Mathematics

MATHEMATICS. - Arithmetic and geometry are the two great trunks of mathematics, from which numerous other branches have sprung forth to weave a tangled interlace where it becomes difficult to distinguish them.

It is not easy to subdivide and list the various developments of mathematics without constant reference being made in each to the others. Not without reason, while Plato with his λέγει ὅτι ὁ θεὸς γεωμετρεῖ recognizes geometrization in the work of God, and while our Galileo sees everything in nature expressed in the language of triangles, the biblical author, on the other hand, with his « Omnia in numero, mensura et pondere disposuisti... », recognizes in the Universe an arithmetizing principle. Two different positions, the one reflecting the Greco-Italic mentality, the other the Oriental, both completing each other in describing the appearance of the Universe.

With the flowing of the centuries, the work of mathematicians, availing themselves of the instrument of a most refined logic, through the three fundamental judgments of belonging, of exclusion and of interference, has discovered ever new horizons, with new entities and new operations, especially in these last three centuries, driven thereto by the necessity of interpreting and foreseeing natural phenomena, whether mechanical or physical or chemical or even biological and indeed social, and in this aided by an ingenious schematization of phenomena and by the devising of a suitable symbolism. Thus, with mutual exaltation, both mathematics and the knowledge of the phenomenology of the Universe have gone on ever more expanding and interpenetrating, especially in recent times and today more than ever.

I. ARITHMETIC

It is founded on the concept of the whole number, symbol of a group of objects; from it one arrives at the concept of the fraction, which symbolizes the part of a group: the subgroup.

The association of two or more groups or subgroups of objects leads to the concept of the sum of wholes and of fractions. The associability of a group with a subgroup, given the character of homogeneity between the one and the other, leads to conceiving the symbols that represent them (wholes and fractions) under a single kind: that of rational numbers.

Both the iteration of the sum and the formation of the subgroup obtained from another subgroup constitute a new operation: multiplication by a whole or by a fraction. Subtraction, on the other hand, is the inverse operation of addition (given the sum of two numbers and given one of these, to find the other) and division is the inverse operation of multiplication (given the product of two numbers and one of these, to find the other). These are the four fundamental operations, the so-called rational operations, because applied to rational numbers they yield as a result still rational numbers.

The iteration of the multiplication of a number by itself yields raising to a power, which is therefore still a rational operation, which however, unlike the first four, gives rise to an inverse operation, which is called extraction of a root, not always possible. That is, there does not always exist a rational number v that is the root of another a; on the other hand there exist rational numbers x which, raised to a power, give as a result rational numbers which, if not coinciding with the given number a, approach it as closely as one wishes. Thus, for example, the square root of 2 does not exist, inasmuch as no rational number, raised to the square, reproduces 2; nevertheless there exist rational numbers which multiplied by themselves give products that approach 2 as closely as one pleases, according to the decimal digits that are calculated. This circumstance suggests the idea of conceiving numbers of a new kind: irrational numbers, each of which, no longer expressed by a whole, or by two (as when it is a matter of a fraction), is only representable by a sequence of infinite decimal digits: tenths, hundredths, thousandths, and so on, that is, by a sum of infinitely many ever smaller terms.

On the other hand, subtraction too suffers an exception: it is not possible to subtract a greater from a whole if the whole is thought of as representing a group. But, just as the extraction of a root is always made possible by the introduction of irrational numbers, so too subtraction becomes practicable, even when it presents an exception, by means of the introduction of negative numbers. Rational and irrational numbers, positive and negative, constitute the class of real numbers.

In turn, the introduction of negative numbers creates a new exception; indeed it is not possible to extract the square root of a negative number, unless by defining numbers of yet another kind: imaginary numbers. No real number, in fact, positive or negative, raised to the square gives for example as a result -25. With the imaginaries, complex numbers are also born, which are the sum of a real with an imaginary. By means of these complex numbers it is then made possible, no longer only to extract the square root of a negative number, but even the extraction of the nth root, whatever the number, even complex, subjected to that operation. Well then, these new numbers, which seem to surpass the limits of abstruseness, render instead the most useful services, not only in the purely mathematical field, but also in numerous applications of electrical engineering and of aerodynamics.

The first three rational operations, sum, product, difference, are called rational integral. With them one can construct a polynomial in x, that is, an expression of the type a x + b, if it is of the first degree, a x² + b x + c, if of the second degree, a x³ + b x² + c x + d, if of the third degree, etc. When, for example in the last polynomial written, the numbers a b c d, coefficients, are given, and x is given, one can find the value assumed by the polynomial, by means of the three above-mentioned operations. The inverse problem, given the value of the polynomial, to find x, constitutes the so-called solution of an algebraic equation of the nth degree, if n is the degree of the polynomial (the highest power of x that appears in it). Well then, this inversion, although not always performable by means of formulas, proves always possible, and admits n solutions (as many as is the degree of the polynomial), without its creating numbers of a new kind, because the solutions all exist in the field of complex numbers.

Those real or complex numbers which solve an algebraic equation with integer coefficients are then called algebraic numbers. But if the equation has fractional or irrational numbers as coefficients, the solutions still fall within the field of algebraic numbers.

A new classification arises when one leaves the rational or algebraic operations to enter the field of operations different from these, and which are therefore called transcendental; the solutions of these are complex numbers that are not algebraic and are therefore called transcendental. Among these is the number π, the ratio between the circumference and its diameter; it is in fact not the solution of any algebraic equation. But together with the extension of the concepts of operation and of number, the possibility has also been studied of obtaining, by the application of rational integral operations carried out on whole numbers, as a result, still whole numbers. In general terms, it has been a matter of determining when and how certain algebraic equations, with integer coefficients, are solvable in whole numbers. The so-called theory of numbers, which deals with such questions and is also linked to problems inherent in other fields of mathematics, has encountered extraordinary difficulties along its way. The determination of prime whole numbers, that is, divisible only by themselves, is likewise still the object of profound studies. To the theory of numbers the names of Fermat and Gauss are particularly linked.

From what has been said, it emerges that fundamental elements are the whole numbers and the fundamental operation is addition. These are the sources from which all the other numbers and all the other operations derive. The mathematician has often substituted in his reasonings, for the purpose of greater generality, a literal symbol for the number, and has thus created classical algebra; more recently, under the symbol, rather than a number he has seen any geometric entity whatsoever, and has then extraordinarily extended the concept of algebra.

II. ALGEBRA

Italy's primacy in classical algebra goes back to the Renaissance, and runs from the names of Leonardo of Pisa, Tartaglia and Cardano to those of Ferrari (eighteenth century) and Ruffini (early nineteenth century). From the solution of the second-degree equation mathematicians had passed to that of the third, then to that of the fourth, and they were still trying to advance further, when Ruffini demonstrated the impossibility of solving in general, by means of algebraic formulas, an equation of degree higher than the fourth, just as had instead been achieved for equations of lower degree.

