Axiom .).—The alteration of motion either generated or destroxed in a body, is proportional to the force applied, and is made in the direction of that right line in which the force acts.
Axiom 3.—Action and reaction are equal between two bodies in opposite directions.
Axiom 4.—Two equal forces acting against each other, or against a body, in opposite directions, destroy each other's effect.
Axiom 5.—If a body is acted upon by two forces in oppo site directions, it is the same thing as if it were only acted upon by one fbree equal to their difference, in the direction of the greater force.
Axiom 6.-1f a body is kept in equilibrio by three or more forces, the sums of the contrary forces, when reduced to parallel directions, are equal.
Axiom 7.—When a right line is drawn in a direction of its length by two forces acting at its extremities, the line may either be flexible or inflexible.
Axiom 8.—When a right line is pressed or pushed by two forces in the direction of its length, and retains its straight ness, the right line is inflexible.
Axiom 9.—When a right line is stretched by two forces, the right line draws each of the forces with the same intensity that the forces stretch the line ; because action and reaction are equal and contrary.
Axiom 1 0.—W hen a right line is pressed by any two forces at its extremities in the direction of the line, it repels the force with the same intensity with which it is pressed by the forces.
Axiom 11.-1f two forces act upon a body and keep it in equilibrio, their lines of direction are in the same right line, and the two forces are equal, and have opposite tendencies.
Axiom I2.—A force pulling by a string or flexible line upon one side of a body, has the same effect in moving or in keeping it in equilibria as an equal force pushing or pressing on the Mine line of direction on the other side.
I3.—A force acting upon a body has the same 1)0\s-or in whatever point of the line of 4111'3.'071On it is applied.
Axiom 14.—If a line he pressed or drawn by two opposite forces in the direction of the line, all its parts will be equally stretched or compressed.
PuNtubite.--t :rant that the intensities of forces may be represented by right lines, as well as their Proposition 1.—Plate 1. Figure 1.-1f any body, A, be moved by any impulse which would cause it to describe the right line A 13, uniformly in a given tune; and if the same body, A, be moved by another impulse, which would cause it to describe the right line A n, uniformly in an equal time : acting at the same instant, would carry the body through the diagonal, A c, of a parallelogram A 13 C D.
For the impulse which is given in the direction A D, will not prevent the body from coming to D c, by the action of the impulse in the direction A D, in an equal time to that in which A D would have been described by the separate impulse : for the same reason the impulse which is given in the A D, will not prevent the body from coniing to C, by the action of the impulse in the direction A n, in an equal time to that in which A 13 would have been described by the separate impulse : therefore, as the body will meet the lines u c and D c at the same time, it will meet in the inter section c ; but because the lines A D 3111d A u, are uniformly described in the same time. any two parts, A a and A F, taken from these lines in the ratio of the lines themselves, will also be described in eqaa! times ; and because 13 C is equal to A D, and a 0 equal 130 AF; A B:BC::AE:EO; therefore the body moves in a straight line which is the diagonal of the parallelogram.
Corollary 1.—Hence, if the direction, intensity, and tendency, of any two forces acting upon a solid are given, a single force may be found, which shall be equivalent to the two.
Corollary 2.—Any single force, whose quantity and line of direction are given, may be resolved into two forces, which shall act at a given point in that line, in two given directions.
Proposition 11.—Given, the tracts, intensities, and ten dencies of two forces making any angle with each other, to find a single force equivalent to them.
Case I. Figure 2.—When each of the two given forces have a tendency, from the points A and c towards B, or from B towards A and c. Complete the parallelogram A B c D, and draw the diagonal 13 n, and it will represent the quantity and direction of the third lbrce, that will be equivalent to A B and c n; and its tendency will be from B towards D, when the extreme forces tend towards II ; but towards n when the extreme forces have a tendency from B.