Remark 3: Further Forms Connected With the Qualitative Determinateness of Magnitude

§ 652

The infinitesimal of the differential calculus is, in its affirmative meaning, the qualitative determinateness of magnitude; and, more precisely, that it is present in the calculus not merely as a power determinateness in general but specifically as the relation of a power function to the power derived from the expansion of the function.

But the qualitative determinateness is also present in another, so to speak, weaker form and this form, together with the use and the meaning of the infinitely small in this connection, are to be the subject matter of this Remark.

§ 653

From the analytical side, the different power determinations appear in the first place as only formal and quite homogeneous, that they signify numerical magnitudes which as such do not possess that qualitative difference from each other.

But in the application to spatial objects, the qualitative determinateness of the analytic relation is fully manifested as the transition from linear to planar determinations, from determinations of straight lines to those of curves, and so on. This application further involves that spatial objects, which by their nature are given in the form of continuous magnitudes, are taken to be discrete, the plane therefore as a multitude of lines, the line as a multitude of points, and so on.

§ 654

The sole interest of this procedure is to determine the points and the lines themselves into which the lines and planes respectively have been resolved, in order that from such determination, progress can be made analytically, that is, strictly speaking, arithmetically; these starting points for the required magnitudes are the elements from which the function and equation for the concrete, that is, the continuous magnitude, is to be derived. For the problems where the employment of this procedure is chiefly indicated, it is requisite that the element forming the starting point should be self-determined — in contrast to the indirect method which, on the contrary, can begin only with limits between which lies the self-determined element as the goal towards which the method advances. But then the result in both methods comes to the same thing if what can be found is only the law for progressively determining the required magnitude without the possibility of reaching the perfect, that is, so-called finite, determination demanded. To Kepler is ascribed the honour of first having thought of this reversal of the process and of having made the discrete the starting point. His explanation of how he understands the first proposition in Archimedes’ cyclometry expresses this quite simply. Archimedes’ first proposition, as we know, is that a circle is equal to a right-angled triangle having one of the sides enclosing the right angle equal to the diameter and the other to the circumference of the circle. Now Kepler takes the meaning of this proposition to be that the circumference of the circle has as many parts as it has points, that is, an infinite number, each of which can be regarded as the base of an isosceles triangle, and so on; he thus gives expression to the resolution of the continuous into the form of the discrete. The expression ‘infinite’ which occurs here is still far removed from the definition it is supposed to have in the differential calculus. When now a determinateness a function, has been found for such discrete elements, they are supposed to be summed up, to be essentially elements of the continuous. But since a sum of points does not make a line, or a sum of lines a plane, the points are already directly taken as linear and the lines as planar. But because these linear elements are at the same time supposed not to be lines, which they would be if they were taken as quantum, they are represented as being infinitely small. What is discrete can only be externally summed up, the moments of such sum retaining the meaning of the discrete one (unit); the analytic transition from these ones is made only to their sum and is not simultaneously the geometrical transition from the point to the line or from the line to the plane and so on; therefore the element which is determined as point or line is at the same time also given the quality of being linear or planar respectively in order that the sum,, as a sum of little lines, may become a line, or as a sum of little planes may become a plane.

