SOLUTION. (Mathematics.) By the solution of a problem should be meant the method of finding that which the problem requires to be found : but the word is frequently understood to apply to the answer A solution is given when the problem is reduced to any other which was supposed to be known before the first was presented : the difficulty peculiar to the given problem is removed as soon as it is shown to be capable of reference to another and a lower class. Thus, though properly speaking a problem is not solved until the answer is presented in numbers, yet it is not thought necessary to require that such a result should be attained, provided the steps which are left arc such as are well known and generally admitted. Thus an equation would be said to be solved were it found that the roots required are those of a given quadratic; for no one is supposed ignorant of the mode of then finding them.
A geometrical solution, in the strict sense of the word, is one in which only the means of construction admitted by Euclid, or others deducible from them, are employed in its attainment. This is the least finished of all solutions; for a mode of laying down the various points which terminate lines is not, generally speaking, a mode of ascertaining the ratio of these lines. Nor must it be forgotten by the admirers of geometry that the most important part of a result, the expression of the ratios which the answer bears to the several data, is only indirectly obtained in their favourite method.
When more means than those allowed by Euclid are employed, the solution used to be called mechanical. It is rarely that such a solution is now employed.
An algebraical solution is one which employs algebra and arithmetic. to the exclusion of geometrical construction ; that is, one in which the answer can always be directly calculated from a formula. Geometrical construction may be necessary for the demonstration of the solution : it is enough that the answer contain no directions to find lines or surfaces by construction.
An approximate solution is one which has an amount of inaccuracy necessarily. Thus if 3 + J2 were the root of an equation, this solu tion would not be called approximate ; for though J 2 cannot be perfectly represented in a finite form, the symbol itself contains the mode of attaining the result with any degree of exactness short of perfection. But if J2 were found to five decimal places, the answer 1.41421 would be called an approximate answer. Most solutions must terminate in an approximate representation. [TRANSCENDENTAL.] SOLVENT. [SOLUTION.)