There remained the possibility of solving an equation with numerical, rather than literal, coefficients, and numerous approximation methods were devised to obtain the solutions. Those methods are now by far superseded by today's electronic machines. But one of the questions that had long since arisen in the applications of mathematics to physics was the solution of systems of first-degree equations with several unknowns. From the eighteenth century onward various methods were born of this, and from it arose the theory of determinants, which facilitates the solution of those systems and also indicates whether or not they can be solved. From determinants there then recently blossomed the theory of matrices, which have proved extremely useful in many geometrical treatments and have lately been applied by Heisenberg in the study of the atom.

The need to solve systems of first-degree equations with numerous unknowns has made itself felt in recent years, both in the calculations of elastic structures and in the solution of the integral equations (cf. no. IX) of mathematical physics. For this purpose methods have been devised with the aim of sparing the arduous labor that the use of determinants would impose; Teofilato as early as 1922 gave one for obtaining the solutions by successive approximations with rapid convergence toward the exact value. It is worth noting here that there is no cause for scandal at the concept of approximation introduced into mathematics. After all, apart from the integer and the fraction, any irrational number, such as, for example, the square root of 2, is attainable only by approximation; the measurement of a magnitude is always affected by errors, even though they may be reduced by the progressive perfecting of instruments, but they are ineliminable. Therefore one should not be impressed if the solutions of a system of equations are exhibited with approximation, all the more so since in most cases the very coefficients of the equations are obtained by means of instrumental measurements.

Moreover, even for a system of one hundred first-degree equations with one hundred unknowns the electronic machine provides the answer in a few seconds; however, one must first set up the perhaps ten thousand coefficients appropriately, in order to make the mechanism work. This mechanism, very delicate and complicated, is equipped with organs that are comparable to what in us are the brain, the nervous system and the muscular system, analogously to another admirable mechanism found aboard aircraft, the so-called autopilot, which takes care of governing the flight automatically. The first organ perceives the setting of the coefficients and transmits the impulse to the second organ, which amplifies it somewhat, and transmits it in turn to the power organ that commands the moving parts.

III. TRANSFORMATIONS

A new type of operation is the transformation, which at first had a meaning limited to substitution, which is the operation that changes a set of elements into the same elements, but arranged in a different order.

To the theory of substitutions, applied to the solutions of an algebraic equation, are linked the names of Galois, Ruffini, Abel; and it is by means of this theory that the aforementioned impossibility of algebraically solving equations of degree higher than the fourth is demonstrated. From the same theory was born the concept of a group of substitutions, by virtue of which it happens that operating successively with two substitutions of the group yields the effect that would be obtained with a single substitution of the same group. If one then passes from a finite number of elements to an infinite number, transformations arise, which Lie studied profoundly at the end of the last century. The displacement of a solid relative to the three edges of a room constitutes a simple example of a transformation, inasmuch as to every point of the solid in its original position there corresponds a point in the second.

With the transformation there is then interwoven the concept of group and of an entity invariant with respect to the transformation itself. For the important Italian contribution to these studies, the names of Luigi Cremona, Ernesto Pascal, Luigi Bianchi, Guido Castelnuovo, Corrado Segre stand out. The theory has assumed enormous importance because it is capable of grand syntheses, both in geometry, where, for example, it is found that projective geometry (cf. n. VII) is a chapter of the theory in question, and in physics, because all electromagnetic phenomena are invariant with respect to the particular Lorentz transformation.

One is tempted to think that every geometric or physical element is linked to an invariant and, conversely, that the discovery of an invariant indicates the existence of a geometric or physical entity.

The concept of transformation and of invariance pervades all mathematical physics, in its most varied fields.

Even in gas dynamics, a science born a few years ago, the invariance of the equation that governs the motion of compressible gases, with respect to a suitable transformation, makes it possible to translate the effect of high velocities in air into the effect of low velocities in tiny water channels, with great ease of study and minimal expense.

It is, at bottom, a particular case of the method widely used in science: transforming one problem into another already studied or one that is simpler to treat.

IV. ELEMENTARY GEOMETRY

Raised to great heights by the work of the Greeks, it remained carved as an imperishable epitaph in Euclid's compilation. Limited in the plane to the study of figures formed by straight lines and circles (and therefore constructible with straightedge and compass) and in space to solids bounded by plane faces, or even to the three round solids (cylinder, cone and sphere), it went on to consider other lines (conchoid, cissoid, strophoid, etc.) and subsequently occupied itself with measurement, which, precisely because of its practical aims, was at first less esteemed by the Greek mentality, finely speculative and aristocratically in search of truth. Measurement was extensively dealt with later also by Archimedes with his fine theorems on pyramids and on round bodies, to which he added his other no less important discoveries on the lever, the movable pulley, the screw, mirrors, and the buoyancy of floating bodies.

Not all plane problems were solved at once with the aid of the two celebrated instruments, straightedge and compass. Some required much research, such as the golden segment of the radius, the side of the regular decagon inscribed in a circle: others were solved approximately, such as those of the rectification and quadrature of the circle (the search for the segment as long as the circumference and the search for the square as large as the circle). The need to resort to approximation depended on the limitation of the instruments employed.

About two centuries ago Mascheroni in his Geometria del compasso demonstrated that all problems that are solvable with straightedge and compass are also solvable with the compass alone: but it is the merit of the last century to have brought to light that with the two aforementioned instruments only those problems are solved which, translated into algebraic questions (as will be better seen below), give rise to equations of the second degree or reducible to these. From the Greeks down to Apollonius, with his theorems on the conic sections (ellipse, parabola and hyperbola), about thirteen centuries passed, without geometry undergoing notable progress. One must go back to the Renaissance to encounter Leon Battista Alberti, Leonardo da Vinci, Piero della Francesca, Vignola, Brunelleschi, Ubaldi dal Monte, who, with their studies on perspective, broadened the field of investigation of the Greeks. Perspective then, between the end of the eighteenth century and the early nineteenth, issued in descriptive geometry, which, by means of a suitable representation on the drawing sheet, makes it possible to recover all the metric particulars of a figure in space.

Descriptive geometry, which at the beginning of the last century had passionate devotees in Naples (Flauti, Fergola, Trudi, Padula), later gave Bellavitis of Padua the cue to found the theory of equipollences, a method useful in the resolution of graphical problems, from which the modern theory of vectors was then born, which in our days has assumed a degree of great generality and power of synthesis, with correspondences also in the aforementioned theory of matrices.

V. NON-EUCLIDEAN GEOMETRY

Elementary geometry had long carried with it an element of doubt, given by the postulate that goes under the name of Euclid: through a point, only one parallel can be drawn to a given line. Already Wallis in the seventeenth century and the Jesuit Father Saccheri in the eighteenth century had tackled the question of the postulate. The former reduced it to another postulate that seemed to him more acceptable; the latter, through the opposite hypothesis, set out to prove its absurdity. Saccheri, however, did not reach the final goal, despite the important results obtained, and so the question remained still open.

