The Elements of the True Arithmetic of Infinites

PROPOSITION I.

1-1 is an infinitesimal infinitesimal: a quantity smaller than any assignable finite quantity, or an infinitely small part of the fraction 1/(1+1), and an infinite series of 1-1 is equal to 1/(1+1). In like manner, 1-2+1 is an infinitely small part of (1-1+2-1+2-1+2-1+2-1)/(1+1), etc., ad infinitum, and an infinite series of 1-2+1 is equal to the same fraction. Similarly, 1-2 is an infinitely small part of (1-1-1-1-1)/(1+1), etc., ad infinitum, and an infinite series of 1-2 is equal to that fraction. Thus, too, 1-3 is the infinitesimal of (1-2-2-2-2)/(1+1), etc., 1-4 of (1-3-3-3-3)/(1+1), etc., and so on for others.

From 1
Subtract 1-1+1-1+1-1+1-1, etc.
And the remainder is +1-1+1-1+1-1+1-1, etc.

But by the second postulate, the remainder added to what is subtracted is equal to the subtrahend. Hence, the series 1-1+1-1+1-1+1-1, etc., added to 1-1+1-1+1-1+1-1, etc., is equal to 1. The series 1-1+1-1+1-1+1-1, etc., is therefore equal to 1/(1+1), and consequently 1-1 is an infinitesimal. For it cannot be 0, since an infinite series of 0 added to an infinite series of 0 can never be equal to 1.

In like manner, if from 1-1-1+2-1-1+2-1-1, etc., be subtracted 1-2+1+1-2+1+1-2+1, etc., the remainder is 1-2+1+1-2+1+1-2, etc., and therefore 1-2+1 is an infinitesimal; and the same applies to the rest.

Corollary 1. Hence, expressions such as 1-1, 1-2+1, 1-2, etc., are neither quantities nor nothings; they are something belonging to number without being a number itself, just as a point, which is the extremity of a line, is something belonging to a line without being a line itself.