The Elements of the True Arithmetic of Infinites

(0+1+11+11+1)/(1-4+6-4+1), and (0+1+26+66+26+1)/(1-5+10-10+5-1). And (0+1)/(1-1) is less than 1/(1-1) by (1-1)/(1-1)=1; (0+1+1)/(1-2+1) is less than (1+1)/(1-1+1) by (1-1)/(1-2+1)=1+2+4+6+8+10, etc.; and so of the rest.

PROPOSITION IV.

In each of the preceding series, whether it begins from 0 or from 1, the last term multiplied by the number of terms is equal to the sum of that series.

For by Proposition II, the number of terms in each of these series must be 1/(1-1). There cannot be a greater number of terms, and the continuity in each of these series is uninterrupted. In the first of these series, therefore, beginning from 1, 1/(1-1) × 1/(1-1) = 1/(1-2+1); in the second, 1/(1-1) × (1+1)/(1-2+1) = (1+1)/(1-3+3-1); in the third, 1/(1-1) × (1+4+1)/(1-3+3-1) = (1+4+1)/(1-4+6-4+1), and so forth. If the series begins from 0, it will be 1/(1-1) × (0+1)/(1-1) = (0+1)/(1-2+1), and so of the rest.

Corollary 1. Similar fractional expressions, whose denominators are the 6th, 7th, 8th, etc., powers of 1-1, will also be found to possess the same property.

Corollary 2. From this proposition and the corollary to the preceding proposition, it follows that these series, when they begin from 0, are infinitely less than when they begin from 1.

PROPOSITION V.

The following propositions in Dr. Wallis's John Wallis (1616–1703), an English mathematician who contributed to the development of calculus. Arithmetic of Infinites are false:

"If a series of numbers in arithmetical progression begins with a cipher cipher: zero, and the common difference be 1, if the last term be multiplied into the number of terms, the product will be double the sum of all the series."