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, &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 rôle 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.
Shaftoe’s face finally came into focus. He looked a little sea-sick. “I’m not such a black-hearted bastard as’d admit a pair of hired killers to spit a helpless professor. There is only one man alive whom I hate enough to wish such an end on him.”
“Thank you,” Daniel said, drawing close enough now that he could feel the candle’s faint warmth on his face.
Shaftoe noticed something, turned sideways to Daniel, and cleared his throat. This was not your delicate pretentious upper-class ’hem but an honest and legitimate bid to dislodge an actual phlegm-ball that had sprung into his gorge.
“You’ve noticed me pissing myself, haven’t you?” Daniel said. “You imagine that it’s your fault—that you put such a terror into me, just now, that I could not hold my urine. Well, you did have me going, it is true, but that’s not why piss is running down my leg. I have the stone, Sergeant, and cannot make water
at times of my own choosing, but rather I leak and seep like a keg that wants caulking.”
Bob Shaftoe nodded and looked to have been somewhat relieved of his burden of guilt. “How long d’you have then?”
He asked the question so offhandedly that Daniel did not get it for a few moments. “Oh—you mean, to live?” The Sergeant nodded. “Pardon me, Sergeant Shaftoe, I forget that your profession has put you on such intimate terms with death that you speak of it as sea-captains speak of wind. How long have I? Perhaps a year.”
“You could have it cut out.”
“I have seen men cut for the stone, Sergeant, and I’ll take death, thank you very much. I’ll wager it is worse than anything you may have witnessed on a battlefield. No, I shall follow the example of my mentor, John Wilkins.”
“Men have been cut for the stone, and lived, have they not?”
“Mr. Pepys was cut nigh on thirty years ago, and lives still.”
“He walks? Talks? Makes water?”
“Indeed, Sergeant Shaftoe.”
“Then, by your leave, Dr. Waterhouse, being cut for the stone is not worse than anything I have seen on battlefields.”
“Do you know how the operation is performed, Sergeant? The incision is made through the perineum, which is that tender place between your scrotum and your anus—”
“If it comes down to swapping blood-curdling tales, Dr. Water-house, we shall be here until this candle has burnt down, and all to no purpose; and if you really intend to die of the stone, you oughtn’t to be wasting that much time.”
“There is nothing to do, here, but waste time.”
“That is where you are wrong, Dr. Waterhouse, for I have a lively sort of proposition to make you. We are going to help each other, you and I.”
“You want money in exchange for keeping Jeffreys’s murderers out of my chamber?”
“That’s what I should want, were I a base, craven toad,” Bob Shaftoe said. “And if you keep mistaking me for that sort, why, perhaps I shall let Bob and Dick in here.”
“Please forgive me, Sergeant. You are right in being angry with me. It is only that I cannot imagine what sort of transaction you and I could…”
“Did you see that fellow being whipped, just before sundown? He would’ve been visible to you out in the dry-moat, through yonder arrow-slit.”
Daniel remembered it well enough. Three soldiers had gone out, carrying their pikes, and lashed them together close to their points, and spread their butts apart to form a tripod. A man had been led out shirtless, his hands tied together in front of him, and the rope had then been thrown over the lashing where the pikes were joined, and drawn tight so that his arms were stretched out above his head. Finally his ankles had been spread apart and lashed fast to the pikes to either side of him, rendering him perfectly immobile, and then a large man had come out with a whip, and used it. All in all it was a common rite around military camps, and went a long way towards explaining why people of means tried to live as far away from barracks as possible.
“I did not observe it closely,” Daniel said, “I am familiar with the general procedure.”
“You might’ve watched more carefully had you known that the man being whipped calls himself Mr. Dick Gripp.”
Daniel was at a loss for words.
“They came for you last night,” said Bob Shaftoe. “I had them clapped into separate cells while I decided what to do with ’em. Talked to ’em separately, and all they gave me was a deal of hot talk. Now. Some men are entitled to talk that way, they have been ennobled, in a sense, by their deeds and the things they have lived through. I did not think that Bob Carver and Dick Gripp were men of that kind. Others may be suffered to talk that way simply because they entertain the rest of us. I once had a brother who was like that. But not Bob and Dick. Unfortunately I am not a magistrate and have no power to throw men in prison, compel them to answer questions, et cetera. On the other hand, I am a sergeant, and have the power to recruit men into the King’s service. As Bob and Dick were clearly idle fellows, I recruited them into the King’s Own Black Torrent Guards on the spot. In the next instant, I perceived that I’d made a mistake, for these two were discipline problems, and wanted chastisement. Using the oldest trick in the book, I had Dick—who struck me as the better man—whipped directly in front of Bob Carver’s cell window. Now Dick is a strong bloke, he is unbowed, and I may keep him in the regiment. But Bob feels about his chastisement—which is scheduled for dawn—the same way you feel about being cut for the stone. So an hour ago he woke up his guards, and they woke me, and I went and had a chat with Mr. Carver.”
