14 Th. If the first have to the second the same ratio which the third has to the fourth; then if the first be greater than the third, the second shall be greater than the fourth; if equal, equal; and if less, less. Given A to B as C to D, and A, C antecedents, B, D consequents. 1. If A be greater than C, the quotient of B divided by A will be less than that of B divided by C (a); but the quotients of B by A and D by C are given equal; for the consequent divided by the antecedent is the ratio (b): therefore the quotient of B divided by C is greater than the quotient of D divided by C; and so, B is greater than D. 2. If A equals C, the two quotients of B divided by A and by C are equal (c); but the quotients of B by A and D by C are given equal, as above (b): therefore the quotients of B and D by C are equal and so, B is equal to D. 3. If A be less than C, the quotient of B by A is greater than that of B by C (a): but the quotients of B by A and D by C are given equal (b): therefore the quotient of B by C is less than that of D by C; and so, B is less than D. Wherefore, if the first have to the second, &c. Recite (a) p. 8, 13, 5; (b) def. 3, 5; (c) p. 9, 5. Q. E. D. NOTE. The ratio of C to B is here introduced as a medium of comparison between B and D; also the magnitudes are so divided in the diagram that each may be taken greater or less as the case may require. 15 Th. Magnitudes have the same ratio to one another which their equimultiples have. Given two magnitudes C, F; and equimultiples of them, AB, DE: C is to F as AB is to DE. A H Because AB is a multiple of C, C is a part of AB (a): for the same reason F is a part of DE. And because AB, DE are equimultiples of C, F; AB contains the measure C as often as DE contains the measure F (b). Apply the measure C, from A to G, from G to H, and from H to B: apply also the measure F, from D to K, from K to L, and from L to E. Then, because the parts AG, GH, HB, are equal; and the parts DK, KL, LE are equal; AG is to DK as GH is to KL, as HB is to LE (c). Now AG, GH, HB are the antecedents, and DK, KL, LE are the consequents; wherefore AG is to DK as AB is to DE (d): but AG is equal to C and DK to F; therefore, C is to F as AB is to DE. Therefore, magnitudes have the same ratio, &c. Recite (a) def. 1, 2 of 5; (c) p. 14, 5 (case 2); (b) def. A, 5; Q. E. D. 16 Th. If four magnitudes of the same kind be proportionals, they shall be proportionals also when taken alternately. If A, B, C, D be four magnitudes of the same kind, and have A to B as C to D; then, alternately, A: C:: B: D. Take E, F equimultiples of A, B, and G, H equimultiples of C, D. Because E, F are equimultiples of A, B, and that magnitudes have the same ratio which their equimultiples have (a); therefore A is to B as E is to F: but A E is to B as C is to D (b); therefore C is to D as E is to F. Again, because G, H are equimultiples of C, D; therefore C is to D as G is to H (a): but C is to D as E is to F, as above; therefore E is to F as G is to H (c); and so, the equimultiples are proportionals (d). Wherefore, if E be greater than G, F is greater than H; if equal, equal; and if less, less (e). But E, F are equimultiples of A, B, and G, H of C, D: therefore A is to C as B is to D. Wherefore, if four magnitudes of the same kind, &c. Recite (a) p. 15, 5; (d) def. 6, 5; Q. E. D. (b) hypothesis; (c) p. 11, 5; 17 Th. If the sum of two magnitudes have to one of them the same ratio which the sum of other two has to one of these, the one left of the former two shall have to the other the same ratio which the one left of the latter two has to the other of these. Let AE, EB be two magnitudes, and CF, FD other two: the sum of the former is AE+EB, of the latter CF+FD. Then, since AE+EB: EB :: CF+FD: FD (a); inversely, EB AE÷EB :: FD: CF+FD (b), And since ratio is the quotient of the consequent divided by the antecedent (c); and division is indicated by writing the AE+EB CF+FD divisor under the dividend: AE EB CF FD FD ; that is, E ; take these equals from the for = EB + EB EB But EB FD mer (d); there will remain line; therefore, EB: AE :: FD: CF; and inversely, AE: EB:: Wherefore, if the sum of two magnitudes, &c. CF: FD (b). Q. E. D. (c1 def. 3, 5; 18 Th. If the first be to the second as the third is to the fourth, the sum of the first and second shall be to the second, as the sum of the third and fourth is to the fourth. Given AE to EB as CF to FD, four magnitudes: then AE+EB: EB:: CF+FD: FD. Because AE is to EB as CF is to FD, by hyp. inversely, EB: AE :: FD: CF (a); and because ratio is the quotient AE CF of the consequent by the antecedent (b), EB FD. EB FD add these equals to the former; then Again, IB D E AE+EB CF-4-FD + (c); that is, Now draw EB EB FD FD these equals into line; therefore EB: AE+EB:: FD: CF+FD; and inversely, (a) AE+EB: EB:: CF+FD: FD, as stated. Wherefore, if the first be to the second, &c. Recite (a) p. B, 5; (b) def. 3, 5; Q. E. D. (e) ax. 2, 1. 