetri until 10 October 1641. He took up residence in Galileo’s house, where Vincenzo Viviani was already living, and stayed there in close friendship with Galileo until the latter’s death on 8 January 1642. While Torricelli was preparing to return to Rome, Grand Duke Ferdinando II of Tuscany, at Andrea Arrighetti’s suggestion, appointed him mathematician and philosopher, the post left vacant by Galileo, with a good salary and lodging in the Medici palace. Torricelli remained in Florence until his death; these years, the happiest of his life, were filled with the greatest scientific activity …

“In 1644 Torricelli’s only work to be published during his lifetime appeared, the Grand Duke having assumed all printing costs … The fame that Torricelli acquired as a geometer increased his correspondence with Italian scientists and with a number of French scholars (Carcavi, Mersenne, F. Du Verdus, Roberval), to whom he was introduced by F. Niceron, whom he met while in Rome. The correspondence was the means of communicating Torricelli’s greatest scientific discoveries but also the occasion for fierce arguments on priority, which were common during that century. There were particularly serious polemics with Roberval over the priority of discovery of certain properties of the cycloid, including quadrature, center of gravity, and measurement of the solid generated by its rotation round the base. In order to defend his rights, Torricelli formed the intention of publishing all his correspondence with the French mathematicians, … but while he was engaged in this work he died” (DSB).

“On p. 9 of the preface [of *Opera geometrica*] the author says that the book was published at the behest (‘non suasit, sed iussit’) of Andrea Arrighetti of Florence. Arrighetti (1592-1672) was a pupil of Benedetto Castelli and held important office in the Tuscan state including that of buildings supervisor. It was he who was responsible for looking after Torricelli in Florence after Galileo's death. There is also mention of a sharp-eyed student of Archimedes, Antonio Nardi of Arezzo, ‘to whom, and to whose learned conversations, I owe whatever there is of geometry in this work (‘scriptura’)’. Nardi again was one of Galileo's pupils along with Magiotti and Torricelli, and indeed he wrote to Galileo about his work on Archimedes” (Macclesfield).

Honeyman VII 2991; Norman 2086; PMM 145; Riccardi II 542; Carli-Favaro 43 (204); Cinti 226 (114). Baron, *The Origins of the Infinitesimal Calculus*, 1969. Cajori, *A History of Mathematics*, 1894.

Three parts in one volume, 4to (227 x 162 mm), pp. [ii], [2, blank], [1-2] 3-243 [1], [1-2], 3-115 (i.e. 151), [1], half-title, part I title with imprint, section titles to parts II & III, dedication to Grand Duke Ferdinand II de’ Medici, part III separately signed & paginated with separate dedication to Prince Leopold de' Medici, imprimatur leaf at end, numerous small woodcut diagrams, one full-page engraving, letterpress tables. Original interim limp boards, uncut. Small stain to first three leaves (probably candle wax). A very fine and entirely untouched copy in its original state.

Item #5610

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Price:
$25,000.00
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