INTRODUCTORY NOTE.¶
The theory of proportions has been established in Book v. in a perfectly general form applicable to all kinds of magnitudes (although the representation of magnitudes by straight lines gives it a geometrical appearance); it is now necessary to apply the theory to the particular case of geometrical investigation. The only thing still required in order that this may be done is a proof of the existence of such a magnitude as bears to any given finite magnitude any given finite ratio; and this proof is supplied, so far as regards the subject matter of geometry, by VI. 12 which shows how to construct a fourth proportional to three given straight lines.
A few remarks on the enormous usefulness of the theory of proportions to geometry will not be out of place. We have already in Books I. and II. made acquaintance with one important part of what has been well called geometrical algebra, the method, namely, of application of areas. We have seen that this method, working by the representation of products of two quantities as rectangles, enables us to solve some particular quadratic equations. But the limitations of such a method are obvious. So long as general quantities are represented by straight lines only, we cannot, if our geometry is plane, deal with products of more than two such quantities; and, even by the use of three dimensions, we cannot work with products of more than three quantities, since no geometrical meaning could be attached to such a product. This limitation disappears so soon as we can represent any general quantity, corresponding to what we denote by a letter in algebra, by a ratio; and this we can do because, on the general theory of proportion established in Book v., a ratio may be a ratio of two incommensurable quantities as well as of commensurables. Ratios can be compounded ad infinitum, and the division of one ratio by another is equally easy, since it is the same thing as compounding the first ratio with the inverse of the second. Thus e.g. it is seen at once that the coefficients in a quadratic of the most general form can be represented by ratios between straight lines, and the solution by means of Books I. and II. of problems corresponding to quadratic equations with particular coefficients can now be extended to cover any quadratic with real roots. As indicated, we can perform, by composition of ratios, the operation corresponding to multiplying algebraical quantities, and this to any extent. We can divide quantities by compounding a ratio with the inverse of the ratio representing the divisor. For the addition and subtraction of quantities we have only to use the geometrical equivalent of bringing to a common denominator, which is effected by means of the fourth proportional.