The Elements of the True Arithmetic of Infinites

the ancient mathematicians, I should conceive that my time was by no means misspent in composing it; but as I presume it will also be found to unfold the nature of the mathematical infinite more satisfactorily than it has hitherto been unfolded, I trust I shall obtain the commendation of the liberal and the wise.

As one of the principal discoveries in this treatise is that in every infinite series of terms, whether integral or fractional, the last term multiplied by the number of terms is equal to the sum of the series, I rejoice to find, as the result of this discovery, that it affords a most splendid instance of the absurdity that may attend reasoning by induction from parts to wholes, or from wholes to parts, when the wholes are themselves infinite. For this contributes to elucidate, in no mean degree, one of the most important dogmas in the philosophy of Plato and Aristotle, to the promulgation of which philosophy I have devoted so considerable a part of my past life, and hope I shall be able to devote the remainder.

In short, it will be found from this treatise that the doctrine of infinite series, as cultivated by mathematicians of the present day, is not to be employed in accurate demonstrations, however useful it may be for practical purposes. For it is here demonstrated that the fractions, from the expansion of which infinite series are produced, are not accurately to each other as one finite number to another finite number. And it is likewise shown in a variety of instances that an infinite series of an infinitely repeating decimal is less than an infinite series of the vulgar fraction A common fraction, such as 1/3, as opposed to a decimal. of which the infinite repetends are the decimal, by the vulgar fraction itself.

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