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ratio, then the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last.
First, let there be three magnitudes A, B, C, and other three D, E, F, which, taken two and two in a cross order, have the same ratio, that is, let A be to B as E to F, and B to C as D to E: then A shall be to C as D to F.
A B C
Take of A, B, D, any equimultiples whatever G, H, K, and of C, E, F, any equimultiples whatever L, M, N : Then, because G, H are equimultiples of A, B, therefore (5. 15) A is to B as G to H: And for the like reason, E is to F as M to N: But A is to B as E to F; therefore G is to H as M to N: And because B is to C as D to E, and that H, K are equimultiples of B, D, and L, M equimultiples of C, E, therefore (5. 4) H is to L as K to M: has been shewn that G is to H as M to N: because there are three magnitudes G, H, L, and other three K, M, N, which, taken two and two in a cross order, have the same ratio, therefore (5. 21) if G be greater than L, K is greater than N, and if equal, equal, and if less, less: But G, K are any equimultiples whatever of A, D, and L, N any whatever of C, F; therefore A is to C as D to F.
And it So then,
A. B. C. D.
Next, let there be four magnitudes A, B, C, D, and other four E, F, G, H, which, taken two and two in a cross order, have the same ratio; that is, let A be to B as G to H, B to C as F to G, and C to D as E to F: then A shall be to D as E to H.
E. F. G. H.
For, because A, B, C are three magnitudes, and F, G, H other three, which, taken two and two in a cross order, have the same ratio, therefore, by the first case, A is to C as F to H: But C is to D as E to F; therefore also, by the first case, A is to D as E to H: and so we may proceed, whatever be the number of magnitudes.
Wherefore, If there be any number &c. Q. E.D.
PROP. XXIV. THEOR.
If the first has to the second the same ratio which the third has to the fourth, and the fifth to the second the same ratio which the sixth has to the fourth, then the first and fifth together shall have to the second the same ratio which the third and sixth together have to the fourth.
Let AB the first have to C the second the same ratio which DE the third has to F the fourth, and let BG the fifth have to C the second the same ratio G which EH the sixth has to F the fourth: then AG, the first and fifth together, shall
have the same ratio to C the second which DH, the third and sixth together, has to F the fourth.
For, because BG is to C as EH to F, therefore, by inversion, C is to BG as F to EH: And, because AB is to C as DE to F, and C is to BG as F to EH, therefore, ex æquali (5. 22), AB is to BG as DE to EH: And therefore jointly (5. 18) AG is to BG as DH to EH: But BG is to C as HE to F; therefore, ex æquali, AG is to C as DH to F.
Wherefore, If the first &c. Q.E. D.
COR. 1. On the same hypothesis, the excess of the first and fifth shall be to the second as the excess of the
third and sixth to the fourth. The demonstration of this is the same with that of the proposition, if division be used instead of composition.
COR. 2. The proposition holds true of two ranks of magnitudes, whatever be their number, of which each of the first rank has the same ratio to the second magnitude that the corresponding one of the second rank has to a fourth magnitude-as is manifest.
PROP. XXV. THEOR.
If four magnitudes of the same kind be proportionals, the greatest and least of them together shall be greater than the other two together.
Let AB be to CD as E to F, and let AB be the greatest of them, and consequently F the least (5. A. & 14) : then AB and F together shall be greater than CD and E together.
Take AG equal to E, and CH equal to F: Then, because AB is to CD as E to F, and that AG is equal to E, and CH to F, therefore AB is to CD as AG to CH, and therefore also (5. 19) the remainder BG is to the remainder DH as AB to CD: But AB is greater than CD; therefore BG is greater than DH (5. A.): And because AG is equal to E, and CH to F, therefore AG and F together are equal to CH and E together: And if to the unequal magnitudes BG, DH, of which BG is the greater, there be added equal magnitudes, viz. AG and F to BG, and CH and E to DH, then AB and F together are greater than CD and E.
Wherefore, If four magnitudes &c.
I. SIMILAR rectilineal figures are those which have their angles equal, each to each, and the sides about the equal angles proportionals.
II. Reciprocal figures, viz. triangles and parallelograms, are such as have their sides about two of their angles proportionals in such a manner, that a side of the first figure is to a side of the other, as the remaining side of this other is to the remaining side of the first.
III. A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater segment as the greater segment is to the less.
IV. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.
PROP. I. THEOR.
Triangles and parallelograms of the same altitude are to one another as their bases.
Let the triangles ABC, ACD, and the parallelograms EC, CF, have the same altitude, viz. the perpendicular drawn from the vertex A to BD: then, as the base BC is to the base CD, so shall the triangle ABC be to the triangle ACD, and the parallelogram EC to the parallelogram CF.
Produce BD both ways, and take any number of straight lines BG, GH, each equal to BC, and, any number, DK, KL, each equal to CD; and join AG, AH, AK, AL.
HGBC D K
Then, because CB, BG, GH are all equal, the triangles ABC, AGB, AHG, are all equal; and therefore, whatever multiple the base HC is of the base BC, the same is the triangle AHC of the triangle ABC: For the like reason, whatever multiple the base CL is of the base CD, the same is the triangle ACL of the triangle ACD: And, if the base HC be equal to the base CL, the triangle AHC is equal to the triangle ACL, and if greater, greater, and if less, less: Therefore, since there are four magnitudes, viz. the two bases BC, CD, and the two triangles ABC, ACD, and of the first and third, viz. the base BC and the triangle ABC, have been taken any equi multiples whatever, viz. the base HC and the triangle AHC, and of the second and fourth, viz. the base CD and triangle ACD, have been taken any equimultiples whatever, viz. the base CL and triangle ACL, and that it has been shewn that, if the base HC be greater than the base CL, the triangle AHC is greater than the triangle ACL, and if equal, equal, and if less, less-therefore (5. Def. 5), as the base BC is to the base CD, so is the triangle ABC to the triangle ACD.
And, because the parallelogram CE is double of the triangle ABC, and the parallelogram CF double of the triangle ACD, and that magnitudes have the same ratio which their equimultiples have (5. 15), therefore, the triangle ABC is to the triangle ACD as the parallelogram EC to the parallelogram CF: But it has been shewn that the triangle ABC is to the triangle ACD