§ 655

It is the need to acquire this moment of qualitative transition and to have recourse for this purpose to the infinitely small, which must be regarded as the source of all the conceptions which, though they are meant to resolve this difficulty, constitute in themselves the greatest difficulty. Before one could dispense with this expedient, it would have to be possible to show that the analytic procedure itself which appears as a mere summation, in fact already contains a multiplication. But this involves a fresh assumption which forms the basis in this application of arithmetical relations to geometrical figures: the assumption, namely, that arithmetical multiplication is also for the geometrical determination a transition into a higher dimension — that the arithmetical multiplication of magnitudes spatially determined as lines also produces a plane from the linear determination; three times four linear feet gives twelve linear feet, but three linear feet times four linear feet gives twelve superficial feet and, in fact, square feet, since the unit in both factors as discrete quantities is the same. The multiplication of lines by lines at first sight appears meaningless in so far as multiplication concerns simply numbers, i.e. is an alteration of a subject matter that is perfectly homogeneous with what it passes over into, with the product, only the magnitude being altered. On the other hand, what was called multiplication of a line as such by a line — it has been called ductus lineae in lineam, like plant in planum, and is also ductus puncti in lineam — is an alteration not merely of magnitude, but of magnitude as a qualitative determination of spatial character, of a dimension; the transition of the line into a plane must be understood as the self-externalisation of the line, and similarly the self-externalisation of the point is a line, and of the plane a whole space [volume]. This is the same as the representation of the line as the motion of a point, and so forth; but motion includes a determination of time and thus appears in this representation rather as merely a contingent, external alteration of state. The transition must be grasped from the standpoint of the Notion which was expressed as a self-externalisation — the qualitative alteration which arithmetically is the multiplication of unit (point, etc.) by amount (line, etc.). We may here further remark that with the self-externalisation of the plane, which would appear as a multiplication of the plane by a plane, there is seemingly a difference between the arithmetical and geometrical operations such that the self-externalisation of the plane as ductus plani in planum would give arithmetically a multiplication of a two-dimensional factor by another such, and consequently a four-dimensional product which, however, is reduced by the geometrical determination to three. Although on the one hand number, because its principle is the one, yields the fixed determination for the external, quantitative element, yet equally, the result of operating with it is formal. Taken as a numerical determination 3 x 3 when it reproduces itself is 3 x 3 x 3 x 3; but this same magnitude as a plane, when it reproduces itself, is restricted to 3 x 3 x 3, because space, represented as an expansion outwards from the point, from the merely abstract limit, has its true limit as a concrete determinateness beyond the line in the third dimension. The difference referred to could prove itself effective as regards free motion in which one side, the spatial side has a geometrical significance (in Kepler’s law, s3 : t2) and the other, the temporal side, is an arithmetical determination.

§ 656

It will now be evident, without further comment, how the qualitative element here considered differs from the subject of the previous Remark. There, the qualitative element lay in the determinateness of power; here, like the infinitely small, it is only the factor as arithmetically related to the product, or as the point to the line or the line to the plane, and so on. Now the qualitative transition which has to be made from the discrete (into which continuous magnitude is imagined to be resolved), to t e continuous, is effected as a process o summation.

§ 657

But that the alleged pure summation does in fact include a multiplication and therefore the transition from linear to planar dimensions, this comes to view most simply in the way in which, for example, it is shown that the area of a trapezium is equal to the product of the sum of the two opposite parallel lines and half the height. This height is represented as being merely the amount of a multitude of discrete magnitudes which must be summed up. These magnitudes are lines which lie parallel between the said limiting parallels; there are infinitely many of them for they are supposed to constitute the plane, and yet are lines which therefore, in order to possess the character of a plane, must at the same time be posited with negation. In order to escape the difficulty that a sum of lines is supposed to give a plane, the lines are directly assumed to be planes but also as infinitely narrow, for they have their determination solely in the linear quality of the parallel limits of the trapezium. As parallel and bounded by the other pair of rectilinear sides of the trapezium, these lines can be represented as the terms of an arithmetical progression, having a simply uniform difference which does not, however, require to be determined, and whose first and last terms are these two parallel lines; as we know, the sum of such a series is the product of the parallels and half the amount or number of terms. This last quantum is called amount or number simply and solely with reference to the conception of infinitely many lines; it is simply the specific magnitude of something which is continuous — the height. It is clear that what is called a sum is at the same time a ductus lineae in lineam, a multiplication of lines by lines, and so according to the above determination the result is something having the quality of a plane. In the simplest case of any rectangle AB, each of the two factors is a simple magnitude; but even in the further, still elementary example of the trapezium, only one of the factors is simple as half of the height. The other, on the contrary, is determined by a progression; it is also linear but its specific magnitude is more complex; and since it can be expressed only by a series, the problem of summing it is called analytical, i.e. arithmetical; but the geometrical moment in it is multiplication, the qualitative element of the transition from the dimension of line to that of plane; the one factor was taken to be discrete only for the arithmetical determination of the other; by itself it is, like the other, the magnitude of a line.

