_a_ _c_ Let --- = --- _b_ _d_ _a_ _c_ Then (114) 1 + --- = 1 + --- _b_ _d_ _a_ + _b_ _c_ + _d_ or --------- = --------- _b_ _d_ or _a_ + _b_: _b_ ∷ _c_ + _d_: _d_ That is, the sum of the first and second is to the second as the sum of the third and fourth is to the fourth. For brevity, we shall not state in words any more of these proportions, since the pupil will easily supply what is wanting. Resuming the proportion _a_: _b_ ∷ _c_: _d_ _a_ _c_ or --- = --- _b_ _d_ _a_ _c_ _a_ 1 - --- = 1 - ---, if --- be less than 1, _b_ _d_ _b_ _b_ - _a_ _d_ - _c_ or --------- = --------- _b_ _d_ that is, _b_-_a_: _b_ ∷ _d_-_c_: _d_ or, _a_-_b_: _b_ ∷ _c_-_d_: _d_, _a_ if --- be greater than 1. _b_ _a_ + _b_ _c_ + _d_ Again, since --------- = --------- _b_ _d_ _a_ - _b_ _c_ - _d_ _a_ and --------- = --------- (--- being greater than 1) _b_ _d_ _b_ _a_ + _b_ _c_ + _d_ dividing the first by the second we have --------- = ----------, _a_ - _b_ _c_ - _d_ or _a_ + _b_ : _a_ - _b_ ∷ _c_ + _d_ : _c_ - _d_ and also _a_ + _b_ : _b_ - _a_ ∷ _c_ + _d_ : _d_ - _c_, _a_ if --- be less than 1. _b_ 185. Many other proportions might be obtained in the same manner. We will, however, content ourselves with writing down a few which can be obtained by combining the preceding articles. _a_ + _b_ : _a_ ∷ _c_ + _d_ : _c_ _a_ : _a_ - _b_ ∷ _c_ : _c_ - _d_ _a_ + _c_ : _a_ - _c_ ∷ _b_ + _d_ : _b_ - _d_. In these and all others it must be observed, that when such expressions as _a_-_b_ and _c_-_d_ occur, it is supposed that _a_ is greater than _b_, and _c_ greater than _d_. 186. If four numbers be proportional, and any two dissimilar terms be both multiplied, or both divided by the same quantity, the results are proportional. Thus, if _a_: _b_ ∷ _c_: _d_, and _m_ and _n_ be any two numbers, we have also the following: _ma_ : _b_ ∷ _mc_ : _d_ _a_ : _mb_ ∷ _c_ : _md_ _a_ _c_ --- : _mb_ ∷ --- : _md_ _n_ _n_ _ma_ : _nb_ ∷ _mc_ : _nd_ _a_ _b_ _c_ _d_ --- : --- ∷ --- : --- _m_ _m_ _m_ _m_ _a_ _b_ _c_ _d_ --- : --- ∷ --- : --- _m_ _m_ _n_ _n_ and various others. To prove any one of these, recollect that nothing more is necessary to make four numbers proportional except that the product of the extremes should be equal to that of the means. Take the third of those just given; the product of its extremes is _a_ _mad_ --- × _md_, or -----, _n_ _n_