Defintion 15

Table of Contents

Definition 15

Hereafter we will call the height corresponding to the speed that height, from which a weight falling to the surface of the earth, acquires that same speed.

Corollary 1

This height must therefore be as the square of the speed, to which it refers. With the speed c arising and with the due height v, v shall be as c2.

Scholium 1

This far we have expressed the speed on a straight line, which can be traversed by the speed in a given time. But in the following it will be more convenient to introduce the corresponding height in place of this

On account of this we put v = cc and vc = .

We will therefore have in the preceding problem this equation …

Corollary 2

Therefore in what follows, in place of the speed c it will be permitted to put v or the square root of height that corresponds to the speed.

Corollary 3

If the force g denotes that of gravity itself, then x will be the height corresponding to the speed c, and thus v = x. For indeed A .. from which it therefore follows that 2g An = .

From this we have gained a convenience, as we have determined the value of the letter n, which must maintain the same value in all cases (155).

Scholium 2

Since g signifies the force of gravity [i. e. the weight of the body; do not confuse g with our symbol for the acceleration of gravity, which it does not represent], then A g is a constant quantity (197). Therefore we can put this as 1, as that is allowed, since the force to the [mass] does not [yet] have a defined ratio. And hence it easily shows the ratio of A g in all cases, or the value of the applied force to the [mass of the] body.

The ratio A g to 1 or g : A as the force g, acting on the body A, is to the weight [this should be mass], that the same body may have in our part of the world.

Therefore the letter A will no longer denote the quantity of matter, but the weight A of the body itself, [since A.1 is the weight] if it should be placed on the surface of the earth. In this way we will compare all forces with weights, since that will add a great deal of light to the measurement of forces.

Corollary 4

Since g An 2 = , g denotes the force of gravity [i. e. the weight of the body and not the acceleration] and if 1=g A , then 2 1=n . That value will always be retained, if the speeds are to be expressed in terms of the appropriate square roots of the heights. And thus in our situation, this becomes A gx A gdx vdv == and . [p. 82. Thus, Euler chooses as his working dimensions not distance and time but acceleration and time. The time is measured in seconds, and with the acceleration of gravity A/g = 1, then 2 1=n .]

Corollary 5

On this account in the general law A .. npds cdc = (157), if the height v of the corresponding speed is c, then 2 dvcdc = , and thus on account of 2 1=n this law is obtained A pds dv = , where p is to A as the force p is to the weight of the body A.

Corollary 6

In a like manner, indeed the equations set out in (161) and (163) : ,and 2dsAcnprdxnpdyAcdc == by substituting v in place of c 2 and 2 1 in place of n, are transformed into ,2and AvdsprdxpdyAdv == where p to A has the ratio in the manner given.

Corollary 7

And in (165) it is found that 2 p Avr = or pr = 2Av. Likewise in (165) there is obtained ,pdsAdv = and in the case of (167) there is obtained .pdsAdv −= And in this manner we have reduced the previous variable quantities n and c to fixed values.