which have the same ratio, taken two and two, but in a cross order, viz., as A: B:: E: F, and as B: C:: D: E. If A be greater than C, D is greater than F; if equal, equal; and if less, less.

A

D

B

C

E F

Because A is greater than C, and B is any other magnitude, A has to B a greater ratio (V. 8.) than C has to B: but as E: F:: A: B: therefore (V. 13.) E has to F a greater ratio than C to B. And because B : C : : D : E, by inversion, C: B:: E: D: and E was shown to have to F a greater ratio than C to B ; therefore (V. 13. cor.) E has to F a greater ratio than E to D: but the magnitude to which the same has a greater ratio than it has to another, is (V. 10.) the less of the two: F therefore is less than D; that is, D is greater than F. Secondly, let A be equal to C; D shall be equal to F. Because A and C are equal, (V.7.) A: B:: C: B: but A:B::E:F; and C: B:: E: D; wherefore (V. 11.) E: F::E: D; and therefore (V. 9.) D is equal to F.

Next, let A be less than C; D is also less than F. For Cis greater than A, and, as was shown, C: B:: E : D, and in like manner B: A::F:E; therefore F is greater than D, by case first; that is, D is less than F. Therefore, if there be three magnitudes, &c.

2

3

A

B

C

A B C

D

E

F

D E F

PROP. XXII. THEOR.

If there be any number of magnitudes more than two, and as many others, which, taken two and two in order, have the same ratio; the first has to the last of the first magnitudes the same ratio which the first has to the last of the others. N.B. This is usually cited by the words "ex æquo," or "ex æquali."

First, let there be three magnitudes A, B, C, and as many others D, E, F, which, taken two and two, have the same ratio, that is, such that A: B:: D: E; and B: C::E: F; then A: C:: D: F. Take of A, D, any like multiples whatever, G, H; of B, E, any whatever K, L; and of C, F any whatever M, N. Then, because A: B:: D: E, and that G, H are like multiples of A, D, and K, L of B, E; (V.4.) as G: K:: H: L. For the same reason, K: M:: L: N: and (V. 20.) because there are three magnitudes G, K, M, and other three H, L, N, which, two and two, have the same ratio; if G be greater than M, H is greater than N; if equal, equal; and if less, less; and G, H are any

B C

D E P

M

H L N

A
G K

like multiples whatever of A, D, and M, N of C, F: therefore (V. def. 5.) as A: C:: D: F.

Next, let there be four magnitudes, A, B, C, D, and other four, E, F, G, H, which two and two have the same ratio, viz., as A: B:: E: F; as B: C:: F: G; and as C: D::G: H: then A: D:: E: H.

A. B. C. D.

E. F. G. H.

Because A, B, C are three magnitudes, and E, F, G other three, which, taken two and two, have the same ratio; by the foregoing case, A: C:: E: G. But C: D:: G: H; wherefore again, by the first case, A: D:: E: H; and so on, whatever is the number of magnitudes. Therefore, if there be any number of magnitudes, &c.

PROP. XXIII. THEOR.

If there be any number of magnitudes, and as many others, which, taken two and two, in a cross order, have the same ratio; the first has to the last of the first magnitudes the same ratio which the first has to the last of the others. N.B. This is usually cited by the words, "ex æquali in proportione perturbata;" or "ex æquo perturbato;" or "ex æquo inversely."

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, such, that A: B:: E: F; and B: C:: DE: then A: C:: D: F.

A

B

C

D E F

H L

Κ M N

Take of A, B, D any like multiples whatever G, H, K; and of C, E, F any like multiples whatever L, M, N. Then, because G, H are like multiples of A, B, and that magnitudes (V. 15.) have the same ratio which their like multiples have; as A: B::G: H. For the same reason, as E: F:: M: N. But as AB:: E:F; as therefore (V. 11.) G: H:: M: N. G And because B: C:: D: E, and that H, K are like multiples of B, D, and L, M of C, E; then (V. 4.) as H: L::K: M: and it has been shown that G: H:: M: N. Then, (V. 21.) 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; if equal, equal; and if less, less; and G, K are any like multiples whatever of A, D; and L, N of C, F; as, therefore, (V. def. 5.) A: C:: D: F.

