Mathematics and the New Sciences
Abstract and Keywords
This article examines the mutual influences between mathematics and the new sciences that emerged in the long seventeenth century, whereby new scientific enterprises fostered the development of new mathematical methods and mathematical developments in turn paved the way for new scientific research. It begins with an overview of the revolutions in mathematics in the long seventeenth century and the status of the mathematical sciences in the Late Renaissance, followed by a discussion on the work of mathematicians in the late sixteenth century including Galileo, René Descartes, Gottfried Wilhelm Leibniz, Jacob Bernoulli, Gerhard Mercator, and Edmond Halley. It also describes the work done in the areas of organic geometry and mechanical curves, infinitesimals, and mechanics.
8.1 Revolutions in Mathematics in the Long Seventeenth Century
A few decades ago it was customary to define the years between the publication of Copernicus’s De Revolutionibus in 1543 and that of Newton’s Principia in 1687 as the period of the ‘Scientific Revolution’. However, the view according to which a radical transformation and momentous leap forward occurred between the mid-sixteenth and late-seventeenth century due to the work of a few giants has been criticized recently. The historian of mathematics, however, is reluctant to abandon this view completely, since it is indeed the case that in the years in question the mathematical sciences underwent changes so deep—in terms of both scope and method—to deserve to be called revolutionary.1
The advent of symbolic algebra in the late Renaissance, of analytic geometry with the works of Fermat and Descartes—among others—in the early decades of the seventeenth century, and then the discovery of the calculus in the age of Newton and Leibniz enabled the mathematization of phenomena that had been regarded as well beyond the scope of mathematical treatment by the generation of natural philosophers before Galileo. To give a measure of this change one need only consider the fact that in the late sixteenth century one of Galileo’s mentors and one of the most important promoters of the mathematical approach to nature (including work on falling bodies and percussion), Guidobaldo dal Monte (1545–1607), concluded that (p. 227) the motion of bodies was subject to too many irregularities for it to be considered the province of mathematical science (Bertoloni Meli 2006, 33–34), whereas in the early decades of the eighteenth century mathematicians with an expertise in the techniques of the calculus were mathematizing—amongst other things—the motion of projectiles in resisting media, the ebb and flow of tides, the shape of planets, the bending of beams, and the motion of fluids. The present chapter is devoted to this momentous progress—or revolution, if we prefer to call it so—and to the actors responsible for it.
The mathematical revolution in natural philosophy also elicited much criticism and resistance. Indeed, many found the new mathematical methods unrigorous, or were sceptical about the possibility of applying the results achieved by mathematicians to natural phenomena. Mathematicians were subject to criticism from experimentalists who argued that natural phenomena, in all their complexity, could hardly be submitted to mathematical laws. Those I have in mind here are thinkers as removed in time and diverse in their methodologies as Francis Bacon, who depicted nature as a silva defying geometrization, Robert Boyle, who favoured experimental philosophy over mathematized natural philosophy, and Alessandro Volta, who had a poor opinion of Charles Augustin Coulomb’s success in discovering an inverse-square law for electric and magnetic phenomena.
As we shall see, in overcoming these criticisms the promoters of the use of mathematics had a profound impact on disciplines such as physics, astronomy, cosmology, and optics: they changed the methods used in these disciplines and their very nature, making their results almost inaccessible to laymen. Actors with different intellectual backgrounds and social and academic qualifications contributed to this process: professors of mathematics working in universities, mathematical humanists active at princely courts, engineers, astronomers, and even instrument-makers. Before delving into the complexities of the mutual influences between mathematics and the new science that emerged, whereby new scientific enterprises fostered the development of new mathematical methods and mathematical developments in turn paved the way for new scientific research, it will be worth examining the work of mathematicians in the late sixteenth century.
8.2 The Status of the Mathematical Sciences in the Late Renaissance
Since the publication of Alexandre Koyré’s seminal works it has become common-place to consider the ‘mathematization of nature’ a key factor in the development of sciences such as mechanics, astronomy, and optics, that took place in the period under consideration in this chapter—a factor as important as the turn toward experimentation praised in the work of Francis Bacon. Indeed, mathematics and mathematicians (p. 228) enjoyed renewed interest and prestige in the second half of the seventeenth century. While Koyré has attributed the rising status of the mathematical sciences in the late Renaissance to a crisis in Aristotelian philosophy and a concomitant shift towards neo-Platonism, Richard Westfall has suggested that the economic importance of technologies such as ballistics, fortifications, water management, navigation, and cartography stimulated mathematical research and offered mathematicians new jobs and opportunities (Koyré 1968) (Westfall 2001).
If we turn our attention first to universities, we find that mathematics was taught in the context of a medieval curriculum based on the Aristotelian subordination of the mathematical sciences to natural philosophy. One of the main tasks of natural philosophy was to determine the causes of change: for instance, those of local motion. We should keep in mind that the notions of cause and change in Aristotelian philosophy possessed a much broader semantic spectrum than those defined by seventeenth-century thinkers such as Descartes, who reduced all change to the local motion of particles, and interpreted change of motion as caused by localized impacts between corpuscles. What is relevant for us is that the language in which Aristotelian natural philosophy was couched was not mathematical but logical: for the natural philosopher’s discourse was based on syllogisms. According to Aristotle, it is syllogistic reasoning and not mathematics that reveals the causes of change in nature.
There is no better example to illustrate the subordinate status of mathematics compared to natural philosophy than the relationship existing between cosmology and astronomy according to many Aristotelians. Astronomy was considered part of the so-called ‘mixed’ mathematical sciences, along with harmonics, mechanics, optics, and so on. The ‘pure’ mathematical sciences were, of course, arithmetic and geometry—two clearly distinct disciplines, one dealing with discrete multitudes, the other with continuous magnitudes. According to Aristotelians, astronomy as a mathematical science could not—and should not—be confused with cosmology, a part of natural philosophy. In their view, the task of astronomy was to predict the positions of the heavenly bodies via the application of mathematical models. Astronomy did not describe the heavens, nor explain the causes of heavenly motions. Therefore, it was conceived of as a mixed mathematical science, with a status subordinate to that of philosophical sciences like cosmology. It is this subordination of astronomy that was called into doubt by Copernicus, who stated it was mathematics that revealed the true structure of the planetary system. The heliocentric system, Copernicus argued, can be described through a mathematical model that is superior to the geocentric one essentially because the parameters of the planetary orbits are interrelated in the heliocentric system, whereas in the geocentric system each planet is treated separately. The mathematical superiority of the heliocentric model was seen as a sign of its truth. It is undeniable that, as Thomas Kuhn has underlined, Copernicus and the few Copernicans active in the latter half of the sixteenth century saw themselves as restorers of a Platonic position that assigns mathematics a role unthinkable for the Aristotelians (Kuhn 1957). Indeed, contra Aristotle, early Copernicans believed that, as Plato had taught, nature was essentially mathematical in structure—hence the power of mathematics to reveal the true nature of the planetary system, even when this (p. 229) conflicts with accepted physics and common sense. The debate on the status of mathematical sciences was thus closely linked to the contested reception of the heliocentric planetary system.
The above debate erupted in a different context in 1547, following the publication of Alessandro Piccolomini’s (1508–1579) commentary on pseudo-Aristotle’s Problemata Mechanica, a text devoted to so-called ‘simple machines’ (sometimes listed as the lever, the balance, inclined plane, wedge, pulley, and the screw, which attracted the attention of professors of mathematics as well as of engineers) (Piccolomini 1547). Actually, Problemata Mechanica played an important role in reviving interest in the applications of mathematics to mechanics, but Piccolomini, like other Aristotelian philosophers, opposed the rising status of mathematics. In his commentary he maintained that mathematics did not possess the deductive purity of syllogistic logic and was not a science because it did not disclose causal relations (Jardine 1997). Piccolomini, Benito Pereira (1535–1610) and others entering this heated science war often stressed the fact that mathematical demonstrations link premisses and consequences in ways that are not unique, for usually more than one demonstration can be found for the same theorem and more than one construction for the same problem. Furthermore, geometrical constructions are carried out by deploying auxiliary figures that do not belong to the figures whose properties are under investigation. Arbitrariness in definitions, plurality of methods, and auxiliary figures and lemmas characterize mathematical practice—a clear sign, for Aristotelians, of the lower scientific status of mathematics compared to natural philosophy, a discipline that instead reveals unique causal relationships by focusing on the essential properties of the objects examined. Typically, Jesuits such as Christoph Clavius (1538–1612), who were giving pride of place to mathematics in their innovative ratio studiorum, defended the scientific status of mathematics. It is hardly surprising that this academic debate on the status of mathematics took place in a moment in which Copernicanism and an upsurge of neo-Platonic philosophies threatened the Aristotelian subalternatio of mathematics to natural philosophy. Yet, well-paid professors of philosophy—in the University of Pisa in the 1580s professors of philosophy were paid many times more than mathematics professors—had to worry about two additional factors that raised mathematics above the status it was assigned in the Aristotelian curriculum: the development of engineering and new trends in mathematical humanism.
Mathematicians were not at work only in universities. We must turn our gaze towards dockyards, arsenals, battlefields, the banks of rivers and canals, and workshops of instrument-makers and cartographers if we wish to find traces of the interesting developments that were taking place in the period of great innovations in the science of war, the regimentation of rivers, and long-distance travels by land and sea. Engineers and polymaths such as the Dutchman Simon Stevin (1548/49–1620), the Italian Benedetto Castelli (1578–1643), the Portuguese Pedro Nuñez (1502–1578), and the English Thomas Digges (1546–1595) found patronage chiefly outside the universities. The mathematics they practiced served a different purpose from that of the discipline that was taught at universities (Biagioli 1989). Their methods were praised for their usefulness, not for their rigour or scientific merits.
