(1.) See N. * 1 Def. Dat. 7. 5, (2.) See N. VII. Segments of circles are said to be given in magnitude, when the angles in them, and their bases, are given in magnitude. VII. Segments of circles are said to be given in position and magnitude, when the angles in them are given in magnitude, and their bases are given both in position and magnitude. IX. A magnitude is said to be greater than another by a given magnitude, when this given magnitude being taken from it, the remainder is equal to the other magnitude. X. A magnitude is said to be less than another by a given magnitude, when this given magnitude being added to it, the whole is equal to the other magnitude. PROPOSITION I. The ratio of given magnitudes to one another is given. Let A, B be two given magnitudes: the ratio of A to B is given. Because A is a given magnitude, there may be found one equal to it; let this be C: and because B is given, one equal to it may be found; let it be D; and since A is equal to C, and B to D; therefore* A is to B, as C to D; and consequently the ratio of A to B is given, because the ratio of the given magnitudes C, D, which is the same with it, has been found. PROPOSITION II. C D If a given magnitude have 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 propor"tional 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. The figures between parentheses in the margin, opposite the head of each proposition, shew the number of that proposition in the other editions. A B C D Because A is given, a magnitude may be found equal to it *; let this be C: and because the ratio of A to B is given, *1 Def. 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 to B, as C to D. But A is equal to C; therefore* B is equal to D. The magnitude B is therefore given *, because a magnitude D equal to it has been found. E F 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. PROPOSITION 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. Because AB is given, a magnitude equal 11.5. 14.5. 1 Def. (3.) A B с * 1 Def. D E F to it may be found; let this be DE: and PROPOSITION IV. (4.).. If 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 C B E F ⚫ 1 Def. DF and because AC is given, one equal to it may be found; let this be DE: wherefore, because AB is equal to DF, and • 1 Def. (12.) See N. * 4 Dat. (5.) AC to DE, the remainder CB is equal to the remainder FE. CB is therefore given *, because FE, which is equal to it, has been found. PROPOSITION V. 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 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. away the common A B C D First, let them be equal, and because AC But if they be unequal, let AC be greater than BD, and AE B C D king 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. PROPOSITION VI. See N. * 2 Def. * 4 Dat. ⚫ E. 5. If a magnitude have a given ratio to 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 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, the remainder FE is * given: and because AB is to AC, as DE to DF, by conversion * AB is to BC, as DE to EF. Therefore, the ratio of AB to BC is A CB FE D 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 * AC is to CB, as DF to FE; and DF, FE are given; therefore* the ratio of AC to CB is given. PROPOSITION VII. • 17. 5. 2 Def. (6.) If two magnitudes which have a given ratio to one another be See N. 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 3 Dat. D E F *18. 5. Because the ratio of AB to BC is given, a ratio may be found which is the same with it; let this be the ratio of the * 2 Def. given magnitudes DE, EF and because DE, EF are given, the whole DF is given*: and because as AB to BC, so is DE to EF; by composition* AC is to CB as DF to FE; and by conversion *, AC is to AB, as DF to DE: wherefore E. 5. 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 *. PROPOSITION VIII. * 2 Def. (7.) If a given magnitude be divided into two parts which have See N. 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, D AC and CB are each of them given. A B FE Because the ratio of AC to CB is given, the ratio of AB to BC is given *, therefore a ratio, which is the same with it, can be found *; let this be the ratio of the given magnitudes DE, * 2 Def. EF: and because the given magnitude AB has to BC the 7 Dat. |