Open the Principia expecting equations and you will not find them. There are no derivatives, no integral signs, and almost no algebra. What you find is diagrams, triangles, ratios of line segments and propositions that read like Euclid.
This is odd, because Newton had invented calculus around 1665 and 1666, more than twenty years before the Principia appeared in 1687. He had the tool. He did not use its notation.
The Explanation Everybody Gives, and Why It Is Wrong
The standard story is that Newton worked everything out in calculus privately and then translated the results into geometry, either to conceal his method or to make the book respectable.
It is an attractive story and it has been repeated since the eighteenth century. It does not survive contact with the manuscripts.
Newton’s working papers were catalogued and published in the twentieth century, above all by D. T. Whiteside in eight volumes, and they show him reasoning geometrically while doing the work. The drafts are geometric. The corrections are geometric. There is no hidden analytic version of the Principia that somebody later dressed up in classical clothing, and Newton’s own claim to the contrary, made late in life during his dispute with Leibniz, is not supported by his papers.
So the geometry is not a disguise. It is how he thought about this particular subject, and there were good reasons for it.
Reason One: Geometry Was Proof, Algebra Was Not
In the seventeenth century the Euclidean tradition was the standard of mathematical certainty, and it had held that position for two thousand years. A geometric demonstration was a demonstration. Algebraic manipulation was regarded as a method of discovery rather than a method of proof, useful for finding results that then had to be established properly.
Newton held that view more strongly than most of his contemporaries. He spent a great deal of effort trying to reconstruct lost works of Greek geometry, he admired Apollonius and Pappus, and he was rude about algebra in private, describing it at one point as the analysis of the bunglers in mathematics. He thought the ancients had possessed a superior method and that modern algebra had traded certainty for convenience.
He was proposing something enormous in the Principia: that one law governs apples, the Moon, the planets and the tides. A claim of that size had to be unassailable in its form. Presenting it in the notation of certainty was a rhetorical decision as much as a mathematical one, and it was the right one.
Reason Two: Nobody Had Seen His Calculus
There is also a blunt practical point. In 1687 Newton’s method of fluxions was unpublished. He had written it up in the early 1670s and left it in a drawer. It did not appear in print until 1736, nine years after his death.
A book whose proofs depended on an unpublished notation invented by its author would have been unverifiable. No reader could have checked it, and the fellows of the Royal Society had no obligation to take Newton’s private symbols on trust.
Writing in geometry meant that any competent mathematician in Europe could follow the argument, step by step, using tools they already had. Given that the point of the book was to convince people, this was not a sacrifice. It was the whole strategy.
Reason Three: The Calculus Is In There
The most important correction to make is that the Principia does contain calculus. It simply does not contain calculus notation.
Book I opens, before any physics, with Section I: eleven lemmas on what Newton calls ultimate ratios of vanishing quantities. Lemma I states that quantities which converge continually to equality in a finite time become ultimately equal. This is a definition of a limit, written in words, a hundred and thirty years before Cauchy.
The lemmas that follow establish that areas under curves can be found as the limit of sums of inscribed and circumscribed rectangles as the rectangles become infinitely narrow. That is integration. It is presented as geometry because the objects are areas and lines, but the logical content is the content of a first calculus course.
The clearest case is Proposition 1 of Book I, the proof of Kepler’s law of areas. Newton takes the orbit, divides the time into equal intervals, replaces the curved path with a polygon of straight segments, shows that the triangles swept out in equal times have equal areas, and then lets the number of segments increase without limit so that the polygon becomes the curve. The result is exact.
Read it once and it looks like classical geometry. Read it again and it is a limiting process applied to an infinite sum, which is calculus doing the work while wearing Euclid’s coat.
What Newton actually invented for this book was neither Greek geometry nor the calculus of the textbooks. It was a hybrid, a geometry of limits, built for the specific problem of motion under a continuously varying force, and it has no real successor because the analytic notation replaced it within a century.
Reason Four: He Wanted It to Be Difficult
Newton told William Derham that he had made the Principia abstruse on purpose, in order to avoid being baited, as he put it, by little smatterers in mathematics.
It is worth taking that at face value. He had been through the 1672 dispute over his paper on light and colours, he loathed public argument, and he had concluded that an impenetrable book would be read only by people capable of understanding it. The strategy worked. The Principia was recognised as monumental immediately and understood by perhaps a few dozen people in Europe for a generation.
What It Cost, a Century Later
The style of the Principia had one serious long term consequence, and it fell on Newton’s own country.
After the priority dispute with Leibniz turned poisonous, British mathematicians treated Newton’s fluxional notation and his geometric style as a matter of national loyalty, while the Continent adopted Leibniz’s differentials, which are far easier to manipulate.
The results are not subtle. Euler, the Bernoullis, Clairaut, d’Alembert, Lagrange and Laplace rebuilt mechanics in analytic form and solved problems Newton had left open. Laplace’s Mécanique Céleste, published between 1799 and 1825, is essentially the Principia rewritten in calculus, and it goes considerably further because the notation lets it.
British mathematics spent the eighteenth century falling behind, defending dots against d’s. The situation was only broken in 1812 when a group of Cambridge students including Charles Babbage and John Herschel founded the Analytical Society to campaign for Continental notation, a campaign they summarised as the principles of pure d-ism against the dot-age of the university.
So the geometry that made the book persuasive in 1687 helped to stall a mathematical tradition for a hundred years. Newton is not to blame for that. His successors are, for confusing a notation with a flag.
Reading It Now
This is why the Principia has a reputation for being unreadable, and the reputation is half deserved. The physics is not the obstacle. The obstacle is that the proofs are conducted in a mathematical language nobody has been taught since about 1820, so a modern reader with three semesters of calculus is worse equipped for Proposition 1 than a competent geometer of 1700.
Two things help. The first is to read the diagrams rather than the text, since Newton’s arguments are largely about what happens to particular lines and triangles as points approach one another. The second is to know that Subrahmanyan Chandrasekhar spent his last years producing Newton’s Principia for the Common Reader, which restates the propositions in modern terms alongside Newton’s originals.
Our edition of the Principia keeps the structure Newton built: the definitions and axioms, then Section I with its lemmas on vanishing quantities, then the three books. Seeing the eleven lemmas sitting in front of the physics, where Newton put them, makes the design of the book obvious in a way that no summary conveys. He is telling you, before he starts, exactly which mathematical tool he is about to use for four hundred pages.
There is a fuller account of how the work is organised in our post on the three books of the Principia.
No products in the cart. 