Every now and again I take a random book down from my bookshelf and read it. Sometimes there is stuff worth talking about.
This time it is V. Arnol’d’s book Huygens and Barrow, Newton and Hooke. This was written for the 300th anniversary of the appearance of Newton’s Principia, and despite the title, Newton is the hero of the book. This is definitely not academic history. Arnol’d is clear about Newton’s defects: the way he treated Hooke, his unethical behaviour over the commission on the invention of calculus, and so on. But, nevertheless, Newton had outstanding geometric intuition, which Arnol’d values.
For example, according to Arnol’d, Leibniz proceeded formally. He knew that d(x+y) = dx+dy, and assumed that the same sort of rule would hold for multiplication (that is, d would be a ring homomorphism). He only realised he was wrong when he had worked out the unpleasant consequences of this assumption. Newton, who thought geometrically, saw the rectangle with small increments of both sides, showing immediately that d(xy) = x.(dy)+(dx).y.
Indeed, he is often ready to give Newton the benefit of the doubt based on his geometric intuition. For example, how did Newton show that the solutions to the inverse square force law were conics? He showed that, for any initial conditions, there is a unique conic which fits those conditions. But doesn’t this assume a uniqueness theorem for the solution of the differential equation? No problem. Newton knew that the solutions depend smoothly on the initial conditions, and this fact obviates the need for a uniqueness theorem. (“This … can be proved very easily”, and Arnol’d gives a proof for our edification. The assumption is that, either Newton knew this, or it was so obvious to him that he felt no need to give a proof.)
One of the most valuable parts of this thought-provoking book is that modern developments are discussed in some detail. For example, Newton calculated the evolvent of the “cubic parabola” y = x3. (Arnol’d prefers the term “evolvent” to the more usual “involute”, according to the translator. This is the curve obtained in the following way: take a thread of fixed length attached to the curve at one end and lying along the curve; then “unroll” it, so that at any stage the thread follows the curve for a while and is then tangent to it. The evolvent is the curve traced out by the other end of the string.) The evolvent turns out to have the same equation as the discriminant of the icosahedral group H3. This leads us to the occurrence of five-fold symmetry in quasicrystals, which has been discovered in nature (these discoveries were quite recent when Arnol’d was writing).
Another highlight is the discussion of Newton’s remarkable proof that an algebraically integrable oval (one for which the area cut off by a secant is an algebraic function) cannot be smooth; at least one of the derivatives must have a singularity. This implies that the position of a planet following Kepler’s laws cannot be an algebraic function of time. There is much more too; Newton’s analysis went much further, but was a forerunner of “impossibility proofs” in mathematics, which flowered much later.