On the path traced by Saccheri, Bolyai and Lobachevsky took up the study again. Having therefore supposed that through a point two parallels can be drawn to a given line, one on one side and one on the other, converging toward the line without ever reaching it, a marvelous logical edifice was constructed; but, passing from deduction to deduction, one never arrived at a contradiction with the starting postulate. There thus arose the suspicion, supported by the authority of Gauss and Riemann, that Euclid's postulate was unprovable. Gauss's celebrated study Disquisitiones circa superficies curvas suggested to Riemann that he extend those considerations to space; thus, having assumed for the distance between two points a form that departs from the one given by the Pythagorean theorem, and which is founded on Euclid's postulate, he gave a glimpse of the possibility of the non-existence of lines parallel to a given line. A fact analogous to what happens on the sphere, where any two meridians (which on the sphere are like the straight lines of the plane, lines of minimum path) always meet (at two points). Whereas Wallis had reduced Euclid's hypothesis to that of the existence of similar triangles, Lobachevsky, with the hypothesis of the two parallels, verified the non-existence of a similarity, so that two triangles with respectively equal angles are equal, just as also happens on the sphere for spherical triangles (formed by arcs of meridians). Indeed, the sum of the angles of a triangle in Lobachevsky's geometry is less than a straight angle, and the deficiency is proportional to the area of the triangle; in spherical triangles the sum of the angles is greater than a straight angle, and the excess is proportional to the area. By now the idea of the simultaneous logical validity of all three geometries — that of Euclid, that of Lobachevsky and the other of Riemann — was wavering. It was only a question of seeing which of the three was actually verified in the physical world. It was therefore necessary to be able to measure the angles of large triangles, to evaluate their excess or deficiency proportional to their size. Well, the result was a sum of angles close to a straight angle, now more, now less, following the so-called law of errors. One had to conclude that in the physical world the Euclidean had the greatest probability of holding compared with the other two geometries, each of which, if it were ever verified, would have to deviate so little from the Euclidean as not to give any way of noticing it.

And in fact, only after the great scientific evolution of this century regarding our physical knowledge, crossing the limits of ordinary experience, has it been observed that when one passes to the world of the atom or to that of the nebulae, Euclidean geometry no longer accords perfectly with the physical world, and instead a pseudo-Riemannian geometry (geometry of the chronotope) is to be held valid, on which it is not the task here to dwell. Space should be considered unlimited, but finite, just as to a being physically endowed with only two dimensions the surface of a perfectly smooth and very large sphere would appear, on which it could move.

In any case, as early as 1865 the Gordian knot that tied such grave doubts to geometry was definitively cut by Eugenio Beltrami, who succeeded in giving the proof of the impossibility of proving Euclid's postulate. On the other hand, if the Universe is finite, its dimensions must be assignable; and then, just as the being living on the aforementioned sphere, if it could appreciate the curvature of a part, even a small one, could also evaluate the entire extent without traversing it all, so it should be no different for us. And, in fact, on this basis the estimate of the dimensions of the Universe, at least as an order of magnitude, has been carried out, and we now have an idea of it comparable, moreover, by approximation, to that which Eratosthenes could have formed about the dimensions of the earth with the measurement from carried out by him of the curvature of the meridian passing through
Alexandria.

VI. PROJECTIVE GEOMETRY

Alongside non-Euclidean geometry, other branches developed: projective geometry, differential geometry, algebraic geometry.

The first, born at the beginning of the last century, took its starting point from the aforementioned theorems of Apollonius, and while it connected itself to descriptive geometry, it directed its attention to the transformations that a plane figure undergoes when it is projected from a point onto another plane (section), proceeding thus through as many successive operations of projection and section as one wishes.

These transformations change points of one plane into points of another, straight lines into straight lines, points aligned on a straight line into points aligned on a straight line, straight lines concurrent at one and the same point into concurrent straight lines; they are called projective transformations, and they can also be extended to the transformation of one space into another. In the transformations there figure, as has already been noted, the invariant entities, and indeed among these the fundamental invariant of projective transformations is the so-called cross-ratio of four points ABCD on a straight line, which is expressed by the quotient (AC/BC) : (AD/BD), and is not altered when, by means of repeated operations of projection and section, one passes from the points A B C D of one straight line to the corresponding points A'B'C'D' of another. If then, by repeated operations of this kind, one returns to the original straight line, then there exist two points of it which, despite the operations performed, come to coincide with their correspondents (so-called two fixed points). When then, from a plane, after repeated operations of projection and section, one returns to the same plane, there are three fixed points, and moreover there exists a conic which comes to coincide with itself (invariant conic) without, however, any point coinciding with its correspondent, but in such a way that each point of the conic, considered before performing the said operations, comes to fall upon another point of the conic itself.

Now, in 1872, Klein, availing himself of Cayley's metric, showed that the three geometries named in the preceding paragraph can be considered as a single geometry from the projective point of view, which depends exclusively on the projective transformation adopted to transform the plane into itself (that is, motion of the plane upon itself, understood in a broader sense). The projective transformation that leaves invariant two special imaginary points, called circular points, and the straight line determined by them (containing the inaccessible points) expresses, with an appropriate metric, the motions of the Euclidean plane (ordinary motions). In Lobachevsky's non-Euclidean plane, the inaccessible is constituted by a real conic which the motions in that plane, that is, the appropriate projective transformation, leave unchanged. In Riemann's plane the inaccessible does not exist, or, better said, is formed by imaginary points. Imaginary points, although not drawable on the plane, can be indirectly represented, as, toward the middle of the last century, Staudt did, who, with synthetic methods of extraordinary elegance, showed the connection of those points with real points, analogously as in projective geometry one handles inaccessible points (points at infinite distance) by way of those that are instead accessible.

But the great advances of geometry came from its union with analysis, as will be seen in the following paragraphs.

VII. ANALYTIC GEOMETRY

At the foundation of analytic geometry lies the representation of a point by means of numbers. In the narrower sense these numbers, called coordinates, are the distances of the generic point under consideration in a plane from two fixed lines O X, O Y of the plane itself, mutually perpendicular, called Cartesian axes. Or else, if the object is to study figures in space, the coordinates, three in number, are the respective distances from three fixed planes, mutually perpendicular, like two adjoining walls and the floor of a rectangular room, whose edges O X, O Y, O Z are the three Cartesian axes of space.

Thus, given a point, one can determine its coordinates by constructing the aforesaid distances, and conversely, given three numbers and interpreting them as distances, one can trace back to the point having the three given numbers as coordinates.