“Sergeant, you are so industrious that I almost cannot follow everything you are about.”
“He told me that Jeffreys personally ordered him and Mr. Gripp to cut your throat. That they were to do it slow-like, and that they were to explain to you, while you lay dying, that it had been done by Jeffreys.”
“It is what I expected,” Daniel said, “and yet to hear it set out in plain words leaves me dizzy.”
“Then I shall wait for you to get your wits back. More to the point, I shall wait for you to become angry. Forgive me for presuming to instruct a fellow of your erudition, but at a moment like this, you are supposed to be angry.”
“It is a very odd thing about Jeffreys that he can treat people abominably and never make them angry. He influences his victims’ minds strangely, like a glass rod bending a stream of water, so that we feel we deserve it.”
“You have known him a long time.”
“I have.”
“Let’s kill him.”
“I beg your pardon?”
“Slay, murder. Let us bring about his death, so he won’t plague you any more.”
Daniel was shocked. “It is an extremely fanciful idea—”
“Not in the least. And there is something in your tone of voice that tells me you like it.”
“Why do you say ‘we’? You have no part in my problems.”
“You are high up in the Royal Society.”
“Yes.”
“You know many Alchemists.”
“I wish I could deny it.”
“You know my lord Upnor.”
“I do. I’ve known him as long as I’ve known Jeffreys.”
“Upnor owns my lady love.”
“I beg your pardon—did you say he owns her?”
“Yes—Jeffreys sold her to him during the Bloody Assizes.”
“Taunton—your love is one of the Taunton schoolgirls!”
“Just so.”
Daniel was fascinated. “You are proposing some sort of pact.”
“You and I’ll rid the world of Jeffreys and Upnor. I’ll have my Abigail and you’ll live your last year, or whatever time God affords you, in peace.”
“I do not mean to quail and fret, Sergeant—”
“Go ahead! My men do it all the time.”
“—but may I remind you that Jeffreys is the Lord Chancellor of the Realm?”
“Not for long,” Shaftoe answered.
“How do you know?”
“He’s as much as admitted it, by his actions! You were thrown in Tower why?”
“For acting as go-between to William of Orange.”
“Why, that is treason—you should’ve been half-hanged, drawn, and quartered for it! But you were kept alive why?”
“Because I am a witness to the birth of the Prince, and as such, may be useful in attesting to the legitimacy of the next King.”
“If Jeffreys has now decided to kill you, what does that signify then?”
“That he is giving up on the King—my God, on the entire dynasty—and getting ready to flee. Yes, I understand your reasoning now, thank you for being so patient with me.”
“Mind you, I’m not asking you to take up arms, or do anything else that ill suits you.”
“Some would take offense at that, Sergeant, but—”
“E’en though my chief grievance may lie with Upnor, the first cause of it wa
s Jeffreys, and I would not hesitate to swing my spadroon, if he should chance to show me his neck.”
“Save it for Upnor,” Daniel said, after a brief pause to make up his mind. In truth, he’d long since made it up; but he wanted to put on a show of thinking about it, so that Bob Shaftoe would not view him as a man who took such things lightly.
“You’re with me, then.”
“Not so much that I am with you as that we are with most of England, and England with us. You speak of putting Jeffreys to death with the strength of your right arm. Yet I tell you that if we must rely on your arm, strong as it is, we would fail. But if, as I believe, England is with us, why, then we need do no more than find him and say in a clear voice, ‘This fellow here is my lord Jeffreys,’ and his death will follow as if by natural law, like a ball rolling down a ramp. This is what I mean when I speak of revolution.”
“Is that a French way of saying ‘rebellion’?”
“No, rebellion is what the Duke of Monmouth did, it is a petty disturbance, an aberration, predestined to fail. Revolution is like the wheeling of stars round the pole. It is driven by unseen powers, it is inexorable, it moves all things at once, and men of discrimination may understand it, predict it, benefit from it.”
“Then I’d best go find a man of discrimination,” muttered Bob Shaftoe, “and stop wasting the night with a hapless wretch.”
“I simply have not understood, until now, how I might benefit from the revolution. I have done all for England, naught for myself, and I have lacked any organizing principle by which to shape my plans. Never would I have dared to imagine I might strike Jeffreys down!”
“As a mudlark, Vagabond soldier, I am always at your service, to be a bringer of base, murderous thoughts,” said Bob Shaftoe.
Daniel had receded to the outer fringes of the light and worried a candle out of a bottle on his writing-table. He hustled back and lit it from Bob’s candle.
Bob remarked, “I’ve seen lords die on battlefields—not as often as I’d prefer, mind you—but enough to know it’s not like in paintings.”