§ 658

But the method of representing planes as sums of lines is also often employed when multiplication as such is not used to produce the result. This happens when the problem is to indicate the magnitude in the equation not as a quantum but as a proportion.

§ 659

It is, for example, a familiar way of showing that the area of a circle bears the same proportion to the area of an ellipse, the major axis of which is the diameter of the circle, as the major axis does to the minor axis, each of these areas being taken as the sum of the relative ordinates; each ordinate of the ellipse is to the corresponding ordinate of the circle as is the minor to the major axis; therefore, it is concluded, the sums of the ordinates, i.e. the areas, are also in the same proportion. Those who want to avoid here the representation of an area as a sum of lines resort to the usual, quite unnecessary expedient of making the ordinates into trapezia of infinitely small breadth; since the equation is only a proportion, only one of the two linear elements of the plane comes into the comparison. The other, the abscissa axis, is assumed as equal in ellipse and circle, as a factor therefore arithmetically equal to i, and consequently the proportion depends solely on the relation of the one determining moment. The two dimensions are necessary to the representation of a plane, but the quantitative determination required to be indicated in this proportion affects only the one moment; to be swayed by the representation of a plane, or to help it out by adding the idea of sum to this one moment, is really to fail to recognise the essential mathematical element here involved.

§ 660

The foregoing exposition also contains the criterion for Cavalieri’s method of indivisibles referred to above which equally is justified by it and does not need to be helped out by the infinitely small. These indivisibles are lines when he is considering a plane, and squares or plane circles when he is considering a pyramid or a cone, etc. The base line or basic plane which is assumed as determined, he calls the regula; it is the constant, and with reference to a series it is its first or last term. The indivisibles are regarded as parallel with the regula and therefore as having the same determination as this with respect to the figure. Now Cavalieri’s general fundamental proposition is ’that all figures, both plane and solid, are proportionate to all their indivisibles, these being compared with each other collectively and, if there is a common proportion in the figures, distributively’.’ For this purpose, he compares in figures of the same base and height, the proportions between lines drawn parallel to the base and equidistant from it; all such lines in a figure have one and the same determination and constitute its whole content. In this way, too, Cavalieri proves for example the elementary proposition that parallelograms of equal height are proportional to their bases; any two lines drawn equidistant from and parallel to the base in both figures are in the same proportion as the base lines and so therefore are the whole figures. The lines do not in fact constitute the whole content of the figure as continuous, but only the content in so far as it is to be arithmetically determined; it is the line which is the element of the content and through it alone must be grasped the specific nature of the figure.

§ 661

This leads us to reflect on the difference which exists with respect to that feature into which the determinateness of a figure falls; this is either the height, as here, or an external limit. Where this determinateness is an external limit, it is admitted that the continuity of the figure, so to speak, follows upon the equality or the proportion of the limit; for example, the equality of figures which coincide follows from the fact that their boundaries coincide. But in parallelograms of equal height and base, only the latter determinateness is an external limit; the height (not the parallelism as such), on which is based the second main determination of the figures, their proportion, introduces a second principle of determination additional to the external limits. Euclid’s proof of the equality of parallelograms having the same height and base reduces them to triangles, to continuous figures limited externally; in Cavalieri’s proof, primarily in that of the proportionality of parallelograms, the limit is simply a quantitative determinateness as such, which is explicated in every pair of lines drawn at the same distance from each other in both figures. These lines, which are equal or in an equal ratio with the base, taken collectively give figures standing in the same ratio. The conception of an aggregate of lines is incompatible with the continuity of the figure; the essential determinateness is completely exhausted by a consideration of the lines alone. Cavalieri frequently answers the objection that the conception of indivisibles involves the comparison of lines or planes as if they were infinite in amount;’ he makes the correct distinction that he does not compare their amount, which we do not know (and which is, as we have remarked, merely an empty idea assumed in support of the theory), but only the magnitude, i.e. the quantitative determinateness as such which is equal to the space occupied by these lines; because this space is enclosed within limits, its magnitude too is enclosed within the same limits; the continuous figure is nothing other than the indivisibles themselves, he says; if it were something apart from them it would not be comparable; but it would be absurd to say that bounded continuous figures were not comparable with each other.

