PROP. XVI. THEOR.

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

Let A, B, C, D be four magnitudes of the same kind, which are proportionals, viz. as A to B, so C to D: they shall also be proportionals when taken alternately; that is, A shall be to C, as B to D.

Take of A and B any equimultiples whatever E and F; and of C and D take any equimultiples whatever G and H: and because E is the same multiple of A, that F is of B, and that *15. 5. magnitudes have the same ratio to one another* which their equimultiples have; therefore A is + Hyp. to B, as E is to F: but as A is to B sot is C to D; wherefore as C is to E

*11.5. D, so* is E to F: A again, because G, H

11. 5.

* 14.5.

are equimultiples of

B

C, D, therefore as C F

-G

C

D

-H

* 15.5. is to D, so* is G to H: 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. But when four magnitudes are proportionals, * if the first be greater than the third, the second is greater than the fourth; and if equal, equal; if less, less; therefore, if E be greater than G, F likewise is greater than H; and if equal, equal; if less, less: and E, F +Const. are any equimultiples whatever of A, B; and *5Def.5. G, H any whatever of C, D: therefore* A is to C as B to D. If, then, four magnitudes, &c. Q. E. D.

If magnitudes, taken jointly, be proportionals, they shall also be proportionals when taken separately: that is, if two magnitudes together have to one of them the same ratio which two others have to one of these, the remaining one of the first two shall have to the other the same ratio which the remaining one of the last two has to the other of these.

Let AB, BE, CD, DF be the magnitudes taken jointly which are proportionals; that is, as AB to BE, so let CD be to DF: they shall also be proportionals taken separately, viz. as AE to EB, so shall CF be to FD.

Take of AE, EB, CF, FD any equimultiples whatever GH, HK, LM, MN; and again, of EB, FD take any equimultiples whatever KX, NP: and because GH is the same multiple of AE, that HK is of EB, therefore GH is the same multiple* of AE, that GK is of AB: but 1.5. GH is the same multiple of AE, that LM is of. CF; therefore GK is the same multiple of AB, that LM is of CF. Again, because LM is the same multiple of CF, that MN is of FD; therefore LM is the same multiple* of CF, that LN *1.5. is of CD: but LM was shown to be the same multiple of CF, that GK is of AB; therefore GK is the same multiple of AB, that LN is of CD; that is, GK, LN are equimultiples of AB, CD. Next, because HK is the same multiple of EB, that MN is of FD; and that KX is also the same multiple of EB, that NP is of FD; therefore HX is the same multiple of EB, that MP is of FD. And because AB is to BE+ as CD is to DF, and that of AB and CD, GK and LN are equimultiples, and of EB and FD,

X

K

HB

E

P

N

DM

GACL

#2.5

+Hyp

5.

* 5 Def. HX and MP are equimultiples*; therefore if GK be greater than HX, then LN is greater than MP; and if equal, equal; and if less, less: but if GH be greater than KX, then, by adding +4Ax.1. the common part HK to both, GK is greater+ than HX; wherefore also LN is greater than MP; and by taking away MN from both, LM +5Ax.1. is greater + than NP; therefore, if GH be greater than KX, LM is greater than NP. In like manner it may be demonstrated, that if GH be equal to KX, LM is equal to NP; and if less, less: but GH, LM are any equimultiples what† Const. ever of AE, CF,† and KX, NP are any what*5 Def. ever of EB, FD: therefore,* as AE is to EB, so is CF to FD. If then, magnitudes, &c. Q. E. D.

5.

If magnitudes, taken separately, be proportionals, they shall also be proportionals when taken jointly: that is, if the first be to the second, as the third to the fourth, the first and second together shall be to the second, as the third ana fourth together to the fourth.

Let AE, EB, CF, FD be proportionals; that is, as AE to EB, so let CF be to FD: they shall also be proportionals when taken jointly; that is, as AB to BE, so shall CD be to DF.

Take of AB, BE, CD, DF any equimultiples whatever GH, HK, LM, MN; and again, of BE, DE, take any equimultiples whatever KO, NP and because KO, NP are equimultiples of BE, DF, and that KH, NM are likewise equimultiples of BE, DF; therefore if KO, the multiple of BE, be greater than KH, which is a multiple of the same BE, then NP, the multiple of DF, is also greater than NM, the multiple of the same DF; and if KO be equal to KH, NP is equal to NM; and if less, less.

First, let KO be not greater than KH; there

K

M.

*3Ax.5.

F

N

fore NP is not greater than NM: and because
GH, HK, are equimultiples of H1
AB, BE, and that AB is greater
than BE, therefore GH is 0
greater than HK; but KO is
not greater than KH; there-
fore GH is greater than KO.
In like manner it may be shown,
that LM is greater than NP.
Therefore, if KO be not greater
than KH, then GH, the multi-
ple of AB, is always greater

L

than KO, the multiple of BE; and likewise LM, the multiple of CD, is greater than NP, the multiple of DF.

Next, let KO be greater than KH; therefore, as has been shown, NP is greater than NM: and 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; which is the same that LM 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 K same multiple of the remainder

H

CF,* that the whole LM is of E
the whole CD: but it was shown
that LM is the same multiple G A
of CD, that GK is of AE; there-

P

* 5.5.

*5.5.

fore GK is the same multiple of AE, that LN is of CF; that is, GK, LN are equimultiples of AE, CF. And because KO, NP are equimultiples of BE, DF, therefore if from KO, NP there be taken KH, NM, which are likewise equimultiples of BE, DF, the remainders HO, MP are either equal to BE, DF, or equimultiples of them.* First, let HO, MP be equal *6. 5. to BE, DF: then because † AE is to EB, as CF +Hyp to FD, and that GK, LN are equimultiples of

5.

*Cor. 4. AE, CF; therefore GK is to EB,* as LN to FD: but HO is equal to EB, and MP to FD; wherefore GK is to HO, as LN to MP: therefore if GK be greater than HO, LN is greater * A. 5. than* MP; and if equal, equal; and if less, less.

But let HO, MP be equimultiples of EB, +Hyp. FD: then because AE is to EB, as CF to FD, and that of AE, CF are taken equimultiples GK, LN; and of EB, FD, the equimultiples HO, MP; if GK be greater than HO, LÑ is greater than MP; and if equal,

5.

*5Def. equal; and if less, less ;* which
was likewise shown in the pre-
ceding case. But if GH be H
greater than KO, taking KH
+5Ax.1. from both, GK is greater than
HO: wherefore also LN is
greater than MP; and conse-
quently adding NM to both, LM

t4Ax.1. is greater than NP: therefore,
if GH be greater than KO,

KB

G!

P

M

N

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. And in the case

in which KO is not greater than KH, it has been shown that GH is always greater than KO, and likewise LM greater than NP: but GH, LM are +Const. any equimultiples whatever of AB, CD,† and KO, NP are any whatever of BE, DF; there*5 Def. fore,* as AB is to BE, so is CD to DF. If then magnitudes, &c. Q. E. D

5.

PROP. XIX. THEOR.

if a whole magnitude be to a whole, as a magnitude taken from the first is to a magnitude taken from the other; the remainder shall be to the remainder as the whole to the whole.

Let the whole AB be to the whole CD, as AE a magnitude taken from AB is to CF a magni