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205. If four quantities are in proportion, they are in proportion by Division; that is, the difference of the first and second is to the first as the difference of the third and fourth is to the third.
a: b = c: d.
ab: a=c-d: c.
a: b = c:d,
ad = bc.
To prove If
then (by 199)
Subtract both members of this equation from ac, then
ac ad = ac - bc,
a (cd) = c(a - b).
Therefore (by 201), a − b : a = c — d : c.
a: b = c: d.
206. If four quantities are in proportion, they are in proportion by Composition and Division; that is, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference.
and (by 205)
a+b: a b
c + d
C- - d
207. The products of the corresponding terms of two or more proportions are proportional.
abc:d, and e:f=g: h.
ae: bf = cg: dh.
Writing the proportions in another form,
α C and
Multiplying these equations member by member,
ae: bf cy: dh.
208. COR. If the corresponding terms of the proportions are equal; that is, if ea, fb, gc, and hd, the result of the preceding theorem becomes
And in general in any proportion like powers of the terms are in proportion.
PROPOSITION IX. THEOREM.
209. In a series of equal ratios, any antecedent is to its consequent as the sum of all the antecedents is to the sum of all the consequents.
Let r be the value of the equal ratios, that is,
=r, =r, and
From these equations,
or by addition,
But by hypothesis,
a = br, c=dr, e=fr,
a+c+e=br+ dr + fr
a + c + e
In like manner,
r = = 1;=<
b+d+f b a+c+e:b+d+f: = a: b = c d = e : f. Q.E.D.
PROPOSITION X. THEOREM.
210. In any proportion, if the antecedents are multiplied by any quantity, as also the consequents, the resulting terms will be in proportion.
a: bc: d.
Multiplying both members of the equation by
ma: nb = mc: nd.
a b c d
Either m or n may be unity.
1. Show that equimultiples of two quantities are in the same ratio as the quantities themselves.
2. Show that if four quantities are in proportion, their like roots are in proportion.
Two straight lines are said to be divided proportionally when their corresponding segments, or parts, are in the same ratio as the lines themselves.
Thus the lines AB and CD are divided proportionally at E and F if
AB: AE = CD: CF.
AD : DB = AE : EC.
CASE I. When AD and DB are commensurable.
Take AF, any common measure of AD and DB, and suppose it to be contained 4 times in AD and 3 times in DB.
212. When a finite straight line, as AB, is cut at a point X between A and B, it is said to be divided internally at X, and the two parts AX and BX are called segments. But if the straight line AB
is produced, and cut at a point Y A beyond AB, it is said to be divided externally at Y, and the parts AY and BY are called segments. The given line is the sum of two internal segments, or the difference of two external segments.
When a straight line is divided internally and externally into segments having the same ratio, it is said to be divided harmonically.
PROPOSITION XI. THEOREM.
213. A straight line parallel to one side of a triangle divides the other two sides proportionally.
In the triangle ABC let DE be parallel to BC.
Through the several points of division of AB draw lines parallel to BC; then since these parallels cut off equal lengths on AB, they will (by 110) cut off equal lengths on AC.
Therefore, AE will be divided into 4 equal parts and EC into 3; that is,
Hence (by 28),
AD: DB = AE : EC.
CASE II. When AD and DB are incommensurable.
In this case we know (170) that we may always find a line AG as nearly equal as we please to AD, and such that AG and GB are commensurable.
Draw GH parallel to BC; then
As these two ratios are always equal while the common measure is indefinitely diminished, they will be equal as GH approaches DE.
214. COR. By composition (204),
AD+DB: AD = AE + EC: AE,
AB: AD AC: AE.
Likewise (by 204), AB: DB = AC: EC, and (by 202), AB: AC
Therefore, this equality of ratios will exist (by 172) when the limiting position DE is reached; that is,
or AD: DB = AE : EC.