THE ELEMENTS OF THE TRUE ARITHMETIC OF INFINITES, &c.

$\frac{0+1+11+11+1}{1-4+6-4+1}$, and $\frac{0+1+26+66+26+1}{1-5+10-10+5-1}$. And $\frac{0+1}{1-1}$ is less than $\frac{1}{1-1}$ by $\frac{1-1}{1-1} = 1$; $\frac{0+1+1}{1-2+1}$ is less than $\frac{1+1}{1-1+1}$ by $\frac{1 \cdot 1}{1-2+1} = 1+2+4+6+8+10$, &c. and so of the rest.

PROP. 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 $\frac{1}{1-1}$. For 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, $\frac{1}{1-1} \times \frac{1}{1-1} = \frac{1}{1-2+1}$; in the second, $\frac{1}{1-1} \times \frac{1+1}{1-2+1} = \frac{1+1}{1-3+3-1}$; in the third, $\frac{1}{1-1} \times \frac{1+4+1}{1-3+3-1} = \frac{1+4+1}{1-4+6-4+1}$; in the fourth, $\frac{1}{1-1} \times \frac{1+11+11+1}{1-4+6-4+1} = \frac{1+11+11+1}{1-5+10-10+5-1}$; and, in the fifth, $\frac{1}{1-1} \times \frac{1+26+66+26+1}{1-5+10-10+5-1} = \frac{1+26+66+26+1}{1-6+15-20+15-6+1}$. But if the series begin from 0, it will be $\frac{1}{1-1} \times \frac{0+1}{1-1} = \frac{0+1}{1-2+1}$, $\frac{1}{1-1} \times \frac{0+1+1}{1-2+1} = \frac{0+1+1}{1-3+3-1}$, and so of the rest.

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

COROL. 2. Second Corollary 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.

PROP. V.

The following propositions in Dr. Wallis's Arithmetic of Infinites Arithmetica Infinitorum, a 1656 work by John Wallis that influenced the development of calculus are false, viz.

"If a series of numbers in arithmetical progression a sequence where the difference between terms is constant begin with a cypher the digit 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."