178. When the first of four numbers is the same multiple-part of the second which the third is of the fourth, the four are said to be _geometrically[27] proportional_, or simply _proportional_. This is a word in common use; and it remains to shew that our mathematical definition of it, just given, is, in fact, the common notion attached to it. For example, suppose a picture is copied on a smaller scale, so that a line of two inches long in the original is represented by a line of one inch and a half in the copy; we say that the copy is not correct unless all the parts of the original are reduced in the same proportion, namely, that of 2 to (1½). Since, on dividing two inches into 4 parts, and taking 3 of them, we get (1½), the same must be done with all the lines in the original, that is, the length of any line in the copy must be three parts out of four of its length in the original. Again, interest being at 5 per cent, that is, £5 being given for the use of £100, a similar proportion of every other sum would be given; the interest of £70, for example, would be just such a part of £70 as £5 is of £100. [27] The same remark may be made here as was made in the note on the term ‘arithmetical proportion,’ page 101. The word ‘geometrical’ is, generally speaking, dropped, except when we wish to distinguish between this kind of proportion and that which has been called arithmetical. Since, then, the part which _a_ is of _b_ is expressed by the fraction _a_/_b_, or any other fraction which is equivalent to it, and that which _c_ is of _d_ by _c_/_d_, it follows, that when _a_, _b_, _c_, and _d_, are proportional, _a_/_b_ = _c_/_d_. This equation will be the foundation of all our reasoning on proportional quantities; and in considering proportionals, it is necessary to observe not only the quantities themselves, but also the order in which they come. Thus, _a_, _b_, _c_, and _d_, being proportionals, that is, _a_ being the same multiple-part of _b_ which _c_ is of _d_, it does not follow that _a_, _d_, _b_, and _c_ are proportionals, that is, that _a_ is the same multiple-part of _d_ which _b_ is of _c_. It is plain that _a_ is greater than, equal to, or less than _b_, according as _c_ is greater than, equal to, or less than _d_. 179. Four numbers, _a_, _b_, _c_, and _d_, being proportional in the order written, _a_ and _d_ are called the _extremes_, and _b_ and _c_ the _means_, of the proportion. For convenience, we will call the two extremes, or the two means, _similar_ terms, and an extreme and a mean, _dissimilar_ terms. Thus, _a_ and _d_ are similar, and so are _b_ and _c_; while _a_ and _b_, _a_ and _c_, _d_ and _b_, _d_ and _c_, are dissimilar. It is customary to express the proportion by placing dots between the numbers, thus: _a_ : _b_ ∷ _c_ : _d_ 180. Equal numbers will still remain equal when they have been increased, diminished, multiplied, or divided, by equal quantities. This amounts to saying that if