(p. 230) The latter half of the sixteenth century was also the period in which humanist philology gave birth to decisive translations and editions of mathematical texts from the Classical tradition. Of particular importance is the work of Federico Commandino (1509–1575), a Humanist, medical adviser, and mathematician who was active in Urbino (Italy) in the service of its Duke and in Rome as the personal physician of a cardinal. Commandino produced commented editions of a great number of works by Apollonius, Archimedes, Euclid, and other Classical authors (Napolitani 1997) (Bertoloni Meli 1992). Relevant here are his commentary of Archimedes’ work on floating bodies (1565), which improved upon the previous edition by Tartaglia (1543), his original work on centres of gravity (1565), and his editions of Heron’s Spiritalium liber (Pneumatics) (1575) and Pappus’s Mathematicae Collectiones (Mathematical Collections) (1588).
The result of this philological work was the rediscovery of Greek mixed mathematics in the fields of statics and fluid equilibrium. Against Aristotle, Archimedes’ works and the compilation of Greek studies in ‘rational mechanics’ featured in the eighth book of Pappus’s Collectio (mainly devoted to the results of Philo of Byzantium and Heron) which dal Monte brought to print after Commandino’s death, showed that mathematics could successfully be applied to nature, at least in some well-defined cases of static equilibrium. The problems treated in this Archimedean tradition were the equilibrium in simple machines and that of bodies (shaped, for instance, as paraboloids of revolution) in a fluid. In this context it was important to determine the centre of gravity of solids. Galileo (whose work will be examined in Section 8.3) was introduced to the Archimedean tradition cultivated in Urbino—a tradition that opened new vistas beyond the Aristotelian canon of mixed mathematical sciences—by his mathematics teacher Ostilio Ricci (probably via Tartaglia’s edition (1543) of Archimedes’ works (Drake 1995, 4)), and by one of Commadino’s most gifted pupils, Guidobaldo dal Monte, the author of the most comprehensive and authoritative treatise on the theory of simple machines, Mechanicorum Liber (1577). Guidobaldo further developed the Archimedean tradition by publishing two works: In Duos Archimedis Aequeponderantium Libros Paraphrasis (1588), which Galileo received as a gift, and De Cochlea Libri Quatuor (1615). Around 1590, Galileo and dal Monte carried out a joint experiment on projectile motion by throwing an ink-stained ball over an inclined plane. Guidobaldo concluded that the trajectory described by the ball rolling over the plane was that of a suspended chain, ‘similar to a parabola or hyperbola’. Motion perhaps was not as mathematically intractable as he had formerly thought.
8.3 Galileo and Proportion Theory
8.3.1 Early Work on Motion in Pisa
Galileo began developing a mathematical science for the motion of bodies in his early years in Pisa. In 1589, thanks to the support of Cardinal Francesco Maria dal Monte, (p. 231) Guidobaldo’s brother, he was elected ‘lettore di matematica’ at the local university.2 Galileo’s very first works were devoted to improving Archimedes’ treatise on floating bodies (La bilancetta (1586)) and to the study of centres of gravity. This initial research located Galileo within a mathematical tradition cultivated by scholars interested in mathematical statics such as Clavius and dal Monte. The project of mathematizing the motion of bodies was considered by many Archimedeans as lying beyond the scope of mathematics. Attempts to mathematize projectile motion had already been made, most notably by the polymath and abacus master Niccolò Tartaglia (1499?–1557) (Fig. 8.1), but the results achieved hardly attained the same level of rigour and exactness as ancient Greek scientists such as Archimedes and Pappus in the field of statics. This Classical tradition is what Galileo’s correspondents and mentors were striving to match. Galileo probably benefited from his exchanges in Pisa with professors of philosophy—most notably, Girolamo Borro and Francesco Buonamici—who were experimenting on falling bodies. Guibobaldo himself, while sceptical about the possibility of mathematizing phenomena as variable and irregular as falling of bodies, experimented on the force of falling bodies hitting an obstacle. The so-called ‘forza della percossa’ was indeed taken to be a measure of instantaneous speed (Drake 1995).
The question young Galileo asked himself was: ‘Is it possible to mathematically describe the fall of bodies?’ His answer can be found in a set of manuscripts known as De motu antiquiora, which he composed before moving to Padua as Professor (p. 232) of Mathematics in 1592. Aristotelians assumed that bodies move upwards or downwards depending on their levitas (‘lightness’) or gravitas (‘heaviness’). Those moving downwards (heavy bodies mostly composed of earth) do so with a speed that is proportional to their heaviness and inversely proportional to the density of the medium through which they move. Perfect void cannot exist, according to this view, since density equal to zero would imply an infinite speed for falling bodies. Galileo’s answer was as follows. Pace Aristotle, all bodies are heavy. Some move upwards, some downwards, because of Archimedes’ law of buoyancy. Tartaglia (1551) and Benedetti (1553) had already conjectured that the speed of falling bodies is proportional to the difference between their specific gravity and that of the medium in which they find themselves. This makes the existence of the void possible. Furthermore, Galileo noted that the rarer the medium, the closer the speeds of vertically falling bodies with different specific weights. He was not far from realizing that in the void all bodies fall with the same velocity (Bertoloni Meli 2006, 45–60).
So how does speed vary? In De motu, Galileo assumes that a body falling from rest, after an initial period in which it accelerates, reaches a state in which it falls with constant speed. Acceleration is here conceived of as a transient irregular accident that should not be taken into account in the mathematical treatment of motion. Constant velocity Galileo regarded as the natural state of motion of falling bodies. Of course, uniform rectilinear motion is the easiest to mathematize. As far as ‘violent’ motion against gravity caused by projection (for instance, the throwing of a stone), Galileo accepted an impetus theory–familiar in the Middle Age–according to which the projected body moves against gravity in the direction of projection because it is endowed with a slowly decaying impetus. When the impetus is completely consumed, the cause of violent motion is no longer present and vertical natural motion occurs. The principle of inertia was yet to come. However, as Galileo realized, the results obtained by applying this theory did not find experimental confirmation. De motu was left unfinished, and it is in Padua that Galileo focused his mathematical and experimental efforts on acceleration, no longer seen as a transient, accidental phenomenon, but as a key feature of the rectilinear fall of bodies. Before turning to the results that Galileo achieved in Padua in the period from 1592 to around 1609, it is worth considering the mathematical tools at his disposal, since these determined the theoretical possibilities and constraints within which his well-known mechanics of projectile motion developed.3
8.3.2 Proportion Theory
Time, distance and speed—the magnitudes Galileo sought to subject to mathematical treatment—are all continuous magnitudes. Since antiquity, continuous magnitudes had been approached through the theory of proportions, codified in Book 5 of (p. 233) Euclid’s Elements. The Euclidean theory of proportions was the main mathematical tool employed in the mathematization of natural phenomena in the first half of the seventeenth century (Bos 1986).
The theory of proportions applies to magnitudes, where a magnitude is a general concept covering all instances in which operations of adding and comparing have meaning. For instance, a magnitude might be a given length, volume, or weight. Ratios can be formed between two magnitudes of the same ‘kind’ (for example, between two areas or volumes), while a proportion is a ‘similitude’ (an analogy) between two ratios (‘A ratio is a sort of relation in respect of size between two magnitudes of the same kind’). So two volumes and might be said to have the same ratio as two weights and . Employing algebraic notation not available to Galileo:(8.1)
Two magnitudes are ‘homogeneous’—that is, of the same kind—when ‘they are capable, when multiplied, of exceeding one another’.4 Note that the theory of proportions does not allow the formation of a ratio between two heterogeneous magnitudes. This is particularly important for kinematics, since it is not possible, for instance, to define speed as a ratio between distance and time.
It should be stressed that ratios are not magnitudes. We tend to read ratios as numbers and proportions as equations (as in eq. (8.1)) , but this was not the case in the old Euclidean tradition. The discovery of incommensurable ratios (such as that between the lengths of the side and diagonal of a square) was interpreted as implying that the concept of number was unable to cover the extension of the concept of ratio. Nonetheless, operations on ratios are possible. An operation on ratios that will concern us was called ‘compounding’. When A, B and C are homogeneous magnitudes, was said to be the ‘compound ratio’ of and . Thus, ratios between homogeneous magnitudes in continued proportion could be compounded: that is, ratios such that the last term of the former, , is equal to the first of the latter, . In order to compound two ratios, and , one has to determine the ‘fourth proportional’ to C, D and B, that is, a magnitude F, homogeneous to B and A, such that . The compound ratio of and is equal to the compound ratio of and , that is, (Bos 2001, 119–134).
