III. If _a_ be to _b_ as _c_ is to _d_, and if _a_ have a greater ratio to _b_ than _c_ has to _x_, _d_ is less than _x_; and if _a_ have a less ratio to _b_ than _c_ to _x_, _d_ is greater than _x_. IV. _a_ has to _b_ a greater ratio than _ax_ to _bx_ + _y_, and a less ratio than _ax_ to _bx_- _y_. 200. If _a_ have to _b_ a greater ratio than _c_ has to _d_, _a_ + _c_ has to _b_ + _d_ a less ratio than _a_ has to _b_, but a greater ratio than _c_ has to _d_; or, in other words, if _a_/_b_ be the greater of the two fractions _a_/_b_ and _c_/_d_, _a_ + _c_ --------- _b_ + _d_ will be greater than _c_/_d_, but less than _a_/_b_. To shew this, observe that (_mx_ + _ny_)/(_m_ + _n_) must lie between _x_ and _y_, if _x_ and _y_ be unequal: for if _x_ be the less of the two, it is certainly greater than _mx_ + _nx_ ----------- or than _x_; _m_ + _n_ and if _y_ be the greater of the two, it is certainly less than _my_ + _ny_ -----------, or than _y_. _m_ + _n_ It therefore lies between _x_ and _y_. Now let _a_/_b_ be _x_, and let _c_/_d_ be _y_: then _a_ = _bx_, _c_ = _dy_. Now _bx_ + _dy_ ----------- _b_ + _d_ is something between _x_ and _y_, as was just proved; therefore _a_ + _c_ --------- _b_ + _d_ is something between _a_/_b_ and _c_/_d_. Again, since _a_/_b_ and _c_/_d_ are respectively equal to _ap_/_bp_ and _cq_/_dq_, and since, as has just been proved, _ap_ + _cq_ ----------- _bp_ + _dq_ lies between the two last, it also lies between the two first; that is, if _p_ and _q_ be any numbers or fractions whatsoever, _ap_ + _cq_ ----------- _bp_ + _dq_ lies between _a_/_b_ and _c_/_d_. 201. By the last article we may often form some notion of the value of an expression too complicated to be easily calculated. Thus, 1 + _x_ 1 _x_ 1 -------- lies between --- and ----, or 1 and ---; 1 + _xx_ 1 _xx_ _x_ _ax_ + _by_ _ax_ _by_ -------------- lies between ----- and ------, _axx_ + _bbyy_ _axx_ _bbyy_ that is, between 1/_x_ and 1/_by_. And it has been shewn that (_a_ + _b_)/2 lies between _a_ and _b_, the denominator being considered as 1 + 1. 202. It may also be proved that a fraction such as _a_ + _b_ + _c_ + _d_ --------------------- _p_ + _q_ + _r_ + _s_ _a_ _b_ _c_ _d_ always lies among ---, ---, ---, and ---, _p_ _q_ _r_ _s_ that is, is less than the greatest of them, and greater than the least. Let these fractions be arranged in order of magnitude; that is, let _a_/_p_ be greater than _b_/_q_, _b_/_q_ be greater than _c_/_r_, and _c_/_r_ greater than _d_/_s_. Then by (200)