Euler insists that the relation of equality holds between any infinitesimal and zero. Similarly, Leibniz worked with a generalized relation of “equality” which was an equality up to a negligible term. Leibniz codified this relation in terms of his transcendental law of homogeneity (TLH), or lex homogeneorum transcendentalis in the original Latin. Leibniz had already referred to the law of homogeneity in his first work on the calculus: “the only remaining differential quantities, namely dx, dy, are found always outside the numerators and roots, and each member is acted on by either dx, or by dy, always with the law of homogeneity maintained with regard to these two quantities, in whatever manner the calculation may turn out.”
The TLH governs equations involving differentials. Bos interprets it as follows:
A quantity which is infinitely small with respect to an- other quantity can be neglected if compared with that quantity. Thus all terms in an equation except those of the highest order of infinity, or the lowest order of infinite smallness, can be discarded. For instance,
(here the expression ddx denotes a second-order differential obtained as a second difference). Thus, formulas like Euler’s
a + dx = a —– (2)
(where a “is any finite quantity”; (Euler) belong in the Leibnizian tradition of drawing inferences in accordance with the TLH and as reported by Bos in formula (1) above. The principle of cancellation of infinitesimals was, of course, the very basis of the technique. However, it was also the target of Berkeley’s charge of a logical inconsistency (Berkeley). This can be expressed in modern notation by the conjunction (dx ≠ 0) ∧ (dx = 0). But the Leibnizian framework does not suffer from an inconsistency of type (dx ≠ 0) ∧ (dx = 0) given the more general relation of “equality up to”; in other words, the dx is not identical to zero but is merely discarded at the end of the calculation in accordance with the TLH.
Relations of equality: What Euler and Leibniz appear to have realized more clearly than their contemporaries is that there is more than one relation falling under the general heading of “equality”. Thus, to explain formulas like (2), Euler elaborated two distinct ways, arithmetic and geometric, of comparing quantities. He described the two modalities of comparison in the following terms:
Since we are going to show that an infinitely small quantity is really zero (cyphra), we must meet the objection of why we do not always use the same symbol 0 for infinitely small quantities, rather than some special ones…
[S]ince we have two ways to compare them [a more pre- cise translation would be “there are two modalities of comparison”], either arithmetic or geometric, let us look at the quotients of quantities to be compared in order to see the difference. (Euler)
If we accept the notation used in the analysis of the infi- nite, then dx indicates a quantity that is infinitely small, so that both dx = 0 and a dx = 0, where a is any finite quantity. Despite this, the geometric ratio a dx : dx is finite, namely a : 1. For this reason, these two infinitely small quantities, dx and adx, both being equal to 0, cannot be confused when we consider their ratio. In a similar way, we will deal with infinitely small quantities dx and dy.
Having defined the two modalities of comparison of quantities, arithmetic and geometric, Euler proceeds to clarify the difference between them as follows:
Let a be a finite quantity and let dx be infinitely small. The arithmetic ratio of equals is clear:
Since ndx = 0, we have
a ± ndx − a = 0 —– (3)
On the other hand, the geometric ratio is clearly of equals, since
(a ± ndx)/a =1 —– (4)
While Euler speaks of distinct modalities of comparison, he writes them down symbolically in terms of two distinct relations, both denoted by the equality sign “=”; namely, (3) and (4). Euler concludes as follows:
From this we obtain the well-known rule that the infinitely small vanishes in comparison with the finite and hence can be neglected [with respect to it].
Note that in the Latin original, the italicized phrase reads infinite parva prae finitis evanescant, atque adeo horum respectu reiici queant. The term evanescant can mean either vanish or lapse, but the term prae makes it read literally as “the infinitely small vanishes before (or by the side of ) the finite,” implying that the infinitesimal disappears because of the finite, and only once it is compared to the finite.
A possible interpretation is that any motion or activity involved in the term evanescant does not indicate that the infinitesimal quantity is a dynamic entity that is (in and of itself) in a state of disappearing, but rather is a static entity that changes, or disappears, only “with respect to” (horum respectu) a finite entity. To Euler, the infinitesimal has a different status depending on what it is being compared to. The passage suggests that Euler’s usage accords more closely with reasoning exploiting static infinitesimals than with dynamic limit-type reasoning.