If a body moves straight parallel to the surface of another stable body, whether it touches it without compression or does not touch it, no percussion will occur.

Let a body A move straight at any velocity along the line AD. The direction of this movement AD is parallel to the surface BC of a body at rest (Table 7.​1, Fig. 31). Either A touches the surface BC or not. A lacks any compressive impulse towards the plane BC. I claim that A does not carry out any percussion against the plane BC. If A does not touch the plane BC, it will not at all touch it during its movement, as a result of the parallelism of the movement AD to the underlying plane BC. Without contact no percussion occurs. If the body A touches the plane BC at B, since a simple contact without any compression is supposed, through all the progress along AD parallel to BC, there is simple contact without mutual compression. But percussion is a compressive effort carried out with some violence which the body subjected to resists. Consequently, since here there are no compressive movement and impetus, nor resistance of the underlying plane, no percussion will occur. Q.E.D.

As must be noted here, the movement of the body A is true rather than relative either to the space of the world or even to the underlying plane BC. Actually, the body A moves from one place of the space of the world and of the underlying plane BC to another by successive contacts. However, as was shown, no percussion occurs. Consequently, this is a third way of preventing percussion.

If two bodies move on parallel planes either in the same direction or in opposite directions or transversely, although they touch each other without compression, they do not strike each other.

Let two bodies A and B move at any velocities on the parallel planes AD and BC (Table 7.​1, Fig. 32). Whether the courses of the movements AD and BC are in the same direction or whether they are in opposite directions or whether they intersect transversely, the two bodies A and B meet and touch each other without any compression. I say that the body A does not carry out any percussion against the body B and the body B resists in no way. This results from the fact that, when they meet, when mutual contact of the bodies occurs, since the impulses always are exerted in parallel planes, a body never compresses the other nor resists it. Consequently, no percussion occurs.

Here also the movement of each of the bodies A and B performed in the space of the world remains true although the movement does not always remain relative. However, there is no percussion. This is a fourth way by which percussion can be prevented.

For the following propositions, the next proposition must be used. Although it is generally assumed to be unquestioned, it can be confirmed by a demonstration. Therefore, it must not be counted by its own right among the unquestioned truths.

If two forces applied at the extremities of the two equal arms of deflected scales, one pulling perpendicularly and the other obliquely, are balanced, the ratio of the absolute force pulling obliquely to its moment in a parallel direction or to the absolute force pulling perpendicularly is equal to the ratio of the arm of the scales to the distance of the oblique direction from the fulcrum.

In the deflected scales ABF, the two arms AB and BF are equal and at the fulcrum they form an angle B (Table 7.​1, Fig. 33). A force D is applied at A and a force GH at F. D pulls the arm BA in the direction DA at right angles to the arm BA of the scales and the other force GH pulls the scales in an oblique direction FG, parallel to AD. The angle BFG of the direction is either acute or obtuse. The straight line AB is prolonged until it intersects the oblique direction FG at K. The moments of the bodies D and GH are equal. I claim that the ratio of the absolute force GH to its moment in the oblique position or to the absolute force D is equal to the ratio of the arm AB of the scales to the distance BK of the direction. A and F are connected by a line. From B the straight line BO is drawn parallel to KF or AD and intersects AF at O. AR is made equal to FO. The force GH is divided so that the ratio of its portion H to its portion G is equal to the ratio RO/OF. By combining: HG/G = RF/OF = AO/OF since RF = AO. The straight scales AF are balanced and immobile about point O, at the end of the vertical line BO drawn from the fulcrum B. Two forces D and GH of equal moments are applied at the ends of the straight scales AF. Consequently, the ratio of the absolute force GH to the absolute force D is equal to the inverse ratio AO/OF. But previously one had: HG/G = AO/OF. Therefore, the ratios of the sum of the forces GH to the two forces D and G are equal. Therefore, the two forces D and G are equal. But the moment of GH was equal to the moment of force D. Thus, the ratio of the absolute force GH to its moment in such an oblique position or to the absolute force of D is equal to AO/OF = AB/BK (since the lines BO and KF are parallel). And BK is actually the distance of the oblique direction from the fulcrum B. Q.E.D.

If a body moves straight and at a constant inclination to the surface of another body absolutely immobile, the ratio of the moment of the impetus in the inclined plane to the total impetus of the body is equal to the sine of the complementary angle of incidence to the trigonometric radius.

Let a body A move along the straight line ABE and collide with the fixed plane DBC at an acute angle of inclination ABD (Table 7.​1, Fig. 34). CF is drawn perpendicular to AE and intersecting AE at E. X is the total impetus of A and Z is the moment of the impetus of the body A, exerted on the inclined plane DC. I claim that Z/X is equal to the ratio of the sine BE of the complementary angle C of the angle ABD to the trigonometric radius BC. From B a straight line BG is drawn perpendicular to DC and intersecting FC at G. FG is equal to GB. A body F equal and homogeneous to A is set at F. Its movement is parallel to AB and its impetus is equal to that of A. A, or B, although pushing in the direction BE, is forced to follow the inclination of the plane along the straight line BC, i.e. along the tangent to the circle of radius GB. Thus, the body A continues to act at B as if it impelled the radius of the scales GB obliquely about the fulcrum G. Two equal and homogeneous bodies provided with an equal motive virtue impel the ends F and B of the equal arms of the deflected scales FGB in directions parallel to each other and perpendicular to FG. F acts directly and perpendicularly, and B very obliquely in the direction BE. Therefore, the ratio of the total impetus of F, or B, or A to the moment of the impetus in the oblique position of B or A is equal to FG/GE = GB/GE or to the ratio BC/BE. Thus, the ratio of the moment of the impetus of A compressing at B the inclined plane BC to the total impetus of A is equal to BE/BC. Q.E.D.