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, viz., A:B:: G: H; B: C:: F: G; and C: D:: E: F: then A: D:: E: H.

E. F. G. H.

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

PROP. XXIV. THEOR.

IF the first have 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 have to the second, the same ratio which the third and sixth together have to the fourth.

Let AB:C:: DE: F; and let BG the fifth have to C the second, the same ratio which EH the sixth has to F the fourth: then, AG, the sum of the first and fifth, has to C the second, the same ratio which DH, the sum of the third and sixth, has to F the fourth.

G

B+

H

E

A

C

Because BG: C:: EH: F; by inversion, C: BG:: F: EH: and because, AB: C:: DE: F; and C: BG:: F: EH; ex æquo (V. 22.) AB: BG:: DE: EH: and, by composition, (V. 18.) as AG: GB:: DH: HE; but as GB: C:: HE: F. Therefore, ex æquo, as AG: C: DH: F: wherefore, if the first, &c.

Cor. 1. If the same hypothesis be made as in the proposition, the difference of the first and fifth shall be to the second, as the difference 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 manifestly holds true of two ranks of magnitudes, whatever is 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.

PROP. XXV. THEOR.

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

Let AB CD:: E: F; and let AB be the greatest, and consequently (V. 14. and A.) F the least. AB and F are together greater than CD, together with E.

Then, because

G

C E P

Take AG equal to E, and CH equal to F. AB: CD:: E: F, and that AG is equal to E, and CH equal to F; AB: CD:: AG: CH. And (V. 19.) because the whole AB is to the whole CD, as AG is to CH, likewise the remainder GB is to the remainder HD, as the whole AB is to the whole CD. But AB is greater than CD, therefore (V. A.) 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. 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, &c.

Cor. The sum of the extremes of three continual proportionals is greater than double the mean.

PROP. F. THEOR.

RATIOS which are compounded of the same ratios, are equal to one another.

:

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

Let A B D : E; and B : C :: E: 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 æquo, (V. 22.) A : C : : D: F.

Next, let A: B:: E: F, and B: C:: D: E; therefore, ex æquo inversely, (V. 23.) A: C:: D: F: and in like manner the proposition may be demonstrated, whatever be the number of ratios. Therefore ratios, &c.

A. B. C.

D. E.-F.

BOOK V.

SUPPLEMENT.

THE theory of proportion has now been delivered in the accurate but prolix method given by Euclid. It may suit the views and convenience, however, of many teachers and learners to have its principal properties established in a more concise and condensed form, as in the following Supplement, so as to save time which may be more profitably devoted to other researches. For this purpose, the notation and some of the simplest principles of algebra may be advantageously employed; and should the student be unacquainted with that science, it will be necessary for him to commence by studying the following explanations and principles.

DEFINITIONS AND PRINCIPLES. *

1. The sign+, called plus, † denotes addition. Thus, the expression A+B signifies the sum of the quantities represented by A and B; so that if A represent 5 feet, and B 3 feet, A+B denotes 8 feet.

2. The sign, called minus, signifies, that the quantity after it is taken from the one before it. Thus, A-B expresses what remains when B is taken from A.

3. The product arising from multiplying one number by another is expressed by writing the letters representing them, one after the other, without any sign between them; and sometimes by placing between them a point, or the sign X. Thus, if A and B represent two numbers, their product is denoted by AB, by A.B, or by AXB; and A and B are called the factors of this product. The product of this result by the number C, is denoted by ABC, by A.B.C, or by AXBXC: and the product of a greater number of factors is expressed in a similar way. In like manner, 2A denotes twice A, 3A three times A, &c.

4. Hence, conversely, if AB be divided by B, the quotient is A. 5. A product is called a power, when the factors are all the same. Thus, AA, or, as it is generally written, A2, is called the

In this Supplement, for brevity, these are referred to, as No. 1, No. 2, &c. A few of these characters have been explained already in page 47. The explanation is repeated here with the view of presenting in connexion all that is necessary on the subject.