Proportion. 
ya 
212 GEOM 
Prov. IV. Tueor. 
If four quantities be proportionals, they shall also be 
proportionals by composition. ° ike 
Let A: B::C:D; then by composition A: A + B 
::C:C+D. 
For let us suppose that A contains five such equal 
parts as B contains three, then also, (Def. 3.) C will con- 
tain five such equal as D contains three. Let 
each of the parts in A and B be X, and let each of the 
parts in C and D be Y ; then because 
A=5X; B=sX;'C=5Y; D= SY. 
It follows that 
A+B=8X Cyed=BY¥: 
Here it is evident that A+-B contains one third of B 
eight times ; and that C+D contains one third of D 
also eight times ; and in general, that A+B will con- 
tain some of B exactly as often as C+D. contains 
a like part of D; therefore (by Def. 3.) A+B: B:: 
C+D: D. 
Prop. V.. Turon. _ 
If four quantities be proportionals, they will also be 
proportionals by division. ' 
Let A: B::C:D; then by division, A—B:B:: 
C—D: D. 
For making the same supposition as in last proposi- 
tion, so that 
A=5X, B=3xX, C=5Y,'D=3Y,. we. have 
A—B=2X, and C—D=2 Y, therefore A-—B con- 
tains one third of B twice, and C—D-contains one- 
third of D also twice: and in general, it is evident that 
A—B will in every case contain a pavt of B. exactly 
as often as C—D., contains.a like part of D: therefore 
(Def. 3.) A—B,: B::C—D: D, 
Prop. VI. 
If four quantities be proportionals, they are also pro- 
portionals by conversion. f 
* Let A: B::C:D 
C:C—D._ ? 
For, making the same supposition as in the two last 
propositions, because 
A=5X, B=3 X, C=5 Y; D=3 Y; 
therefore A—B=2 X, and C—D=2Y; ; 
Hence it appears that A contains one half of A—B five 
times, and that C contains one half of C—D also five 
times, therefore A contains a part of A—B as often as 
C contains a like part of C—D; therefore A: A—B:: 
C:C—D. (Def. 3.) 
THEor. 
; then, by conversion, A : A—B:: 
Prop, Vil. Tueror: 
If four quantities be proportionals, and there be taken 
any equimultiples of the antecedents, and also any 
equimultiples of the consequents ; the resulting quan- 
tities will also be four proportionals, ' 
Let A: B:: C:D; and supposing m and 2 to be an 
two numbers, let the Ts bat and C be iad 
each m times, and the consequents B and D each x 
times ; then shall m A: nB::mC:nD. 
For suppose that A contains two such equal parts as 
/B contains three: and consequently that C contains 
/two such equal parts as D. contains three. (Def. 3.) 
’ and the consequents by”, and. 
ETRY. 
Let.each of the parts contained in A and. B-bé X, and Propo 
each of the parts contained in C and D be Y ;.so that 
A=2X, +. Be SR aly 5 Melby 
Ti: Cm WY; iD =/3Xisiotok: : K 
Then, multiplying the antecedents by the number m, 
observing that mx 2= 
2 xm, and thatnx2=2 X% mn; wehave 9) 5 6 5 
mA=2xmX, eB Biotin Mein saad 
mC=2x mY, nD=3xKnY. 0 
Here it is evident that mA contains one third of x B 
twice; and that mC contains ‘one third of »D also, 
twice; therefore mA:nB::mCinD.. (Defi3.) | 
Prop. VIII. Teor: 
If there be any number of magnitudes ond as many 
others, which taken ow and two have the same ratio; 
the first shall have to the last of the first series, the same. 
ratio which the first has to the last of the other series. 
Filet, (:let - shoot hes aliens 
A: Cau /HsLes : 
For let us suppose that A contains 2 such 
each equal to X, as B contains $, and as C contains 7; 
then, (Def. 3.) H will contain 2 such parts (each of). 
which we shall denote by Y) as K contains 3, and as. 
L contains 7 ; so that we have ; Pte At 
pity He Xe) Bais Ke Co 9 XS tate hae 
then, Eb=@¥,) Ki Sisto 7A = inid 
Here it is evident that A will contain one seventh of C 
twice, and that H will contain one seventh of L also: 
twice ; therefore (Def. 3.) Ai @::H:L. > bpd 
Next, let there be four quantities ; 
itudes | 4, B,C. - 
A, B, C, and other three H, K, L, such,” Digs: 
that A: B:: H: K, and B: C:: K: L, then, H,K,L. 
A, B,C, D,.and other four H, K, L, M;} 4, B,C, D. 
such, that A, B: : Hy: K,and B: C:: |) eri. aba 
K:L, and C: D:1L: M; then shall'| Hy K, L, M.), 
A:D::H:M. For by. the first.case /- 
Piatto ate 
it is evident that A:.C:: H: L; and because: C: Ds) - 
L: M;; therefore, as before, A: D:: H: M. The de- 
monstration applies in the same manner to any number 
of quantities. 7 
Note. Quantities which are proportionals, according 
to the hypothesis of this theorem, are said to be so from 
equality of distance-directly, and the theorem is usually » 
cited by the words ex @quali, or ex @quo. wipe 
7 Raa IX. Tueor. 
ve 
. If there be any number. of quantities, and as 
others, which taken two and two in a cross order have 
the same ratio; the first shall have to the last‘of the 
first series the same ratio that the first has te the last 
of the other series, | No 
) 
, 
First, let there be three quantities A, B,C, 
and. other three H, K,.L, such, that A: B:: 
K: L, and B:C::H:K; then A: Ci: 
H:L. : 
For suppose A to contain two such equal parts as B 
contains three, then K will contain two such equal 
parts as L contains three, (Def. 3.); let each. of the 
equal parts contained in A and B.be,X, and let each of 
the equal parts contained in K and L be Y, so that 
A=2X, B=3X, K=2Y, L=sY. 7 
Also let Z be the same part of C that Y is of L, and let V 
be the same part of H that X is of A ; so that we have. 
C=3Z, H=2V. ; 
Then, because B:C::H:K; that is $X:3Z::2V: 
A, BSG. 
H, K,L. 
~ 