The simple example of uniform rectilinear motion (‘equable motion’) will allow us to understand some characteristics of the mathematization of natural philosophy in Galileo’s work. The relationship between time, distance and speed in equable motion might be expressed as(8.2)
(p. 234) This was not possible within the framework of the theory of proportions, since one had to express a magnitude, speed, as being equal to the ratio between two heterogeneous magnitudes, distance and time. One has instead to state a series of proportions allowing only two of the three magnitudes involved (distance, time and speed) to vary. So it is possible to state that ‘when the speed is the same and we compare two equable motions the distances are as the times’:(8.3)
where () is the distance covered in time (). Or, ‘when the distance covered is the same, the speeds are as the times inversely’:(8.4)
A more elaborate mathematical formulation of uniform motion can be achieved through the operation of compounding. The aim here is to state that the distances are in the compound ratio of the speeds and times. This can be achieved as follows. Let us assume that distance is covered with speed in time , and that distance is covered with speed in time . How can we compare the two motions? The trick consists in considering a third uniform motion which lasts for a time and with speed . We can state:(8.5)
The operation of compounding allows one to state that is the compound ratio of and . This fact was expressed by statements such as ‘the distance is proportional to the speed and to the time conjointly’. However, in order to be purely Euclidean, to compound the right-hand ratios of eqs. (8.5) and (8.6) we first have to determine three homogeneous magnitudes (let us say segments with lengths K, L and M) such that and . So at last we can state that(8.7)
The theory of proportions was not flexible enough to solve the problems of seventeenth-century natural philosophy. Several proposals aimed at relaxing the rigidity of the Euclidean scheme were advanced (Giusti 1993). As we shall see in Section 8.4, Descartes’ geometry replaced the theory of proportions. But this still had to be developed when Galileo was devising his new theory of uniformly accelerated motion—a theory that was disseminated via correspondence and finally printed in the Discorsi (1638).
(p. 235) 8.3.3 Work on Accelerated Motion in Padua
It was in Padua, where Galileo took up the post of Professor of Mathematics in 1692, that the new science of motion was born. Our aim here is to concentrate on the mathematical tools that Galileo deployed in order to mathematize uniformly accelerated vertical fall, a key element in his study of projectile motion (Clavelin 1974) (Bertoloni Meli 2006, 96–104).
One of the basic intuitions that Galileo had was that in vertical fall bodies do not reach a terminal speed; rather, they accelerate continuously. Acceleration is not—as Galileo believed when he was in Pisa—an accidental and irregular phenomenon that only takes place at the beginning of an object’s fall. It is likely that Galileo’s study of pendulums (through which he obtained the famous law of isochronism, approximately valid for small oscillations) persuaded him of the need to study motions with a continuously varying velocity.
It thus became necessary for him to mathematize a variable speed. Galileo initially maintained that in vertical fall speed varies directly with the length of the space covered by the falling body from rest. Only after 1604 did he realize that speed varies linearly with time rather than distance. From the extant manuscripts, it is clear that experiments with bodies falling along inclined planes contributed more to this discovery than what was printed in the Discorsi (and which we shall shortly turn to examine).
The third day of the Discorsi opens with a series of theorems concerning equable (or uniform) motion. Mathematizing constant velocity is indeed an important premiss to Galileo’s science of motion, not only because—as he himself realized—equable motion is a state in which a body perseveres until some external cause, such as an impact or gravitation, changes its state, but also because the study of accelerated motion was reduced by Galileo to equable motion.5 The theorems on equable motion opening the third day of the Discorsi are framed in the language of proportion theory. To acquire a flavour of this language one can turn to theorem VI, which reads:
If two moveables are carried in equable motion, the ratio of their distance will be compounded from the ratio of spaces ran through and from the inverse ratio of times. (Galilei 1974, 152)
In symbols we would write:(8.8)
Several observations are in order. Firstly, speed is not defined as the ratio of distance over time. As previously noted, this definition goes against the rules of proportion theory according to which ratios can only be formed between homogeneous magnitudes. Galileo, therefore, defines the properties of speed via a series of postulates and theorems. Secondly, the fact that four magnitudes are in the same ratio is verified by showing that the conditions stated in definition 5, book 5, of Euclid’s Elements are (p. 236) met—something which requires rather laborious demonstrations.6 Thirdly, the compounding of ratios requires the determination of a fourth proportional. The need to recur to these mathematical techniques makes Galileo’s simple theorems on equable motion painfully cumbersome and difficult to follow. Finally, it should be noted that the theory of proportions is best suited to express linear relations between continuous magnitudes. Even slightly more complicated functional relationships pose difficulties, as we shall see in the case of the quadratic dependence of distance from time in the case of uniformly accelerated motion.
In order to express the dependence of distance from the square of time in the language of proportion theory, Galileo ultimately resorts to the following corollary, which reduces the quadratic relationship to a linear relationship via the determination of a mean proportional:
if at the beginning of motion there are taken any two spaces whatever, run through in any [two] times, the times will be to each other as either of these two spaces is to the mean proportional space between the two given spaces. (Galilei 1974, 170–1)
In symbols we would first determine the mean proportional between the two given distances and :(8.9)
and then translate the corollary as follows:(8.10)7
In order to appreciate the constraints that proportion theory imposes on Galileo’s treatment of naturally accelerated motion, it is worth quoting Theorem 1 at length. This reduces accelerated motion to equable motion via a proposition that had been well known since the Middle Ages:
The time in which a certain space is traversed by a moveable in uniformly accelerated movement from rest is equal to the time in which the same space would be traversed by the same moveable carried in uniform motion whose degree of speed is one-half the maximum and final degree of speed of the previous, uniformly accelerated, motion.
Let line AB [Fig. 8.2] represent the time in which the space CD is traversed by a moveable in uniformly accelerated movement from rest at C. Let EB, drawn in any way upon AB, represent the maximum and final degree of speed increased in the instants of the time AB. All the lines reaching AE from single points of the line AB and drawn parallel to BE will represent the increasing degrees of speed after the instant A. Next, I bisect BE at F, and I draw FG and AG (p. 237) parallel to BA and BF; the parallelogram AGFB will [thus] be constructed, equal to the triangle AEB, its side GF bisecting AE at I.
Now if the parallels in triangle AEB are extended as far as IG, we shall have the aggregate of all parallels contained in the quadrilateral equal to the aggregate of those included in triangle AEB, for those in triangle IEF are matched by those contained in triangle GIA, while those which are in the trapezium AIFB are common. Since each instant and all instants of time AB correspond to each point and all points of line AB, from which points the parallels drawn and included within triangle AEB represent increasing degrees of the increased speed, while the parallels contained within the parallelogram represent in the same way just as many degrees of speed not increased but equable, it appears that there are just as many momenta of speed consumed in the accelerated motion according to the increasing parallels of triangle AEB, as in the equable motion according to the parallels of the parallelogram GB. For the deficit of momenta in the first half of the accelerated motion (the momenta represented by the parallels in triangle AGI falling short) is made up by the momenta represented by the parallels of triangle IEF.
It is therefore evident that equal spaces will be run through in the same time by two moveables, of which one is moved with a motion uniformly accelerated from rest, and the other with equable motion having a momentum one-half the momentum of the maximum speed of the accelerated motion; which was [the proposition] intended. (Galilei 1974, 165–6)
Some observations can be drawn here. Uniformly accelerated motion is graphically displayed by the abscissae AB representing time and ordinates EB representing the linearly increasing degrees of speed. Via this theorem Galileo is able to associate an equable motion to an accelerated motion, and therefore to mathematize uniformly accelerated motion in terms of equable motion. Note that Galileo does not identify the distance CD traversed by the falling body with the area of the triangle AEB. Mathematicians belonging to the generation after Galileo’s, such as Evangelista Torricelli, would here be inclined to conceive of the surface of triangle AEB as constituted by infinitely many parallelograms whose infinitely small areas (in Leibniz’s notation vdt) represent the lengths of the infinitely small spaces (ds) covered by the body in an infinitesimally small interval of time (dt). Galileo never reached this (p. 238) understanding: he was rather comparing the ‘aggregate of all parallels’ contained in the quadrilateral with those contained in the triangle. It appears, he writes, that ‘there are just as many momenta of speed consumed in the accelerated motion … as in the equable motion’. It is this equality of aggregates that justifies Galileo’s ‘evident’ conclusion that equal spaces will be traversed by the two bodies, one accelerated in free fall, the other moving with constant speed. The infinite and infinitesimal lurk behind Galileo’s demonstration, yet its author was anxious about the dangers inherent in reasoning with the infinite, and framed his demonstrations one inch within (or beyond?) the ‘limits’, so to speak, of what has been termed ‘pre-classical’ mechanics (Damerow 2004). As we shall see in Section 8.6, these limits were transgressed by those mathematicians who dared to employ infinitesimal magnitudes.
The success of Galileo’s mathematical treatment of bodies is illustrated in the fourth day of the Discorsi, where, by compounding inertial equable motion in the (horizontal) direction of projection with vertical naturally accelerated motion, he proves that frictionless motion of projectiles takes place along a parabola. A successful answer to those who were sceptical about the possibility of treating the motion of bodies in mathematical terms. Yet, despite Galileo’s breakthrough in the mathematization of motion, it was only possible to move beyond the limits of his Discorsi by abandoning the strictures of proportion theory. This was precisely the move made by René Descartes.
