Page images
PDF
EPUB

* Hyp.

+ 8. 5.

7.5. 11. 5.

a 8. 5.

b 10.5.

viz. a : b :: d : e, and bce: f; then if a be greater than c, d shall also be greater than f; if equal, equal; and if less, less. If a c, then because e f :: bc, by inversion it will be .fe::cb. But therefore > > or r, therefore d <f. In like

с

a

e

a

if a = C.

manner we show, if a > c, also d > f,
Because f: e::cb::‡a: b::d: e§. Whence d = f.

Q. E. D.

PROPOSITION XXI.

THEOREM.

If there be three magnitudes, and others equal to them in number, which taken two and two have the same ratio, and let their proportion be perturbate; if the first magnitude be greater than the third, the fourth, also, will be greater than the sixth; if equal, equal; and if less, less.

Let A, B, C, be three magnitudes, and others D, E, F, equal to them in number, which taken two and two have the same ratio, and let their proportion be perturbate; viz. as A is to B so is E to F, and as в is to c so is D to E; then if A be greater than C, D will also be greater than F; if equal, equal; and if less, less.

For because A is greater than c, and B some other magnitude; therefore A has a greater ratio 10 в than c has to B. ABC DEF But as A is to в so is E to F; also inversely, as c is to в so is E to D; therefore also E has a greater ratio to F than E to D. But to that magnitude, which the same has a greater ratio, is the less; b therefore F is less than D; whence D is greater than F. In like manner, we show if a be equal to C, D will also be equal to F; and if less, less. If, therefore, there be three magnitudes, &c. Q. E. D.

The same by Algebra.

Let a, b, c, be three magnitudes, and others d, e, f, equal to them in number, which taken two and two have the same ratio, viz. a be: f, and bc:: de; if a be greater than c, d is also greater than f; d: if equal, equal; and if less, less.

1. If a c, then because de :: b: c, therefore, inversely, ed: cb, but<; whence, that is, than, therefore d > ƒ.

2. In like manner, if a <c, then is d < f.

3. If a = c, then because e: d::c: b:: a b :: e:f, therefore is d = f. Q. E. D.

PROPOSITION XXII.

THEOREM.

If there be any number of magnitudes, and others equal to them in number, which taken two and two have the same ratio, they shall also be by equality in the same ratio, "that is, the first shall be to the last of the first magnitudes, as the first of the others is to the last."

Let there be any number of magnitudes A, B, C, and others equal to them in number, viz. D, E, F, which taken two and two have the same ratio, as a is to B so is D to E, and as в is to c so is E to F; then by equality, A shall be to c as D to F.

For take G, H, equimultiples of A, D, and K, L, any other equimultiples of в and

E; moreover M, N, any equimultiples of c, F.

B

A

other

[blocks in formation]

And

[blocks in formation]

because A is to в as D is to E; and G, H, are taken equi

multiples of A, D, also K, L,

any other equimultiples of B, E;a therefore as G is to 4. 5. K so is H to L. For the same reason as к is to м so

is L to N. And because G, K, M, are three magnitudes, and others equal to them in number, viz. H, L, N, which taken two and two are in the same ratio; therefore by equality if G exceeds м, H also exceeds N; b 20. 5. if equal, equal; and if less, less. And G, H, are equimultiples of A, D, also M, N, any other equimultiples

b

с

of c, F; therefore it is as a is to c so is D to F. If, 5 Def, 5. therefore, there be any number, &c. Q. E. D.

The same by Algebra.

Let a, b, c, be any magnitudes, and others d, e, f, equal to them in number, which taken two and two have the same ratio; viz. a: b :: d : e, and bcef; then by equality a c :: d: f, or

a

d ===. For because ===, and; multiply these

C

-=

[merged small][merged small][ocr errors][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small]

a 15.5.

b 11. 5.

PROPOSITION XXIII.

THEOREM.

If there be three magnitudes, and others equal to them in number, which taken two and two have the same ratio, and their proportion be perturbate, they will also, by equality, be in the same ratio, "the first shall have the same ratio to the last of the first magnitudes, as the first of the others has to the last."*

Let A, B, C, be three magnitudes, and others D, e, f, equal to them in number, which taken two and two have the same ratio, and let their proportion be perturbate, viz. as A is to в so is E to F; also as в is to c so is D to E; then as A is to c so is D to F.

