TRANSFORMATION, a general term of mathematics, indicating a change made in the object of s problem or the shape of s formula, in such manner that the orginal problem or formula is more easily solved, calculated, or used after the transformation. Thus it frequently happens that the solution of an equation is facilitated by reducing it to another equation having roots which bear a simple relation to the roots of the former : as an instance, we may refer to the solution of the cubic equation in the article IRREDUCIBLE CASE.
All the process of algebra consists in transformation, from and after the point at which the problem to be solved is reduced to an equation : so that to write on this subject in detail would require an article on algebra. A few remarks on the leading points which present them selves in transformations are all we can here undertake to give.
It frequently happens that transformation points out the nature of a consequence in s manner by which the direct reasoning of algebra is strongly confirmed and illustrated. For instance, when we assert that a quantity has two square roots, one positive and one negative, our assertion is easily verified in its positive part : but it does not follow by the same reasoning that s quantity has only two square roots. We may say that xl= 4 is satisfied by x=2, or x= —2, because 2 x 2= 4, and —2 x —2=4; but how are we to say that there are no other values which satisfy this equation / 'When we transform the equation into (x-2) (x+ 2)=- 0, with which it is identical, we then see that this product can only vanish when x-2 or x+ 2 vanishes ; that is, when x is + 2 or —2.
Transformations frequently leave a point unsettled which can only be determined by a subsequent species of experimental test ; or, lest the word experimental as applied to mathematical reasoning should give alarm, by a process of detection which is to choose between alternatives which the process of transformation leaves undecided. This frequently happens when the nature of the transformation is ascertained by means not of the expression to be transformed, but of one of its particular properties. For instance, when we expand a' into a series of powers of x, supposing we proceed upon the property a z x a r = + '',we find that there is no series fit to fulfil this condition except but we also find that this series is equally fit to fulfil the condition, whatever may be the value of A. So far then our transformation is effected : we see that one among the series formed by giving values to A must be the series we want, if there be any such series. If we make x=-1, we then immediately detect the condition which is to give the value of .t, namely, that A must be so taken as to make in many parts of algebra the science will refuse to acknowledge and obey a false assumption of form, yet it is almost impossible to draw the line at which this refusal ends, and the idea that such a power is universal in algebra will lead the student into many a serious difficulty in the higher branches of mathematics.