§ 662

It is evident that Cavalieri means to distinguish what belongs to the outer existence of the continuous figure from what constitutes its determinateness; it is the latter alone to which we must attend when comparing the continuous or constructing theorems about it. The categories he employs in this connection, namely that the continuous is composed or consists of indivisibles, and the like, are of course inadequate since they demand at the same time the intuition of the continuous or, as already said, its outer existence; instead of saying that ’the continuous is nothing other than the indivisibles themselves’, it would be more correct and also directly self-explanatory to say that the quantitative determinateness of the continuous is none other than that of the indivisibles themselves. Calvalieri does not support the erroneous conclusion that there are greater and lesser infinites which is drawn by the schools from the idea that the indivisibles constitute the continuous; further on’ he gives expression to his quite definite awareness that his method of proof by no means forces on him the idea that the continuous is composed of indivisibles; continuous figures follow only the proportion of the indivisibles. He says he has taken the aggregate of indivisibles not as an apparent infinity for the sake of an infinite number of lines or planes, but in so far as they possess a specific kind of limitedness. But then, in order to remove this stumbling block, he does not spare himself the trouble of proving the principal proportions of his geometry (in the seventh book specially added for this purpose) in a way which remains free from any admixture of infinity. This method reduces the proofs to the usual form of the coincidence of figures above mentioned, i.e. as remarked, to the conception of determinateness as an external spatial limit.

§ 663

About this form of coincidence, we may add that it is, on the whole, a so to speak childish aid for sense perception. In the elementary theorems about triangles, two triangles are represented side by side; of their six component parts, three are assumed equal to the corresponding three of the other triangle and it is then shown that such triangles are congruent, i.e. that the remaining three parts of each triangle are also equal to those of the other triangle — because by virtue of the equality of the first three parts, the triangles coincide. If we take the matter more abstractly, then it is just because of this equality of each pair of corresponding parts in both triangles that there is only one triangle before us; in this it is assumed that three parts are already determined and from this follows the determinateness of the three remaining parts. In this way, the determinateness is exhibited as completed in the three parts; hence for the determinateness as such, the three remaining parts are a superfluity, the superfluity of sensuous existence, i.e. of the intuition of continuity. Expressed in this form, the qualitative determinateness stands out in its distinction from what is given in intuition, from the whole as continuous within itself; the form of coincidence does not bring this distinction to notice.

§ 664

With parallel lines and with parallelograms there enters, as we have observed, another factor, partly the equality of the angles only and partly the height of the figures, from which latter their external limits, the sides of the parallelograms, are distinct. This gives rise to uncertainty whether in these figures, besides the determinateness of one side, the base, which is an external limit, we are to take for the other determinateness, the other external limit, namely the other side of the parallelogram, or else the height. In the case of two such figures having the same base and height, one of them being rectangular and the other having very acute angles (the opposite angles therefore being very obtuse), the latter triangle can easily look greater than the former, when its long side is taken as the determinant and, as in Cavalieri’s method, the planes are compared according to the aggregate of parallel lines intersecting them; the longer side can be regarded as a potentiality of more lines than is given by the vertical side of the rectangle. Such a conception, however, is no argument against Cavalieri’s method; the aggregate of parallel lines imagined in the two parallelograms, for the purpose of comparison, also presupposes the equidistance of the lines from each other or from the base; from which it follows that the height, and not the other side of the parallelogram, is the other determining moment. But the case is different again when the comparison is between two parallelograms having the same height and base but not lying in the same plane and making different angles with a third plane; here, the parallel sections which arise, when the third plane is imagined as cutting through the parallelograms and moving parallel to itself are no longer equidistant from each other and the two planes are unequal. Cavalieri is very careful to draw attention to this distinction which he defines as the difference between a transitus rectus and a transitus obliquus of the indivisibles and thus prevents a superficial misunderstanding which could arise on this point.’ I remember that an objection to indivisibles by Tacquet, an acute geometer who was also working at this time on new methods, concerns this very point: it is referred to in the above-mentioned work of Barrow who also uses the method of indivisibles, although with him it is tainted with the assumption (which he passed on to his pupil Newton and other contemporary mathematicians including Leibniz, too), that a curvilinear triangle, like the so-called characteristic triangle, may be equated with a rectilinear triangle if both are infinitely, that is, very small.

