PROP. XXXVI. THEOR. If there be three magnitudes (A, B, C) and as many Fig. 37. others (D, E, F) which if taken two by two are proportional, but perturbate (A to B as E to F, and B to C as D to E), and if there be taken equi-submultiples (a, b, d of the first and second magnitudes in the first series, and of the first in the second series (of A, B and D), and also any equi-submultiples (c, e and f) of the third in the first series, and the second and third in the second series (of C, E and F), these equi-submultiples of the magnitudes in the first and second series are also proportional perturbate (a to bas e to f, and b to c as d to e.) Because A is to B as E to F (1), and a and b are (1) Hypoth. taken equi-submultiples of A and B, and e and f, equisubmultiples of E and F, a is to bas e to ƒ (2). (2) Cor. 2. And since B is to C as D to E (1), and 6 and d are Prop. 28. equi-submultiples of B and D, and c and e equi-sub- B. 5. multiples of C and E (1), b is to c as d to e (2). PROP. XXXVII. THEOR. If there be three magnitudes (A, B, C) and as many Fig. 32. others (D, E, F) which if taken two by two are in the same ratio but perturbate (A to Bas E to F, and Bto C as D to E) and if there be taken equi-submultiples (a and d) of the first magnitude in each series (A and D,) and also any equi-submultiples (c and f) of the last in each series (C and F), and if the submultiple (a) of the first in the first series be greater than the submultiple (c) of the last in the same series, the submultiple (d) of the first in the second series is also greater than the submultiple (f) of the last in the same series. Take b the same submultiple of B that a is of A, and also e the same submultiple of E that ƒ is of F, there are three magnitudes a, b, c, and as many others d, e, f, which if taken two by two are proportional, but perturbate (1), and therefore if a be greater than c, d is greater than f(2). (1)Prop.36. B. 5. PROP. XXXVIII. THEOR. Fig.33834. If there be any number of magnitudes, and as many others, which taken two by two are in the same ratio but perturbate, they are ex æquali in the same ratio. First, let there be three magnitudes, A, B, and CO, and as many others, D, E, F; if A be to B as E to F, and B to CO as D to E, A is to CO as D is to F. For if there be taken any equi-submultiples a and d of A and D, d is contained in F as often as a contained in CO. For, if it be possible, let d be contained in F oftener than a is contained in CO, repeat a as often as d is contained in F, so as to make up CY; CY is greater than (1) Prop.5. CO (1), and a is a submultiple of CY, let x be the equisubmultiple of CO, and also z of F. B. 5. (2) Ax. 3. B. 5. (3) Constr. Because x and a are equi-submultiples of CO and CY, and CO is less than CY, x is less than a (2); and since z is the same submultiple of F that x is of CO (3), and d is contained in F as often as a is contained in CO (3), d is contained in F as often as z, a submultiple of F, is contained in the same F, therefore d is either (4) Cor. 5. less than ≈ or equal to it (4). Therefore there are taken Prop.7.B.5. equi-submultiples a and d of A and D, and x and z also equi-submultiples of CO and F, and a is greater than î, but d not greater than z, which is impossible (5) (5)Prop.37. B. 5. Fig. 34. (6) Prop. proc. Therefore no submultiple of D is contained in F oftener than the equi-submultiple of A is contained in CO; and in the same manner it can be demonstrated that no submultiple of A is contained in CO oftener than the equi-submultiple of D is contained in F: therefore A is to CO as D is to F. Next, let there be four magnitudes, A, B, C, G, and as many others, D, E, F, H, and let A be to B as F to H, B to C as E to F, and C to G as D to E; A is to G as D is to H. Because there are three magnitudes A, B, C, and as many others E, F, H, which if taken two by two are proportional perturbate, A is to C as E to H (6), but C is to Gas D to E (7); and therefore A is to Gas D is (7) Hypoth. to H (6). In the same manner it can be demonstrated whatever be the number of magnitudes given. PROP. XXXIX. THEOR. If there be three magnitudes proportional (A to B as Fig. 1, B to C), and as many others also proportional (D to E Plate 7. as E to F); and if the first magnitude in the first series be to the last (A to C), as the first to the last in the second series (D to F); the first is to the second in the first series (A to B) as the first to