Given then in the plane a line, the two coordinates a, b of a generic point of it cannot both be arbitrary, because if one is known, the other is deduced. For example, if the abscissa a is known, which is the distance of the point P from the y-axis, the point can be determined as the intersection of the given line with the line parallel to the y-axis, situated at the distance a from it; and then, once the position of P on the line is known, the other coordinate b will also be knowable. Hence a line entails a connection between the coordinates of its points, which will be expressed by an equation among the latter. Thus an equation of the first degree between the coordinates represents a straight line, one of the second degree a conic (ellipse or parabola or hyperbola) and so on, while from the coordinates of two points, by means of the Pythagorean theorem, one obtains the distance between them, and from the equations representing two straight lines one obtains the angle they form with each other. In this way graphical problems can be transformed, sometimes advantageously, into algebraic problems or, as is now said with greater generality, into analytic problems.

At bottom, trigonometry too, which can be traced back to Ptolemy, translates into numbers the problems concerning triangles and polygons in general, but from a less general point of view than analytic geometry. By means of trigonometry and with the aid of algebra, many problems of elementary geometry are translated into the solution of equations, often facilitating a search which, if resolved through the purely synthetic method in a more elegant way, sometimes entails greater labor of investigation. Naturally, in the choice of reference axes, to avoid complications, it is advisable to use elements intrinsically connected with those of the problem. This is the view that guided Ernesto Cesaro in his ingenious treatment of Intrinsic Geometry, published in 1896. Nevertheless, the economy of logical work that occurs in employing the analytic method in the solution of a geometric question finds as its counterpart a greater amount of work in the development of the calculations.

Analytic geometry, in its development, has gradually established important links with projective geometry and has brought to light peculiar properties of lines and surfaces, but its grand progress it had when it was wedded to infinitesimal analysis to give rise to differential geometry.

VIII. INFINITESIMAL ANALYSIS

The concept of a magnitude thought of as however small, or as it is also usually said indefinitely small, is essentially dynamic, in that no sooner has one magnitude been fixed than it is urged to fix another smaller one, for example smaller than half the preceding one, and so on in a continuous process. From this dynamism of diminution there springs the concept of the (actual) infinitesimal.

It is already found, albeit hiddenly, in elementary geometry (in the theorem on the equivalence of pyramids and in the theory of the rectification and quadrature of the circle), while it interpenetrates with the existence of the limit toward which a magnitude tends, in that the difference between the magnitude and its limit can become infinitesimal. With Newton there intervenes the comparison, and more precisely the ratio, between infinitesimals, when instantaneous velocity is thought of as the ratio between an infinitesimal space and the infinitesimal time employed in traversing it. Thus arise the concepts of function, of incremental ratio and of derivative.

A function is nothing other than the relation of dependence between two variables, as, for example, the relation between the space traversed by a moving body and the time employed, the relation between the length of a bar and its temperature. The second of the two variables is called independent, the first is said to depend on it or also to be a function of it; but the role of dependence can also be inverted.

The incremental ratio is the ratio between the increment of the dependent variable and the corresponding increment of the independent variable. Thus if the space traversed, from 8 o'clock to 11 o'clock, has gone from 280 to 490 km., the increment of space will be 210 km., the increment of time will be 3 hours, and the incremental ratio (average velocity) will be expressed by 210 : 3 = 70. When these increments are infinitely small one has the so-called derivative of the function. Thus, for example, the instantaneous velocity cited above is the derivative of space (function) with respect to time (independent variable).

It is often convenient to give a geometric representation of a function, by plotting on the plane of the drawing the points whose coordinates are, the one the value of the independent variable x, the other the corresponding value y of the function; and thus one obtains a curve which is called the graph of the function. The function y is denoted by f(x) to recall its dependence on the variable x; hence one writes y = f(x). The independent variable x can be considered, not for all possible values, but only, for example, for those comprised between the two numbers a, b, and then one says that the function f(x) is defined only in the interval (a, b). In the same way the motion of a train is considered only in the interval of time from departure to arrival. The derivative of f(x) calculated point by point, for all the points of the interval (a, b), constitutes in turn a new function, because to every value of x comprised in that interval there corresponds a value of the derivative. This is denoted by f'(x) or also, when one wishes to bring out its origin from the incremental ratio, by df/dx or by dy/dx (infinitesimal difference of y divided by dx, the corresponding infinitesimal difference of x).

But then upon f'(x) the operation of derivation can generally be iterated, and one then obtains a new function: the second derivative, which is denoted by f''(x), and so on.

To the operation of derivation there is linked the inverse operation: indefinite integration, which consists in the search for the function F(x) which admits as its derivative a given function f(x); a search not always easy.

Let us now consider the area bounded by the graph of a function f(x), by the segment (a, b) of the x-axis comprising the points x for which the function is defined, and moreover bounded by the two perpendiculars drawn to the x-axis at the points a, b. To seek this area, first let the segment (a, b) be divided into infinitesimal parts which will be denoted generically by dx; then let the areas of all the small rectangles be summed, having as base an element dx, and as height the segment HK, which is of length f(x), if x is the measure of OH (one of the coordinates of the point K of the graph). Each small area will be given by the product f(x) . dx and therefore the sum will be represented by the customary symbol:

∈t_{b}ᵃ f(x) · dx

The calculation of this summation of infinitely many terms, each infinitesimal, is purely ideal if one considers it as the limit of a sum of a finite number of finite terms, however small. Often, however, it can be carried out by means of the indefinite integration mentioned above. And indeed, if one succeeds in finding that function F(x) which has f(x) as its derivative, then the difference F(b) - F(a), that is between the value that F assumes at b and that which it assumes at a, is precisely the value of that summation. This takes the name of definite integral; its concept was established for the first time with rigor by Mengoli (1659).

In the last century and in the early years of the present one, the possibility of performing operations of derivation and integration gave rise to a series of questions which clarified and extraordinarily extended the concept of function. Thus were born the theory of functions and the theories relating to all the infinitesimal operations applied to them (derivation, integration, differential equations, integral equations, integrodifferential equations). Only a few of the numerous host of mathematicians who have worked in these fields will be cited: Cauchy, Jacobi, Riemann, Briot Bouquet, Weierstrass, Betti, Brioschi, Casorati, Dini, Mittagheffler, Arzelà, Peano, Du Bois Reymond, Pincherle; and closer to us Volterra, Vivanti, Borel, Lebesgue, Denpois, Caratheodory, De La Vallée Poussin, Tonelli, Ricci.

Ordinary differential equations link the derivatives of one or more functions of a single variable (unknown functions) to other known functions. All the problems of the dynamics of mechanical systems, constituted by a discrete number of material points, depend on equations of this kind. These are such that, if the position of a material point is known, together with its velocity, at a given instant, and if the forces acting on the point are known, one can describe the motion that it can perform from that instant onward, and also know the motion that it has traversed before that instant. A result, this, which, in the times when the mechanism of the Universe reigned supreme, gave rise to determinism. Since the Universe is composed of mobile material particles, one can, once the position and velocity of the individual material points at a given instant are known, reconstruct the history of the past and prophesy the development of the future.