§ 665

The difficulty raised by Tacquet likewise concerns the question which line, in the calculation of conical and spherical surfaces, should be taken as the basis of determination in the method based on the employment of the discrete. Tacquet’s objection to the method of indivisibles is that in the calculation of the surface of a right-angled cone, this atomistic method represents the triangle of the cone to be composed of straight lines parallel to the base and perpendicular to the axis, which are at the same time radii of the circles of which the surface of the cone consists. Now if this surface is defined as a sum of the circumferences and this sum is determined from the number of their radii, i.e. from the length of the axis or the height of the cone, then, says Tacquet, such a result clashes with the truth formerly taught and demonstrated by Archimedes. Now Barrow counters this by showing that to determine the surface it is not the axis but the side of the triangle of the cone which must be taken as the line the revolution of which generates the surface; consequently, it is this line and not the axis which must be assumed as the specific magnitude for the aggregate of the circumferences.

§ 666

Objections and uncertainties of this kind have their origin solely in the indefinite idea employed of the infinite aggregate of points of which the line (or of lines of which the plane) is supposed to consist; this idea obscures the essential determinateness of the magnitude of the lines or planes.

§ 667

The intention of these Remarks has been to bring to notice the affirmative meanings which, in the various applications of the infinitely small in mathematics, remain so to speak in the background, and to lift them out of the nebulosity in which that category, merely negatively held, has concealed them. In infinite series, as in Archimedes’ cyclometry, the meaning of the infinite is nothing more than this: that the law determining the series being known, but the so-called finite — i.e. arithmetical — expression not being given, the reduction of the arc to the straight line cannot be effected; this incommensurability is their qualitative difference. The qualitative difference between the discrete and the continuous generally, equally contains a negative determination which makes them appear as incommensurable and introduces the infinite in the sense that the continuous which is to be taken as discrete, is no longer to possess, as continuous, any quantum. The continuous which is to be taken arithmetically as a product, is therefore posited as in its own self discrete, i.e. it is analysed into the elements which are its factors and in these lies its quantitative determinateness; just because they are these factors or elements, they are of a lower dimension, and as powers are concerned, are of a lower power than the magnitude of which they are the elements or factors. This difference appears arithmetically as a purely quantitative one, that of the root and power, or whatever degree of powers it may be; however, if the expression is to be taken only quantitatively, for example, a : a2 or d.a2 = 2a : a2 = 2 : a, or for the law of descent of a falling body, t : a2, then it yields the meaningless ratios of 1 : a, 2 : a, 1 : at; in supersession of their merely quantitative aspect, the sides would have to be held apart by their different qualitative significance, as s = at2, the magnitude in this way being expressed as a quality, as a function of the magnitude of another quality.

§ 668

Here then, we are faced merely with a quantitative determinateness; there is no difficulty in operating with this in accordance with its own manner and no objection can be offered if the magnitude of one line is multiplied by the magnitude of another; but the multiplication of these same magnitudes at the same time results in the qualitative alteration of the transition from line into plane; and to that extent a negative determination comes into play. It is this that occasions the difficulty, a difficulty which is resolved by an insight into its peculiarity and into the simple nature of the matter; but the introduction of the infinite which is meant to remove the difficulty only serves to aggravate it and prevent its solution.