These things agree wonderfully with the phenomena. Kepler found that in general the areas of the orbits of the planets described by radii drawn from the sun to the orbits are in proportion to the times of the revolutions around the sun, and I have demonstrated an important general proposition, namely, that all those bodies which revolve in harmonic motion (that is to say, so move that their distances from the center are in arithmetical progression, while their velocities are in harmonic progression or inversely as the distances), and moreover, if these bodies have a paracentric motion (that is to say, are heavy or light as regards the same center, whatever law this attraction or repulsion may obey)- all such bodies describe areas which vary necessarily as the times, just as Kepler observed in the case of the planets. I conclude that the deferent fluid orbs of the planets revolve harmonically, and I give an a priori reason for this. Now, empirically observing that in fact this motion is elliptical, I find that the law of paracentric motions, which when combined with the harmonic revolutions describe ellipses, ought to be such that the attraction is inversely as the squares of the distances, that is, exactly the same as what we found above to be true a priori by the laws of radiation. From this I then deduce special characteristics and the whole was broached in my publication in the Acts of Leipsic some time ago.

I will say nothing of my calculus of increments or differences, by which I determine the tangents without eliminating irrationals and fractions even when unknown quantities are involved in them and by which I subject quadratics and transcendental problems to analysis. Neither will I speak of an entirely new analysis confined to Geometry and differing entirely from Algebra, and even less of certain other subjects which I have not yet had the time to develop. I should have liked to be able to explain them all to you in a few words, so as to have upon them your opinion, which would be of infinite service to me, had you as much leisure as I have deference for your criticism. Your time, however, is too precious, and my letter is already quite long. Therefore I bring it to an end here, and am sincerely, etc.

THE END

Footnotes

*001 Leibniz always used the form Arnaud. - Trans.

*002 Leibniz remarks on the margin of Arnauld's letter: "I have always endorsed this sentiment." Interesting as indicating Leibniz's attitude toward Catholicism.- Editor.

*003 Cause exemplaire in the original.

*004 Leibniz's note: "I do not remember having said that."