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Take of A, B, D any equimultiples whatever G, H, K; and of C, E, F any equimultiples whatever L, M, N:1

And because G, H are equimultiples o A, B, and that magnitudes have the same ratio which their equimultiples have (15. v.); as A is to B, so is G to H:

And for the same reason, as E is to F, so c is M to N:

But as A is to B, so is E to F; as therefore G is to H, so is M to N (11. v.).

And because as B is to C, so is D to E, and that H, Kare equimultiples of B, D, and L, M of C, E; as H is to L, so is (4. v.) K to M.

D E F

H

K

M N

And it has been shown that G is to H, as M to N;

Then, because there are three magnitudes G, H, L, and other three K, M, N, which have the same ratio taken two and two in a cross order; if G be greater than L, K is greater than N; and if equal, equal; and if less, less (21. v.):

A. B. C. D.

And G, K are any equimultiples whatever of A, D; and L, N any whatever of C, F; as, therefore, A is to C (5 Def. v.), so is D to F. 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, viz., A to B, as G to H; B to C, as F to G; and C to D, as E to F: A is to D, as E to H. Because A, B, C are three magnitudes, and F, G, H which, taken two and two in a cross order, have the same ratio; by the first case, A is to C, as F to H:

E. F. G. H.

other three,

But C is to D, as E is to F; wherefore again, by the first case, A is to D, as E to H: and so on, whatever be the number of magnitudes. Therefore, if there be any number, etc. Q. E. D.

PROPOSITION 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, 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 which EH the sixth has to F the fourth: AG the first and fifth together, shall have to C the second, the same ratio which DH the third and sixth to. gether, has to F the fourth.

Because BG is to C, as EH to F; by inversion, C is to BG (B. v.), as F to EH:

And because, as AB is to C, so is DE to F; and as C to BG, so F to EH; ex æquali (22. v.), AB is to BG, as DE to EH:

B

HI

E

A C D F

And because these magnitudes are proportionals, they shall likewise be proportionals when taken jointly (18. v.); as therefore AG is to GB,

so is DH to HE: but as GB (Hyp.) to C, so is HE to F: therefore ex æquali (22. v.), as AG is to C, so is DH to F. Wherefore, if the first, etc. Q. E. D.

COR. 1. If the same hypothesis be made as in the proposition, 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 to the second magnitude, the same ratio that the corresponding one of the second rank has to a fourth magnitude; as is manifest.

PROPOSITION XXV.

THEOR. If four magnitudes of the same kind are proportionals, the greatest and least of them together are greater than the other two together.

Let the four magnitudes AB, CD, E, F be proportionals, viz. AB to CD, as E to F; and let AB be the greatest of them, and consequently F the least (A. and 14. v.): AB together with F, are greater than CD together with E.

Take AG equal to E, and CH equal to F:

Then because as AB is to CD, so is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG to CH:

And because AB the whole, is to the whole CD, as AG is B

to CH, likewise the remainder GB shall be to the re- cmainder HD, as the whole AB is to the whole (19. v.) CD:

But AB is greater than CD; therefore (A. v.) GB is greater than HD:

And because AG is equal to E, and CH to F; AG and F together, are equal to CH and E together:

D

H

A CE F

If therefore to the unequal magnitudes GB, HD, of which GB is the greater, there be added equal magnitudes, viz. to GB the two AG and F, and CH and E to HD; AB and F together, are greater than CD and E. Therefore, if four magnitudes, etc. Q. E. D.

PROPOSITION F.

THEOR. Ratios which are compounded of the same ratios, are the same with one another.

A. B. C. D. E. F.

Let A be to B, as D to E; and B to C, as E to F: the ratio which is compounded of the ratios of A to B, and B to C, which, by the definition of compound ratio, is the ratio of A to C, is the same with the ratio of D to F, which, by the same definition, is compounded of the ratios of D to E, and E to F. Because there are three magnitudes A, B, C, and three others D, E, F, which, taken two and two in order, have the same ratio, ex æquali A is to C, as D to F (22. v.).

Next, let A be to B, as E to F, and B to C, as D to E; therefore

A. B. C.

ex æquali in proportione perturbatâ (23. v.), A is to C, as D to F; that is, the ratio of A to C, which is compounded of the ratios of A to B, and B to C, is the same with the ratio of D to F, which is compounded of the ratios of D to E, and E to F. And in like manner the proposition may be demonstrated, whatever be the number of ratios in either case.

D. E. F.

EXERCISES ON BOOK V.

