The elements of the true arithmetic of infinities

PROPOSITION I.

1-1 is an infinitesimal an infinitely small quantity, or infinitely small part of the fraction $\frac{1}{1+1}$, and an infinite series of 1-1 is equal to $\frac{1}{1+1}$. In like manner, also, 1-2+1 is an infinitely small part of $\frac{1-1+1-2+1-1+1-2+1-1, &c.}{1+1}$ ad infin. to infinity and an infinite series of 1-2+1 is equal to $\frac{1-1+1-2+1-1+1-2+1-1, &c.}{1+1}$. And 1-2 is an infinitely small part of $\frac{1-1-1-1-1, &c.}{1+1}$ ad infin. and an infinite series of 1-2 is equal to $\frac{1-1-1-1-1, &c.}{1+1}$. Thus, too, 1-3 is the infinitesimal of $\frac{1-2-2-2-2, &c.}{1+1}$ 1-4 of $\frac{1-3-3-3-3, &c.}{1+1}$ and so of others.

From 1
Subtract 1-1+1-1+1-1+1-1, &c.
And the remainder is . +1-1+1-1+1-1+1-1, &c.

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, &c. added to 1-1+1-1+1-1, &c. is equal to 1. The series 1-1+1-1+1-1, &c. is therefore equal to $\frac{1}{1+1}$, 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, &c.
be subtracted 1-2+1+1-2+1+1-2+1, &c.
the remainder is . 1-2+1+1-2+1+1-2, &c.

and therefore 1-2+1 is an infinitesimal; and so of the rest.

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