_c_ _mbc_ while that of the means is _mb_ × ---, or -----. _n_ _n_ But since _a_ : _b_ ∷ _c_ : _d_, by (181) _ad_ = _bc_, _mad_ _mbc_ whence, by (180), _mad_ = _mbc_, and ----- = -----. _n_ _n_ _a_ _c_ Hence, ---, _mb_, ---, and _md_, are proportionals. _n_ _n_ 187. If the terms of one proportion be multiplied by the terms of a second, the products are proportional; that is, if _a_: _b_ ∷ _c_: _d_, and _p_: _q_ ∷ _r_: _s_, it follows that _ap_: _bq_ ∷ _cr_: _ds_. For, since _ad_ = _bc_, and _ps_ = _qr_, by (180) _adps_ = _bcqr_, or _ap_ × _ds_ = _bq_ × _cr_, whence (182) _ap_: _bq_ ∷ _cr_: _ds_. 188. If four numbers be proportional, any similar powers of these numbers are also proportional; that is, if _a_ : _b_ ∷ _c_ : _d_ Then _aa_ : _bb_ ∷ _cc_ : _dd_ _aaa_ : _bbb_ ∷ _ccc_ : _ddd_ &c. &c. For, if we write the proportion twice, thus, _a_ : _b_ ∷ _c_ : _d_ _a_ : _b_ ∷ _c_ : _d_ by (187) _aa_ : _bb_ ∷ _cc_ : _dd_ But _a_ : _b_ ∷ _c_ : _d_ Whence (187) _aaa_ : _bbb_ ∷ _ccc_ : _ddd_; and so on. 189. An expression is said to be homogeneous with respect to any two or more letters, for instance, _a_, _b_, and _c_, when every term of it contains the same number of letters, counting _a_, _b_, and _c_ only. Thus, _maab_ + _nabc_ + _rccc_ is homogeneous with respect to _a_, _b_, and _c_; and of the third degree, since in each term there is either _a_, _b_, and _c_, or one of these repeated alone, or with another, so as to make three in all. Thus, 8_aaabc_, 12_abccc_, _maaaaa_, _naabbc_, are all homogeneous, and of the fifth degree, with respect to _a_, _b_, and _c_ only; and any expression made by adding or subtracting these from one another, will be homogeneous and of the fifth degree. Again _ma_ + _mnb_ is homogeneous with respect to _a_ and _b_, and of the first degree; but it is not homogeneous with respect to _m_ and _n_, though it is so with respect to _a_ and _n_. This being premised, we proceed to a theorem,[28] which will contain all the results of (184), (185), and (188). [28] A theorem is a general mathematical fact: thus, that every number is divisible by four when its last two figures are divisible by four, is a theorem; that in every proportion the product of the extremes is equal to the product of the means, is another.