Descartes himself studied the way balls move and collide, how they gather speed as they go down ramps, et cetera, and tried to explain all of his data in terms of a theory that was purely geometrical in nature. The result of his lucubrations was classically French in that it did not square with reality but it was very beautiful, and logically coherent. Since then our friends Huygens and Wren have expended more toil towards the same end. But I need hardly tell you that it is Newton, far beyond all others, who has vastly expanded the realm of truths that are geometrickal in nature. I truly believe that if Euclid and Eratosthenes could be brought back to life they would prostrate themselves at his feet and (pagans that they were) worship him as a god. For their geometry treated mostly simple abstract shapes, lines in the sand, while Newton’s lays down the laws that govern the very planets.

I have read the copy of Principia Mathematica that you so kindly sent me, and I know better than to imagine I will find any faults in the author’s proofs, or extend his work into any realm he has not already conquered. It has the feel of something finished and complete. It is like a dome-if it were not whole, it would not stand, and because it is whole, and does stand, there’s no point trying to add things on to it.

And yet its very completeness signals that there is more work to be done. I believe that the great edifice of the Principia Mathematica encloses nearly all of the geometrickal truths that can possibly be written down about the world. But every dome, be it never so large, has an inside and an outside, and while Newton’s dome encloses all of the geometrickal truths, it excludes the other kind: truths that have their sources in fitness and in final causes. When Newton encounters such a truth-such as the inverse square law of gravity-he does not even consider trying to understand it, but instead says that the world simply is this way, because that is how God made it. To his way of thinking, any truths of this nature lie outside the realm of Natural Philosophy and belong instead to a realm he thinks is best approached through the study of alchemy.

Let me tell you why Newton is wrong.

I have been trying to salvage something of value from Descartes’ geometrickal theory of collisions and have found it utterly devoid of worth.

Descartes holds that when two bodies collide, they should have the same quantity of motion after the collision as they had before. Why does he believe this? Because of empirical observations? No, for apparently he did not make any. Or if he did, he saw only what he wanted to see. He believes it because he has made up his mind in advance that his theory must be geometrickal, and geometry is an austere discipline-there are only certain quantities a geometer is allowed to measure and to write down in his equations. Chief among these is extension, a pompous term for “anything that can be measured with a ruler.” Descartes and most others allow time, too, because you can measure time with a pendulum, and you can measure the pendulum with a ruler. The distance a body travels (which can be measured with a ruler) divided by the time it took covering it (which can be measured with a pendulum, which can be measured with a ruler) gives speed. Speed figures into Descartes’ calculation of Quantity of Motion-the more speed, the more motion.

Well enough so far, but then he got it all wrong by treating Quantity of Motion as if it were a scalar, a simple directionless number, when in fact is is a vector. But that is a minor lapse. There is plenty of room for vectors in a system with two orthogonal axes, we simply plot them as arrows on what I call the Cartesian plane, and lo, we have geometrickal constructs that obey geometrickal rules. We can add their components geometrickally, reckon their magnitudes with the Pythagorean Theorem, amp;c.

But there are two problems with this approach. One is relativity. Rulers move. There is no fixed frame of reference for measuring extension. A geometer on a moving canal-boat who tries to measure the speed of a flying bird will get a different number from a geometer on the shore; and a geometer riding on the bird’s back would measure no speed at all!

Secondly: the Cartesian Quantity of Motion, mass multiplied by velocity (mv), is not conserved by falling bodies. And yet by doing, or even imagining, a very simple experiment, you can demonstrate that mass multiplied by the square of velocity (mv²) is conserved by such bodies.

This quantity mv² has certain properties of interest. For one, it measures the amount of work that a moving body is capable of doing. Work is something that has an absolute meaning, it is free from the problem of relativity that I mentioned a moment ago, a problem unavoidably shared by all theories that are founded upon the use of rulers. In the expression mv² the velocity is squared, which means that it has lost its direction, and no longer has a geometrickal meaning. While mv may be plotted on the Cartesian plane and subjected to all the tricks and techniques of Euclid, mv² may not be, because in being squared the velocity v has lost its directionality and, if I may wax metaphysical, transcended the geometrickal plane and gone into a new realm, the realm of Algebra. This quantity mv² is scrupulously conserved by Nature, and its conservation may in fact be considered a law of the universe-but it is outside Geometry, and excluded from the dome that Newton has built, it is another contingent, non-geometrickal truth, one of many that have been discovered, or will be, by Natural Philosophers. Shall we then say, like Newton, that all such truths are made arbitrarily by God? Shall we seek such truths in the occult? For if God has laid these rules down arbitrarily, then they are occult by nature.

To me this notion is offensive; it seems to cast God in the role of a capricious despot who desires to hide the truth from us. In some things, such as the Pythagorean Theorem, God may not have had any choice when He created the world. In others, such as the inverse square law of gravity, He may have had choices; but in such cases, I like to believe he would have chosen wisely and according to some coherent plan that our minds-insofar as they are in God’s image-are capable of understanding.

Unlike the Alchemists, who see angels, demons, miracles, and divine essences everywhere, I recognize nothing in the world but bodies and minds. And nothing in bodies but certain observable quantities such as magnitude, figure, situation, and changes in these. Everything else is merely said, not understood; it is sounds without meaning. Nor can anything in the world be understood clearly unless it is reduced to these. Unless physical things can be explained by mechanical laws, God cannot, even if He chooses, reveal and explain nature to us.

I am likely to spend the rest of my life explaining these ideas to those who will listen, and defending them from those who won’t, and anything you hear from me henceforth should probably be viewed in that light, Daniel. If the Royal Society seems inclined to burn me in effigy, please try to explain to them that I am trying to extend the work that Newton has done, not to tear it down.

Leibniz

P.S. I know the woman Eliza (de la Zeur, now) whom you mentioned in your most recent letter. She seems to be attracted to Natural Philosophers. It is a strange trait in a woman, but who are we to complain?

“Dr. Waterhouse.”

“Sergeant Shaftoe.”

“Your visitors have arrived-Mr. Bob Carver and Mr. Dick Gripp.”

Daniel rose from his bed; he had never come awake so fast. “Please, I beg you, Sergeant, do not-” he began, but he stopped there, for it had occurred to him that perhaps Sergeant Shaftoe’s mind was already made up, the deed was all but done, and that Daniel was merely groveling. He got to his feet and shuffled over the wooden floor towards Bob Shaftoe’s face and his candle, which hung in darkness like a poorly resolved binary star: the face a dim reddish blob, the flame a burning white point. The blood dropped from Daniel’s head and he tottered, but did not hesitate. He’d be nothing more than a bleating voice in the darkness until he entered the globe of light balanced on that flame; if Bob Shaftoe had thoughts of letting the murderers into this room, let him look full on Daniel’s face first. The brilliance of the light was governed by an inverse square law, just like gravity.