third, has to D the fourth; but C the chird a greater ratio to D the fourth than E the fifth, has to F the sixth. The first A has to the second B, a greater ratio than the fifth E, has to the sixth F.

A.

M

G

H

C

E

B

D

F

K

L

N

Take G and H equimultiples of C and E, and K and L equimultiples of D and F, such that G may be greater than K, but H not greater than L (V. Def. 7). Whatever multiple G is of C, take M the same multiple of A; and whatever multiple K is of D, take N the same multiple of B.

Because A is to B, as C is to D (Hyp.), and M and G are equimultiples, of A and C; and N and K are equimultiples of B and D. Therefore, if M be greater than N, G is greater than K; if equal, equal; and if less, less (V. Def. 5). But G is greater than K (Const.). Therefore M is greater than N. But H is not greater than L (Const.); and M and Hare equimultiples of A and E; and N and L equimultiples of B and F. Therefore A has a greater ratio to B, than E has to F (V. Def. 7). Wherefore, if the first, &c. Q. E. D.

COROLLARY.-If the first have a greater ratio to the second, than the third has to the fourth, but the third the same ratio to the fourth, which the fifth has to the sixth; it may be demonstrated, in like manner, that the first has a greater ratio to the second, than the fifth has to the sixth.

If the first has the same ratio to the second which the third has to the fourth; and if the first be greater than the third, the second is greater than the fourth; if equal, equal; and if less, less.

Let the first A have the same ratio to the second B, which the third C, has to the fourth D.

First, if A be greater than C (fig. 1), B is greater than D.

Because A is greater than C, and B is another magnitude of the same kind, A has to B a greater ratio than C has to B (V. 8). But, as A is to B, so is C to D (Hyp.). Therefore also C has to D a greater

Fig. 1.

Fig. 2.

Fig. 3.

ratio than C has to B (V. 13). But of two magnitudes, that to which another of the same kind has the greater ratio is the less (V. 10). Therefore D is less than B; that is, B is greater than D.

Secondly, if A be equal to C (fig. 2), B is equal to D.

For A is to B, as C, that is, A is to D. Therefore B is equal to D (V.9).

Thirdly, if A be less than C (fig. 3), B is less than D.

For C is greater than A, and C is to D, as A is to B. Therefore D is greater than B, by the first case; that is, B is less than D. Therefore, if the first, &c. Q. E. D.

Magnitudes have the same ratio to one another which their equimultiples

have.

Let A B be the same multiple of C, that D E is of F. C is to F, as A B is to D E.

Divide A B into magnitudes, each equal to C, viz., A G, GH, and HB; and D E into as many (Hyp.) magnitudes, each equal to F, viz., DK, KL, and L E.

AGH B

D K LE

F

Because the magnitudes AG, GH, H B, are all equal to one another, and the magnitudes DK, KL, LE, are also equal_to_one_another. Therefore AG is to DK as GH to K L, and as HB to LE (V. 7). But as one of the antecedents is to its consequent, so are all the antecedents together to all the consequents together (V. 12). Therefore, as A G is to Ď K, so is A B to D E.ˆ But A G is equal to C, and D K to F (Const.). Therefore as C is to F, so is A B to DE. Therefore, magnitudes, &c. Q. E. D.

If four magnitudes of the same kind be proportionals, they are also proportionals when taken alternately.

Let A, B, C, and D be four magnitudes of the same kind, and let A be to B, as C is to D. They are also proportionals when taken alternately (Def. 13); that is, A is to C, as B to D.

Take of A and B, any equimultiples what- Eever, E and F; and of C and D`any equimultiples whatever, G and H.

G

A

C

B

D

H

Because E is the same multiple of A, that F is of B, and that magnitudes have the same ratio to one another which their equimulti- F ples have (V. 15). Therefore A is to B as E is to F. But as A is to B, so is C to D (Hyp.). Therefore as C is to D, so is E to F (V. 11). Again, because B and H are equimultiples of C and D. Therefore as C is to D, so is G to H (V. 15). But it was proved that as C is to D, so is E to F. Therefore, as E is to F, so is G to H (V. 11). But when four magnitudes are proportionals, if the first be greater than the third, the second is greater than the fourth: if equal, equal; if less, less (V. 14). Therefore, if E be greater than G, F likewise is greater than H; if equal, equal; and if less, less. But E and F are any equimultiples whatever of A and B (Const.) and G and H any whatever of C and D. Therefore A is to C as B to D. (V. Def. 5.) If then four magnitudes, &c. Q. E. D.

