or = ; therefore a : c :: b : d. PROP. XVII.—Theorem. If four magnitudes be proportional, the difference between the first and second : the second :: the difference between the third and fourth : the fourth (dividendo). Let a : b :: c : d : then a − b : b :: c − d : d; Dem.—Since a : b :: c : d, = ; therefore − 1 = − 1, or = ; therefore a − b : b :: c − d : d. PROP. XVIII.—Theorem. If four magnitudes be proportionals, the sum of the first and second : the second :: the sum of the third and fourth : the fourth (componendo). Let a : b :: c : d; then a + b : b :: c + d : d. Dem.—Since a : b :: c : d, = ; therefore + 1 = + 1, or = ; therefore a + b : b :: c + d : d. PROP. XIX.—Theorem. If a whole magnitude be to another whole at a magnitude taken from the first it to a magnitude taken from the second, the first remainder : the second remainder :: the first whole : the second whole. Let a : b :: c : d, c and d being less than a and b;then a − c : b − d :: a : b. Dem.—Since a : b :: c : d, then a : c :: b : d [alternando], and c : a :: d : b [invertendo]; therefore = , and 1 − = 1 −, or = . Hence a − c : b − d :: a : b. Prop. E.—Theorem (Simson). If four magnitudes be proportional, the first : its excess above the second :: the third : its excess above the fourth (convertendo). Let a : b :: c : d; then a : a − b :: c : c − d. Dem.—Since a : b :: c : d, = ; therefore = [Dem. of xvii.], therefore ÷ = ÷, or = , therefore a : a − b :: c : c − d. PROP. XX.—Theorem. If there be two sets of three magnitudes, which taken two by two in direct order have equal ratios, then if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less. Let a, b, c; a′, b′, c′ be the two sets of magnitudes, and let the ratio a : b = a′ : b′, and b : c = b′ : c′; then, if a be greater than, equal to, or less than c, a′ will be greater than, equal to, or less than c′. Dem.—Since a : b :: a′ : b′, we have = , In like manner, = , Hence × = ×, or = . Therefore if a be greater than c, a′ is greater than c′; if equal, equal; and if less, less. PROP. XXI.—Theorem. If there be two sets of three magnitudes, which taken two by two in transverse order have equal ratios; then, if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less.