me2Y, and because 3X : 8Z::X:Z, and 2V:2Y::V: Y, 
"(1.)} therefore X :Z::V:Y, (Ax. 3-); hence, 
, 8Y=L; therefore A: C:: Hs: L. 
Next, let there be four quantities, 
A, B,C, D, and other four H,K,L, M, | A, B,C, D. 
such that A: B::L:M, and B:C:: | 1, L.M 
_ K: LU, and: Di:H:K; then A:D:: »K,L, M, 
H:M;; for it is evident by the first case, 
that A: C::K:M; and because C:D:: H: K, there- 
fore, as before, A: D::H:M.. The.same mode of de- 
monstration will apply to any number of quantities. 
Note. The guseties in this proposition are said to 
be proportion 
Prop..X. Tueror.. 
If the first have to the second the same ratio which the 
third has tothe fourth, and the fifth have to the second 
the same ratio which the sixth has to the fourth; the first 
and gly eg shall have.to the second the same ratio 
which the third and sixth together have to the fourth. 
Let A: B::C: D,andalsoE: B::F:D,then A+E 
| :B::C+F:D. a 
_ . Because E:B::F:D, by inversion B: E::D:F, 
(22): But _by peveiete A:B::C:D; therefore ex 
{ equali, (8.) A: E::C:F, and by composition A +E: 
_ E::C+F:F. Now again by hypothesis, E: B:: F: D, 
therefore ex equali, (8.) A4+-E:B::C+4F:D. 
zx st - 
co) 1 O4 «gro Prop. XL: Taror. 
If four quantities be proportionals, asthe sum of one 
antecedent and its consequent is to their difference, so is 
___ the sum of the othér antecedent and consequent to their 
| difference. 
| | LetA:B::C:D,then A+B: A—B::C4+D:C—D. 
For by composition, A+B: B::C+4D: D (4.) 
7s And by Div. and Inver. B: A—B:: D: C—D(5and 2. 
~ Therefore ex equo A+B: A—B::C4+D:C—D (8) 
__ Note. Proportionals formed in this manner, are said 
to be so by mixing. 
Prop, XII. Turor. 
If there be any number of proportionals, as one ante- 
tedent is to its consequent, so is the sum of all the an- 
tecedents to the sum of all the consequents. 
LetA:B::C:D::E:F, thn A:B::A+C+ 
E:B+D4+F. — 
_ For suppose that A contains two such parts, each = 
X, as B contains three; and that C contains two such 
parts, each =Y, as D contains three ; and that E con- 
tains two such parts, each = Z, as F contains. three ; 
and so on, then 
A=2X,B = 8X, 
C=2Y,D=3Y, 
E=2Z, F =8Z. 
(by, 
Prop. 7.) 2X:8Z'1:2V:8Y; but 2X=A; 3Z=C, sd 
from equality of distance, but in a cross. 
order ; and the theorem is usually cited by the words. 
ex ii in proportione perturbala, or ex.a@quo per-. 
: oe proport Pp y qua’ per 
GEOMETRY: 213° 
Proportion: 
Prop. XIII) Pros. wrest i, 
To find the numerical ratiovof two. straight lines AB, Fig. 65. 
CD, supposing them to have a common measure. 
Take the lesser of the two lines on the greater as of= 
tee possible ; for example twice, with a remainder 
Take the remainder BE on the line CD as often as 
possible ; once, for example, with a remainder DF. 
Take the second remainder DF on the first BE as of- 
a possible; once, for example, with a remainder 
Take the third remainder BG on the second DF as 
often as possible, and continue this “alee until a re- 
mainder is found, which is contained an exact number 
of times in that going before it. ‘Then the last remain- 
der shall be thé common measure of the proposed lines; 
and considering it as unity, we shall easily find the va- 
lues of the preceding remainders, and at. last those of 
the two proposed lines ; that is, we shall know how of- - 
ten each contains the unit, so that if AB contain it m 
times, and CD contain itn times, then AB: CD:: m:n. 
For example, if it is found that GB is contained ex-. 
actly twice in-FD, BG shall be the common measure of 
the twolines.. Let BG-=1, thenF D=2; but EB=FD+. 
GB, therefore EB=3;CD= EB4-FD,therefore CD=5; 
lastly, AB=2 CD+EB, therefore AB=13 :. therefore. 
the ratio of AB to CD is that of 13 to 5. 
_ ScuoniuM. This operation is evidently the same as .- 
that by which the common measure of two numbers is 
found. Its demonstration is given in Aucrepra, Art.. 
72.and 73.. If the operation terminate, and the lines 
have a common measure, they are said to. be commensu-, 
rable; but.the lines may be.such that the operation will. 
never terminate, and as then the quantities have no com- 
mon measure, they are said to be zncommensurable. The 
side of a square AB, and its diagonal AC, are of this na- . 
ture, (Fig. 86.) For if we take AD=AB, and draw Fig. 86. 
DE perpendicular to AC, to meet CB in, E, and join. 
AE, the triangles ABE, ADE will be equal, (18. 1.) 
and BE=DE. But. the angle DEC=DAB (1 Cor.. 
24. 1.) =DCB, (12. 1.) therefore; DE=DC, (13. 1.) 
and hence BE=DC. ~ Now to.determine whether AB 
and AC have a common measure, we first take AB out 
of AC, and DC will remain ; we next take DC out of. 
CB, and get it once, with a remainder CE ; but as CE. 
is still greater than DC, we must again take CD out. 
of CE, and then proceeding exactly as before, we must 
take the last remainder out of CD as often as we can, and 
so on. Now CE is evidently the diagonal of a square, 
of which DC is a side ; therefore it appears, that.in 
seeking the common measure, we must make the very. 
same kind of construction in this second square that 
was made upon the first ; and again, in pursuing the 
operation, we must make a like construction on a third 
square, and so on continually, so that the operation can 
never come to.an end: therefore the quantities AC, 
AB can have no common measure. ; 
On the subject of incommensurable quantities, see 
also Atcesra, Sect. VI. 
In the theory of proportion, we have, with a view to 
brevity and perspicuity, treated only of commensurable 
ratios ; that is, such as can be accurately expressed by 
numbers, Although the ratio of incommensurable quan- 
tities cannot be so expressed, yet a ratio may be al- 
ways assigned in numbers, which shall be as near to the 
true ratio as we please, For let A and B be any two 
