The Elements of the True Arithmetic of Infinites

1+1+1+1+1, &c.
1+1+1+1+1, &c.

1+1+1+1+1
1+1+1+1+1
1+1+1+1+1
1+1+1+1+1
1+1+1+1+1

1+2+3+4+5+4+3+2+1

In this multiplication, only the part of the product 1+2+3+4+5 is true. This is because when the multiplication continues further—that is, when more than five terms are multiplied by five—none of the terms in this part of the product receive any increase.

If the multiplication involved a finite number of terms by a finite number, as in this instance of five terms by five, then all terms of the product (1+2+3+4+5+4+3+2+1) would be true. However, the case is far different when the multiplication involves an infinite number of terms.

If, however, someone looking at the properties of finite series and assuming they are the same as those of infinite series (though the contrary will be abundantly shown in what follows) should still contend that the series 1+2+3+4+5, etc., ad infinitum Latin: to infinity, is not the true product of 1+1+1+1+1, etc., ad infinitum multiplied by 1+1+1+1+1, etc., ad infinitum, 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 multiplicand: the number to be multiplied is always equal to the multiplier multiplier: the number by which another is multiplied. But 1+2+3+4, etc., divided by 1+1+1+1+1, etc., gives 1+1+1+1+1, etc.

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.