The assertion, which, within certain narrow limits, represents a fine conquest of science (so much so as to allow many astronomical phenomena to be predicted and to permit the designing of mechanisms whose actual functioning adheres to the predictions), becomes extremely proud and fallacious when it is assumed in too extended a sense. Even granting the hypotheses of classical mechanics, it is demonstrated that in some material systems there exist moments of indeterminacy (a kind of dead points such as are also encountered in trivial mechanisms) from which one can emerge without effort and in various directions.

Partial differential equations concern links between the derivatives of one (or more) unknown function, taken with respect to each of its two or more variables, and other known functions. The problems of motion with infinitely many variables, such as pertain to continuous media (fluids, elastic bodies, etc.) are connected with these equations. To them is linked the majority of the questions of mathematical physics.

In integral equations, the unknown functions are, together with known functions, constrained under the operation of integration; ordinarily the relations are very simple (linear) and then the solution of an integral equation can be carried out by approximation, for example, by reducing the equation to a system of algebraic equations of the first degree (cf. no. III). To the theory of integral equations are linked the names of Abel, Volterra, Fredholm, Schmidt, Picard, Goursat, Hilbert, Fantappiè and others.

In integrodifferential equations, studied for the first time by Volterra, the unknown function appears not only subjected to operations of integration, but also of differentiation. Such equations present themselves in physical problems of interest, which arise when materials subjected to strong stresses, if they are elastic, or to strong magnetizations, if they are paramagnetic, retain the memory of the history of the treatments undergone from the beginning.

The search for the solutions of ordinary differential equations was placed on rigorous foundations, ascertaining first of all the existence of solutions and then their constructibility, by Cauchy. Very notable studies followed throughout the last century and up to today, especially as concerns the dependence of the solutions on the so-called boundary conditions, that is, on the conditions that must be verified at the boundaries of the domain in which the unknown functions are sought. The boundary conditions sometimes cannot be given entirely at will, because they require a certain compatibility which can be ensured only when a parameter of the equation, that is, a variable that does not appear in the operations of differentiation, assumes particular numerical values called eigenvalues, to each of which generally corresponds a solution of the equation. The particular solution takes the name of eigenfunction. Immense is the interest of eigenvalues and eigenfunctions. The former, in many problems of motion that occur in mathematical physics, give the vibration frequencies of the phenomenon, and one knows how much the knowledge of these matters, when, in projects of refined engineering, it is necessary to guard against the dangerous phenomenon of resonance.

IX. SERIES EXPANSIONS

Mention has already been made of the impossibility of expressing a rational number by means of a limited number of digits and of the impossibility of expressing certain operations by means of a limited number of others. One is thus often faced with procedures without end; among these, a very important and frequent one is that of summation by series.

Add to a number its half, then the half of the half, and so on; one will have one of the simplest examples of a convergent series (in this case the sum tends toward twice the initial number). Not always, however, can the limiting result be reached in a simple way, and then one must be content to sum only the first terms of the series, generally committing an error that is the smaller the greater the number of terms actually added. There are, however, series that converge more or less rapidly toward a limit, and there also exist divergent series, that is, such that as the number of terms actually added grows, the resulting sum grows beyond every limit. There also exist indeterminate or oscillating series, so called because, as the number of added terms increases, the sum does not show any tendency to tend toward a determined limit, as, for example, happens for the series: -I + I - + I - I + ...

The concept of a numerical series (sum of infinitely many numbers) extends to literal series and therefore to series of functions. A series of functions synthesizes an infinity of numerical series, one for each point of the interval (a, b) where the functions are defined. Symbolically one will write: f₁(x) - f₂(x) + f₃(x) - f₁(x) + ... where the subscript index of the letter f designates the diversity of the functions of x that are added, and where to to every value of x there corresponds, in the indicated sum, a numerical series.

In very general terms it may be said that every operation performed on a variable x is expressible in the form of a series of functions. A polynomial of degree n, such as for example a₀ + a₁x + a₂x² + a₃x³ + a₄x⁴ + ... + aₙxⁿ, is the simplest operation that can be performed on the variable x; a₀ a₁ a₂... are fixed numbers, while x varies (the a's are the coefficients of the polynomial). If the number of terms of this polynomial, and hence its degree, increases beyond every limit, then one has the so-called power series, which was one of the first types considered for series of functions. By means of them many so-called elementary functions are expressed, such as the trigonometric functions (sine, cosine, tangent and cotangent), the logarithm, the exponential, e, etc. With polynomials, or with elementary functions, other series are composed, among which important ones are the series of Legendre, Laguerre and Hermite polynomials, the trigonometric series, which are sums of sines and cosines of multiples of an angle x, the series of Bessel, of Sturm, Liouville, etc.

The properties of series have been the object of study on the part of Frullani, Bonacci, Abel, Kummer, Riemann, Dirichlet, Dini, Peano, Cesaro and, in this century, among the most recent, Borel, Tonelli, Picone. The questions treated concern both the representability of functions by means of the various kinds of series and the nature of their convergence.

There exist series that are not directly summable, because they oscillate, but which can be used, because they are summable in the mean. That is, the sum becomes performable by grouping the terms, taking the sum for each group and finally adding these partial sums, after applying to each of them a suitable coefficient. This method is due to Cesaro; further developments and considerations are owed to Féjer and to Teofilato, who, placing himself from a very general point of view, has shown how Cesaro's method is connected to the integrals of Poisson and of Dirichlet. The study of trigonometric series and the representability of a generic function by means of them, and, as is usually said, the developability of the function in a Fourier series, was deepened by Dirichlet and subsequently extended and elaborated by Tonelli and by Picone.

From the generalization of developments in trigonometric series have come those in series of orthogonal functions, which generally are nothing other than the eigenfunctions of which mention was made above and which are of such great importance in questions of mathematical physics. With regard to orthogonal functions, the names of Schmidt, Tamarkine, Picone, Sansone are cited.

X. FUNCTIONAL ANALYSIS AND THE CALCULUS OF VARIATIONS

It arose around 1880-90 through the work of some young Italian mathematicians, who laid the foundations of a study of functions of a line, called functionals, such as, for example, the definite integral expressing the length of a curve, or the area of the diagram of the intensity of a current passing through a given circuit, considered as a function of time, each of which quantities depends simultaneously on all the values that a function assumes in a given interval.

