Chapter 1b

Proposition 1

Motion is the translation of a body from the place it occupies to another place. True rest is a body remaining at the same place.

Euler
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PROPOSITION 1. Theorem.

  1. A body or a point, which is moving on a given line and is acted on by no forces, always keeps the same speed, only if any two adjoining elements of this line nowhere constitute a finite angle.

Demonstration

Since the body, while it is moving on the line AM (Fig. 1), is acted on by no force and neither does the curve have any friction, and the motion of the body is unable otherwise to be changed, except in as much as it is impeded by the line AM , on which a small body is able to move freely; from which any change in the speed which should arise is to be investigated. Let the speed which the body has at M be equal to c; here therefore with this speed, the body progresses along the tangent Mv, if it could move freely ; which now, since the body is unable to leave the curve AM, is unable to happen, for the body is forced to progress along Mm. Hence for this reason the motion of the body along Mv can be considered to be resolved into the motion along Mm and the motion along Mn, with the right- angled parallelogram Mmvn arising. It is evident that this motion along Mn, the direction of which is normal to the element of the curve Mm, is to be absorbed by the curve unless there is no change in the motion. Therefore the body progresses with another motion along Mm with a speed, which is to the former speed as Mm to Mv; whereby the speed, .c

Since truly with which the body describes the element of the curve Mm, is equal to Mm Mv Mvm is the triangle for the rectangle mn, thus Mm < Mv, [p. 6] this speed is less than the previous speed c and the decrease in the speed is equal to ( Mv − Mm )c. For the value of

…

this can be found MO, the radius of osculation of the curve at M which is equal to r and the element Mm is equal to ds; and this is, on account of the angle O = angle mMv, MO : Mm = Mm : mv , from which there comes about mv = dsr and …

From this the decrement in the speed can now be obtained, while the element of the curve ds is traversed, equal to cds2 , of which with the whole gives the decrement of the speed, ..

while the body traverses the finite portion of the curve AM. But the expression cds2 is

..

equivalent to a differential of the second order ; therefore the integral of this is a differential of the first order. On account of which the decrease of the speed, after the body has traversed an arc of some given size, is infinitely small and the body is carried with a uniform motion along the whole curve AM, only if the radius of osculation r was nowhere infinitely small. Q.E.D.

[We meet this kind of geometrical argument time after time in Euler’s work, where we would now refer to the principle of conservation of energy; in the present case, no work is done on the particle.]

Corollary 1

  1. Therefore in any curve, in which the radius of osculation is nowhere infinitely small, the body moves uniformly, if indeed it is not allowed to be acted on by any forces or friction.

Corollary 2

  1. If the radius osculation is infinitely small, then cds2 is either a finite quantity or is a .. differential of the first order. [p. 7] In the first case the body parts with a finite change in the speed, in the other truly it is infinitely small.

Corollary 3

  1. Moreover since points of this kind are rare in all curves and are widely scattered between each other, and the body still travels uniformly along the arc intercepted by two such points.

Scholium 1

  1. The case, in which the body suffers a sudden finite decrease in speed, is only possible where the curve has cusps. For with these in place the body is forced to turn back directly and normally on the point of the cusp it strikes. Therefore the body then not only loses a finite step in its speed, but it must lose all of its motion entirely, except perhaps is put to be elastic, in which case it may be reflected with the speed with which it arrived, and thus the uniform motion is conserved. In a cusp, two elements of an infinitely acute angle are put in place.

Scholium 2

  1. Truly as well as cusps, other points can be given on curves, in which the radius of curvature is infinitely small ; because any two touching elements are placed in almost the same direction and following this the angle is infinitely small, but it cannot happen that the body suffers a finite decrease in the speed, as the above demonstration shows. On account of which, since points of this kind are rare, the body nevertheless moves with a uniform motion.

Corollary 4

  1. Therefore if the motion of the body were elastic, then the motion is always carried on uniformly on any curve ; but if it is not elastic, then the cusps only upset the motion, while evidently they destroy it.

Scholium 3

  1. In order that these may be made clearer, let two elements of the curve be AB and BC (Fig. 2) and following the angle ABC that they make, CBD is next put in place infinitely small, and the sine of this angle is dz, with the total sine put equal to 1. Because the body, after it describes the element AB, by the force of inertia, [vi insita] tries to progress along BD with the previous speed which was c, this motion is taken in two parts, the one in the direction BC, and the other in the direction normal to BC, which cannot be effected.

Therefore by sending the perpendicular DC from D to BC, the body moves along BC with another motion, with a speed which is to the first speed as BC to BD, i. e. as ( 1 − dz 2 ) to 1. Therefore the speed along BC is equal to c ( 1 − dz 2 ) or c − cdz

; whereby the decrement of the speed is cdz , which is equivalent to a … differential of the second order. From which it is understood, as long as the angle CBD of the curve is infinitely small, the motion of the body progresses at a steady rate. But on the curve the angle CBD is either infinitely small or the angle ABC itself is infinitely small, when the point falls on cusps. Consequently only cusps disturb the uniformity of the motion, unless the body was elastic, in which case the motion nevertheless is conserved.

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