The elements of the true arithmetic of infinities

$= 1-3+6-12+24-48,$ etc. For $1-3$ is equal to $-2$, and $-2$ added to $6$ is $+4$, and $+4-12$ is $-8$, and $-8$ added to $24$ is $+16$, and so of the rest; and $\frac{1-1}{1+2} \times \frac{1}{1-1} = \frac{1}{1+2}$.

COR. 4. Corollary 4 Hence, also, in every infinite series, whether fractional or integral consisting of whole numbers, the terms of which have an uninterrupted continuity a sequence without gaps, the last term multiplied by the number of terms will be equal to the sum of the series. For the last term of every infinite series is equal to the fraction by which that series is produced, multiplied by $1-1$; and as the number of terms is $\frac{1}{1-1}$, it is evident that this last term multiplied by $\frac{1}{1-1}$, will be equal to the sum of the series. Hence, too, all infinite series are to each other as their last terms, namely as the terms which are obtained by multiplying those series by $1-1$.

COR. 5. Corollary 5 As when any finite number of terms of these infinite series is taken, the last term multiplied by the number of terms will by no means be equal to the sum of the terms, but when the series begin from unity, will continually diverge move further away from such equality, as will be found upon trial to be the case; hence, in infinite series, we cannot always reason from the parts to the whole, or from the whole to the parts; and this perspicuously clearly shows that wholes, when they possess infinite power, have properties very different from their parts. Dr. Wallis John Wallis (1616–1703), an English mathematician who contributed to the development of infinitesimal calculus, from not perceiving this truth, founded his Arithmetic of Infinites on an induction inference of general laws from particular instances from the parts of infinite series, and in consequence of this his leading propositions are, as we have demonstrated, false.

Dr. Cheyne George Cheyne (1671–1743), a Scottish physician and mathematician, also, in consequence of having no conception of this truth, asserts, in his Philosophical Principles of Religion, page 148, that the series $1+2+3+4, \text{etc.} = \frac{1}{1-2+1}$, is but half the square of $1+1+1+1, \text{etc.} = \frac{1}{1-1}$, though $\frac{1}{1-2+1}$ is the square of $\frac{1}{1-1}$. It might be easy to confute disprove this assertion by merely observing, that if $\frac{1}{1-2+1}$ is the square of $\frac{1}{1-1}$, the series $1+2+3+4+5, \text{etc.}$ must be the square of the series $1+1+1+1, \text{etc.}$; for if it is not, the algebraic rules of subtraction and division are false, and also the common mode of multiplication; and it is strange that any man in his senses, who understood the elements of arithmetic