COMPOSITION
, is a species of reasoning by which we proceed from things that are known and given, step by step, till we arrive at such as were before unknown or required; viz, procceding upon principles self-evident, on definitions, postulates, and axioms, with a previously demonstrated series of propositions, step by step, till it gives a clear knowledge of the thing to be known or demonstrated. Composition, otherwise called the synthetical method, is opposed to Resolution, or the analytical method, and is chiefly used by the ancients, Euclid, Apollonius, &c. See Pappus; also the term Analysis.
Composition of forces, or of motion, is the union or assemblage of several forces or motions that are oblique to one another, into an equivalent one in another direction.
1. When several forces or motions are united, that act in the same line of direction, the combined force or motion will be in the same line of direction still. But when oblique forces are united, the compounded force takes a new direction, different from both, and is either a right line or a curve, according to the nature of the forces compounded.
2. If two compounding motions be both equable ones, whether equal to each other or not, the line of the | compound motion will be a straight line. Thus, if the one equable in the direction AB be sufficient to carry a body over the space AB in any time, and the other motion sufficient to pass over AC in the same time; then by the compound motion, or both acting on the body together, it would in the same time pass over the diagonal AD of the parallelogram ABDC. For because the motions are uniform, any spaces Ab, Ac passed over in the same time, are proportional to the velocities, or to AB and AC; and consequently all the points A, d, d, D, of the path are in the same right line.
3. And though the compounding motions be not equable, but variable, either accelerated or retarded, provided they do but vary in a similar manner, the compounded motion will still be in a straight line. Thus, suppose, for instance, that the motions both vary in such a manner, as that the spaces passed over in the same time, whether they be equal to each other or not, are both as the same power n of the time; then ABn : ACn :: Abn : Acn, and hence AB : AC :: Ab : Ac, and therefore, as before, AddD is still a right line.
4. But if the compounding motions be not similar to each other, as when the one is equable and the other variable, or when they are varied in a dissimilar manner; then the compounded motion is in some curve line. So if the motion in the one direction EF be in a less proportion, with respect to the time, than that in the direction EG is, then the path will be a curve line EiH concave to wards EF; but if the motion in EF be in a grearer proportion than that in EG, then the path of the compound motion will be a curve EhH convex towards EF: that is, in general, the curvilineal path is convex towards that direction in which the motion is in the less proportion to the time. Hence, for a particular instance, if the motion in the direction EF be a motion of projection, which is an equable motion, and the motion in the direction EG that arising from gravity, which is a uniformly accelerated motion, or in proportion to the squares of the times; then is EG as GH2, and Eg as gh2, that is EG : Eg :: GH2 : gh2, which is the property of the parabola; and therefore the path EhH of any body projected, is the common parabola.
5. If there be three forces united, or acting against the same point A at the same time, viz, the force or weight B in the direction AB, and the forces or tensions in the directions AC, AD; and if these three forces mutually balance each other, so as to keep the common point A in equilibrio; then are these forces directly proportional to the respective sides of a triangle formed by drawing lines parallel to the directions of these forces; or indeed perpendicular to those directions, or making any one and the same angle with them. So that, if BE be drawn parallel, for instance, to AD, and meet CA produced in E, forming the triangle ABE; then are the three forces in the directions AB, AC, AD, respectively proportional to the sides AB, AE, BE.
And this theorem, with its corollaries, Dr. Keil observes, is the foundation of all the new mechanics of M. Varignon: by help of which may the force of the muscles be computed, and most of the mechanic theorems in Borelli, De Motu Animalium, may immediately be deduced.
See more of the Composition of Forces under the article Collision.
Composition of Numbers and Quantities. See COMBINATION.
Composition of Proportion, according to the 15th definition of the 5th book of Euclid's Elements, is when, of four proportionals, the sum of the 1st and 2d is to the 2d, as the sum of the 3d and 4th is to the 4th: as if it be a : b :: c : d, then by composition a+b : b :: c+d : d. Or, in numbers, if 2 : 4 :: 9 : 18, then by composition 6 : 4 :: 27 : 18.
Composition of Ratios, is the adding of ratios together: which is performed by multiplying together their corresponding terms, viz, the antecedents together, and the consequents together, for the antecedent and consequent of the compounded ratio; like as the addition of logarithms is the same thing as the multiplication of their corresponding numbers. Or, if the terms of the ratios be placed fraction-wise, then the addition or composition of the ratios, is performed by multiplying the fractions together. Thus, the ratio of a : b, or of 2 : 4, added to the ratio of c : d, or of 6 : 8, makes the ratio of ac : bd, or of 12 : 32; and so the ratio of ac to bd is said to be compounded of the ratios of a to b, and c to d. So likewise, if it were required to compound together the three ratios, viz, of a to b, c to d, and e to f; then are the terms of the compound ratio; or the ratio of ace to bdf is compounded, or made up of the ratios of a to b, c to d, and e to f.
Hence, if the consequent of each ratio be the same as the antecedent of the preceding ratio, then is the ratio of the first term to the last, compounded, or made up of all the other ratios, viz, the ratio of a to e, equal to the sum of all the ratios of a to b, of b to c, of c to d, and of d to e; for the terms or exponents of the compounded ratio. |
Hence also, in a series of continual proportionals, the ratio of the first term to the third is double of the ratio of the first to the second, and the ratio of the 1st to the 4th is triple of it, and the ratio of the 1st to the 5th is quadruple of it, and so on; that is, the exponents are double, triple, quadruple, &c, of the first exponent: as in the series 1, a, a2, a3, a4, &c; where the ratio 1 to a2 is double, of 1 to a3 triple, &c, of the ratio of 1 to a; or the exponent of a2, a3, a4, &c, double, triple, quadruple, &c, of a.