19 Th. If one magnitude be to another as a part of the one is to a part of the other, the parts left have the same ratio as the whole magnitudes. Let the two magnitudes AB, CD have the same ratio as the parts AE, CF; then the other parts EB, FD have the same ratio as AB to CD. Becanse ABAE+EB, and CD=CF+FD: therefore F AE+EB: CF+FD::AE: CF (a); and, alternately, AE+ EB AE CF+FD: CF (b): but these are joint proportionals, which may be taken separately: therefore AE: EB :: CF: FD (c); and, alternately, AE: CF:: EB: FD (b). But AE is to CF as AB is to CD (a); therefore, also, EB is to FD as AB is to CD (d). Wherefore, if one magnitude be, &c. C ΛΙ Q. E. D. E E Th. Of four proportionals, the first is to its excess above the second as the third is to its excess above the fourth. Given AB BE:: CD: DF; then, by hyp., AB: AB-BE:: CD: CD-DF. Let the parts of AB be AE, EB, and the parts of CD be Then, since AB: EB:: CD: FD; by division (a) AE: с AL E 20 Th. If there be three magnitudes, and other three, which, taken two and two, in order, have the same ratio: if the first be greater than the third, the fourth will be greater than the sixth; if equal, equal; and if less, less. Take A, B, C and D, E, F, six magnitudes: then, by hypothesis, A: B:: D:E, and B C:E: F. Wherefore, since, in compound ratios, the product of the antecedents are antecedents, and the product of the con- A sequents are consequents (a); AXB: BXC:: DXÉ: EXF. But since magnitudes have the same ratio as their equimultiples (b), the co-factors B, E may be rejected (c). Therefore A: C:: D: F. In which case, if Å be greater than C, D will be greater than F; if equal, equal; and if less, less (d). Wherefore, if there be three magnitudes, &c. Recite (a) def. 10, 5; (c) Note 3, page 32; (b) p. 15, 5; Q. E. D. 21 Th. If there be three magnitudes, and other three, which, taken two and two, out of order, have the same ratio; if the first be greater than the third, the fourth will be greater than the sixth; if equal, equal; and if less, less. Take A, B, C and D, E, F, six magnitudes: then, by Ahypothesis, A: B::E: F, and B:C::D: E. B Wherefore, since in compound ratios, the product of cthe antecedents are antecedents, and the product of the consequents are consequents (a); AXB: BXC::D×E:EXF. D E But, since magnitudes have the same ratio as their F equimultiples (b), the co-factors B, E may be rejected (c). Therefore A:C::D: F. In which case, if A be greater than C, D will be greater than F; if equal, equal; and if less, less (d). Wherefore, if there be three magnitudes, &c. Recite (a) def. 10, 5; (c) Note 3, page 32; (b) p. 15, 5; Q. E. D. NOTE.-The lines are so divided that the magnitudes A, B, C-and D, E, F may be taken greater, equal, or less, as the case may require. 22 Th. If there be any number of magnitudes, and as many others; which, taken two and two in order, have the same ratio: then the first is to the last of one rank, as the first is to the last of the other rank. Take the magnitudes A, B, C, D and E, F, G, H, in such wise, that A: B::E: F B:C::F: G and CD::G: H. Then, since, in compound ratios, the product of the antecedents are antecedents, and the product of the consequents are consequents (a); A G B K Therefore AXBXC: BXCXD::EXFXG: FXGXH. And, since magnitudes have the same ratio as their equimultiples (6), the co-factors B, C and F, G may be rejected (c): Therefore A: D:: E: H. In which case, if A be greater than D, E will be greater than H; if equal, equal; and if less, less (d). Wherefore, if there be any number, &c. Recite (a) def. 10, 5; (d) p. A of b 5. (b) p. 15, 5; Q. E. D. (c) Note 3, page 32; N. B. This is cited ex æquo, simply; or, ex æquali distantia. In our example, page 70, it is given, "from equal distance in order ;" that is, the homologous terms are equidistant in the order of the series of which the proportion is a part. 23 Th. If there be two ranks of magnitudes, which, taken two and two, out of order, have the same ratio; then, the first is to the last of one rank, as the first is to the last of the other. Take the magnitudes A, B, C, D and ▲ — E, F, G, H, in such wise, that A: B::G: H, B:C::F: G, C: D:: E: F. Then, since, in compound ratios, the products of the antecedents are antecedents, and the products of the con sequents are consequents (a); B H Therefore AXBXC: BXCXD:: EXFXG: FXGXH. And since magnitudes have the same ratio as their equimultiples (b), the co-factors B, C and F, G may be rejected (c). Therefore A: D:: E: H. In which case, if A be greater than D, E will be greater than H; if equal, equal; and if less, less (d). Wherefore, if there be two ranks, &c. Recite (a) def. 10, 5; (d) p. A of b 5. N. B. This is cited ex æquo perturbato; or, ex æquali distantia in proportione inordinata. In our example, page 70, it is given, "from equal distance out of order," as exemplified above. |