A uniform transverse impetus and the angle of incidence of an oblique and transverse impetus are given. Find the oblique and the downwards impetus.

The straight line AB represents the uniform transverse impetus and the line CA the oblique impetus with which a mobile falls on the underlying plane AB at A (Table 7.​1, Fig. 43). CAB is the angle of incidence. From the point B, BC is drawn perpendicular to BA and intersects AC at C. The impetus of the oblique movement is the resultant of the two velocities of the transverse and downwards movements. The three movements or velocities form a right-angled triangle ABC. In this triangle one side AB, the angle CAB comprising this side and the hypotenuse are given. Consequently, the kind of the triangle ABC is also given. Thus, the ratio of the side AB to the side BC and that of the side AB to the hypotenuse AC are known. Therefore, the oblique impetus AC is given in length as well as the downwards impetus CB.

Moreover, the transverse and the downwards impetus are given. Find the oblique impetus and the angle of incidence of the oblique impetus on the transverse one.

Again the straight line AB represents the uniform transverse impetus and the normal straight line BC represents the downwards impetus (Table 7.​1, Fig. 43). Obviously, the oblique impetus is equal in power to these two impulses. The ratio of the transverse impetus AB to the downwards impetus BC is given. Thus the triangle ABC is a right-angled triangle. Therefore, the oblique impetus AC is known as well as the angle of inclination of the incidence CAB.

It is demonstrated that the same is true when the impetus is uniformly accelerated.

The same mobile moves away from D provided with two velocities, one uniform transverse BA, and the other uniformly accelerated downwards (Table 7.​1, Fig. 44). Its impetus increases continuously until it arrives at the contact of the underlying plane AB with a maximum impetus acquired during the descent. The extreme force of this downwards movement thus is that with which the mobile falls at A. Since it is combined with the transverse impetus BA, in its incidence at A the mobile is carried by an oblique impetus which is then equal in power to the sum of the transverse impetus AB and the maximum downwards impetus at B, or A. Moreover, because of the curved route DA, its incidence on the underlying plane AB occurs at the inclination at which it finally meets the plane. But this last inclination is measured by the inclination of the straight line AC tangential to the curve AD at A. Consequently, the inclination of the incidence is measured by the angle CAB. Thus, the mobile strikes the underlying plane AB, following a curved route as if it was carried straight along the tangent CA in an uniform movement. The two impulses are perpendicular to each other, the uniform transverse BC and the downwards one through DB which at its incidence at A is the greatest of all the preceding continuously increasing velocities and is measured by the straight line BC. The oblique impetus is the resultant of these two impulses. It is thus known as well as the angle of incidence CAB.

The energy of an oblique percussion is measured by the sine of the angle of incidence.

The curved impetus DA forms an angle of incidence CAB with the underlying plane AB (Table 7.​1, Fig. 44). The impetus is the resultant of a uniform transverse movement AB and the greatest and final degree of velocity of the movement at the incidence perpendicular to the underlying plane. Thus, the ratio of the force and energy of the percussion which would be carried out perpendicularly on the stable plane sustaining the blow AC to the oblique percussion on the plane AB is equal to the ratio of the hypotenuse AC to the upright CB or to the ratio of the trigonometric radius to the sine of the angle of incidence. Actually, the resultant curved movement carries out a percussion at an oblique incidence by an angle CAB. Consequently, the energy of its oblique percussion will be measured by CB, the sine of the angle of incidence.

From this it is deduced that projectiles thrown in a horizontal direction carry out a combined movement, made of a uniform horizontal movement with a constant impetus and a uniformly accelerated downwards movement which is that of heavy objects which fall. This is shown by experience. Since the downwards movements of heavy objects, as far as our senses can perceive, occur along parallel lines BC because of the huge distance from the centre of the earth and the comparative smallness of the fall of the projectile, the combined movement is carried out along a parabola.

This being supposed, I claim that the same body, whatever the velocity at which it is impelled horizontally, in its oblique incidence on a horizontal underlying plane carries out percussions always equally strong and of the same energy.

Let the same body D be impelled from the position D by any motive virtue horizontally, i.e. in a direction parallel to the underlying horizontal plane BA (Table 7.​1, Fig. 44). In this way, the transverse impetus transmitted by the throwing subject is one and the same. It is thus uniform. But, after the transverse blow delivered by the throwing subject, the fall of the body resulting from its innate gravity immediately begins. It is uniformly accelerated. Consequently, the combination of these movements results in the oblique movement DA in a parabola the amplitude of which is larger or smaller depending on whether the uniform velocity impressed by the throwing subject is larger or smaller. But the height or axis of the parabola DB remains the same. Consequently, the very last impetus acquired at the end of the fall will be absolutely the same either in an ampler or in a shorter parabola. This impetus is measured by the uniform movement carried out along the straight line CB which is twice the height of the parabola DB. Therefore, CA is a tangent to the parabola and indicates the inclination of the incidence and the hypotenuse AC shows the impetus of the body in the act of incidence. The projectile falls obliquely at A provided with a double impetus, one horizontal AB and the other perpendicular CB. It acquired the latter at the end of the fall at B. The transverse impetus of the falling projectile does not carry out any percussion on the underlying plane BA since the direction of its movement is parallel to the underlying plane BA. Only the other perpendicular impetus remains able to carry out a percussion. Consequently, the energy of the oblique percussion of the projected body along the curved line DA is measured precisely by the sine of the angle of incidence, i.e. by CB or by the impetus of the fall, i.e. the impetus which was acquired at the very end of the descent. Since, whatever the amplitude of the parabola, its height DB always remains the same, for any amplitude of the parabola, the impetus at the end of the descent also remains the same. Consequently, whatever the velocity at which the projectile is impelled horizontally, it always strikes the underlying plane AB with the same energy. Q.E.D.