The Elements of the True Arithmetic of Infinites

metic and algebra should make such an assertion. But lest someone should still stubbornly contend that Dr. Cheyne is right, the following is a complete demonstration of the contrary.

To multiply 1+1+1+1, etc., by itself is the same thing (by the 4th postulate) as to add this series to itself 1+1+1+1, etc., times; namely, it will be equal to 1/(1-1) + 1/(1-1) + 1/(1-1), etc., infinitely, or to (1+1+1+1+1, etc.) / (1-1). But (1+1+1+1+1, etc.) / (1-1) is equal to 1+2+3+4+5, etc., or rather (1+1+1+1+1, etc.) / (1-1) is equal to 1/(1-1) = 1/(1-2+1) = 1+2+3+4, etc. This, however, is owing to the wonderful nature of the infinite. That the infinite, indeed, has a power very different from the finite was not unknown in some instances to Dr. Wallis, as is evident from the following observation made by him in his Arithmetica Infinitorum, pp. 131–132: original: "Si series subsecundanorum aliquousque continueter, puta √0+√1+√2+√3+√4+√5+√6, ipsius ratio ad maximum toties positum, puta 7√6, non videtur alias explicabilis quam (√0+√1+√2+√3+√4+√5+√6) / 7√6 vel (0+1+√2+√3+√2+√5+√6, &c.) / 7√6. Verum si eadem series supponatur in infinitum continuanda, prodibit tandem ratio 2/3, vel 2 ad 3, aut 1 ad 1 1/2, ut dictum est prop. 53, 54. ipsa quidem infinitate (quod mirum videatur) irrationabilitatem destruente." Translation: "If a series of square roots is continued for some time, for example √0+√1+√2+√3+√4+√5+√6, its ratio to the maximum term multiplied by the number of terms, for example 7√6, does not seem explicable in any other way than (√0+√1+√2+√3+√4+√5+√6) / 7√6... But if the same series is supposed to be continued to infinity, the ratio 2/3 will eventually emerge, or 2 to 3, or 1 to 1 1/2, as stated in propositions 53 and 54, with the infinity itself (which may seem wonderful) destroying the irrationality." Dr. Wallis, however, had no conception of the difference between the power of an infinite and a finite whole in the way we have already explained.

PROP. IX.

Numbers connected together by an affirmative or negative sign are different from the same numbers when actually added together, or subtracted, and expressed by one number.

Thus 1+1 is not the same as 2; for 1+1 subtracted from 2 leaves the infinitesimal 1-1. Thus, too, 1+1+1 subtracted from 3 leaves 2-1-1; 1+1+1+1 subtracted from 4 leaves 3-1-1-1. And further still, 2+1 is not the same as 1+2. For 2+1 subtracted from 3 leaves 1-1, but 1+2 subtracted from 3 leaves