PROPOSITION I. THE ratios of given magnitudes to one another are given.† Let A, B be two given,magnitudes, the ratio of A to B is given. (1. def. Dat.) be found one equal to it; let this *1. Ir a given magnitude has a given ratio to another magnitude, "and if unto the two magnitudes, by which the given ratio is exhibited, and the given magnitude, a fourth proportional can be found;" the other magnitude is given.† Let the given magnitude A have a given ratio to the magnitude B; if a fourth proportional can be found to the three magnitudes above named, B is given in magnitude. A B C E F Because A is given, a magnitude may be found equal to it (1. def.); let this be C: and because the ratio of A to B is given, a ratio which is the same with it may be found; let this be the ratio of the given magnitude E to the given magnitude F: unto the magnitudes E, F, C find a fourth proportional D, which, by the hypothesis, can be done. Wherefore because A is to B, as E to F; and as E to F, so is C to D; A is (11. 5.) to B, as C to D. But A is equal to C; therefore (14. 5.) B is equal to D. The magnitude B is therefore given (1. def.) because a magnitude D equal to it has been found. The limitation within the inverted commas is not in the Greek text, but is now necessarily added; and the same must be understood in all the propositions of the book which depend upon this second proposition, where it is not expressly mentioned. See the note upon it. * The figures in the margin show the number of the propositions in the other editions. + See Note. PROP. III. IF any given magnitudes be added together, their sum shall be given. Let any given magnitudes AB, BC be added together, their sum AC is given. B C Because AB is given, a magnitude equal to it may be found (1. def.); let this be DE: and because BC is given, one equal to it may be A found; let this be EF: wherefore, because AB is equal to DE, and BC equal to EF; the whole AC is equal to the D whole DF: AC is therefore given, because DF has been found, which is equal to it. PROP. IV. E F Ir a given magnitude be taken from a given magnitude, the remaining magnitude shall be given. From the given magnitude AB, let the given magnitude AC be taken; the remaining magnitude CB is given. Because AB is given, a magnitude equal to it may (1. def.) be found; let this be DE: and because AC is given, one equal to it may be A found; let this be DF: wherefore because AB is equal to DE, and AC D F E to DF; the remainder CB is equal Ir 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 one of them is greater than the other by a given magnitude.* Let AB, BC, CD be three magnitudes, of which AB together with BC, 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 AC is A B equal to BD, take away the common part BC; therefore the remainder AB is equal to the remainder CD. C D 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; there fore (4. dat.) AE the remainder is A E given. And because EC is equal to * See Note. B C D - -| BD, by taking BC from both, the remainder EB is equal to the remainder CD. And AE is given; wherefore AB exceeds EB, that is CD, by the given magnitude AE. PROP. VI. Ir a magnitude has a given ratio to a part of it, it shall also have a given raito 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 (2. def.) which is the same to it: let this be the ratio of DE a given magnitude to the given magnitude DF. And because DE, DF are given, A C B F E the remainder FE is (4. dat.) given and because AB is to AC, as DE to DF, by D conversion (E. 5.) AB is to BC, as DE to EF. Therefore the ratio of AB to BC is given, because the ratio of the given magnitudes DE, EF, which is the same with it, has been found. 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 division (17. 5.) AC is to CB, as DF to FE: and DF, FE are given; therefore (2. def.) the ratio of AC to CB is given. IF two magnitudes which have a given ratio to one another, be added together: the whole magnitude shall 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.. A B C Because the ratio of AB to BC is given, a ratio may be found (2. def.) which is the same with it; let this be the ratio of the given magnitudes DE, EF: and because DE, EF are given, the whole DF is given (3. dat.) and because as AB to BC, so is DE to EF; by composition (18. 5.), AC is to CB, as DF to FE; and by conversion (E. 5.), AC is to AB, as DF to 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 given (2. def.). PROP. VIII. Ꭰ E F 7. Ir the given magnitude be divided into two parts which have a given ratio to one another, and if a fourth proportional can be * See Note. 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 each of them given. Α C B A Because the ratio of AC to CB is given, the ratio of AB to BC is given (7. dat.); therefore a ratio which is the same with it D can be found (2. def.); let this be the ratio of the given magnitudes, DE, EF: and because the given magnitude AB has to BC the given ratio of DE to EF, if unto DE, EF, AB a fourth proportional can be found, this which is BC is given (2. dat.); D and because AB is given, the other part AC is given (4. dat.) F E 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 which have given ratios to the same magnitude, have also a given ratio to one another. 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 (2. def.); 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 found (2. def.); 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 ratio of A to C is A B C D given (2. def.) because the ratio of E 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 E, and F to G. * See Note. Ir 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 magnitudes; these other magnitudes shall also have given ratios to one another. Let two or more maguitudes 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 (9. dat.): but the ratio of A B to E is given, therefore (9. dat.) Bthe ratio of D to E is given: and be- C. cause the ratio of B to C is given, D E F and also the ratio of B to E; the ratio of E to C is given (9. dat.): 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. Ir two magnitudes have each of them a given ratio to another magnitude, both of them together shall have a given ratio to that other. A B -C Let the magnitudes AB, BC have a given ratio to the magnitude D; AC has a given ratio to the same D. Because AB, BC have each of them a given ratio to D, the ratio of AB to BC is given (9. dat.): and by composi tion the ratio of AC to CB is given; (7. dat.): but the ratio of BC to D is given; therefore (9. dat.) the ratio of AC to D is given. PROP. XII. D 23. Ir 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 one 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 FG is * Sce Note. |