. Annals of Philosophy. 212 Mr. Her'apath on the Law of Continuity. [March", relations and properties to the distance, some interesting con- clusions may be deduced ; but we shall at present only notice one or two well known theorems of motion, and then proceed to the demonstration of the saltation of states. Suppose the perpendicular distance to be always moving parallel to itself, and as the velocity of a moving body, the straight line which it cuts off from a given point being as the time, then will the area described by the distance be as the space or distance described by the moving
. Annals of Philosophy. 212 Mr. Her'apath on the Law of Continuity. [March", relations and properties to the distance, some interesting con- clusions may be deduced ; but we shall at present only notice one or two well known theorems of motion, and then proceed to the demonstration of the saltation of states. Suppose the perpendicular distance to be always moving parallel to itself, and as the velocity of a moving body, the straight line which it cuts off from a given point being as the time, then will the area described by the distance be as the space or distance described by the moving body. For the incre- ment or fluxion of this area will be equal to the fluxion of the right line drawn into the perpendicular distance, that is, to the fluxion of the time drawn into the velocity. But the product of the velocity into the fluxion of the time is well known to be equal to the fluxion of the space. Therefore the space and this area have always the same rate of increase, and are consequently, equal, or in a given ratio. The same conclusion may be a» easily obtained by the method of exhaustions. Again, let the perpendicular R S be drawn any where in the line N O, and let its intersection with the curve be S. Upon R S construct the little rectangle R T, which is the fluxion of the space described by the moving body. Let W be the inter- section of the curve and of the side X T, and let S V be taken equal to twice S T ; and with T X, T V, and T W, T V, construct the rectangles V X, V W. Then because the time flows equably, the equals R X, X Y, repre- sent the fluxions of the time at the moments R X; and the rectangles S X, W Y, the corresponding fluxions of the spaces. Consequently the rectangle W V (= + WT x R X), is the second fluxion of the space; T \V being the fluxion of the velocity. And since (v) the fluxion of the velocity is equal to the force (f) multiplied by the fluxion (/) of the time, we shall have + x = ft*; an equation well known in the higher branches
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