[ocr errors]

For take G, H, K, equimultiples of A, B, D, also L, M, N, any other equimultiples

GHL ABC

DEFKMN

And because it is

of C, E, F. And because G, H, are equimultiples of A, B, also that magnitudes have the same ratio which their equimultiples have; therefore as A is to в SO is G to H. By the same reason E is to F as M is to N; but it is as A is to в so is E to F; therefore also as G is to H so is м to N. as B is to c so is D to E, and alternately, as в is to D so is c to E. Also, because н, K, are equimultiples of B, D; and magnitudes have the same ratio which their equimultiples have; therefore as в is to D so is H to K; but as в is to D so is c to E; whence also, as н is to K so is c to E. Again, because L, M, are equimultiples of c, E; therefore it is as c is to E so is L to M. But as c is to E so is H to K; whence, also, as H is to K so is 1 to м, and, alternately, as H is to L so is K to M. But it has been shown as G is to H so is м to N; and because there are three magnitudes G, H, L, and others, K, M, N, equal to them in number, taken two and two, have the same ratio, and their proportion is perturbate, therefore, by equality, if G exceeds L, K also exceeds N; if equal, equal; and if less, less. And G, K, are equimultiples of A, D, also L, N, of c, F; therefore it • 5 Def. 5. is as a is to c so is D to F. If, therefore, there be three magnitudes, &c. Q. E. D.

c 15. 5.

a 21.5.

* Euclid has demonstrated this proposition with proposing three magnitudes only; but this, as also the two following, will hold good with any number of maguitudes whatever.

The same by Algebra.

Let a, b, c, be three magnitudes, and d, e, f, as many others, which taken two and two have the same ratio; viz. a: b::e: f, and b:c::de; then a c

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors]

tiply these two equations together, and it will be

[blocks in formation]

If the first magnitude have the same ratio to the second which the third has to the fourth; and the fifth, the same ratio to the second, which the sixth has to the fourth; then the first and fifth taken together shall have the same ratio to the second which the third and sixth together have to the fourth.

For let the first magnitude, AB, have the same ratio to the second, c, which the third, DE, has to the fourth, F; and let the fifth, BG, have the same ratio to the second, c, which the sixth, EH, has to the fourth, F; then shall AG, the first and fifth taken together, have the same ratio to the second, c, which DH, the third and sixth together, have to the fourth, F.

B

G

H

a 22.5.

For because it is as BG to c so is EH to F; by inversion, therefore, as c is to BG so is F to EH. And because it is as AB to c so is DE to F, but as c to BG so is F to EH; therefore, by equality, it is as AB to BG so is DE to EH. And because magnitudes divided are proportional, they shall also be proportional when compounded; therefore as AG is to BG so is DH to HE. But it is as BG to c so is EH to F; therefore, by equality, it is as AG to c so is DH to F. If, therefore, the ↳ 18. 5. first magnitude, &c. Q. E. D.

The same by Algebra.

ACD F

Let a the first magnitude have the same ratio to b the second, as c the third has to d the fourth; and let e the fifth have the same ratio to b the second, as ƒ the sixth to d the fourth; or a b :: cd, and e: b a + e c + ƒ

:: fd; then a +eb::c+f:d, or

b

=

d

[ocr errors]
[blocks in formation]

a 19.5.

PROPOSITION XXV.

THEOREM.

If four magnitudes be proportional, the greatest and least together are greater than the remaining two together.

Let the four magnitudes AB, CD, E, F, be proportional; viz. as AB is to CD so is E to F; let AB be the greatest, and consequently, F the least, then AB and F together are greater than CD and E together.

ACE F

For make AG equal to E, also CH equal to F. Therefore because it is as AB is to CD so is E to B F, but AG is equal to E and CH to F; therefore it is as AB is to CD so is AG to CH. And because it is as the whole magnitude AB is to the whole CD so is AG to CH, also the remainder GB shall be to the remainder HD as the whole AB is to the whole CD. But AB is greater than CD; therefore GB is also greater than HD. And because AG is equal to E, also CH to F; therefore AG and F together are equal to CH and E together. And because, if equals be added to unequals, the wholes are unequal; if, therefore, GB, HD, being unequal, and CB being the greater; to GB add AG, F; also to HD add CH, E, therefore AB and F together will be greater than GD, E. If, therefore, four magnitudes, &c. Q. E. D.*

Deduction.

If the three magnitudes be proportional the two extremes shall be greater than double of the mean.

* From this it is manifest if the first term of the proportion be a maximum, the last will be a minimum.

« PreviousContinue »