1.7 Inﬁnitesimals in Greek geometry?
Maybe it was Antiphon, a Greek mathematician and contemporary of Socrates, who for the ﬁrst time contemplated the existence of inﬁnitesimals. According to Heath  he, Antiphon, stated that, in Heath’s words:
“If one inscribed any regular polygon, say a square, in a circle, then inscribed an octagon by constructing isosceles triangles in the four segments, then inscribed isosceles triangles in the remaining eight segments, and so on ‘until the whole area of the circle was by this means exhausted, a polygon would thus be inscribed whose sides, in consequence of their smallness, would coincide with the circumference of the circle’.”
There are at least two interpretations in modern terminology of this. One is that the end product of Antiphon’s construction is a polygon with a hyperlarge number of sides, so that the length of each side is a positive inﬁnitesimal. But this would imply that the end product would not coincide with the circumference of the circle, that is, not exactly. The other one is that the end product is the circumference of the circle itself. But this would imply that the end product no longer was a polygon. Either interpretation contains a contradiction, so it is diﬃcult to say what really was in Antiphon’s mind.
Anyway, Antiphon’s idea was not accepted by his fellow mathematicians. Again in Heath’s words:
“The time had, in fact, not come for the acceptance of Antiphon’s idea, and, perhaps as the result of the dialectic disputes to which the notion of the inﬁnite gave rise, the Greek geometers shrank from the use of such expressions as inﬁnitely great and inﬁnitely small and substituted the idea of things greater or less than any assigned magnitude. Thus, as Hankel says, they never said that a circle is a polygon with an inﬁnite number of inﬁnitely small sides; they always stood still before the abyss of the inﬁnite and never
ventured to overstep the bounds of dear conceptions. They never spoke of an inﬁnitely close approximation or a limitingvalue of the sum of a series extending to an inﬁnite number of terms.”
Note that the two interpretations mentioned above are also present in this quotation (‘inﬁnitely close’ and ‘limiting value’).
Nevertheless, the Greek geometers solved many problems involving limits. They managed to do so by means of the so-called method of exhaustion. Given the problem to determine, say, the area of some ﬁgure, it is the method to ﬁnd a sequence of inscribed ﬁgures as well as a sequence of circumscribed ﬁgures, each of known area, such that the given ﬁgure is approximated better and better by the terms of either sequence. But this does not mean that they thought in terms of limits. From the areas of the terms of both sequences they derived (guessed?) the area of the given ﬁgure, and a rigorous proof was obtained by showing that the proposed area of the given ﬁgure always lied between the areas of corresponding terms of both sequences. All that we can criticize is that they took the existence of the desired area for granted. In fact they managed to determine many limits without ever presenting a deﬁnition of limit.
Perhaps in his ‘Methods’ Archimedes comes closer to the use of inﬁnitesimals. For example (see , Supplement, p. 15), when showing that the area of a segment ABC of a given parabola is 4/3 of the area of the triangle ABC, if, with D the middle point of the chord AC, BD is parallel to the axis of the parabola, Archimedes begins with some sort of plausible reasoning, where he states that the segment is made up of line segments between the parabola and the chord of the segment, all parallel to the axis of the parabola. Apparently, in his mind all these line segments together make up the entire segment of the parabola. It is tempting to conclude that the line segments were treated by him as parallelograms of hypersmall but positive breadth. At any rate, Heath (, Supplement, p. 8) writes that the line segments are
“... of course ... indeﬁnitely narrow strips (areas) ...; but the breadth ... (dx, as we might call it) does not enter into the calculation because it is regarded the same in each of the two corresponding elements which are separately weighed against each other, and therefore divides out.”
If this would be correct Archimedese would have continued his plausible reasoning by showing that the parallelograms could each be ‘weighed’ (letting the area of a parallelogram be its weight) against one of the parallelograms making up a certain ﬁgure F. But the area of F could easily be shown to be equal to 4/3 of the area of triangle ABC.
There is an alternative, however, similar to the second interpretation mentioned earlier when discussing Antiphon’s idea, where not parallelograms but line segments are weighed against each other (letting the length of a line segment be its weight). In fact Archimedes neither mentioned something like breadth, nor discussed dividing something out at all. Instead, he considered line segments making up certain areas, not thin parallelograms. True, in this case the number of line segments is inﬁnite, so a limit is involved, but when working with parallelograms each individual comparison of weights is not exact. And since (as Archimedes remarks himself) the reasoning is not to be regarded as a rigorous one, it is not clear which interpretation is the right one. Anyway, Archimedes later on presented a rigorous proof – based on the method of exhaustion – where he could use the ratio 4/3 that he found by plausible reasoning.
Let us close this discussion with Heath’s remark that Archimedes’ ‘Method’ is a rare instance where a Greek mathematician shows how his intuition has led him to the solution of some problem by means of plausible reasoning. Usually, in Greek mathematics any trace of the intuitive machinery used was completely cleared away.
Open question: Have inﬁnitesimals been wandering through the minds of some Greek mathematicians, or didn’t they?
更新时间： 2019-10-09 10:06:28