Congress of Arts and Sciences (Boston and New York, 1905), Vol. 1, p. 476.
[1938]. Zeno was concerned with three problems.... These are the problem of the infinitesimal, the infinite, and continuity.... From him to our own day, the finest intellects of each generation in turn attacked these problems, but achieved broadly speaking nothing.... Weierstrass, Dedekind, and Cantor, ... have completely solved them. Their solutions ... are so clear as to leave no longer the slightest doubt of difficulty. This achievement is probably the greatest of which the age can boast.... The problem of the infinitesimal was solved by Weierstrass, the solution of the other two was begun by Dedekind and definitely accomplished by Cantor.—Russell, Bertrand.
International Monthly, Vol. 4 (1901), p. 89.
[1939]. It was not till Leibnitz and Newton, by the discovery of the differential calculus, had dispelled the ancient darkness which enveloped the conception of the infinite, and had clearly established the conception of the continuous and continuous change, that a full and productive application of the newly-found mechanical conceptions made any progress.—Helmholtz, H.
Aim and Progress of Physical Science; Popular Lectures [Flight] (New York, 1900), p. 372.
[1940]. The idea of an infinitesimal involves no contradiction.... As a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares.—Pierce, C. F.
The Law of Mind; Monist, Vol. 2 (1891-1892), pp. 543, 545.
[1941]. The chief objection against all abstract reasonings is derived from the ideas of space and time; ideas, which, in common life and to a careless view, are very clear and intelligible, but when they pass through the scrutiny of the profound sciences (and they are the chief object of these sciences) afford principles, which seem full of obscurity and contradiction. No priestly dogmas, invented on purpose to tame and subdue the rebellious reason of mankind, ever shocked common sense more than the doctrine of the infinite divisibility of extension, with its consequences; as they are pompously displayed by all geometricians and metaphysicians, with a kind of triumph and exultation. A real quantity, infinitely less than any finite quantity, containing quantities infinitely less than itself, and so on in infinitum; this is an edifice so bold and prodigious, that it is too weighty for any pretended demonstration to support, because it shocks the clearest and most natural principles of human reason. But what renders the matter more extraordinary, is, that these seemingly absurd opinions are supported by a chain of reasoning, the clearest and most natural; nor is it possible for us to allow the premises without admitting the consequences. Nothing can be more convincing and satisfactory than all the conclusions concerning the properties of circles and triangles; and yet, when these are once received, how can we deny, that the angle of contact between a circle and its tangent is infinitely less than any rectilineal angle, that as you may increase the diameter of the circle in infinitum, this angle of contact becomes still less, even in infinitum, and that the angle of contact between other curves and their tangents may be infinitely less than those between any circle and its tangent, and so on, in infinitum? The demonstration of these principles seems as unexceptionable as that which proves the three angles of a triangle to be equal to two right ones, though the latter opinion be natural and easy, and the former big with contradiction and absurdity. Reason here seems to be thrown into a kind of amazement and suspense, which, without the suggestion of any sceptic, gives her a diffidence of herself, and of the ground on which she treads. She sees a full light, which illuminates certain places; but that light borders upon the most profound darkness. And between these she is so dazzled and confounded, that she scarcely can pronounce with certainty and assurance concerning any one object.—Hume, David.
An Inquiry concerning Human Understanding, Sect. 12, part 2.