See N. IF of three magnitudes, the first together with the second be given, and also the second together with the third; either the first is equal to the third, or cne of them is greater than the other by a given magnitude. Let AB, BC, CD be three magnitudes, of which AB together with CC, that is AC, is given ; and also BC together with CD, that is BD, is given. Either AB is equal to CD, or one of them is greater than the other by a given magnitude. Because AC, BD are each of them given, they are either equal to one another, or not equal. First, let them be equal, and because A B C D AC is equal to BD, take away the common part B C; therefore the remainder AB is equal,to the remainder CD. But if they be unequal, let AC be greater than BD, and make CE equal to BD. Therefore CE is given, because BD is given. And the whole AC is given ; a 4. dat. therefore a AE the remainder is A E B C D given. And because EC is equal -1 I See N. a 2. def. : IF a magnitude has a given ratio td a part of it, it shall also have a given ratio to the remaining part of it. Let the magnitude AB have a given ratio to AC a part of it; it has also a given ratio to the remainder BC. Because the ratio of AB to AC is given, a ratio may be found a which is the same to it : let this be the ratio of DE a given magnitude to the given magni A tude DF. And because DE, DF are gi. с в ven, the remainder FE is b given : and because AB is to AC, as DE to DF, by D conversion c AB is to BC, as Di to EF. F E Therefore the ratio of AB to BC is given, because the ratio of the given magnitudes DE, EF, wbich is the same with it, has been found. b 4. dat. CE. 5. Cor. From this it follows, that the parts AC, CB have a given ratio to one another: because as AB to BC, so is DE to EF; by divisiond, AC is to CB, as DF to FE: and DF, FE are d 17. 5. given ; therefore a the ratio of AC to CB is given. a 2. def. IF two magnitudes which have a given ratio to one See N. another, be added together; the whole magnitude shail have to each of them a given ratio. Let the magnitudes AB, BC which have a given ratio to one another, be added together; the whole AC has to each of the magnitudes AB, BC a given ratio. Because the ratio of AB to BC is given, a ratio may be found a which is the same with it; let this be the ratio of the a 2. def. given magnitudes DE, EF: and because DE, EF are given, the whole A B С DF is given b: and because as AB to b 3. dat. BC, so is DE to EF; by composition D E F CAC is to CB, as DF to FE; and by c 18. 5. conversiond, AC is to AB, as DF to d E. 5. DE: wherefore because AC is to each of the magnitudes AB, BC, as DF to each of the others DE, EF; the ratio of AC to each of the magnitudes AB, BC is givena. IF a given magnitude be divided into two parts See Note which have a given ratio to one another, and if a fourth proportional can be found to the sum of the two magnitudes by which the given ratio is exhibited, one of them, and the given magnitude; each of the parts is given. Let the given magnitude AB be divided into the parts AC, CB which have a given ratio to one another; if a fourth proportional can be found to the above named magnitudes; AC and CB are A с B each of them given. Because the ratio of AC to CB is D F E given ; the ratio of AB to BC is givena; therefore a ratio which is a 7. dat. b 2. def. the same with it can be found b, let this be the ratio of the given magnitudes DE, EF: and с B to BC the given ratio of DE to EF, if -1 unto DE, EF, AB a fourth propor- D F E tional can be found, this which is BC c 2. dat. is givenc; and because AB is given d 4. dat. the other part AC is givend. In the same manner, and with the like limitation, if the difference AC of two magnitudes AB, BC which have a given ratio, be given; each of the magnitudes AB, BC is given. MAGNITUDES whch have given ratios to the same magnitude, have also a given ratio to one another. a 2. def. Let A, C have each of them a given ratio to B; A has a given ratio to C. Because the ratio of A to B is given, a ratio which is the same to it may be found a ; let this be the ratio of the given magnitudes D, E: and because the ratio of B to C is given, a ratio which is the same with it may be founda; let this be the ratio of the given magnitudes F, G: to F, G, E find a fourth proportional H, if it can be done; and because as A is to B, so is D to E; and as B to C, so is (F to G, and so is) E to H; ex æquali, as A to C, so is D to H; therefore the ra- À B C D È H tio of A to C is given a, because the F G ratio of the given magnitudes D and H, which is the same with it, has been found : but if a fourth proportional to F, G, E cannot be found, then it can only be said that the ratio of A to C is compounded of the ratios of A to B, and B to C, that is, of the given ratios of D to L, and F to G, a IF two or more magnitudes have given ratios to one another, and if they have given ratios, though they be not the same, to some other magnitudies; these other magnitudes shall also have given ratios to one another. Let two or more magnitudes A, B, C have given ratios to one another; and let them have given ratios, though they be not the same, to some other magnitudes D, E, F: the magnitudes D, E, F have given ratios to one another. Because the ratio of A to B is given, and likewise the ratio of A to D; therefore the ratio of D to B is given a ; but A D the ratio of B to E is given, B. E therefore a the ratio of D to C F a 9. dat. E is given : and because the ratio of B to C is given : and also the ratio of B to E ; the ratio of E to C is givena ; and the ratio of C to F is given; wherefore the ratio of E to F is given : D, E, F have therefore given ratios to one another. IF two magnitudes have each of them a given ratio to another magnitude, both of them together shall have a given ratio to that other. Let the magnitudes AB, BC have a given ratio to the magnitude D; AČ has a given ratio to the same D. Because AB, BC have each of them a given ratio to D, the ratio B of AB to BC is given a ; and by A -C a 9. dat. composition, the ratio of AC to D. CB is given b: but the ratio of b 7. dat. BC to D is given ; therefore a the ratio of AC to D is given. See N. IF the whole have to the whole a given ratio, and the parts have to the parts given, but not the same, ratios, every one of them, whole or part, shall have to every cne a given ratio. Let the whole AB have a given ratio to the whole CD, and the parts AE, EB have given, but not the same, ratios to the parts CF, FD, every one shall have to every one, whole or part, a given ratio. Because the ratio of AE to CF is given, as AE to CF, so make AB to CG ; the ratio therefore of AB to CG is given; wherefore the ratio of the remainder EB to the remainder & 19. 5. FG is given, because it is the same a with the ratio of AB to B b 9. dat to FG is givenb; and by conversion, the ratio of FD to DG is C F G D c 6. dat. givenc; and because AB has to each of the magnitudes CD, CG a given, wherefore b the ratio of CD to DF is given, and consed cor. 6. quently d the ratio of CF to FD is given ; but the ratio of CF to dat. e 10. dat. AE is given, as also the ratio of FĎ to EB, wherefore e the ra tio of AE to EB is given; as also the ratio of AB to each of f7. dat. themf: the ratio therefore of every one to every one is given. See N. IF the first of three proportional straight lines has a given ratio to the third, the first shall also have a giveni ratio to the second. Let A, B, C be three proportional straight lines, that is, as A to B, so is B to C; if A has to C a given ratio, A shall also have to B a given ratio. Because the ratio of A to C is given, a ratio which is the a 2. def. same with it may be founda; let this be the ratio of the given b 13. 6. straight lines D, E; and between D and E find a b mean |