8.4 Descartes and Analytic Geometry
Few works in the history of mathematics have been more influential than Descartes’ Géométrie (1637) (Bos 2001). The canon defined in this revolutionary essay was to dominate the scene for many generations, and its influence on seventeenth-century mathematicians cannot be overestimated. In the very first lines of this work, its author claims to be in possession of a method capable of ‘reducing any problem in geometry’.8
Book 1 of the Géométrie explains how to translate a geometrical problem into an equation. Descartes was able to attain this goal by making a revolutionary break with the tradition of the theory of proportions to which Galileo adhered. Indeed, Descartes interpreted algebraic operations as closed operations on segments. For instance, if a and b are segments, the product ab is not conceived by Descartes as representing an area, but rather another segment. Prior to the Géométrie the multiplication of two segments of lengths a and b would have been taken as the representation of the area of a rectangle with sides measuring a and b. Descartes’ interpretation of algebraic operations was a huge innovation (Fig. 8.3). From Descartes’ new viewpoint, ratios are considered as quotients, and proportions as equations. Consequently, the direct (p. 239) multiplication of ratios is allowed, contrary to what occurred in the theory of proportions. The homogeneity of geometrical dimensions is no longer a constraint for the formation of ratios and proportions. As far as physical magnitudes are concerned, once a unit is specified for each magnitude, these can be represented through numbers or segments. Since on numbers and segments the four operations are closed, the constraints on the homogeneity of physical dimensions characteristic of the theory of proportions are overcome. So multiplication of time and speed or division of weight by surface’s area are all possible. Descartes’ approach won great success.
Descartes’ method of problem solving was divided into an analytical part and a synthetic (Bos 2001, 287–9). The analytical part was algebraic and consisted in reducing problems into polynomial equations. If the equation was in one unknown, the problem was determinate. The equation’s real roots would correspond to the finitely many solutions of the problem. The analysis, or resolution, was not, according to early-modern standards, the synthesis, or solution, of the problem. The solution of the problem must be a geometrical construction of what is sought in terms of legitimate geometrical operations. Descartes accepted the traditional idea that constructions must be performed through the intersection of curves. Therefore, he devised procedures aimed at geometrically constructing the roots of algebraic equations in one unknown via the intersection of plane curves. For instance, Descartes explained how one can construct segments, the length of which represent the roots of third- and fourth-degree equations, via the intersection of circle and parabola.
The synthetic part of the process of problem solving opened up a series of questions. Which curves are admissible in the solution of problems? When can these be considered as known or given? Which curves, among those admissible and (p. 240) constructible, are to be preferred in terms of simplicity? In asking himself these questions Descartes was continuing—albeit on a different plane of abstraction and generality—a long debate on the function and classification of curves in the solution of problems, which from Antiquity had reached early-modern mathematicians such as François Viète, Marino Ghetaldi (Getaldić), and Pierre de Fermat by way of Pappus’s Mathematical Collections (1588) (Bos 2001, 287–9).
Descartes prescribed that in the construction of problems one had to use what he called ‘geometric’ (that is algebraic) curves of the lowest possible degree. By contrast, he banished what he called ‘mechanical’ (that is transcendental) curves such as the spiral and cycloid.
One of the high points of the Géométrie is its method for drawing normals to algebraic plane curves. Descartes described this as ‘the most useful and most general problem in geometry that I know, but even that I ever desired to know’ (Descartes 1954, 95). In order to determine the normal at point C to an algebraic curve, Descartes considered the sheaf of circles passing through C and with centre P located on the axis of the abscissae. Descartes observed that the circles having their centre close to the point where the normal cuts the axis intersect the curve in a second point near C, whereas the circle having its centre at the point where the normal cuts the axis touches the curve at C. Descartes was able to translate this geometric condition into algebraic terms (see Fig. 8.4) (Andersen 1994, 293–4).
The determination of normals to algebraic curves played an important role in Descartes’ optics. In Dioptrique (1637), published alongside the Géométrie as one of the essays accompanying Discours de la Méthode, Descartes sets out to discover what shape of lens can eliminate spherical aberrations. His answer is a lens with a hyperbolic surface. The problem Descartes sets himself in Book 2 of the Géométrie is to find a lens the surface of which is shaped in such a way that light rays starting from one (p. 241) point may be refracted when striking it and converge in a second point. By applying the sine law of refraction, Descartes found that surface of this kind are obtained by revolution of the so-called ovals of Descartes around their axis. Cartesian ovals are fourth-degree algebraic curves. Determining the normal to the ovals was, of course, important in order to apply the law of refraction and determine the direction of the refracted rays after hitting the surface.
The success achieved by Descartes in mathematizing plane curves in algebraic terms and applying algebraic tools, such as his method of normals, to optics cannot be overestimated. Still, the Géométrie was also marked by some tensions and left certain question open. Descartes’ insistence that algebra could be applied to geometry when variables and constants occurring in equations are interpreted as finite segments and the fact that he only deployed polynomial equations seemed to bar the way for the development of techniques of vital importance for seventeenth-century mathematicians—particularly those interested in applying mathematics to natural philosophy. In particular, infinitary techniques and infinitesimals, which proved essential to many innovative works in the seventeenth century, are noticeably absent from the Géométrie (or perhaps only obliquely visible in Descartes’ method of normals). Consequently, Descartes had little to say about the rectification of curves, the calculation of areas, surfaces, and volumes, and the calculation of centres of gravity. One of the prescriptions of Descartes’ that proved to be a serious limitation was his banishment of mechanical (transcendental) curves. In his correspondence with Marin Mersenne in 1638, Descartes was prompted by the Minim friar to consider the cycloid and, in solving a simple mechanical problem, the logarithmic spiral. Descartes discussed these curves, but he could not handle them with the algebraic methods of the Géométrie, which were confined to polynomial equations (Jesseph 2007). As the century progressed, the importance of mechanical curves became more and more evident. As we shall see in Section 8.6, these curves naturally resurfaced as solutions for what are nowadays identified as problems of integration, as well as for the solution of differential equations. They proved useful in mechanics, optics, and astronomy. Therefore, a need was felt to overcome the limitations inherent in the mathematical canon of the Géométrie. Furthermore, Descartes’ physics, unlike his geometrical optics, remained largely qualitative. Most notably, when dealing with planetary motions, Descartes made recourse to explanations in terms of vortices swirling around the Sun that were difficult to translate into any mathematical model.
8.5 Organic Geometry and Mechanical Curves
It is interesting to observe that Descartes devoted great effort not only to the algebraic study of curves, but also to their mechanical generation via tracing mechanisms (Burnett 2005). This interest in what has often been referred to as ‘organic (p. 242) geometry’ is rooted in a number of exigencies that shaped the Cartesian conception of geometry and its relation to algebra (Bos 2001). Here I would like to underline the fact that in the Dioptrique Descartes provided a detailed description of curve-tracing devices and lens-grinding machinery (Fig. 8.5)—topics he discussed extensively with high-ranking men of letters such as Constantijn Huygens, mathematicians such as Florimond de Beaune, and well-known artisans such as Jean Ferrier.9
In his autobiography, the Savilian Professor of Geometry at Oxford, John Wallis, claimed that in early seventeenth-century England mathematics was practiced not as an academic pursuit but as something useful for trade and technology. As Feingold has shown, Wallis’s often quoted assertion greatly underestimates the role of Cambridge and Oxford in promoting the mathematical sciences (Feingold 1984, 1, 21). Nonetheless, Wallis’s recollections are interesting since they suggest that as historians of mathematics we should search for evidence of the development and application of the mathematical sciences also in the work of traders and artisans; that we should study not only Descartes’ peers but also Ferrier’s colleagues. Wallis, referring to his studies in Cambridge in the 1630s, wrote:
For Mathematicks, (at that time, with us) were scarce looked upon as Accademical studies, but rather Mechanical; as the business of Traders, Merchants, Seamen, Carpenters, Surveyors of Lands, or the like; and perhaps some Almanak-makers in London. And amongst more (p. 243) than Two hundred Students (at that time) in our College, I do not know of any Two (perhaps not any) who had more of Mathematicks than I, (if so much) which was then but little; And but very few, in that whole University. For the Study of Mathematicks was at that time more cultivated in London than in the Universities. (Scriba 1970, 27)
What kind of mathematics was deployed and improved upon by the practitioners Wallis mentioned, of whom Ferrier was a French representative? And how did these men interact with more theoretically inclined mathematicians, often employed in the universities and later in scientific academies? Historians are just beginning to explore the shores of a continent that has remained hidden in contemporary accounts of the history of the mathematical sciences in the long seventeenth century (Bennett 1986).
Descartes and Ferrier were discussing a topic—the mechanical generation of curves (Braunmühl 1892)—that constituted a trading zone in the commerce of mathematical ideas between humanists, painters, instrument-makers, mechanicians, and pure mathematicians. In this section we shall briefly explore this trading zone and review some of the research that was carried out at the time in the field of organic geometry. Other choices would of course be possible, since other trading zones could easily be identified. Mathematicians were interacting with practitioners in the fields of ballistics, fortification, navigation, and horology, for instance, in many ways. Here it is worth recalling in passing that astronomers’ expertise was sought for the determination of longitude; that the study of the motion of falling bodies was not only an academic subject, but one which also aroused the interest of ballistic engineers; and that mechanical theories on corpuscolarism and cohesion addressed issues vital for alchemist, architects and ship-builders. Indeed, as mentioned previously, a complex network of interactions existed between pure mathematicians and practitioners of mathematics in the period under consideration in this chapter, one that represents a huge continent whose exploration is still patchy.
The mechanical generation of curves was clearly already part of the classic geometrical canon: mechanical generations of the conics, conchoids, quadratrix, and spirals occurred in geometrical constructions provided in Greek and Islamic treatises for theoretical purposes. For the mathematical practitioners active in the early modern period a curve-tracing device was often seen not so much as a theoretical construct as an instrument to be applied in one’s workshop. The curve traced by an instrument could serve as the conic surface of a lens, the hyperboloid surface of the fusee of a clock, the cycloidal shape of the teeth of a wheel, or the stereographic projection of the lines of equal azimuth of the celestial sphere.
