Abstract
II. IN dealing with the falling body I had to ask you to think what is the speed at any moment of a body which is changing its speed every moment, every half moment, every hundredth part of a moment or what we call continuously. It is easy to see that it has some speed at every point, and that the speed at every point is quite definite. I indicated a way in which we could fix this approximately, by taking the average speed over short intervals. A similar question is raised in considering the path of the projectile. Its direction changes from point:o point. The bullet is shot towards the east, and, for the sake of picturing its path, I imagine the lines vertically upward to be called northwards, as on a vertical map. At first the particle starts off, let me say, in a direction N.N.E. When it has reached the top of its path it is going horizontally—due east—when it has got back to the level the Northing has been turned into Southing, and it is going S.S.E. In its upward motion it changes continuously from N.N.E. to E. At a certain position it is half a point more to the east and less to the north; further on, a point more; further on again, the Northing has disappeared. The path has curved away; it is curving away at every point of it. A particle moving at a uniform rate in a circle changes its direction; but at every point the amount of curvature or immediate bending away from the direction in which the particle moves at any moment is the same. In a small circle the curve bends away faster than in a larger one from the line which represents the direction of motion at any point, but in each separate circle the measure of bending must at every point be the same. How will it be in a different kind of curve, such as an ellipse, or the path of a projectile, a parabola? As the speed of falling changes from moment to moment continuously, the curvature changes from moment to moment.
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Galileo and the Application of Mathematics to Physics 1 . Nature 21, 58–61 (1879). https://doi.org/10.1038/021058a0
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DOI: https://doi.org/10.1038/021058a0