The elements of the true arithmetic of infinities

In this multiplication, the part of the product 1+2+3+4+5 alone is true, because when the multiplication is continued farther, meaning, when more than five terms are multiplied by five, no one of the terms of this part of the product receives any increase.

If, indeed, the multiplication was of a finite number of terms by a finite number, as in this instance, of five terms by five, then all the terms of the product, namely 1+2+3+4+5+4+3+2+1 would be true; but the case is far otherwise when the multiplication is of an infinite by an infinite number of terms.

If, however, some one looking to the properties of finite series, and supposing them to be the same with those of infinite series though the contrary will abundantly be shown to be true in what follows should still contend that the series 1+2+3+4+5, &c. ad infinitum to infinity, is not the true product of 1+1+1+1+1, &c. ad infin. to infinity multiplied by 1+1+1+1+1, &c. ad infin., he may be fully convinced of his error, by considering that wherever the product of the multiplication of two quantities is true, the quotient of the product divided by the multiplicand the number to be multiplied is always equal to the multiplier the number by which another is multiplied. But 1+2+3+4, &c. divided by 1+1+1+1+1, &c. gives 1+1+1+1+1, &c.

4. That to multiply one number, or one series of numbers, by another, is the same thing as to add either of those numbers, or series of numbers, to itself, as often as there are units in the other.