a 15.5. byp.

d 6. def.5.

* 22. and 23. 5.& 20, def.1.

@byp.

PROP.

XXIII.

If there be three magnitudes A,B, C, and others D, E, F, equall to them in number, which taken two and two are in the fame ratio, and their proportioA B C D E F and B.C:: D. E.) they shall be nality inordinate (A.B:: E. F. GH KILM in the fame ratio alfo by equa

lity.

Take G, H, I equimultipli-
ces of A,B,D; and also K, L,
Mequimultiplices of C,E,F.
Then G. H:: a A. BE. F
:: L.M.Moreover because B.
C: D.E,thence is H.K :: I.
L; therefore G, H, K, and I,
L,M are according to 21. 5.
Wherefore if G be,,
K, then is likewife I
M. and fo'd confequently A.C:: D.F. w.w.
to be Dem.

If there be more magnitudes then three, this way of demonftration holds good in them alfo.

*

Coroll.

From hence it follow's that ratio's compounded of the fame ratio's, are among themselves the fame; as also that the fame parts ofthe fame ratio's, are among themselves the fame.

CPF

D

F

PROP.

B

E H

X XIV.

If the first magnitude A B G have the fame ratio to the fecond C,which the third DE hath to the fourth F; if the fifth BG have the fame ratio to the fecond C,which the fixth E H bath to the fourth F, then shall the first compounded with the fifth (AG) have which the third comthe fame ratio to the fecond C pounded with the fixth (D H) hath to the fourth F. For because A B. C:: DE. F, and by the Hyp.

and

b 22.5.

and inversion C. BG:: F. EH; therefore by be.
quality AB. BG:: DE. EH. whence by compound-
ing, AG. BG:: DH, EH. Alfo BG. C: EH. F. byp.
Therefore again by bequality A G. C:: D H. F.
W.W.to be Dem.

D

PRO P. XXV.

Iffour magnitudes be proportionall (AB.
CD:: E.F) the greatest AB and the leaft
Ffball be greater then the reft CD,and Ě.

a bypa

Make AG E, and CHF. Be
caufe AB. CD:: a E. F:: b AG. CH,
thence is AB.CD:: GB.HD. but AB
CD.e therefore GBHD. But AGb 7.5.
C 19. S.
+FE+CH,therefore AG+F+dbye.
GBE+CH + HD, that is, AB efebol. 14.5.

ACEF+FE+CD. w.w.to be Dem.
Thefe propofitions which follow are not
Euclid's, but taken out of other Authors, and here sub-
joyned because of their frequent ufe.
PRO P. XXVI.

If the firft have a

greater proportion to the
fecond; then the third to
the fourth, then contrary-
wife, by converfion, the fe-

cond fhall have a leffe proportion to the first, then the fourth to the third.

Let A C I fay that BD For conceive

BD.

E b whence ACE. c there- a

B B

E..

C

If the first have a greater proportion to the fecond, then the third to the fourth; then alternatly the firft fhall have a greater proportion to the thirdsthen the fecond to the fourth.

Let

13. §.

b 10.5. c 85. d cor. 4

Let A

B

C then I fay A B For conceive
D.

С- Б.

Catherefore AE, band therefore A

for B W. W. to be Dem.

B C D E F

슬플

If the firft have a greater proportion to the fecond then the third to the fourth, then the first compounded with the fecond fhall have a greater proportion to the Second, then the third compounded with the fourth to the fourth.

Let AB

BC

DE I fay that AC DE For con

EF

BC EF,

ceive GB____DE a therefore is ABGB. adde BC

BC

EF.

C

to each part, then b will ACGC. c therefore AC GC that is DF w.w.to be Dem.

If the firft compounded with the fecond have a greater proportion to the fecond, then the third compounded with the fourth bath to the fourth; then by divifion the firft fhall have a greater proportion to the fecond, then the third to the fourth.

Let ACDF then I fay AB

BC EF.

DE For conBC EF.

ceive GC DE 4 therefore ACGC, Take away

BC EF.

BC, that is common; then remains ABGB.
Therefore AB, GB d or DE w.w. to be Dem.

E

If the first compounded with the fecond have a greater proportion to the fe

cond, then the third compounded with the fourth bath to the fourth,then by converfe ratio fhal the first compounded with the fecond have a leffer ratio to the first, then the third compounded with the fourth fhall have to the third. DF Then I fay that ACDF For

Let AC BC EF. becaufe that AC a

BC

A B DE.

a hyp.

DF b therefore by divifion b 19.5.

E F.

AB. DE by converfionc therefore BC,

BC EF.

A B

EF and
DE.

therefore by compofition ACDF w. w. to be

AB DE.
XXXI.

If there be three magnitudes A,B,C,&' 0thers alfo D,E,F equall to them in number;if there be a greater pro

portion of the first of the former to the fecond, then there

is of the first of the last to their second (3) and

and there, be alfo a greater proportion of the fecond of the first magnitudes to the third, then there is of the fecond of the last magnitudes to their third

(응답)

Then by equality alfo fhall the ratio of the first of
the former magnitudes to the third be gre ter then
greater
the ratio of the first of the latter magnitudes to the third

C 26.5. d 28. 5.

Conceive C Fa therefore is BC7G, & there- a 10 5. fore AA Again conceive HD therefore с ៩ B.

By R. 27.5. & 30.5.

F.

and alfo the ratio of the fecond of the former to the third be greater then the ratio of the first of the latter to the fecond (BD) then by equality alfo fhall the pro

fecond (BD)

portion of the first of the former to the third,
then that of the first of the latter to

(끝)

be greater

the third

The demonstration of this propofition is altogether like that of the precedent.

PRO P. XXXIII.

E

F

mainder E B to

If the proportion of the whole A B to the whole CD be greater then the proportion of the part taken away AE to the part taken away CF; then shall alfo the ratio of the rethe remainder F D be greater then that

of the whole A B to the whole CD.

Because that A Ba

CD

AE therefore by permu

EF,

tation AB CD therefore by converfe ratio

AE CF,

ABCD and by permutation again ABEB

EB FD,

w.w.to be Dem.

CD

FD.

PROP.