191. If the two means of a proportion be the same, that is, if _a_ : _b_ ∷ _b_: _c_, the three numbers, _a_, _b_, and _c_, are said to be in _continued_ proportion, or in _geometrical progression_. The same terms are applied to a series of numbers, of which any three that follow one another are in continued proportion, such as 1 2 4 8 16 32 64 &c. 2 2 2 2 2 2 2 --- --- --- --- ---- ---- &c. 3 9 27 81 243 729 Which are in continued proportion, since 2 2 2 1 : 2 ∷ 2 : 4 2 : --- ∷ --- : --- 3 3 9 2 2 2 2 2 : 4 ∷ 4 : 8 --- : --- ∷ --- : --- 3 9 9 27 &c. &c. 192. Let _a_, _b_, _c_, _d_, _e_ be in continued proportion; we have then _a_ _b_ _a_ : _b_ ∷ _b_ : _c_ or --- = --- or _ac_ = _bb_ _b_ _c_ _b_ _c_ _b_ : _c_ ∷ _c_ : _d_ --- = --- _bd_ = _cc_ _c_ _d_ _c_ _d_ _c_ : _d_ ∷ _d_ : _e_ --- = --- _ce_ = _dd_ _d_ _e_ Each term is formed from the preceding, by multiplying it by the same number. Thus, _b_ _c_ _b_ = --- × _a_ (180); _c_ = ---× _b_; _a_ _b_ _a_ _b_ _b_ _c_ _b_ and since --- = ---, --- = --- or _c_ = --- × _b_. _b_ _c_ _a_ _b_ _a_ _d_ _d_ _c_ _b_ Again, _d_ = --- × _c_, but --- = ---, which is = ---; _c_ _c_ _b_ _a_ _b_ therefore, _d_ = --- × _c_, and so on. _c_ _b_ If, then, --- _a_ (which is called the _common ratio_ of the series) be denoted by _r_, we have _b_ = _ar_ _c_ = _br_ = _arr_ _d_ = _cr_ = _arrr_ and so on; whence the series _a_ _b_ _c_ _d_ &c. is _a_ _ar_ _arr_ _arrr_ &c. Hence _a_ : _c_ ∷ _a_ : _arr_ (186) ∷ _aa_ : _aarr_ ∷ _aa_ : _bb_ because, _b_ being _ar_, _bb_ is _arar_ or _aarr_. Again, _a_ : _d_ ∷ _a_ : _arrr_ (186) ∷ _aaa_ : _aaarrr_ ∷ _aaa_ : _bbb_ Also _a_ : _e_ ∷ _aaaa_ : _bbbb_, and so on; that is, the first bears to the _n_ᵗʰ term from the first the same proportion as the _n_ᵗʰ power of the first to the _n_ᵗʰ power of the second.