Much research was carried out in the sixteenth and seventeenth centuries on the mechanical construction of conics. Here the probable influence of Islamic mathematicians should be noted, since in the Islamic culture much effort was paid to the construction of trammels and compasses that, amongst other things, were essential in the drawing of sundials and astrolabes (Raynaud 2007). The tracing of conics was of interest to sundial-makers, painters and architects, clockmakers and cartographers. Francesco Barozzi, whose edition of Heron’s mechanics (1572) was highly appreciated, described machines for tracing conic sections. Barozzi’s work was complemented (p. 244) by Jacques Besson’s (1540?–1573) Theatrum Instrumentorum et Machinarum (1571–72) (Fig. 8.6). Besson was highly esteemed for his description of several ingenious lathes. His discipline, the ars tornandi, was further pursued by Agostino Ramelli in Le Diverse et Artificiose Machine (1588), by Salomon de Caus in Les Raisons des Forces Mouvantes avec Diverses Machines (1615), and by George Andrea Böckler in Theatrum Machinarum Novum (1662). Some of this literature on lathes and curve-tracing machines was well-known to Descartes and his acolytes. The young Descartes might have admired many of these machines in the workshop of the famous Rechenmeister, Johann Faulhaber, in Ulm.
Descartes promoted interest in curve-tracing among his Dutch acolytes, who did not fail to stress the usefulness of mathematical machines. One of Descartes’ Dutch (p. 245) mathematical disciples, Frans van Schooten (1615–1660), devoted a whole treatise (1646) to the mechanical generation of conics. Right from its title-page he made it clear that he was not simply interested in pure geometry: for his work, he emphasized, was useful not only to geometers, but also to opticians, designers of gnomons, and mechanicians. Van Schooten was Professor of Mathematics at Leiden; he edited Viète’s works, translated Descartes’ Géométrie into Latin, and promoted mathematical research in the Low Countries. One of his disciples was Christiaan Huygens. Van Schooten included Jan de Witt’s ‘Elementa Curvarum Linearum’ in the second Latin edition of Descartes’ Géométrie (1659–61). De Witt’s work was to inspire Isaac Newton, whose mechanism for the tracing of conics—devised in the early 1670s—anticipated an intuition of Steiner’s theorem on the homographic generation of conic sections (Guicciardini 2009, 93–101). Alongside professors of mathematics such as Newton and van Schooten stood practitioners developing mechanism for the generations of curves, from Hans Lencker, a Nuremberg goldsmith who in Perspectiva (1571) devised an instrument for orthographic projections (Fig. 8.7), to Benjamin Bramer, who authored a treatise in the vernacular entitled Apollonius Cattus oder Kern der gantzen Geometriae (1684).
For many centuries the tracing of curves had been the métier of sundial-makers such as Ignazio Danti (1536–1586). The makers of sundials studied the curves traced (p. 246) out by the shadow of the gnomon. Danti is a good example of how broad-ranging the activity of polymaths could be in the late Renaissance. He designed large-scale gnomons for the cathedrals in Florence and Bologna. In addition, he also worked as a cartographer and cosmographer for the Medici and the Pope, and lent his services as an architect, canal- and harbour-engineer, and painter and writer on perspective. Danti taught practical mathematics in Florence and Bologna, and was ultimately rewarded by the Pope with a Bishopric (Biagioli 1989). The design of gnomons—an important topic for the construction of the meridian lines necessary for the reform of the calendar the Catholic Church was promoting—had been dealt with in the ninth book of Vitruvius’s De Architectura, which was published in two famous editions by Cesare Cesariano (Italian translation 1521) and Daniele Barbaro (Italian translation 1556, Latin edition 1567). The design of sundials occupied many mathematicians in this period, such as Commadino (who edited Ptolemy’s Planisphere (1558), adding his own treatise on the calibration of sundials), Giovanni Battista Benedetti, and Christoph Clavius. This topic, which required the stereographic projection of circles on the celestial sphere onto the plane of the equator, was related to that of perspective and the generation of curves per umbras. The study of the shadows of curves was of course also of interest to painters such as Albrecht Dürer. Pure mathematicians, such as Newton and later Patrick Murdock, studied the shadows generated by the central projection of cubics. Just as conic sections can be generated by centrally projecting a circle, so all cubics—as Newton stated in his Enumeratio Linearum Tertii Ordinis (1704) and Murdoch proved in Neutoni Genesis Curvarum per Umbras (1746)—can be generated as shadows of one of the five divergent parabolas. An important chapter in the history of projective geometry is therefore related to the practical mathematics promoted by sundial-makers like Danti.
The study of the mechanical generation of curves obliterated the distinction between algebraic and transcendental curves that was central to Descartes’ Géométrie. Once a curve is defined as the locus traced by a mechanism, the fact that its algebraic representation is not possible via a polynomial equation cannot be taken as a good enough reason to banish it from geometry, as stated by Descartes. The need to study transcendental curves emerged from many sectors of natural philosophy.
One of these curves, the cycloid, deserves our attention. A cycloid is the curve defined by the path of a point on the edge of a circular wheel as the wheel rolls along a straight line without sliding. Its properties attracted the attention of architects and mechanicians. For example, in the 1670s Ole Rømer discussed the use of the cycloid in designing toothed wheels with Leibniz. In the 1630s Mersenne called the attention of mathematicians, including Descartes, to the cycloid, and circulated a method for determining its area and tangent, which he had derived from the work of Gilles Personne de Roberval. Analogous methods are also featured in Torricelli’s Opera Geometrica (1644). For the finding of the tangent, Roberval had decomposed the motion of the tracing point P, which generates the curve, into two components, and applied the parallelogram law to the instantaneous component velocities of P. The kinematic tracing of tangents played an important role in Newton’s early researches on the drawing of tangents to mechanical lines.
(p. 247) In 1658, Blaise Pascal rekindled the interest of mathematicians in the properties of the cycloid with a prize challenge requiring a way of determining the quadrature, cubature, and centres of gravity of plane and solid figures bound by cycloidal arcs. Huygens, François de Sluse, Michelangelo Ricci, and Christopher Wren held back after some initial success. Wallis and Antoine de Lalouvere (1600–1664) vied for the prize but Pascal did not regard their solutions as satisfactory. Many searched for the quadrature and rectification of the cycloid: the results of Pascal and Wren proved particularly decisive.10 The study of the cycloid was actually the motivation for protracted mutual challenges and accusations between French, Italian, and English mathematicians. The fact that the cycloid attracted so much attention and elicited such heated reactions can be explained partly by the fact that the exigency of dealing with transcendental curves was particularly felt in the mid-seventeenth century.
What is most interesting about the cycloid is that it proved useful not only for mechanical applications, as De La Hire aimed to prove in his Traité des Epicycloïdes et de Leurs Usages dans les Méchaniques (1694), but also in the study of natural philosophy. Indeed, in 1659 Huygens discovered that a cycloidal clock is exactly isochronous, while in 1697 the brothers Jacob and Johann Bernoulli, Leibniz, Ehrenfried Walther von Tschirnhaus, Guillaume de l’Hospital, and Newton proved that a cycloidal arc answers the problem of determining the ‘brachistochrone’—a curve between two points that is covered in the least possible time by a body that starts at the first point with zero speed and is forced to move along the curve to the second point under the action of constant gravity and assuming no friction.
One of the most important examples of the interaction between mechanical engineering, mechanics, horology, navigation, and natural philosophy is Huygens’ Horologium oscillatorium (1673), a treatise where the cycloid plays a prominent role. In this work the great Dutch polymath and astronomer used geometrical methods that were comparable in rigour to those employed by Classical Greek mathematicians. Huygens deployed proportion theory and exhaustion techniques reminiscent of the work of Archimedes (Yoder 1988).
Huygens (1629–1695)—an aristocrat whose family was intimate with Descartes, who had studied in Leiden under Frans van Schooten, and who spent many years in Paris as one of the most eminent members of the Académie des Sciences—was much interested in navigation. A clock carried on board a ship and able to keep time in a precise enough way would have enabled the determination of longitude at sea: that is, it would have solved one of the most acutely felt problems that remained open in the art of navigation in the seventeenth century. After devising a balance spring clock, (p. 248) Huygens found a way of correcting the dependence on amplitude of a simple circular pendulum clock. Galileo’s assumption that circular pendulums are isochronous was already known to be wrong, for the period of swing of these clocks slightly increases with amplitude.
Huygens proved that a body sliding without friction along an inverted cycloidal arc has a period of oscillation that does not depend on amplitude. He also studied the evolute and involute of curves and proved a remarkable result: he showed that the evolute of a cycloid is another cycloid. Involutes, it is worth bearing in mind, are mechanically generated as follows: in order to find the involute B of the evolute A, one should wrap a string tightly against A, on its convex side. Then, keeping one end of the string fixed, the other end is pulled away from the curve. If the string is kept taut, then the moving end of the string will trace out the involute B. The evolute of a curve is the locus of all its centres of curvature. Equivalently, it is the envelope of the normals to a curve. The theory of evolutes found many applications in the rectification of curves.
