that the multiple of AB fhall be greater than that of D, but the multiple of BC not greater.

For of BC, CA take any equimultiples GF, FE fuch, that they may be each greater than D; and of D take the multiples K and L fuch, that L may be that which is first greater than GF, and K that which is next lefs than L.

Then, because L is that multiple of D which is the first that becomes greater than GF, the next preceding multiple K will not be greater than GF; that is GF will not be less than K.

And, fince FE is the fame multiple of AC that GF is of BC (by Conft.), GF will also be the fame multiple of BC that EG is of AB (V. 1.)

The magnitudes EG and GF are, therefore, equimultiples of the magnitudes AB and BC, and L is a multiple of

D.

And, fince GF is not less than K, and EF is greater than D (by Conft.), the whole EG will be greater than K and D taken together.

But K and D, taken together, are equal to L (by Conft.); therefore EG will be greater than L, and FG not greater than L, as was to be fhewn.

PROP.

PROP. V. THEOREM.

If four magnitudes be proportional, any equimultiples whatever of the antecedents will be proportional to any equimultiples whatever of the confequents.

Let A be to B as c

is to D, and of A and C take any and of B and D any equimultiples GM, HN; then will EK be to GM, as FL is to HN.

equimultiples EK, FL;

For of EK and FL take any equimultiples whatever EP, FR; and of GM and HN any equimultiples whatever

GS, HT:

Then, fince EK is the fame multiple of A, that FL is of c (by Conft.), and of EK, FL have been taken the equimultiples EP, FR, EP will be the fame multiple of A, that FR is of c (V. 3.)

And, in the fame manner, it may be fhewn, that Gs is the fame multiple of B, that HT is of D.

But A has the fame ratio to B that c has to D (by Hyp.); and of A and C have been taken the equimultiples EP, FR; and of B and D the equimultiples GS, HT.

If, therefore, EP be greater than GS, FR will also be greater than HT; and if equal, equal; and if less, less (V. Def. 5.)

And, fince EP, FR are any equimultiples whatever of EK, FL; and GS, HT are any equimultiples whatever of GM, HN; EK will have the fame ratio to GM, that FL has to HN (V. Def. 5.) Q.E.D.

PROP. VI. THEOREM.

If four magnitudes be proportional, and the first be greater than the second, the third will alfo be greater than the fourth; and if equal, equal; and if lefs, lefs.

Let A have to Bв the fame ratio that c has to D; then if A be greater than B, C will alfo be greater than D; and if equal, equal; and if less, less.

For, of A and C take any equimultiples E and G, and of B and D the same equimultiples F and H.

Then, because A is to B, as C is to D (by Hyp.) if e be greater than F, G will alfo be greater than H; and if equal, equal; and if less, less (V. Def. 5.),

And, fince E, F, G, H are the fame multiples of a, b, C, D, each of each, these last magnitudes will also observe the fame agreement of equality, excefs, or defect with their equimultiples.

If, therefore, a be greater than B, C will also be greater than D; and if equal, equal; and if less, less.

Q.E.D.

PROP.

PRO P. VII. THEOREM.

If four magnitudes be proportional, they will be proportional alfo when taken inverfely,

A

H

Br

E

If A has to в the fame ratio that c has to D; then, in-◄ verfely, в will have to A the fame ratio that D has to c.

For, of B and D take any equimultiples whatever E and F; and of a and c any equimultiples whatever G and H:

Then, fince A is to B as C is to D (by Hyp.), and G, H are equimultiples of A, C, and E, F of B, D (by Conft.), if G be greater than E, H will be greater than F; and if equal, equal; and if lefs, lefs (V. Def. 5.)

And, because G has with E the fame agreement of equality, excefs, or defect, that H has with F, E will have with G the fame agreement of equality, excess, or defe&t, that F has with H.

If, therefore, E be greater than G, F will also be greater than H; and if equal, equal; and if lefs, lefs.

But E and F are any equimultiples whatever of B and D (by Conft.); and G and H are any equimultiples whatever of A and c; therefore, в is to A as D is to c (V. Def. 5.)

Q.E.D.

PROP. VIII. THEOREM.

If the first of four magnitudes be the fame multiple or part of the second as the third is of the fourth; the firft will have the fame ratio to the fecond as the third has to the fourth,

Let A the first, be the fame multiple of в the second, that c the third is of D the fourth; then will A have to B the fame ratio that C has to D.

For of A and C take any equimultiples whatever E and G; and of B and D any equimultiples whatever F`and H:

Then, because A is the fame multiple of в that c is of D (by Hyp.), and E is the fame multiple of A that G is of c (by Conft.), E will also be the fame multiple of в that G is of D (V. 3.)

And, fince E is the fame multiple of B that G is of D, and F is the fame multiple of в that H is of D (by Conft.), if E be greater than F, G will be greater than H; and if equal, equal; and if less, less.

But E and G are any equimultiples whatever of a and c; and F and H are any equimultiples whatever of в and D; therefore, A will have to в the fame ratio that c has to D (V. Def. 5.)