_a_ = _b_ and _p_ = _q_, _a_ + _p_ = _b_ + _q_, _a_ - _p_ = _b_ - _q_, _ap_ = _bq_, _a_ _b_ and --- = ---. _p_ _q_ It is also evident, that _a_ + _p_-_p_, _a_ -_p_ + _p_, _ap_/_p_, and _a_/_p_ × _p_, are all equal to _a_. 181. The product of the extremes is equal to the product of the means. Let _a_/_b_ = _c_/_d_, and multiply these equal numbers by the product _bd_. Then, _a_ _abd_ --- × _bd_ = ----- (116) = _ad_, _b_ _b_ _c_ _cbd_ and --- × _bd_ = ----- = _cb_: hence (180), _ad_ = _bc_. _d_ _d_ Thus, 6, 8, 21, and 28, are proportional, since 6 3 3 × 7 21 --- = --- = ------ = --- (180); 8 4 4 × 7 28 and it appears that 6 × 28 = 8 × 21, since both products are 168. 182. If the product of two numbers be equal to the product of two others, these numbers are proportional in any order whatever, provided the numbers in the same product are so placed as to be similar terms; that is, if _ab_ = _pq_, we have the following proportions:-- _a_ : _p_ ∷ _q_ : _b_ _a_ : _q_ ∷ _p_ : _b_ _b_ : _p_ ∷ _q_ : _a_ _b_ : _q_ ∷ _p_ : _a_ _p_ : _a_ ∷ _b_ : _q_ _p_ : _b_ ∷ _a_ : _q_ _q_ : _a_ ∷ _b_ : _p_ _q_ : _b_ ∷ _a_ : _p_ To prove any one of these, divide both _ab_ and _pq_ by the product of its second and fourth terms; for example, to shew the truth of _a_: _q_ ∷ _p_: _b_, divide both _ab_ and _pq_ by _bq_. Then, _ab_ _a_ _pq_ _p_ ---- = ---, and ---- = ---; hence (180), _bq_ _q_ _bq_ _b_ _a_ _p_ --- = ---, or _a_ : _q_ ∷ _p_ : _b_. _q_ _b_ The pupil should not fail to prove every one of the eight cases, and to verify them by some simple examples, such as 1 × 6 = 2 × 3, which gives 1: 2 ∷ 3: 6, 3: 1 ∷ 6: 2, &c. 183. Hence, if four numbers be proportional, they are also proportional in any other order, provided it be such that similar terms still remain similar. For since, when _a_ _c_ --- = ---, _b_ _d_ it follows (181) that _ad_ = _bc_, all the proportions which follow from _ad_ = _bc_, by the last article, follow also from _a_ _c_ --- = ---, _b_ _d_ 184. From (114) it follows that _a_ _b_ + _a_ 1 + --- = ---------, _b_ _b_ _a_ and if --- be less than 1, _b_ _a_ _b_ - _a_ 1 - --- = ---------, _b_ _b_ _a_ while if --- be greater than 1, _b_ _a_ _a_ - _b_ --- - 1 = ---------. _b_ _b_ _a_ + _b_ _a_ - _b_ Also (122), if --------- be divided by --------- _b_ _b_ _a_ + _b_ the result is ---------. _a_ - _b_ Hence, _a_, _b_, _c_, and _d_, being proportionals, we may obtain other proportions, thus: