Archimedes in Boyer’s History of the Calculus, 1, counterbalanced parabola; 2, parabola exhaustion

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Carl Boyer

The History of the Calculus and Its Conceptual Development (The Concept of the Calculus)

Chapter 1

Selection, first two Archimedes demonstrations (pp. 48-53)

[The following is quotation, except for material between double brackets]

The greatest mathematician of antiquity, Archimedes of Syracuse, displayed two natures, for he tempered the strong transcendental imagination of Plato with the meticulously correct procedure of Euclid. He "gave birth to the calculus of the infinite conceived and brought to perfection successively by Kepler, Cavalieri, Fermat, Leibniz, and Newton,"¹³⁴  [ft 134: Chasles, A perçu historique sur l’origine et le développement des méthods en géométrie, p.22.] and so made the concepts of the derivative and the integral possible. In the demonstration of his results, however, he adhered to the clearly visualized details of the Eudoxian procedure, modifying the method of exhaustion by considering not only the inscribed figure but the circumscribed figure as well. The deductive method of exhaustion was not a tool well adapted to the discovery of new results, but Archimedes combined it with infinitesimal considerations toward which Democritus and the Platonic school had groped. The freedom with which he handled these is shown most clearly in the treatise to which we have already referred, the Method.¹³⁵ [ft. 135: For the works of Archimedes in general, see Heiberg, Archimedis opera omnia and T. L. Heath, The Works of Archimedes. For Archimedes' Method, see T. L. Heath, The Method of Archimedes, Recently Discovered by Heiberg; Heiberg and Zeuthen, "Eine neue Schrift des Archimedes"; and Smith, "A Newly Discovered Treatise of Archimedes.'']

This work, addressed to Eratosthenes the geographer, astronomer, and mathematician of Alexandria, was lost and remained largely un- | -known until rediscovered in 1906. (Boyer 48|49)  In it Archimedes disclosed the method which is presumably that which he employed in reaching many of his conclusions in problems involving areas and volumes. Realizing that it is advantageous to have a preliminary notion of the result before carrying through a deductive geometrical demonstration, Archimedes employed for this purpose, in conjunction with his law of the lever, the idea of a surface as made up of lines. For example, he showed that the truth of the proposition that a parabolic segment is 4/3 the triangle having the same base and vertex (the vertex of the segment being taken as the point from which the perpendicular to the base is greatest) is indicated by the following considerations from mechanics.¹³⁶ [Ft 136: See T. L. Heath, The Method of Archimedes, Proposition I, pp. 15-18.]

In the diagram given in Figure 2 [[shown above]], in which V is the vertex of the parabola,

BC is tangent at B,

BD = DP,

and X is any point on AB,

we know from the properties of the parabola that for any position of X we have the ratio

But X" is the center of gravity of XX"',

so that from the law of the lever we see that XX', if brought to P as its midpoint, will balance XX"' in its present position.

This will be true for all positions of X  on AB.

Inasmuch as the triangle ABC consists of the straight lines XX"' in this triangle, and since the parabolic segment AVB is likewise made | up of the lines XX', we can conclude that the triangle ABC in its present position will be in equilibrium at D with the parabolic segment when this is transferred to P as its center of gravity. (Boyer 49|50)

But the center of gravity of ABC is on BD and is 1/3 the distance from D to B, so that the segment AVB is the triangle ABC, or 4/3 the triangle AVB.

This method of Archimedes indicates an anticipation of the use of the concept of the indivisible which was to be made in the fourteenth century and which, when developed again more freely in the seventeenth century, was to lead directly to the procedures of the calculus. The basis of the method is to be found in the assumption of Archimedes that surfaces may be regarded as consisting of lines. We do not know in precisely what sense he intended this to be understood, for he did not speak of the number of elements in each figure as infinite, but said rather that the figure is made up of all the elements in it. That he probably thought of them as mathematical atoms is indicated not only by this manner of expression, but also by the highly suggestive fact that he was led to many new results by a process of balancing, in thought, elements of dissimilar figures, using the principle of the lever precisely as one would in weighing mechanically a collection of thin laminae or material strips.

