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.

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

a

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 в 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 H 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 L to м, and, alternately, as H is to L so is K to M.C But it has been shown as G is to н 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,d 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.

• 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 For because=and =

::d:f, or =

с

d

b

с

d

*; mul

e

tiply these two equations together, and it will be

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.

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.a And because magnitudes divided are proportional, they shall also be proportional when compounded;

B

G

H

* 1 Ax. 5.

a 22.5.

therefore as AG is to BG so is DH to HE. ACD F 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.

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 ab :: c: d, and e: b a + e c + ƒ

:: fd; then a + e : b :: c +f:d, or = 1

b

1

For, because=and; add the two equations

a

e

* 2 Ax. 1. together and it will be + &

. 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.

B

G D

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 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 maxithe last will be a minimum.

mum,

EUCLID'S ELEMENTS.

BOOK VI.

DEFINITIONS.

1. Similar rectilineal figures are those which have their angles equal each to each, and the sides about the equal angles proportional.

2. Reciprocal figures are such as have their sides about two of their angles proportional 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.

3. A right 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 perpendicular drawn from the vertex to the base.

PROPOSITION I.

THEOREM.

Triangles, and parallelograms, which have the same altitude, are to one another as their bases.

Let there be the triangles ABC, ACD, also the parallelograms EC, CF, which 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.

Produce BD both ways to the points H, L, and take BG, GH, any number of times equal to the base BC; also DK, KL, any number of times

equal to the base CD; and join AG, AH, AK, AL. Therefore because CB, BG, GH, are equal to one another, the triangles AHG, AGB, ABC, will be equal to one another. Therefore the base нc

E

A

HG B C D K L

. 38. 1.

is the same multiple of the base BC, as the triangle AHC is of the triangle ABC. For the same reason the base LC is the same multiple of the base CD as the triangle ALC is of the triangle ACD; and if the base HC be equal to the base CL, the triangle AHC is equal to the triangle ALC: if the base HC be greater than the base CL, the triangle AHC is also greater than the triangle ALC; and if less, less. Therefore there are four magnitudes; viz. the bases BC, CD, and the two triangles ABC, ACD, such that if equimultiples of the base BC and the triangle ABC be taken, viz. the base нc and the triangle AHC; also of the base CD and the triangle ACD any other equimultiples, viz. the base CD and the triangle ALC. And it has been shown that if the base HC be greater than the base CL, the triangle AHC will be greater than the triangle ALC, if equal, equal, and if less, less; therefore as the base BC is to the base CD so 5 Def. 5. is the triangle ABC to the triangle ACD.

41. 1.

d 15. 5.

• 11. 5.

And because the parallelogram EC is double of the triangle ABC also the parallelogram Fc is double of the triangle ACD; and magnitudes have the same ratio which their equimultiples have ; therefore as the triangle ABC is to the triangle ACD So is the parallelogram Ec to the parallelogram Fc. Hence because it has been shown, that as the base BC is to the base CD so is the triangle ABC to the triangle ACD; but as the triangle ABC is to the triangle ACD so is the parallelogram Bc to the parallelogram Fc; and, therefore, as the base BC is to the base CD so is the parallelogram EC to the parallelogram Fc. Therefore, triangles, &c. q. E. d.

Deductions.

1. Triangles and parallelograms having equal bases, are to each other as their altitudes.

2. If four right lines be proportional, their squares shall also be proportional.