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4. The first terms of the ratios (that is, the first and third terms of the proportion) are called antecedents, and the second and fourth, consequents.

5. The first and fourth terms are cailed extremes, and the second and third, means.

6. Quantities are in continued proportion when a consequent of one ratio is the same as the antecedent of the next.

Thus, A : B. :: B: C::C: D, etc.

7. The second quantity is then said to be a mean proportional between the first and third ; and the third is a third proportional to the first and second.

8. In a proportion the fourth term is said to be a fourth proportional to the other three taken in order.

9. Quantities are in proportion alternately, when antecedent is compared with antecedent, and consequent with consequent.

If A :B :: C :D, then, alternately, A :C:: B : D.

10. Quantities are in proportion inversely, when antecedent is made consequent, and consequent, antecedent.

If A :B :: C:D, then, inversely, B : A :: D: C.

11. Quantities are in proportion by composition, when the sum of antecedent and consequent is compared with either antecedent or consequent. If A :B :: C: D, then, by composition,

A+B : A or B :: C+D: C or D. 12. Quantities are in proportion by division, when the difference of antecedent and consequent is compared with either antecedent or consequent.

If A :B:: C:D, then, by division,

A-B : A or B :: C-D : C or D.

Proposition 1. Theorem.If four quantities be in proportion, the product of the extremes is equal to the product of the means.

Let

A : B :: C: D, then we wish to prove ARD=Bx C.

А С
For, because multiplying by BxD, we have

B

AxD=Bx C.

D'

Corollary 1.-If three quantities be in continued proportion, the product of the extremes is equal to the square of the

теап.

If

A:B::B:C, then, by this proposition,

Ax C=BxB = BP.

Corollary 2.-A mean proportional between two quantities is the square root of their product; for B= VAR C.

Scholium.- When we speak of the product of two quantities, one of them at least must be a number. We cannot have the product of two lines, or of two surfaces, or of two solids, using these terms in their geometrical sense. When we, for convenience, use the expression product of two lines, we mean the number of units in the length of one, multiplied by the number of units of the same kind in the length of the other. Expressed in this way, lines become numerical quantities, and may be used as factors. If A and B be two lines, and C the common unit of measure, and if A contain C, m times, and

A B contain C, n times, then evidently

Hence,

B when we have the ratio of two lines, we may substitute the ratio of their numerical measures, and the same is true of surfaces and solids. The Theorems of this book which involve

m

т C nc

n

simply ratio are therefore directly applicable to all cases of geometrical magnitudes; while the remainder are rendered so by the considerations mentioned in Scholium 2, page 117. In the above m and n may represent any numbers integral, fractional, or mixed.

Proposition 2. Theorem.-If the product of two quantities be equal to the product of two others, these four quantities may form a proportion, one set being taken for the extremes and the other for the

means.

If
then
For, because

AxD=BxC
A:B::C: D.
AxD-BxC

Dividing both sides by Bx D, we have

B
A : B :: C: D.

С
D

or

Scholium.-From this it is manifest that we can form a number of proportions from the equation A x D=Bx C. The only limitation being, that if A be made a mean or an extreme, D must be the other mean or extreme. Thus we might have

A: 0 :: B : D,
B:A :: D: C,
C:A :: D: B, etc.

Proposition 3. Theorem.-Equimultiples of two quantities are proportional to the quantities.

Let A and B be two quantities, and mA and mB equimultiples of them. Then

A : B :: mA : mB.

A mA
For

B :: MA : mB.
B

:

mB'

or A

Corollary.-Equal measures of two quantities are proportional to the quantities.

This may be proved as the above by supposing m to be fractional.

Proposition 4. Theorem.-If equimultiples be taken of the first and second terms of a proportion, and also of the third and fourth, the resulting terms are proportional.

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Proposition 5. Theorem.-If equimultiples be taken of the first and third terms of a proportion, and also of the second and fourth, the resulting terms are proportional.

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Proposition 6. Theorem.-If four quantities be in proportion, if the first be any multiple of the second, the third is the same multiple of the fourth.

If

A:B::C:D, and A = mB, then

C=mD.

A С
For, because

and A = mB,
B D
mB C

С therefore,

m, whence C=mD. B

D

or

D

Proposition 7. Theorem.If four quantities of the same kind be in proportion, they are in proportion when taken alternately. If

A : B:: 0 :D, then

A: C:: B: D. From (IV.1) AD-BxC therefore (IV. 2), A:C::B:D.

Proposition 8. Theorem. If four quantities be in proportion, they are in proportion when taken inversely. If

А B then

B:A :: D: C. From (IV. 1) AXDBx C therefore (IV.2), B:A ::D: C.

C: D,

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Proposition 9. Theorem.-If four quantities be in proportion, they are in proportion by composition. If

A : B::C :D, then

A+B:B:: C+D :D, and

A+B: A:: C+D: C.

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