Using this heuristic method, Archimedes was able to anticipate the integral calculus in achieving a number of remarkable results. He discovered, among other things, the volumes of segments of conoids and cylindrical wedges and the centers of gravity of the semicircle, of parabolic segments, and of segments of a sphere and a paraboloid.¹³⁷ [ft 137: See The Works of Archimedes, Chap. VII, "Anticipations by Archimedes of the Integral Calculus"; also Method.] However, to assert that here "for the first time one can correctly speak of an integration"¹³⁸ is to misinterpret the mathematical process known by this name. [ft 138: Hoppe, "Zur Geschichte der Infinitesimalrechnung," p. 154; cf. also p. 155.] The definite integral is defined in mathematics as the limit of an infinite sequence and not as the sum of an infinite number of points, lines, or surfaces.¹³⁹ [ft 139: T. L. Heath (The Method of Archimedes, pp. 8-9) correctly points out that the method here used is not integration; but he gratuitously imputes to Archimedes the concept of the difierential of area.] Infinitesimal considerations, similar to those in the M etlzod, were at a later period to furnish perhaps the strongest incentive to the development of the calculus, but, | as Archimedes realized, they lacked in his time all basis in rigorous thought. (Boyer 50 | 51). This they continued to do until the concepts of variability and limit had been carefully analyzed. For this reason Archimedes considered that this method merely indicated, but did not prove, that the result is correct.¹⁴⁰ [ft. 140:   T. L. Heath, History of Greek Mathematics, II, 29.]

Archimedes employed his heuristic method, therefore, simply as an investigation preliminary to the rigorous demonstration by the method of exhaustion, It was not a generous gesture that led Archimedes to supplement his "mechanical method" by a proof of the results in the rigorous manner of the method of exhaustion; it was, rather, a mathematical necessity. It has been asserted that Archimedes' method "would be quite rigorous enough for us today, although it did not satisfy Archimedes himself."141 [ft 141: T. L. Heath, The Method of Archimedes, p. 10.] Such an assertion is strictly correct only if we ascribe to him our modern doctrines on number, limit, and continuity. This ascription is hardly warranted, inasmuch as Greek geometry was concerned with form rather than with variation. It was, as a result, necessarily unable to frame a satisfactory definition of the infinitesimal, which of necessity was to be regarded as a fixed quantity rather than as an auxiliary variable. Archimedes was probably well aware of the lack of any sound basis for his method and for this reason recast all of his analysis by infinitesimals in the orthodox synthetic form, much as Newton was to do almost nineteen hundred years later after the methods of the calculus had been discovered but still lacked adequate foundation.

[[Demonstration 2]]

The suggestive analysis of the problem of determining the area of a parabolic segment had been given by Archimedes in the Method. However, formal proofs (both mechanical and geometrical) of the proposition were carried out by the method of exhaustion in another treatise, the Quadrature of the Parabola [[see especially propositions 23-24]].¹⁴² [ft 142: See The Works of Archimedes, and T. L. Heath, History of Greek Mathematics, II, 85- 91. A good adaptation of the geometrical proof is given in Smith, History of Mathematics, II, 680-83.] In these proofs Archimedes followed his illustrious predecessors in omitting all reference to the infinite and the infinitesimal. In the geometrical demonstration, for example, he inscribed within the parabolic segment a triangle of area A , having the same base and vertex as the segment. Then within each | of the two smaller segments having the sides of the triangle as bases, he similarly inscribed triangles. Continuing this process, he obtained a series of polygons with an ever-greater number of sides, as illustrated (fig. 3).

He then demonstrated that the area of the nth such polygon was given by the series

where A is the area of the inscribed triangle having the same vertex and base as the segment. The sum to infinity of this series is 4/3A, and it was probably from this fact that Archimedes inferred that the area of the parabolic segment was also 4/3A.¹⁴³ [ft. 143: T. L. Heath, "Greek Geometry with Special Reference to lnfinitesimals."]

However, he did not state the argument in this manner. Instead of finding the limit of the infinite series, he found the sum of n terms and added the remainder, using the equality

As the number of terms becomes greater, the series thus "exhausts" 4/3A only in the Greek sense that the remainder,

can be made as small as desired. This is, of course, exactly the method of proof for the existence of a limit,¹⁴⁴ but Archimedes did not so interpret the argument. [ft.  144 As Miller pointed out in "Some Fundamental Discoveries in Mathematics."] He did not express the idea that there is no remainder in the limit, or that the infinite series is rigorously equal to 4/3A .145 [ft. 146: The Works of Archimedes, p. cxliii.]] Instead, he proved, by the double reductio ad absurdum of the method of exhaustion, that the area of the parabolic segment could be neither greater nor less than4/3A. In order to be able to define 4/3A as the sum of the infinite series, it would have been necessary to develop the general concept of real number. Greek mathematicians did not possess this, so that for them there was always a gap between the real (finite) and the ideal (infinite).

(Boyer, 52)

Above quotation and unmodified images from:

Boyer, Carl. The History of the Calculus and Its Conceptual Development (The Concept of the Calculus). New York: Dover, 1949.