1. Certain of the following properties have been proved in the Fifth Book, respecting four proportional magnitudes A, B, C, D: let the student point out which and where; and prove the remaining ones.

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2. Some of these are true when A, B are of a species different from that of C, D: let the student say which, and tell why.

3. Let m and n be any numbers, rational or irrational: then

mAnB::m C:n D.

4. If m A = A', n B = B', mC = C', and n D = D' show that the proportions (1) to (9) are true of A', B', C', D'.

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5. If A: B:: B: C; then A+B is greater or less than B±C, according as A is greater or less than B.

6. If there be several ratios, each of which is greater or less than another ratio, then the ratio of the sum of the antecedents of the first named ratios to the sum of the consequents will be respectively greater or less than that other ratio.

7. If the first of four magnitudes have a greater ratio to the second than the third has to the fourth; then,

(a) The second will have to the first a less ratio than the fourth to the third :

(b) The first will have to the third a greater ratio than the second has to the fourth :

(c) If the first be the greatest, the fourth will be the least, and vice versa:

(d) The first and second together will have to the second a greater ratio than the third and fourth together have to the fourth. 8. Prove the converses of these; and in all the cases show whether the truth be confined to the case of the four magnitudes being of the same kind or not.

9. If two magnitudes which have a ratio be each increased by equal magnitudes; then the ratio of the magnitudes thus composed will be

increased or diminished according as the original antecedent is less or greater than the consequent.

10. Show also what takes place if the original antecedent and consequent be diminished by equal magnitudes.

11. (a.) If any number of magnitudes be continued proportionals, their successive differences will also be proportionals.

(b.) Are they continued proportionals?

12. The ratio compounded of the ratios in the preceding proposition is greater than any of the component ratios.

13. (a.) Generally, a ratio compounded of ratios the whole of which are of greater or less inequality will be itself a ratio of respectively greater or less inequality; and likewise respectively greater or less than each of the component ratios.

(b) What is the ratio compounded of several ratios of equality? 14. If four magnitudes be proportionals, the ratio compounded of the ratio of one pair and the inverse ratio of the other pair is a ratio of equality.

15. If three magnitudes be proportionals, the two extreme terms together are greater than double the mean term.

[This is sometimes expressed by saying that the arithmetical mean is greater than the geometrical.]

16. Let any number of pairs of magnitudes have the same ratio :— Also, let any number of the antecedents be combined in any possible way by addition or subtraction, or both; and the corresponding consequents be combined in the same order :

And likewise, let any other combination of other antecedents, or of the same, or of some of the former and some of the other, be similarly made, and the same combination of the corresponding consequents ;Then the first combined antecedents will have to the corresponding combined consequents the same ratio that the latter has to the latter. 17. Of what proposition in the Fifth Book is the preceding an extension? and what conditions are essential amongst the magnitudes so combined, as regards species? : N be continued porpor

18. Let A:B:C:D:

tionals; then also will

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be continued proportionals.

.: M±N

19. Make the same hypothesis as in the preceding proposition; let also a series of magnitudes be taken increasing or decreasing by the difference of A and B, according as A is less or greater than B, of which A is the first: if these be A, B, C, D', N', then throughout the series C', D',... N' will respectively be greater than C, D,. . . N.

20. If two multiplicate ratios of the same number of ratios (as both duplicate, both triplicate, etc.), be the same, then the fundamental ratios will be also the same.

[This theorem is important, and must be demonstrated by the pupil. The method ex absurdo is perhaps the most simple.]

BOOK VI.

DEFINITIONS.

1. SIMILAR rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportionals.

2. "Reciprocal figures, viz. triangles and parallelograms, are such 66 as have their sides about two of their angles proportionals in such a 66 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."

3. 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.

4. The altitude of any figure is the straight line drawn from its vertex perpendicular to the base.

PROPOSITIONS.

PROPOSITION I.

THEOR. Triangles and parallelograms of the same altitude are one to another as their bases.

Let the triangles ABC, ACD, and the parallelograms EC, CF, have the same altitude, viz. the perpendicular drawn from the point A to BD; then, as the base BC is to the base CD, so is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF.

E A

F

Produce BD both ways to the points H, L, and (3.1.) take any number of straight lines BG, GH, each equal to the base BC; and DK, KL, any number of them, each equal to the base CD; and join AG, AH, AK, AL:

Then, because CB, BG, GH are all equal, the triangles AHG, AGB, ABC are all equal (38. 1.): therefore, whatever

B C

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K

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