If four magnitudes be proportionals; by division the excess of the first above the second is to the second, as the excess of the third above the fourth is to the fourth.

Let A B, BE, CD, and D F, be proportionals; that is, as A B to B E,

so is CD to DF. And let AE be the excess of AB above BE; and CF the excess of CD above DF. As AE is to EB, so is CF to FD (V. Def. 16).

Take of AE, EB, CF, and FD any equimultiples whatever G H, HK, LM, and MN; and of EB and FD any other equimultiples whatever KX and NP.

Because GH is the same multiple of AE, that HK is of EB (Const.). Therefore, GH is the same multiple of AE, that GK is of AB (V. 1). But GH is the same multiple of AE that LM is of CF (Const.). Therefore GK is the same multiple of AB, that LM is of CF. Again, because L M is the same multiple of CF, that MN is of FD (Const.). Therefore LM is the same multiple of CF, that LN is of CD (V. 1). But LM was shown to be the same multiple of CF, that G K is of AB. Therefore GK is the same multiple of AB, that L N is of CD; that is, GK and LN are equimultiples of AB and CD. Next, because HK is the same multiple of E B, that MN is of FD (Const.), and KX the same multiple of EB, that NP is of FD (Const.). Therefore HX is the same multiple of EB, that MP is of FD (V. 2). Because AB is to BE as CD is to DF (Hyp.), and G K and LN are equimultiples of AB and CD, and H X and MP are equimultiples of E B and FD. Therefore if GK be greater than HX, LÑ is greater than MP; if equal, equal; and if less, less (V. Def. 5). But if G H be greater than KX, by adding the common part HK to these unequals, GK is greater than HX (I. Ax. 4). Wherefore also LN is greater than MP. By taking away MN from these unequals, LM is greater than NP (I. Ax. 5). Therefore, if G H be greater than K X, LM is greater than N P. In like manner it may be shown that if GH be equal to KX, LM is equal to NP; and if less, less: but GH and LM are any equimultiples whatever of AE and CF (Const.), and KX and NP are any whatever of EB and FD. Therefore, AE is to EB, as CF is to FD (V. Def. 5). If then magnitudes, &c. Q. E. D.

The term division used in the enunciation of this proposition, is not used in the arithmetical sense of that term, but in that of separation or subtraction.

PROP. XVIII. THEOREM.

If four magnitudes be proportionals; by composition, the first and second together are to the second, as the third and fourth together are to the fourth. Let AE, EB, CF, and F D be proportionals; that is, as AE to EB, so let CF be to FD. And, let A B be the sum of AE and EB; and

CD the sum of CF and FD. As AB to BE, so is CD to DF (V. Def.

15).

Take of AB, BE, CD, and DF any equimultiples whatever GH, HK, LM, and MN; and of BE, DF, any other equimultiples whatever K O and NP.

Because KO and NP are equimultiples of BE and DF; and KH and NM are likewise equimultiples of BE and DF. If KO, the multiple of BE, be greater than KH, which is also a multiple of B E. Therefore, NP, the multiple of DF, is also greater than NM, the multiple of the same DF; if KO be equal to K H, NP is equal to N M ; and if less, less.