Next, Huygens developed his basic idea for the construction of an isochronous pendulum. He noted that if a pendulum is swung between two cheeks shaped as the arcs of a cycloid, then the bob of the pendulum will trace out the involute of a cycloid—that is, a cycloid (Fig. 8.8). Since the cycloid is an isochrone curve, the pendulum will have a period independent of amplitude. In his late years Huygens devised experiments with clocks carried on ships sailing the Atlantic to the Cape of Good Hope, in order to measure variations in local gravity that might have confirmed his theory of gravitation and disproven the one promoted by Newton in his Principia (1687).
Huygens deserves an important place in the history of mechanics in the period between Galileo and Newton. He studied the motion of compound pendulums, found a mathematical expression for acceleration in uniform circular motion, stated the principle according to which the centre of the mass of a system of bodies can rise up to the height from which it was left falling, he applied the principle of Galilean (p. 249) relativity to the laws governing the impact of bodies,11 he developed a wave theory of light propagation based on what is still called Huygens’ principle (valid for the determination of the advancement of wave-fronts), and developed a theory on the phenomenon of double refraction.
In his last years Huygens corresponded with Leibniz about mathematical topics. Leibniz had first met Huygens in Paris in 1672, and Huygens had been of great help in introducing the young German diplomat to the most advanced mathematical discoveries. But it was Leibniz who now had something new to teach his old mentor: the differential and integral calculus. At first Huygens was sceptical about the new formalism that Leibniz had developed by 1676 and published in 1684–86, but soon he had to admit that the calculus was more powerful compared to the geometrical methods he had privileged all his life. Leibniz was able to show Huygens how to easily find the catenary, the shape that a hanging chain will assume when supported at its ends and acted upon only by its own weight, a transcendental curve as the cycloid.12 This curve proved useful in architecture, since it is the ideal curve for an arch which supports only its own weight. The correspondence between Huygens and Leibniz in the 1690s is indicative of the change that occurred with the invention of the calculus. The geometrical methods that Huygens promoted in Horologium oscillatorium gave way to a more abstract, algorithmic, and general mathematical method that Newton termed the method of series and fluxions and Leibniz the differential and integral calculus. However, even after the creation of calculus, well into the eighteenth century the mechanical tracing of curves continued to play a certain role in the integration of differential equations, as has been shown by Tournès (2009).
8.6 Infinitesimal Analysis
In the mid-seventeenth century, several problems in physics and astronomy called for the development of new mathematical tools capable of handling continuously varying magnitudes in terms of infinitesimals. Acceleration (in uniformly accelerated motion in the case of Galileo, and in uniform circular motion in the case of Huygens) was geometrically represented in terms of infinitesimal deviations from inertial motion acquired in an infinitesimal interval of time. What Galileo had merely hinted at also became clear: that the area subtended by the graph of speed in function of time measures the distance covered. Keplerian astronomy also demanded the deployment of infinitesimals and infinite summations. Johannes Kepler in Astronomia nova (1609) proved that planets move in ellipses having the Sun placed at one focus. He (p. 250) also discovered that each planet moves in such a way that the radius vector joining it to the Sun sweeps equal areas in equal times. When the elliptic orbit is known, the position of the planet in function of time can thus be found by calculating the area of the focal sector. This is the so-called Kepler problem, and is equivalent to the solution for x of the Kepler equation (e and z given). Approximation techniques were sought either in terms of geometrical constructions (Christopher Wren proposed one implying the use of the cycloid) or in terms of numerical iterative procedures. In his Principia, Newton published a procedure equivalent to the so-called Newton–Raphson method.
Mathematicians interested in methods applicable to continuously varying magnitudes broached problems such as the determination of tangents and of radii of curvature to plane curves; the calculation of curvilinear areas and volumes; the determination of centres of gravity; the rectification of curves, and similar. Other problems that attracted much attention were solutions of what were called ‘inverse tangent problems’, in which it was required to determine a curve given its local properties—typically the properties of tangents or of osculating circles. We have seen previously how Descartes tackled an inverse tangent problem when he determined the shape of a refracting surface that focuses light rays. Nowadays, inverse tangent problems are solved through differential equations.
Before Newton and Leibniz all these problems were dealt with by a variety of methods that were not ruled by any single theory (Andersen 1994). The contributions made by Kepler, Grégoire de Saint-Vincent, Cavalieri, Torricelli, Fermat, Roberval, Sluse, Hudde, Pascal, Wallis, Neil, J. Gregory, Mercator, and Barrow are generally grouped under the heading of ‘pre-calculus’. The diversity of approaches, notations, and methods among these authors, however, renders any such grouping problematic. Still, some shared featured in their research can be identified. Curves were conceived of as polygonals constituted by infinitely many infinitesimal sides, while curvilinear surfaces were seen as composed of infinitely many infinitesimal rectilinear surfaces. These views enabled the determination of tangents and curvatures via limiting procedures (often disguised, as in Descartes’ method of tangents, for instance) and that of the areas of curvilinear surfaces and of the volumes of curvilinear solids via infinite summations. Some of these problems were perceived as being easier: thus a variety of well-behaved methods were known for tracing the tangent to some classes of plane curve, while the determination of the area of the surface bounded by a curve or its rectification proved thornier tasks. It was observed that the equation of the tangent to an algebraic curve is algebraic, while the area of the surface bounded by an algebraic curve—or the arclength of an algebraic curve—is often transcendental: consider the area of the surface subtended to the hyperbola: in Leibnizian terms, . Indeed, the discovery, due to the Jesuits Grégoire de Saint-Vincent and Alphonse Antonio de Sarasa, of the relationships between the hyperbola and logarithms highlighted the importance of quadrature techniques for the art of navigation, geography, and astronomy. Logarithms, simultaneously introduced by the Scottish nobleman John Napier (1550–1617) and the Swiss craftsman Joost Bürgi (1552–1632) in the second decade of the seventeenth century, greatly improved the composition of numerical (p. 251) tables, as Kepler first appreciated in his Tabulae Rudolphinae (1627). Undoubtedly, the need to handle logarithms and trigonometric magnitudes was one of the chief motivations behind the birth of calculus.
A mixture of geometry and algebra characterizes the works of the above-mentioned authors. Their understanding of limit arguments and the convergence of infinite summations was often based on geometrical intuition. These were some of the major results they attained: infinite summations were deployed in order to find curvilinear areas and volumes (Cavalieri, Torricelli, Pascal, J. Gregory); the area subtended to the hyperbola was calculated and it was understood that it measures the natural logarithm (Grégoire de Saint-Vincent, de Sarasa, Mercator); the length of the arc of certain curves was calculated (Wren, Neil); tangents were drawn to algebraic and mechanical lines (Roberval, Fermat, Sluse, Hudde); Gregory and Barrow intuited the inverse relation between tangent and area problems; and finally Wallis developed an arithmetical approach to the quadrature of curves leading to his famous discovery of the infinite product for the calculation of (Stedall 2002).
It is due to two great thinkers, Newton and Leibniz, that from the multifaceted and unsystematic heritage of the precalculus era two equivalent formalisms surfaced, which enabled—at least in principle—the problems concerning continuously varying magnitudes to be tackled. Newton developed his method of series and fluxions in the mid-1660s, whereas Leibniz discovered differential and integral calculus during his stay in Paris between 1672 and 1676.
The innovations brought by Newton and Leibniz can be illustrated briefly by considering three aspects of their mathematical work: problem-reduction, the calculation of areas subtended to plane curves through an inversion of the process for calculating tangents, and the creation of algorithms. The invention of calculus can thus be seen to lie in these three contributions.
Newton and Leibniz realized that a whole variety of problems concerning the calculation of centres of gravity, areas, volumes, tangents, arclengths, the radii of curvatures, and so on, which had occupied mathematicians in the first half of the seventeenth century, could in fact be traced back to two basic problems (the calculation of the tangent to a plane curve and the calculation of the area of the surface subtended by a plane curve). Furthermore, they fully realized that these two problems were one the inverse of the other (and this is the ‘fundamental theorem’ of the calculus). They understood that the solution of the former, and easier, problem could be used to answer the latter. Last but not least, Newton and Leibniz developed two efficient algorithms that could be applied in a systematic and general way. It is through these contributions that Newton and Leibniz overcame the limits of precalculus methods.
Newton conceived of geometrical magnitudes as generated by a continuous flow in time. For instance, the motion of a point generates a line, and the motion of a line generates a surface. The quantities x, y, z, generated by flow are called fluents. Their instantaneous speeds , , , are called fluxions. The moments of the fluent quantities (p. 252) are the infinitely small additions by which those quantities increase in each infinitely small interval of time. By conceiving of a plane curve as generated by the motion of a point , where and are Cartesian coordinates, Newton determined the slope of its tangent as the ratio between and ; by generalizing the results contained in the appendices to the Latin edition of Descartes’ Géométrie, he then found the relevant algorithm. This algorithm contains the rules for the differentiation of the sum (), product (xy), power (), and quotient (), as well as the chain rule (Guicciardini 2009).
Newton deployed several techniques of power (even fractional power) expansion of the fluent quantities. The most famous is the series for the binomial elevated to a fractional power that he obtained, generalizing Wallis’s results in Arithmetica infinitorum, during the winter of 1664–5. Infinite series allowed Newton to express transcendental functions and integrate by termwise integration. A few examples illustrate this fact.
