The Arithmetic of Infinitesimals

Here for you at last (most distinguished gentleman) is now the whole of that work of which I gave hope in that proposition on circle measurement that I gave you in its stead in print last Easter (see Figure 1). For since, by custom, when one puts something out in public, it ought to be dedicated to someone, I thought to seek not only a great gentleman but a great mathematician to whom I might offer it. And therefore I saw that to none other greater than you can that easily be done, who is among mathematicians most deserving, and also by whose writings I readily confess that I have profited: who indeed in your Clavis mathematicae, though not a large work, have there taught both briefly and clearly, what we seek in vain in the large volumes of others.¹

That proposition (most famed Gentleman) that I have shown before to you, concealed in shape and in the form of a problem, and also to not a few other mathematicians, to whom I held out the thing some years ago, hiding for the most part (though it was discovered by several) the target it was aimed at: here at last I declare ahead openly, in the form of a Theorem (which was previously buried).

If there is proposed a series,¹ of quantities in arithmetic proportion (or as the natural sequence of numbers)² continually increasing, beginning from a point or 0 (that is, nought, or nothing),³ thus as 0, 1, 2, 3, 4, etc., let it be proposed to inquire what is the ratio of the sum of all of them, to the sum of the same number of terms equal to the greatest.