Thus there also arose the so-called functional operations, that is, those correspondences that carry one from one functional to another, and with them there also arose a symbolic calculus, the first to occupy themselves with which were Pincherle and Amaldi. The point of view of these two authors was particularly qualitative; alongside it came another, due to Volterra, who made use of the conceptual passage from the discrete to the continuous, in order to transform the indices (that is, the ordinal number) by which the functions on which one operates are characterized, into a continuous variable (parameter). In this way, instead of speaking of a first function, second, third, etc., one will speak of a continuous infinity of functions, distinguished by the value assigned to the parameter, which ends up intervening as one more variable to be added to those already possessed by the functions. Derivation with respect to the index, already adopted by Euler and by Laplace, was thus extended by Volterra to partial derivation (cf. no. IX), to the differential of the functional, to the Taylor series expansion, thereby obtaining algorithms very useful for the treatment of integral equations. With a different view, basing himself on Dini's results concerning the theory of functions of a real variable, and basing himself on set theory, Arzelà laid the foundations of the structure of general analysis, constituted later, in recent years, by Fréchet and by others. Tonelli, in turn, occupying himself with the calculus of variations, that is, the calculus that studies the variation of a functional (such as the variation of the length of a curve that undergoes an infinitesimal deformation), introduced the concept of the semicontinuity of the functional, a fruitful concept that allowed him, among other things, to account for the success of certain procedures used previously, and that led him to new developments and applications. In this direction there are notable works by Cinquini, Caccioppoli, Cimmino.

Fantappiè has followed a new path in the study of functionals, developing in the complex field the results that Volterra had obtained in the real field. In this way Fantappiè achieved a coordination of various theorems, which was lacking in Volterra's exposition, and found brilliant correlations between properties of functions of a complex variable and functionals; lastly he applied his theory to differential equations, to integral equations and to matrices.

Finally, mention deserves to be made of the symbolic operational calculus, which, in opportune circumstances, allows one of the operations of derivation or integration to be replaced by a variable, thereby often facilitating the treatment of the equations of electrical engineering. To be remembered in this regard are the names of Heaviside, Giov. Giorgi and Aldo Ghizzetti.

XI. ALGEBRAIC GEOMETRY

The application of algebra to geometry has already been discussed in the paragraph devoted to analytic geometry. From that union algebraic geometry was also born.

In the first half of the last century, this had valiant cultivators in Italy thanks to the Neapolitan school and to Tortolini and Chelini in Rome; in England, through the work of Salmon, Cayley, Sylvester. In the second half of the nineteenth century, Battaglini continued in the English direction, while Cremona drew inspiration rather from the synthetic direction followed in Germany by Steiner and Staudt, instilling in the young mathematicians of his time a love of geometry. In numerous works he brought to light, in geometric problems, that qualitative aspect which, in years closer to our own, has taken on a functional character. To Cremona are owed the birational transformations, later called cremonian in his honor, which were studied from the point of view of groups and invariants (cf. no. IV) and constitute a generalization of projective transformations. A vast array of disciples followed in the master's footsteps; one may cite De Paolis, Caporali, Bertini, Corrado Segre, Montesano, Del Pezzo, Guccia.

Alongside the evolution of projective geometry, which was assuming an ever more general character in the Cremonian sense, and which might be called a birational character, hyperspace geometry was developing, which in Italy had cultivators in D'Ovidio, Veronese, Corrado Segre, Del Re. The concept of hyperspace corresponds to the representation of a set of geometric elements, each determined by a set of n numbers (n greater than 3), the coordinate of the element in its hyperspace, which are to be understood in a very general sense that we shall now attempt to adumbrate.

A point of the plane is identified by two numbers, which may be either its familiar Cartesian coordinates or, for example, the lengths of the radii of the two circles passing through the point under consideration and through the origin of the Cartesian axes, and having their respective centers on those axes themselves. Indeed these two aforementioned lengths are capable of permitting the construction and therefore the identification of the point. Thus the lines of a plane can each be identified by the lengths of the two segments that the line cuts off on the axes, starting from the origin. But then, in the plane, a line can be represented by a point, namely the one to which the same pair of numbers corresponds, and vice versa; and indeed there exists the so-called principle of duality, by which from all graphic theorems (in which, that is, exclusively the concept of incidence enters) concerning lines and points one deduces just as many theorems, by the simple exchange of the words: lines and point.

In ordinary space, while a point is identified, as is known, by three numbers, a line is determined by four numbers, in such a way that the lines of ordinary space can be made to correspond to the points of a four-dimensional hyperspace; and then, since projective transformations can be extended to the hyperspace of points, so from the projective properties of the points of the four-dimensional hyperspace one deduces projective properties concerning the lines of ordinary space.

The most general meaning attributed to coordinates, exploited with success from Lamé onward, in so many questions of mathematical physics, and the concept of hyperspace became fruitful in important results. Castelnuovo and Segre applied the hyperspace method to the study of linear series on a curve and sought the properties invariant with respect to Cremonian transformations. New studies arose, such as the geometry on a surface and enumerative geometry, while to the algebraic methods of research, followed until 1900, there were added transcendental methods, by means of which Severi, availing himself of the original concept of the characteristic series of a continuous system, established interesting connections between the characters of a surface and the integrals that refer to it. A numerous array of Italian mathematicians has devoted itself to the study of this connection between geometric properties and transcendental functions, so much so that one can confidently assert Italy's primacy in algebraic geometry.

now become, through the abundance of means of research and of harvest gathered, one of the most superb branches of m.

XII. DIFFERENTIAL GEOMETRY

The first origins, which go back to the 17th century, are linked to the names of Cavalieri and Torricelli; a century later the names of Euler, Monge, Dupin appear; at the beginning of the 19th century, those of Gauss and Lamé. There followed, in Italy, Chelini, Codazzi, Betti, Brioschi, Casorati and, brilliant above all others for importance and elegance of results, Beltrami; outside Italy, Riemann and later Klein made considerable contributions to the new discipline.

Differential geometry made use of the methods of infinitesimal calculus in order to study the peculiarities of lines, surfaces and hyperspaces, and more generally to direct its investigations wherever the consideration of continuity, that of elements infinitely close to one another, came into play. Thus the curvature of lines and surfaces has been studied, and moreover the applicability of one surface onto another by simple bending, without stretching, just as when the surface of a cylinder, cut along a generatrix, lies perfectly flat upon a plane without its having to be stretched, as happens if one must lay a spherical rubber cap upon a flat table. Particular lines lying on the surface and enjoying important properties have also been studied; among them, the geodesic lines or lines of shortest path, which, when the surface lies upon another in the sense described above, provided it is applicable, are superimposed upon the geodesics of the other, just as the labels of a cylinder, which on the cylinder are lines of shortest path, lie upon the straight lines of the plane onto which the cylinder is developed. These concepts, which are fundamental for cartography (whose purpose is to represent the surface of the earth upon a map), were applied by Beltrami to a particular surface, the pseudosphere, deriving from it brilliant connections with non-Euclidean geometry. In fact, the geometry constructed with the geodesics of the pseudosphere is nothing other than the geometry that holds for the straight lines of the Lobachevskian plane.

