Mathematical Principles of Natural Philosophy

Overview

Nine hundred and seventy pages in the modern edition, in three books, written in Latin, published in 1687 after Halley asked Newton what curve a planet would follow under an inverse-square force and Newton said an ellipse, he had calculated it, and could not find the paper.

You should know before you begin that this is a mathematics book, and that the mathematics is not the calculus Newton had invented. It is classical geometry: ratios, limits of vanishing quantities, and propositions proved with diagrams in the manner of Euclid. Newton chose that deliberately, partly so the results would be unimpeachable to readers trained on the Greeks. The effect is that a modern physicist finds it harder than a modern classicist expects.

Nobody outside a specialist course reads it whole. What is assigned is the framing material and a handful of propositions, and that is a complete and honest way to meet it.

The argument

The front matter is where most readers spend their time and it is where the revolution is. Newton opens with definitions: quantity of matter, quantity of motion, inertia, impressed force, centripetal force. Then the scholium on absolute space and time, which asserts that true time flows equably without relation to anything external, an assumption Leibniz attacked immediately and Einstein eventually dismantled. Then the three Laws of Motion, stated in half a page, with corollaries giving the parallelogram of forces.

Book I works out the motion of bodies under centripetal forces in a void. The key results: a body moving under a force directed at a fixed centre sweeps equal areas in equal times, which is Kepler’s second law; and if the orbit is an ellipse with the force directed at a focus, the force varies as the inverse square of the distance. He proves the converse too, which is what lets him move from observation to law.

Book II treats motion in resisting media, and is the part now read least. It matters historically for a negative reason: Newton shows that Descartes’ vortices, the reigning explanation of planetary motion, cannot produce Kepler’s laws. He is demolishing the competition.

Book III, the System of the World, applies all this to the sky. It opens with the Rules of Reasoning in Philosophy, four short statements that are the methodological heart of the whole enterprise. Then the phenomena: the satellites of Jupiter and Saturn, the planets, the Moon. Then the argument, moving from those observations to a single universal gravitation, proportional to mass and inversely to the square of distance, acting between every pair of bodies in the universe. From it he derives the orbits, the shape of the Earth, the precession of the equinoxes, the tides, and the paths of comets.

The General Scholium was added to the second edition in 1713, in response to complaints that gravity acting at a distance was an occult quality. Newton’s answer is to decline the question. He has deduced gravity from the phenomena and has no hypothesis about its cause, and refuses to offer one, because hypotheses have no place in experimental philosophy.

How to read it

Read the Definitions, the scholium on space and time, the Laws of Motion and their corollaries, the Rules of Reasoning at the start of Book III, and the General Scholium. That is perhaps forty pages and it contains the method, the metaphysics and the argument about God. If your course assigns Newton, this is almost certainly the assignment.

Then attempt one or two propositions from Book I. Proposition 1, the equal-areas result, is the right one: it is elegant, it needs no calculus, and working through the diagram shows you exactly how Newton reasons with vanishing quantities.

Use Cohen and Whitman. Their volume opens with Cohen’s Guide to the Principia, a book-length introduction that is the best thing written on how to read it, and their translation is the modern standard. Motte’s 1729 English is free and is what the Great Books set prints, but the archaism adds a difficulty Newton did not put there.

Do not expect F equals ma. The formula is not in the book, and neither is the inverse-square law as an equation; both are later algebraic restatements of propositions Newton gives geometrically.

The common misreading is that the General Scholium is a pious afterthought. Newton thought the argument from the arrangement of the solar system to an intelligent Being was as sound as anything else in the book, and he wrote far more on theology than on physics.

Questions it raises

• Newton declines to say what gravity is. Is that scientific restraint, or an admission that the theory describes without explaining?
• Absolute space and time are asserted rather than argued. What work do they do, and could the mechanics survive without them?
• The Rules of Reasoning assume nature is simple and uniform. What justifies those assumptions, given that they cannot be derived from experience?
• Newton proves that Cartesian vortices fail. Is disproving a rival the same kind of achievement as establishing your own theory?
• If the system of the world requires an intelligent Being to arrange it, what happens to that argument once later work shows the arrangement could arise naturally?
• Why write in the geometry of the ancients when he had invented a calculus that would have made it easier?