CHAPTER ONE

Corollary 4.

18. Therefore, if a moving body is elastic In 18th-century physics, an elastic body is one that bounces back perfectly without losing energy, whereas a non-elastic body might stop or lose speed upon impact., it will always be carried with uniform motion Constant speed. along any curve whatsoever. But if it is not elastic, only cusps A cusp is a sharp point where a curve suddenly reverses direction, like the point of a "V". will disturb the motion by completely removing it.

Scholion 3.

19. To understand these things more clearly, let there be two elements of the curve Euler treats a curve as if it were made of an infinite number of tiny, straight-line segments. AB and BC, and let the angle they form be ABC. Let CBD be the exterior angle The angle formed by extending the line AB and measuring the deviation of BC., which is infinitely small; let its sine be $dz$, assuming the total sine The radius of the unit circle used for trigonometric calculations. is $1$. Because the body, after it has traversed the element AB, strives to proceed in the direction BD by its innate force What we now call inertia: the tendency of an object to keep moving in a straight line. with its previous speed, which we call $c$; its motion should be conceived as twofold: one part in the direction BC, and another in the direction perpendicular to BC, which cannot be put into effect Because the body is constrained to stay on the track/curve, the sideways component of its motion is blocked.. Therefore, by dropping a perpendicular DC from D onto BC, the body will move through BC with the other component of motion at a speed which is to the previous speed as BC is to BD; that is, as $\sqrt{1-dz^2}$ is to 1. Consequently, it will have a speed through BC equal to $c \sqrt{1-dz^2}$ or $c - \frac{cdz^2}{2}$ Euler is using a mathematical approximation (a Taylor series expansion) where $\sqrt{1-x}$ is roughly $1 - \frac{x}{2}$ when $x$ is very small.. For this reason, the decrease in speed will be $\frac{cdz^2}{2}$, which is equivalent to a differential of the second degree An "infinitesimal of the second order." In calculus, these are so small that they are considered zero when compared to ordinary changes (first-order differentials).. From this, it is understood that as long as the angle CBD is infinitely small in any curve, the body will proceed with uniform motion. However, in every curve, the angle is either infinitely small, or it is the angle ABC itself, which happens at cusps. Consequently, only cusps disturb the uniformity of motion, unless the body is elastic, in which case the uniformity of motion is nonetheless preserved.