Newton’s binomial series is:(8.11)
Application of the binomial series to negative exponents leads to interesting results that had escaped Wallis. Most notably, Newton wrote,(8.12)
a result that he considered valid when x is small. Newton studied the hyperbola for . He knew that the area under the hyperbola and over the interval for (and the negative of this area when ) is . By term-wise integration he could express as a power series:(8.13)
Newton somewhat intuitively understood that this series converges for and . In this period, questions regarding the convergence of infinite series were approached without any general theory of convergence. Mathematicians were simply happy to verify by application to numerical examples that the series (8.13) converged when the absolute value of x was smaller than 1. These series allowed Newton to calculate logarithms, he extended his numerical calculations to more than fifty decimal places! Newton obtained also power series representations of the trigonometric functions such as(8.14)
Infinite series were a basic ingredient for squaring curves (that is, integrating functions). Another approach that Newton adopted consisted in applying the fundamental theorem of the calculus. In modern terms, one might say that Newton (p. 253) was aware of the fact that antiderivatives are related to definite integrals through the fundamental theorem of calculus and provide a convenient means of tabulating the integrals of many functions. Further, Newton listed many functions whose area he could determine via methods we now identify as integration by parts and by substitution (Guicciardini 2009).
In the 1670s, for a number of complex reasons Newton somewhat distanced himself from the algorithmic style of his early researches. He became a great admirer of Huygens, whose Horologium oscillatorium he read in 1673, and of the ancient geometers. In this context he developed a geometric approach to limiting procedures. He termed this geometric approach the ‘method of first and last ratios’. This method is based on postulates or lemmas concerning the limits of ratios and sums of ‘vanishing magnitudes’, and its purpose is to enable the determination of tangents to curves and the calculation of areas of curvilinear surfaces by geometrical arguments based on limiting procedures. The method of first and last ratios played a prominent role in Newton’s Principia and is best illustrated through an example.
One of the main problems that Newton broached in the Principia was the mathematical treatment of central force motion. In order to deal with central forces by using geometrical methods, a geometrical representation of such forces is required. This result is not easy to attain, since the central force applied to an orbiting body changes continuously, both in strength and direction. Before Newton’s work, mathematicians were able to tackle the problem of rectilinear accelerated motion and circular uniform motion. Newton’s intuition was that locally one could approximate the trajectory of a body acted upon by a central force either as a small Galilean parabola traversed in a constant force field or as a small circular arc traversed with constant speed v. Therefore, he locally applied either Galileo’s law of fall or Huygens’ law of circular motion. The second approach required the determination of the radius of curvature of the orbit so that the normal component of the central force F is equal to .
Let us see how Newton applied Galileo’s law of fall to central force motion (Guicciardini 1999). He does so in Proposition 6, Book 1, of the Principia. This proposition applies an hypothesis that had first been suggested to Newton by Robert Hooke. The planet P is accelerated in void space by a central force, and its motion, as Hooke had suggested, is decomposed into an inertial motion along the tangent and an accelerated motion toward the force centre, the Sun S. We thus have a body accelerated by a centripetal force directed toward S (the centre of force) and that describes a trajectory like the one shown schematically in Fig. 8.9. PQ is the arc traversed in a finite interval of time. The point Q is fluid in its position on the orbit, and one has to consider the limiting situation when points Q and P come together. Line ZPR is the tangent to the orbit at P. QR, the deviation from inertial tangential motion, tends to become parallel to SP as Q approaches P. QT is drawn normal to SP. In Lemma 10 Newton states that ‘at the very beginning of the motion’ the force can be considered constant. In the case represented in Fig. 8.9 this implies that as Q approaches P the displacement QR is proportional to force times the square of time. (p. 254) Indeed, in the limiting situation, QR can be considered as a small Galilean fall caused by a constant force.
Newton could now obtain the required geometrical representation of force. Since Kepler’s area law holds for central force motion, as Newton has proven in Proposition 1, the area of SPQ is proportional to time. Moreover, SPQ can be considered a triangle, since the limit of the ratio between the vanishing chord PQ and arc is 1. The area of triangle SPQ is . Therefore, the geometrical measure of force is(8.15)
where the ratio has to be evaluated in the limiting situation where points P and Q come together and ∝ is used to mean ‘is proportional to’.
Proposition 6 is a good example of the application of the method of first and ultimate ratios. The limit to which the ratio tends is to be evaluated by purely geometrical means. Note that SP remains constant as Q tends to P; therefore one has to consider the limit of the ratio .
Newton proved that when the trajectory O is an equiangular spiral (in polar coordinates ) and S is placed at the centre, as Q tends to P,(8.16)
and thus the force varies inversely with the cube of distance. Whereas, when the trajectory O is an ellipse and the centre of force lies at its centre (Proposition 10), , that is, the force varies directly with distance.
Newton considered Keplerian orbits. In Proposition 11, Book 1, of the Principia Newton proved that if the body describes a trajectory O, O is an ellipse and the force is directed toward a focus S, then the force varies inversely as the square of the distance. In Propositions 12 and 13, Newton showed that the force is also inverse-square if O is a hyperbola or parabola. To conclude: when the orbit is a conic section and S is placed at one focus,(8.17)
where L is a constant (the latus rectum), and the ratio is evaluated, as always, as the first or last ratio with the points Q and P coming together. Therefore, the strength of a force that accelerates a body obeying the first two Keplerian laws varies (p. 255) inversely as the square of distance. This is often considered to be the birth of gravitation theory, even though, as experts know, Propositions 11–13, Book 1, played a limited role in Newton’s deduction of universal gravitation from astronomical phenomena (Guicciardini 1999).13
During his productive stay in Paris (1672–6) Leibniz became acquainted with cutting-edge mathematical research (Hofmann 1974). Following the advice of Huygens, who was one of the most eminent members of the Académie des Sciences, the young German diplomat devoted his attention to many recent mathematical works, including treatises by Pascal, Torricelli, Wallis, and Barrow. By the end of his Parisian period, Leibniz had developed differential and integral calculi, which he published starting in 1684 in a journal he had helped found, Acta Eruditorum. In 1684 he published the rules of the differential calculus (, , , and so on), and applied them to some geometrical problems. Most notably, the slope of the tangent of a plane curve whose Cartesian coordinates are x and y was determined by the ratio . Two years later, Leibniz published his first paper on the integral calculus, a technique that enabled him to solve problems of quadrature, cubature, rectification, and so on, in a systematic way. Leibniz’s calculus aroused criticism from geometric purists such as Vincenzo Viviani and Huygens, and from mathematicians such as Bernard Nieuwentijt, Michel Rolle, and Detlev Clüver, who were worried by Leibniz’s puzzling use of symbols for infinitesimal magnitudes, differentials (dx), and even higher-order differentials (). The calculus, however, also had its partisans: initially in Basel, with Jacob and Johann Bernoulli, and later in Paris, with Guillaume F. de L’Hôpital and Pierre Varignon.
Pierre Varignon set himself the task of translating some of the propositions of Newton’s Principia concerning central force motion and motion in resisting media into calculus terms. In 1700 he published three papers in which he formulated the relation between force F, speed v, and displacement x described in Proposition 39, Book 1, of Newton’s Principia(8.18)
and the expression of central force in terms of differentials (r, , polar coordinates, s trajectory’s arc-length),(8.19)
The importance of these translations of Newton’s geometrical results into calculus terms cannot be overestimated. There were two main advantages to the application of the calculus rather than geometry to the science of motion. First. As the readers of the Principia will know well, the geometric methods that Newton employed can often be applied only to the specific problem for which they were devised. Varignon showed that the results obtained in calculus terms were applicable to whole classes of problems. Second. The calculus allowed a systematic handling of higher-order differentials. Newton was often at pains to classify the infinitesimal order of vanishing geometrical magnitudes (a simple example will suffice to illustrate this point: in Fig. 8.9, QR is a second-order infinitesimal, whereas QT is a first-order infinitesimal, so that the limit of eq. (8.15) for Q tending to P is finite). It should also be noted that despite his preference for geometrical methods, in the Principia—and particularly its most advanced parts—Newton himself often employed calculus tools such as integration techniques, the calculation of the radii of curvatures, and infinite series expansions.
Leibniz, Varignon, the Bernoullis, Jacob Hermann, and a group of Italian mathematicians that included Jacopo Riccati and Gabriele Manfredi all proved aware of the advantages of the calculus approach to mechanics, the science of motion, and optics. The calculus problems faced by Leibnizians and their British rivals—including Brook Taylor, David Gregory, Roger Cotes, and Colin Maclaurin—at the beginning of the eighteenth century were often inspired by applications to physical situations. The cases of the brachistochrone and catenaria have already been mentioned. The repertoire of curves that emerged from the solution of differential equations (‘inverse tangent problems’) soon increased. These mathematicians vied for the definition of the shape of a loaded beam (the elastica) or of a sail inflated by the wind (the velaria), and began studying oscillations of extended bodies, such as those of vibrating strings. The solutions to these problems were hardly of much use to engineers interested in designing ships—for instance—or suspension bridges, yet they showed that the calculus could be conceived of as a promising tool for dealing with rather complex physical facts in mathematical terms. The power of calculus in dealing with transcendental curves and in integrating classes of differential equations was greatly appreciated. Jacob Bernoulli was so fascinated with the logarithmic spiral (), the properties of which he studied in a tour de force of mathematical ingenuity in the early 1690s, that he requested this curve be reproduced on his funerary emblem. Bernoulli (and at an earlier date John Collins) also stressed the usefulness of the logarithmic spiral as the stereographic projection to the tangent plane at the pole of the loxodrome: the curve crossing each meridian at a constant angle that was sought by mathematicians interested in navigation, such as Pedro Nuñez, (p. 257) Simon Stevin, Willebrord Snel, Gerhard Mercator, Thomas Harriot, and Edmond Halley (Fig. 8.10).