Meanwhile, Riemann took as his starting point the form assumed by the distance of two infinitely close points upon a surface, expressed by means of geometric and analytic elements referring to the surface itself, in order to break away from the Pythagorean theorem and consider the distance of two points in space or in hyperspace in a new and more general form. Thus the idea of the curvature of space and of hyperspace emerged from this, an idea that in today's theory of relativity has acquired concrete meaning (cf. no. VI).

In the study of the transformations of surfaces, those that preserve areas have then been examined, those that are conformal, which preserve angles (within infinitesimal regions), those that change geodesics into geodesics — all transformations, these, of particular interest for cartography.

The various treatments set out above belong to the classical approach; but the more modern and fruitful approach, which has allowed the development of the theory of relativity, is the one marked out by absolute calculus, invented by Ricci Curbastro and further developed by Levi Civita. This calculus studies the invariance of geometric entities with respect to the most general transformation of coordinates, understanding the latter in the broadest sense, which has already been mentioned in no. XII.

It has already been noted in no. VIII that reference to one system of axes rather than another is not indifferent, because, if the reference is connected with the question being treated, it extraordinarily simplifies its development. For this reason Cesaro, in the aforementioned Geometria intrinseca, had used an appropriate reference, which he called intrinsic and which in modern language would be called anholonomic: a system of axes and therefore a system of coordinates for each point of a line (or surface) in order to study its behavior in the infinitesimal neighborhood of the point considered.

The great success of absolute calculus in its applications to relativistic physics is due precisely to the use of an anholonomic reference, even more advantageous than that of Cesaro, because in the aforementioned applications it presents an invariant character for the entities considered.

Absolute calculus, to which the previous researches of Riemann, Beltrami, Christoffel had paved the way, asserts itself with the so-called parallel transport and the covariant derivative. Parallel transport has a very clear meaning in Euclidean space, when one transports a segment (vector) parallel to itself, but at first sight it is undefined in a curved space (cf. no. VI). However, in a curved space, the curvature makes itself little felt within a small region, in the same way as the curvature of the earth's surface is not perceived in a small mirror of water. Therefore, for every point of the curved space, one can consider an infinitesimal ordinary space, which is, as is usually said, tangent to the curved one, that is, it coincides with the latter in the infinitesimal field, just as a small region of the earth's surface can be confused with a plane. And then, within this limited ordinary space, one can carry out the transport of the vector A B, from the position A B, to the parallel position A' B', infinitely close, according to elementary Euclidean concepts. Subsequently one considers the infinitesimal ordinary space tangent to that neighborhood of the curved space that surrounds A' B', and in this second ordinary space carry out the parallel transport from A' B' to A' B', and so on.

In the same way, on the surface of the sea the displacement of the ship's longitudinal axis A B, parallel to itself by one kilometer, has a meaning. Adopting geographic coordinates (latitude and longitude) and iterating the displacement a thousand times, the ship will have undergone a parallel transport in the sense of absolute calculus, but one cannot assert that the ship's axis in the starting position is effectively parallel to that in the final position, in the Euclidean sense. In fact, considering in every position of the ship the projection of A B onto the meridian and that onto the parallel, these will not vary as long as they refer to nearby positions, but will instead vary greatly for distant positions; it suffices to think of the ship's displacement along the parallel of Naples, while, kilometer by kilometer, it orients the axis A B according to the meridian of the place, instead of keeping itself parallel to itself as it should do in parallel transport. In other words, consider the cone tangent to the terrestrial sphere along the aforementioned parallel, almost a pointed paper cone that contains, touching it, a ball representing the globe, and spread out the sheet of the cone upon a table. On the map thus spread out, draw, time by time, the image of the ship's axis in its parallel displacement. The initial position of this axis will turn out, on the table, to be parallel to the final one, but the parallelism will cease when the sheet is rolled up again onto the sphere, while the various images of the axis A B will make with the respective generatrices of the cone an angle varying from generatrix to generatrix. In short, after a thousand kilometers, the angle between the starting position and the final position will be slightly less than ten degrees.

To establish the vector sum of two vectors, one must construct the diagonal of the parallelogram formed by the two vectors transported parallel to themselves until they have a common origin, just as is done for two forces acting on a solid body when one wishes to find their resultant. One proceeds analogously for the vector difference, which in ordinary space is done by parallel transport, as when, given the resultant of two forces acting on a solid body and given one of the forces, one wishes to find the other. This operation, which in ordinary space has a meaning independent of the position that each of the two vectors occupies, in curved space is instead defined by performing parallel transport in the new sense from neighborhood to neighborhood (let one think of the ship mentioned above). When the difference is infinitesimal, one has the differential (infinitesimal increment of the vector), and when one divides this difference by the differential (infinitesimal increment) of the variable, one has the covariant derivative mentioned earlier.

These are the fundamental entities of the absolute calculus, which are then further deepened and extended, thereby endowing mathematics with a powerful instrument of investigation in the physical field.

Other very recent studies link back to these theories, due to Bortolotti (Enes), Bompiani, Cartan. But the essentially metric study of differential geometry was bound to find, from the very beginning, a point of contact with projective geometry. The connection first established by Klein in 1872 reduced, as was already mentioned in no. VII, the three geometries — those of Euclid, of Lobatchewski, of Riemann — to a single one, from the projective standpoint. These, in turn, could be synthesized with other geometries by considering projective transformations as included among Cremona transformations.

A further step was taken, between 1918 and 1924, by Weyl and by Cartan, with the consideration of space with affine connection, or with conformal connection, or with projective connection. A space is thus constructed, step by step, in such a way that every infinitesimal spatial element can be referred to its neighbor while respecting affinity, or conformity, or projectivity.

In another direction, and precisely in the deepening and extension of analysis situs, glimpsed by Clifford in the last century, topology has been developing, which, wedded to differential geometry, has been the subject of very notable studies by Severi and by Blaschke. The interweaving of the various branches mentioned in no. I is more than ever confirmed, in the interference of differential geometry with the other mathematical disciplines.

XIII. CALCULATION OF PROBABILITIES

Born between 1600 and 1700 with Buffon, Pascal, and elaborated shortly afterward by Bernoulli, Euler, D'Alembert, Condorcet, it was brought to completion by Laplace and, in the first half of the nineteenth century, by Gauss and Poisson; notable contributions and clarifications were later made to it by Bertrand, Tchebichef, Kolmogorov.

The calculus of probabilities aims to determine the laws of chance. The intention is at first sight paradoxical, inasmuch as what is truly fortuitous, the moment it leaves room for predictions, loses that essence of uncertainty with which it is clothed. But what is there that is truly fortuitous? In reality there are present numerous physical causes, imponderable or unweighed, which make the event oscillate about a mean, interfering with one another, now adding together and now canceling each other out; among the causes, however, as was already mentioned apropos of mechanistic determinism, there must also be included the moral ones, which are of indisputable value and capable of directing the event in the most unforeseen direction, or, as it is said in probabilistic language, the least probable.