the second in the second series (D to E). B. 5. B. 5. For, if it be possible, let one of them A have a less ratio to B, than D has to E, and since B is to C as A to B, and E to F as D to E (1), B has a less ratio to C (1) Hypoth. than E has to F (2); therefore there is a submultiple of (2) Prop.19. B which is contained in C oftener than the equi-submultiple of E is contained in F (3); let those submulti- (3) Def.p.7. ples be b and e, and take equi-submultiples a and d of B. 5. A and D; because a and b are equi-submultiples of A and B, a is to b as A to B (4), and also d to e as D (4) Prop.28. to E (4); therefore a has a less ratio to b than d has to e (5), and therefore x can be taken a submultiple of a (5) Hypoth. which is contained in b oftener, than the equi-submultiple z of d is contained in e (6), and since x is con- (6) Def. 7. tained in b oftener than ≈ is contained in d, and b is B. 5. contained in C oftener than d in F, x is contained in C oftener than is contained in F (7), but x and z are (7) Cor. Prop. 1. equi-submultiples of a and d, and a and d are equi-submultiples of A and D, therefore x and x are equi-submultiples of A and D (8); therefore z is contained in (8) Prop. 7. F as often as x is contained in C (9), but it was B. 5. proved before that x was contained in Coftener than (9) Hypoth. & Def. 5. z in F, which is absurd. B. 5. Therefore A has not a less ratio to B than D has to E; and in the same manner in can be demonstrated that D has not a less ratio to E than A has to B; and therefore A is to B as D to E. B. 5. ELEMENTS OF EUCLID. BOOK VI. DEFINITIONS. 1. Similar rectilineal figures are those which have all their angles respectively equal, and the sides about the equal angles proportional. 2. A right line is said to be cut in extreme and mean ratio, when the whole line is to the greater segment, as the greater segment is to the less. 3. The altitude of any figure is a right line drawn from the vertex perpendicular to the base. 4. A parallelogram described upon a right line is said to be applied to that right line. 5. A parallelogram, described upon a part of a right line is said to be applied to that line deficient by a parallelogram, that is by the parallelogram which is described upon the remaining part. 6. When a given right line is produced, the parallelogram described upon the whole line is said to be applied to the given line exceeding by a parallelogram, that is by the parallelogram which is described upon the produced part. PROP. I. THEOR. Triangles (ABC, DBG) and parallelograms (BA Fig. 1. and CF) which have the same altitude, are to each other Plate 5: as their bases. Part 1. Divide the base of the triangle ABC into any number of equal parts AF, FK, and KC, take on the base DG, as often as it is possible, the segments DI, IE equal to AF, and draw BF, BK, BI, and BE. Book 6. See N. B. 1. Since the right lines AF, FK, KC, DI and IE are equal (1), and the triangles constructed upon them have (1) Constr. the same altitude, the triangles are equal (2); and (2) Prop.38, therefore whatever submultiple AF is of AC, the same is the triangle ABF of the triangle ABC; and the triangle ABF is contained in the triangle DBG as often as AF is contained in the base DG; and in the same manner it can be shewn that as often as any other submultiple of AC is contained in DG, so often the equi-submultiple of ABC is contained in DBG: and therefore ABC is to DBG as the base AC B. 5. to the base DG (3). Part 2. The parallelograms BA and CF, which have the same altitude, are as their bases BC and CD. Fig. 2. For draw BA and AD, and since the triangles BAC (4) Prec. and CAD have the same altitude, they are to one ano-part ther as their bases BC and CD (4): but the paral- B. 1. (5) Prop.34. lelograms are double of these triangles (5), and therefore (6) Prop.28. are to one another as BC and CD (6). Cor. 1. Triangles or parallelograms, which have equal altitudes are to one another as their bases. B. 5. For if the bases were placed in directum, the right line joining the vertices is parallel to the line in which the bases are, because the perpendiculars let fall from the vertices upon the bases are parallel, and since they are equal (1), the right lines joining them are parallel (1) Hypoth. (2); and therefore it can be demonstrated, in the same (2) Prop.33. manner as in the proposition, that the triangles or B. 1. parallelograms are to one another as their bases, |