The generality and power of the approach to the science of motion in terms of ordinary differential equations can be appreciated best through an example. In 1711 Johann Bernoulli sent a letter to Varignon in which he discussed a problem that had come to his mind when reading Newton’s Principia (Bernoulli, Johann (1711)). In Proposition 10, Book 2, Newton had considered a body moving under the action of uniform constant gravity and a resistance proportional to the product of the density of the medium times the square of the speed of the body. Given the trajectory traced by the body, Newton sought to determine the density and velocity at each point of the trajectory (Guicciardini 1999, 233–249). Bernoulli faced a more general problem. He assumed that the trajectory was given, and studied the case in which the body is acted upon by a central force (F) and moves in a medium exerting a resistance (R) proportional to the density () times some power of the speed (). In Bernoulli’s case, the density is known and one seeks to determine the central force.
Bernoulli employed formulas to represent central force in calculus terms which were well established by 1711. He stated that the intensity of the tangential component () of the central force is given by:(8.20)
In order to write the differential equation of motion, Bernoulli expressed the tangential component of the total force acting on the body by ‘adding or subtracting’ to the tangential component of central force (equation (8.20)) the force of resistance ‘according whether the body is moving away from the centre [in this case and R have the same sign] or whether it is approaching it [in this case and R have the opposite signs]’ (Bernoulli, J. 1711, p. 503). According to Bernoulli’s sign conventions, the tangential component of the total force acting on the body multiplied by is equal to . Therefore, setting Bernoulli writes:(8.21)
And ‘since the relation between , , , and is given in terms of r and y’, setting and ,16 one has the following differential equation:(8.23)
This is the non-linear (for ) differential equation that Jacob Bernoulli had already solved in 1695, while Johann had proposed an alternative integration method in 1697. By integrating eq. (8.23), Johann obtained v and, since ,(8.24)
One cannot but appreciate the generality of Johann Bernoulli’s approach to the motion of projectiles in resisting media based on the theory of differential equations. (p. 259) In their treatment of the matter, Leibnizian mathematicians deployed the wealth of results reached by the Basel school in the field of differential equations and the calculus of transcendental magnitudes. Their approach proved to be more successful compared to the one followed by the British Newtonians. When in 1718 John Keill proposed as a challenge against the Leibnizians the solution of the inverse problem of resisted motion—that is, the problem of determining the trajectory of a body accelerated by constant gravity and by a resistance proportional to density times the square of velocity—Johann Bernoulli replied in papers published in 1719 and 1721 by solving an even more general problem: that of determining the trajectory of a body resisted by a force proportional to density times some power of the velocity. This problem had indeed already been solved by Hermann in his Phoronomia (1716), due to the use of the differential equation solved by Bernoulli in 1711 (Blay 1992, pp. 322–30).
(p. 260) 8.7 Enlightened Mechanics
I began my chapter by stating that the idea of a ‘scientific revolution’ seems justified from the point of view of the historian of mathematics: one need only compare the qualitative geometric treatment of projectile motion in the works of Tartaglia or dal Monte carried out in the sixteenth century to the calculus approach illustrated in Johann Bernoulli’s solution of eq. (8.21) in 1711.
The achievements of the generation immediately following Leibniz and Newton’s (that of Varignon, Jacob and Johann Bernoulli, and Hermann on the Continent, and of Cotes, Stirling, and Taylor in Britain) made a lasting impact on the mathematical sciences. Ordinary differential equations became the language employed for the mechanics of point masses, as evidenced by Leonhard Euler’s Mechanica (1736). The need to extend mathematical methods to the study of continuous, rigid, elastic, and fluid bodies led to the development of partial differential equations. Extremal principles, such as the least action principle, were framed in terms of the calculus of variations, bringing abstraction one step further.
The establishment of calculus as the language of what came to be described as ‘analytical mechanics’ had consequences that cannot be overestimated. The laws of motion came to be expressed through equations instead of proportions, as had still been the case in the works of Huygens and Newton. Dimensional constants (for example, with a dimension equal to length divided by the square of time) thus became a feature of the equations of eighteenth-century mathematicians, making the expression of explicit solutions much easier. More fundamentally, the calculus enabled the adoption of a systematic approach to the science of motion, making the algorithmic control of approximations possible, as illustrated by the works on planetary perturbations that were carried out in the mid-eighteenth century by d’Alembert, Euler, and Lagrange. Indeed, the series expansions included in the works of these authors on planetary motions enabled systematic control of the order of approximation that could be carried on, say, up to the level of a chosen power of the orbit’s eccentricity.
All these advantages should not make us forget that progress came with a price (Gingras 2001). In the first place, the mathematical sciences became the province of professional experts—gifted mathematicians who acquired extraordinary prestige in Continental academies. While Galileo’s Discorsi could have been read by all cultivated readers interested in natural philosophy, Newton’s Principia, and even more so the eighteenth-century treatises of Clairaut, Euler, and Lagrange, could be understood by only a handful of well-trained experts in the field.
Secondly, the explanatory power of mathematical models—their ability to describe the causes of natural phenomena—was weakened, according to some critics, by the abstraction of symbolism. For mathematicians practicing the analytic mechanics in vogue at the time—especially in Continental academies—solving a problem meant producing a technique for the integration of a differential equation, rather than any explanatory model.
(p. 261) Thirdly, when considering eighteenth-century analytical mechanics we tend to forget how remote from applications that sort of pure mathematics was: before the expansion of mathematical science to heat, magnetism, light, and so on, as described by Kuhn, mathematical physics was of interest at most for the astronomer (Kuhn 1977). In fact, the mechanical problems faced by, say, Johann Bernoulli or by Euler were pursued for the sake of their mathematical interest, even when they were motivated by mechanical or geometric applications. It has often been said—and with good reason—that analytical mechanics provided the main stimulus for eighteenth-century mathematicians. However, problems which aroused much interest, such as the brachistochrone and the vibrating string, were only faintly connected with applications; they were rather pursued because of their mathematical interest. As Terrall notes:
The most rigorous analytical mathematics that filled the pages of the [French] academy’s journal was not necessarily useful in terms of practical applications or public displays. The equations of celestial mechanics or hydrodynamics did not translate easily into engineering design or industrial production. (Terrall 1999, 247)
Again, the example of Johann Bernoulli’s approach to projectile resisted motion can illustrate the point. Bernoulli’s equation (8.21) remained fundamental for the study of exterior ballistic. However, much experimental and theoretical work was still to be done in order to reach a realistic description of the drag force that could be used with some success in practical applications. Indeed, it was only at the end of the nineteenth century that the equations of fluid dynamics could be used on the battlefield.
In the seventeenth century, mathematical giants such as Huygens and Newton preferred to publish their results in a geometrical language. By contrast, few mathematicians resisted the new drive towards abstraction and symbolism in the eighteenth century: it was left to outsiders such as Louis Bertrand Castel to oppose the mathematical abstractions of the powerful académiciens. The turn towards analytical mechanics proved irreversible.
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(3) Galileo built his theory of projectile motion on previous results achieved by Medieval natural philosophers as well as by Renaissance mathematicians and ballistic engineers. We will not deal here with this historiographic issue.
(4) Euclid, The Elements, 5, definition 4. In modern terms, a ratio (when ) can be formed if and only if there is a positive integer n such that .
(5) The peculiarities of Galileo’s formulations of a principle of ‘circular inertia’, and their difference from the modern one, is a much debated issue.
(6) Definition 5 reads: ‘Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever are taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order.’ Translation by D. E. Joyce at http://aleph0.clarku.edu/~djoyce/java/elements/elements.html.
(7) Indeed, allowing ourselves an algebraic language not accessible to Galileo,
(10) Wren provided a method for calculating the cycloid’s arclength. Pascal, in Histoire de la Roulette (1658), reviewed the various attempts made to face his challenge and provided a biased history of the study of the cycloid, while publishing his own research on the matter in Lettre de A. Dettonville à Monsieur de Carcavy (1658). The Lettre proved to be very important for Leibniz’s early researches on the calculus. Pascal’s publications were badly received in Italy because in them allegations of plagiarism were levelled against Torricelli and in England aroused the criticisms of Wallis.
(11) Huygens’ paper on the impact of bodies was printed in the Philosophical Transactions for 1669, while John Wallis and Christopher Wren tackled this issue in papers printed in 1668.
(12) The equation of the catenary in Cartesian coordinates is .
(15) . Note that there is no normal component of the resistive force: it is thus legitimate to set . As customary, the mass term m is ‘absorbed’ in these prototype equations/proportions so that, to all effects, .
(16) Both p and q are functions of r and y, but one of the variables can be eliminated by employing the equation of the curve which is given.
(17) It is worth briefly illustrating Johann Bernoulli’s method to solve eq. (8.23). Cases and considered by Newton (resistance proportional to v or ) lead to an easy linear differential equation. For the non-linear cases Bernoulli searched the solution as a product . Substitution in the differential equation leads to
Since M and N are arbitrary functions, Bernoulli can set
which leads to
For this value of M the equation is transformed into
which is in integrable form. One obtains the integral as
From this the value for N is obtained:
And multiplying M by N one obtains v: