Mr Gregory devoted to it a chapter of his work, and noticed particular — Differential equation

"[W]e find an excellent tract by James Bernoulli concerning the elastic curve, isochronous curves, the path of mean direction in the course of a vessel, the inverse method of tangents, &c. On most of these subjects he had treated already; but here he has given them with additions, corrections, and improvements. His scientific discussions are interspersed with some historical circumstances... This memoir concluded with an invitation to mathematicians, to sum up a very general differential equation, of great use in analysis. The solution which James Bernoulli had found of this problem, as well as those which Leibnitz and John Bernoulli gave of it, were published in the Leipsic Transactions."

"We should speak of a dialectics of the calculus... the problem element in so far as... distinguished from the properly mathematical element of solutions. Following Lautman... a problem has three aspects: its difference in kind from solutions; its transcendence in relation to the solutions... and its immanence in the solutions which cover it, the problem being the better resolved the more it is determined. Thus the ideal connections constitutive of the problematic ([Platonic] dialectical) Idea are incarnated in the real relations which are constituted by mathematical theories and carried over into problems in the form of solutions... like the discontinuities compatible with differential equations."

"A of the Riccati type is derived for the of the optimal filtering error. The solution of this equation completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or non-stationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the . Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The duality principle relating estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of s."

"When [Born and Heisenberg and the Göttingen theoretical physicists] first discovered they were having, of course, the same kind of trouble that everybody else had in trying to solve problems and to manipulate and to really do things with matrices. So they had gone to Hilbert for help and Hilbert said the only time he had ever had anything to do with matrices was when they came up as a sort of by-product of the eigenvalues of the boundary-value problem of a differential equation. So if you look for the differential equation which has these matrices you can probably do more with that. They had thought it was a goofy idea and that Hilbert didn’t know what he was talking about. So he was having a lot of fun pointing out to them that they could have discovered Schrödinger’s wave mechanics six month earlier if they had paid a little more attention to him."

"Science is a differential equation. Religion is a ."

"Newton almost entirely melted down the treatise of Quadratures into another entitled, the Method of Fluxions, and of Infinite Series. This contains only the simple elements of the geometry of infinite, that is to say, the methods of determining the tangents of curve lines, the common maxima and minima, the lengths of curves, the areas they include, some easy problems on the resolution of differential equations, &c. The author had it in contemplation several times to print this work, but he was always diverted from it by some reason or other, the chief of which was no doubt, that it could neither add to his fame, nor even contribute to the advancement of the higher geometry. In 1736, nine years after Newtons death, Dr. Pemberton gave it to the world in english."