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because, as AB is to C, fo is DE to F; and as C to BG, fo F Book V. to EH by equality b, AB is to BG, as DE to EH: and, by compofition, as AG is to GB, fo is DH to HE; but as GB b 22. 5. is to C, fo is HE to F; therefore, by equality, as AG is to c 18. 5. C, fo is DH to F. Wherefore, &. Q. E. D.

COR. 1. If the fame hypothefis be made as in the propofition, the excefs of the first and fifth fhall be to the fecond, as the excefs of the third and fixth to the fourth. The demonftration of this is the fame with that of the propofition, if division be used inftead of compofition.

COR. 2. The propofition holds true, of two ranks of magnitudes, whatever be their number, of which each of the first rank has to the second magnitude the fame ratio that the corre fponding one of the second rank has to a fourth magnitude; as is manifeft.

PROP. XXV. THEOR.

four magnitudes of the fame kind be propor

are greater than the other two together.

Let AB be to CD, as E is to F, and let AB be the greatest

of them, and confequently F the leaft : AB, together with F, is a A. & 14 greater than CD, together with E.

5.

D

B.

b 19. 5.

CA. 5.

Take AG equal to E, and CH equal to F: Then, becaufe as AB is to CD, fo is E to F, and that AG is equal to E, and CH equal to F; AB is to CD, as AG B to CH: wherefore the remainder GB is to the remainder DH, as the whole AB to G the whole CD: But AB is greater than CD; therefore 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. If therefore AG and F be added to the greater magnitude BG, and CH and E to the lefs DH; the whole AB and F together are greater than the whole CD and E together. Wherefore, &c. e 4. Ax. 1. Q. E. D.

e

A C EF

d 2. Ax. 1.

PROP.

BOOK V.

a 22. 5.

b 23. 5.

R

PROP. F. THEOR.

ATIOS which are compounded of the fame ratios, are the fame with one another.

Let the ratios of A to B, and B to C, be the fame with the ratios of D to E and E to F; the ratio of A to A, B, C, C, which is compounded of the first ratios, is the fame with the ratio of D to F, which is compound D, E, F.

ed of the other ratios.

First, Let A be to B, as D to E; and B to C, as E to F; then, by equality 2, as A is to C, fo is D to F.

Next, Let A be to B, as E to F; and B to C, as D to E; therefore, by perturbate equality ", A is to C, as D to F; that is, the ratio compounded of the ratios of A to B, and B to C, is the fame with the ratio compounded of the ratios of D to E, and E to F: and in like manner, the propofition may be demonftrated, whatever be the number of ratios in either cafe.

THE

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Two magnitudes are faid to be reciprocally proportional to two See N. others, when one of the first is to one of the other magni

tudes, as the remaining one of the last two is to the remaining

one of the first.

III.

A straight line is said to be cut in extreme and mean ratio, when the whole is to the greater fegment, as

the greater fegment is to the lefs.

.

IV.

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

PROP.

BOOK VI.

See N.

PROP. I. THEOR.

RIANGLES and parallelograms of the fame altitude are one to another as their bafes.

TRI

Let the triangles ABC, ACD, and the parallelograms EC, CF have the fame altitude, viz. the perpendicular drawn from the point A to BD: Then, as the bafe BC is to the bafe CD, fo is the triangle ABC to the triangle ACD, and the parallelogram EC to the parallelogram CF.

Produce BD both ways to the points H, K, and take any number of ftraight lines BG, GH, each equal to the bafe BC, fo that CH be greater than CD; and join AG, AH: and becaufe the bafes CB, BG, GH are all equal, the triangles ABC, a 38. 1. AGB, AHG are all equal; therefore the triangle AHC is the fame multiple of ABC, that the bafe HC is of BC. In like manner, if CK be taken the leaft multiple of CD that is greater than CH, and AK be joined, it may be proved, that the triangle ACK is the fame multiple of ACD: Wherefore the base DK and the triangle ADK contain CD and ACD equally. And if the base HC be equal to DK, the triangle. AHC is equal to the triangle ADK: But, if not, the base HC is greater

b Def.A.5.

a

E A

F

than DK, but lefs than CK, and the HG BC
triangle AHC is greater than ADK,

DK

but less than ACK; therefore the bafe CH contains CD the fame number of times that DK contains it ; and the triangle AHC contains ACD the fame number of times that ADK does: But the bafe DK and the triangle ADK contain DC and ACD equally; therefore the bafe HC and triangle ABC contain them equally and the bafe HC and tirangle AHC are any equimultiples of BC and ABO: as many times, therefore, as any multiple of the bafe BC contains the bafe CD, fo many times does the fame multiple of the triangle ABC contain the triangle € 5. Def. 5. ACD; therefore as the bafe BC is to the bafe CD, fo is the triangle ABC to the triangle ACD.

And because the parallelogram CE is double of the triangle € 41. 1. ABC, and the parallelogram CF double of the triangle ACD, and that magnitudes have the fame ratio which their equimultiples have ; as the triangle ABC is to ACD, fo is the parallelogram EC to CF: And as the base BC is to CD, fo is the triangle ABC to ACD; therefore, as the bafe BC is to CD, fo is

d15. 5.

d

the

the parallelogram EC to the parallelogram CF. Wherefore Book VI. triangles, &c. Q. E. D.

COR. From this it is plain, that triangles and parallelograms e 11. 5. that have equal altitudes, are one to another as their bafes.

Let the figures be placed so as to have their bases in the fame ftraight line; and having drawn perpendiculars from the vertices to the bafes, the ftraight line which joins the vertices, is paral

lel to their bafes f, becaufe the perpendiculars are both equal f 33. 1. and parallel to one another: Then, if the fame conftruction be made as in the propofition, the demonftration will be the fame.

PROP. II. THEOR.

IF a

Fa ftraight line be drawn parallel to one of the fides of a triangle, it fhall cut the other fides, or those produced, proportionally: And if the fides, or the fides produced, be cut proportionally, the ftraight line which joins the points of fection, shall be parallel to the remaining fide of the triangle.

Let DE be drawn parallel to BC, one of the fides of the triangle ABC: BD is to DA, as CE to EA.

b 7. 5.

C I. 6.

Join BE, CD; then the triangle BDE is equal to the triangle CDE, because they are on the fame bafe DE, and be- a 37. T. tween the fame parallels DE, BC: ADE is another triangle, and equal magnitudes have to the fame, the fame ratio; there fore, as the triangle BDE to the triangle ADE, fo is the triangle CDE to the triangle ADE; but as the triangle BDE to the triangle ADE, fo is BD to DA, because having the fame altitude, viz. the perpendicular drawn from the point E to AB, they are to one another as their bafes; and for the fame reafon, as the triangle CDE to the triangle ADE, fo is CE to EA. Therefore, as BD to DA, fo is CE to EA d. Next, Let the fides AB, AC of the triangle ABC, or thefe produced, be cut proportionally in the points D, E, that is, fo that BD be to DA, as CE to EA, and join DE: DE is parallel to BC.

D

Ꭺ.

E

The fame conftruction being made, becaufe, as BD to DA, fo is CE to EA; and as BD to DA, fo is the triangle BDE to the triangle ADE ; and as CE to EA, so B is the triangle CDE to the triangle ADE; therefore the triangle BDE is to the triangle ADE, as the triangle CDE to the triangle ADE; that is, the triangles BDE,

S

C

CDE

d 11. 5.

e 1. 6.

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