First, if KO be not greater than KH; NP is not greater than NM. Because, GH and HK are equimultiples of AB and BE, and AB is greater than BE. Therefore G H is greater than HK (Ax. 3); but KO is not greater than KH (Hyp.). Therefore G H is greater than K O. In like manner it may be shown, that LM is greater than NP. Therefore if K O be not greater than KH, GH, the multiple of A B, is greater than K O, the multiple of B E; and likewise L M, the multiple of CD, is greater than NP, the multiple of DF. Next, let K O be greater than K H. Therefore, as has been shown, NP

is greater than NM. Because the whole GH is the same multiple of the whole AB, that HK is of BE. Therefore the remainder GK is the same multiple of the remainder AE that GH is of AB (V. 5), which is the same that L M is of CD. In like manner, because LM is the same multiple of CD, that MN is of DF. Therefore the remainder LN is the same multiple of the remainder CF, that the whole L M is of the whole CD (V. 5). But it was shown that LM is the same multiple of CD, that G Kis of A E. Therefore GK is the same multiple of AE, that LN is of CF; that is, GK and LN are equimultiples of AE and CF. But KO and NP are equimultiples of B ́E and DF; and if from K O and NP there be taken ĤK and MN, which are likewise equimultiples of BE and DF. Therefore, the remainders HO and MP are either equal to BE and DF, or equimultiples of them (V. 6).

First, let H O and M P be equal to BE and DF. Because AE is to EB, as CF to FD (Hyp.), and GK and L N are equimultiples of AE and CF. Therefore GK is to EB, as LN to FD (V. 4, Cor.). But HO is equal to EB, and M P to FD. Therefore G K is to HO, as LN to MP. Wherefore if GK be greater than HO, LN is greater than MP; if equal, equal; and if less, less (V. A).

Next, let HO and MP be equimultiples of E B and FD. Because

AE is to EB, as CF to FD (Hyp.), and of AE and CF are taken equimultiples GK and LN; and of EB and FD, the equimultiples

HO and MP. If G K be greater than HO, LN is greater than MP; if equal, equal; and if less, less (V. Def. 5); which was likewise shown in the preceding case. But if GH be greater than K O, taking KH from both, GK is greater than HO (Ï. Ax. 5). Therefore also LN is greater than MP. By adding NM to these unequals, LM is greater than NP (I. Ax. 4). Therefore, if GH be greater than K O, LM is greater than NP. In like manner it may be shown, that if GH be equal to KO, LM is equal to NP; and if less, less. But in the case in which K O is not greater than K H, it has been shown that GH is always greater than K O, and likewise LM greater than NP. And GH and LM are any equimultiples whatever of AB and CD (Const.), also KO and NP are any whatever of BE and DF. Therefore, as AB is to BE, so is CD to DF (V. Def. 5). If four magnitudes, &c. Q. E. D.

The term composition used in the enunciation of this proposition, signifies simply the addition of the two magnitudes, or the finding of one magnitude equal to both.

PROP. XIX. THEOREM.

If there be two magnitudes such that the first is to the second, as a part of the first is to a part of the second; the remainder is to the remainder as the first is to the second.

E

B

Let AB be to CD, as AE a part of AB is to CF a part of CD. The remainder EB is to the remainder FD, as AB is to CD. Because AB is to CD, as AE to CF. There- A fore alternately, BA is to A E, as DC to CF (V. 16). Because EB is the excess of AB above A E, and DF the excess of CD above CF. Therefore, as BE is

to EA, so is DF to FC (V. 17). But alternately, as

C F D

BE is to D F, so is EA to FC (V. 16), and as A E to CF, so is AB to to CD (Hyp.). Therefore BE is to DF, as AB is to CD (V. 11). Wherefore, if there be two magnitudes such, &c. Q. E. D.

COROLLARY.-If there be two magnitudes such that the first is to the second as a part of the first is to a part of the second; the remainder is to the remainder, as the part of the first is to the part of the second. The demonstration is contained in the preceding.

PROP. E. THEOREM.

If four magnitudes be proportionals, they are also proportionals by conversion; that is, the first is to its excess above the second, as the third to its excess above the fourth.

E B

Let A B be to BE, as CD to DF. And let A E be the excess of A B above BE, and CF the excess of CD above D F. AB is to AE, as DC to CF. Because AB is to BE, as CD to DF (Hyp.). Therefore, by division, AE is to EB, as CF to A FD (V. 17). But, by inversion, BE is to EA, as DF to FC (V. B). Wherefore, by composition, BA is to C F D AE, as DC is to CF (V. 18). If therefore four, &c.

Q. E. D.

This proposition was added by Dr. Simson, as a substitute for a corollary to the next proposition given in the original Greek, which he declares to be vitiated.