RESOLUTION

, in Physics, the reduction of a body into its original or natural state, by a dissolution or separation of its aggregated parts. Thus, snow and ice are said to be resolved into water; water resolves in vapour by heat; and vapour is again resolved into water by cold; also any compound is resolved into its ingredients, &c.

Some of the modern philosophers, particularly Boyle, Mariotte, Boerhaave, &c, maintain, that the natural state of water is to be congealed, or in ice; in as much as a certain degree of heat, which is a foreign and violent agent, is required to make it fluid: so that near the pole, where this foreign agent is wanting, it constantly retains its fixed or icy state.

, or Solution, in Mathematics, is an orderly enumeration of several things to be done, to obtain what is required in a problem.

Wolfius makes a problem to consist of three parts: The proposition (or what is properly called the problem), the Resolution, and the demonstration.

As soon as a problem is demonstrated, it is converted into a theorem; of which the Resolution is the hypothesis; and the proposition the thesis.

For the process of a mathematical Resolution, see the following article.

Resolution in Algebra, or Algebraical RESOLUTION, is of two kinds; the one practised in numerical problems, the other in geometrical ones.

In Resolving a Numerical Problem Algebraically, the method is this. First, the given quantities are distinguished from those that are sought; and the former denoted by the initial letters of the alphabet, but the latter by the last letters.—2. Then as many equations are formed as there are unknown quantities. If that cannot be done from the proposition or data, the problem is indeterminate; and certain arbitrary assumptions must be made, to supply the defect, and which can satisfy the question. When the equations are not contained in the problem itself, they are to be found by particular theorems concerning equations, ratios, proportions, &c.—Since, in an equation, the known and unknown quantities are mixed together, they must be separated in such a manner, that the unknown one remain alone on one side, and the known ones on the other. This reduction, or separation, is made by addition, subtraction, multiplication, division, extraction | of roots, and raising of powers; resolving every kind of combination of the quantities, by their counter or reverse ones, and performing the same operation on all the quantities or terms, on both sides of the equation, that the equality may still be preserved.

To Resolve a Geometrical Problem Algebraically.— The same sort of operations are to be performed, as in the former article; besides several others, that depend upon the nature of the diagram, and geometrical properties. As 1st, the thing required or proposed, must be supposed done, the diagram being drawn or constructed in all its parts, both known and unknown. 2. We must then examine the geometrical relations which the lines of the figure have among themselves, without regarding whether they are known or unknown, to find what equations arise from those relations, for finding the unknown quantities. 3. It is often necessary to form similar triangles and rectangles, sometimes by producing of lines, or drawing parallels and perpendiculars, and forming equal angles, &c; till equations can be formed, from them, including both the known and unknown quantities.

If we do not thus arrive at proper equations, the thing is to be tried in some other way. And sometimes the thing itself, that is required, is not to be sought directly, but some other thing, bearing certain relations to it, by means of which it may be found.

The final equation being at last arrived at, the geometrical construction is to be deduced from it, which is performed in various ways according to the different kinds of equations.

Resolution of Forces, or of Motion, is the resolving or dividing of any one force or motion, into several others, in other directions, but which, taken together, shall have the same effect as the single one; and it is the reverse of the composition of forces or motions. See these articles.

Any single direct force AD, may be resolved into two oblique forces, whose quantities and directions are AB, AC, having the same effect, by describing any parallelogram ABDC, whose diagonal is AD. And each of these may, in like manner, be resolved into two others; and so on, as far as we please. And all these new forces, or motions, so found, when acting together, will produce exactly the same effect as the single original one. See also Collision, PERCUSSION, Motion, &c.