Chance refers to physical causes and is governed by two laws, which make it less fortuitous: the law of large numbers and the law of deviations (or of errors); the first due to Bernoulli, Poisson, the second to Gauss.

The first, for example, ensures that, given an urn containing ninety balls all alike, differing only in color, because 89 are white and only one is red, the number of times k that the red ball comes out, following n drawings carried out by each time putting the ball drawn back into the urn, is such that the fraction k/n turns out the closer to 1/90 the greater the number n of draws.

Conversely, if over n draws it is found that the ratio k/n oscillates about 1/90, approaching 1/90 ever more closely as n increases, one may infer that among the 90 balls only one is red, and with a degree of certainty the greater the greater n is. If this circumstance does not occur, it is a sign that among the balls there is some other difference besides that of color. On the other hand, when one goes to measure a magnitude, whatever the precision of the instruments, one finds each time a different value, with deviations the smaller the more perfect the instrument. Well then, the second law states that while the mean of the measurements obtained is the most probable value of the measurement of the magnitude, the deviations from this mean value vary according to the equation y = k · a^{-α²}, where α is the deviation and y the frequency with which it occurs; the base a of the exponential function and the coefficient k depend on the precision of the measurements.

On these foundations, the theory of probabilities has undergone notable developments and applications; from the theory of measurement errors, which offers the surest way to compensate for them, to the theory of betting which studies the fairness of games of chance, the so-called mathematical expectation and risk, thus governing the determination of insurance premiums; from the deduction of natural laws as they manifest themselves in a group of experiments, to the measurement of aggregates, whether they are constituted by physical, biological, demographic, or social elements.

Moreover, even classical physics, at first wholly deterministic, has little by little slipped somewhat into probabilism, when it enunciated the second principle of thermodynamics, or principle of the degradation of energy, according to which nature evolves toward ever more probable forms. Thus, for example, a transparent bottle which contains in its lower half sand colored blue, and in the upper half the same quality of sand, but colored red, if one shakes it to mix the grains of the two colors, will end after no long time by revealing a uniform gray color, so that, taking out a small quantity of sand, which nevertheless contains a great number of grains, one will find that the number of red grains is roughly equal to that of the blue ones, with a deviation the smaller the greater the number of grains taken out. It seems impossible that, continuing to shake the bottle, the separation of the grains of the two colors into the original arrangement could be obtained again; nevertheless this result is not logically impossible, but is as little probable as it would be to reproduce a sonnet of Dante by drawing the words of a vocabulary at random.

The restoration is precisely to be attributed to an intelligence which guides, or which arouses suitable forces to guide; all the more so if, as in the miracles reported by the sacred books and by uninterrupted traditions, the restoration to the initial conditions occurs after a dissolution of an organism has taken place, at the command of a word.

In modern physics, the impossibility of being able to discern the becoming of a single subatomic corpuscle obliges one absolutely to have recourse to probabilistic criteria, just as one would do for a crowd that one is forced to observe from afar; being unable to follow the action of a single individual, one tries to follow the group to which he belongs. With this difference, however, that while in the case of the crowd the judgment is passed on the group for lack of suitable instruments of observation, on the other hand, for the subatomic corpuscle, the judgment is bound, besides to the experimental impossibility, also, at least for now, to a theoretical impossibility; so that in the ultimate elements of matter indeterminism would reign and upon this the calculus of probabilities. Indeterminism has been said; indeterminacy would have been better said, so as not to suggest an ontological indeterminism, which is pure chance, that is, the negation of any law (which instead is found to exist even in chance) or is an animism of matter, which is likewise to be discarded, except insofar as it is connected to vital phenomena. Of this finalistic activity of matter, Fantappiè has attempted to give a possible mathematical interpretation, by means of the so-called accelerated potentials.

XIV. RATIONAL MECHANICS AND MATHEMATICAL PHYSICS

These two important branches of mathematics have previously had frequent occasion to be mentioned, but a brief further mention concerning their developments will not be superfluous.

From Archimedes, who may be considered the founder of the two disciplines, one must go back through the centuries to Galileo, Torricelli and Newton, to see their luxuriant development begin with continuous contributions, to which are linked the names of Huygens, Bernoulli, Euler, D'Alembert, Laplace, to arrive at the classical systematization of rational mechanics accomplished by Lagrange, in his celebrated Mécanique analitique, where all questions of equilibrium and motion are made to descend from two principles: that of virtual work, which goes back to Galileo, and that of the equilibrium of lost forces, due to D'Alembert.

Along with the mechanics of solids, that of fluids also progressed, while new and fruitful general principles were being discovered and, with ever more powerful mathematical means, all the fields of physics were being investigated.

To the general principles and methods are linked the names of Jacobi, Poisson, Hamilton and Hölder; to the classical theory of electromagnetism: Ampère, Maxwell, Kirchhoff; to the modern (relativistic) one: Eisenberg, Schrödinger, De Broglie; to the theory of elasticity: Betti, Volterra, Kelvin, Somigliana, Levi Civita; to fluid dynamics: Euler, Lagrange, Laplace, Kelvin, Helmholtz, Kirchhoff, Levi Civita, Volterra, Prandtl, Cisotti.

The calculation of planar elastic structures (frameworks) has had particular developments in Italy, with general and elegant methods, due to Cremona, Castigliano, Betti, Menabrea. The structures of aircraft, moreover, in which lightness is in perpetual conflict with strength, have given occasion, because of their complication, for important recent works by Broglie, who, having clarified and extended the principles of maximum or minimum work, has found new and ready methods of calculation for three-dimensional structures, hitherto of difficult approach.

From rational mechanics there has then germinated a new and important branch, that of aerodynamics, which studies the stresses exerted by a fluid stream upon a solid immersed in it. Having arisen with the disconcerting Euler-D'Alembert paradox which demonstrated the absence of a resistance to the advance of a solid in the fluid, aerodynamics has had extraordinary developments in the last forty years through the work of Prandtl, Levi Civita, Karman, Lagally, Müller, Pistolesi, Oosen, Cisotti, Teofilato, Silla, Panetti, Ferrari.

The study of the streams of compressible and viscous fluids, of transonic and supersonic streams, still forms the object of numerous researches. Ultrasonic velocities, intimately connected with the principles of thermodynamics, in fact constitute a very new branch: gas dynamics, cultivated in Italy by Ferrari, Luigi Crocco, Teofilato; it intertwines with classical ballistics, to which are linked the names of Euler, Lagrange, Siacci, Signorini, and with superballistics, to which Gaetano Crocco dedicates his fervent activity.

All the equations that govern the various physical phenomena can be made to descend from principles of minimum which in substance reduce to what Maupertuis called the law of the minimum means, almost a law of economy of nature, which is perhaps also the foundation of the gradual passage of one species into another and which gives an appearance to evolution, and is perhaps also the reason for the sweetness of a harmony or the beauty of an architecture, universal - See plate XXVI.

Pietro Teofilato