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Proposition 1. 281. If four quantities are in proportion, the product of the extremes is equal to the product of the means. Hyp. Let

a :=cid.

bc.

To prove

ad =

a

с

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Proof. By (277),

b d* Multiplying by bd, all = bc.

Q. E. D. 282. Cor. If a : b=b:c, .. 7= ac.

(281)

Nac. That is, the mean proportional between two quantities is equal to the square root of their product.

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To prove

a

с

Proposition 2. 283. Conversely, if the product of two quantities be equal to the product of two others, two of them may be made

, the extremes, and the other two the means, of a proportion. Hyp. Let

ad = bc.

a :b=c:d. Proof. Dividing the given equation by bd,

=; b

ài that is,

Q.E.D. In a similar manner it may be shown that the proportions

a:c=b:d, b:a=d:c, 7:d

=d:C,

c:d=a: b, etc., are all true provided that ad = bc.

a : 6

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c:d.

284. Sch. By the product of the extremes, or of the means of a proportion,is meant the product of the numerical measures of those quantities. Hence, the product of two lines will be often used for brevity, meaning the product of the numbers which represent those lines.

(277)

Proposition 3. 285. If four quantities are in proportion, they are in proportion by inversion; that is, the second term is to the first as the fourth term is to the third. Hyp. Let

a:5=c:d.

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286. If four quantities are in proportion, they are in proportion by alternation; that is, the first term is to the third as the second term is to the fourth.

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Proposition 5. 287. If four quantities are in proportion, they are in proportion by composition; that is, the sum of the first and second is to the second as the sum of the third fourth is to the fourth.

a:b=c:d.
a+b:b=c+d:d.

Hyp. Let

To prove

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Proposition 6. 288. If four quantities are in proportion, they are in proportion by division; that is, the difference of the first and second is to the second as the difference of the third and fourth is to the fourth. Hyp. Let

a :=c:d.
a - b:b=c-d:d.

To prove

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Proposition 7. 289. 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. Hyp. Let

a:b=c:d.
a a+b:a-b=c+d:c- d.

:

a +3 c+d Proof. By (287),

d

;

To prove

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Proposition 8. 290. The products of the corresponding terms of two or more proportions are proportional. Hyp. Let

a:b=c:d, and e: f =g:h.

ae: bf = cg:dh.

To prove

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cg

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bf dh
.. ae: óf = cg: dh.

Q.E.D. 291. A greater quantity is said to be a multiple of a less, when the greater contains the less an exact number of times.

Equimultiples of two quantities are quantities which contain them the same number of times. Thus, ma and mb are equimultiples of a and b.

Proposition 9. 292. When four quantities are in proportion, if the first and second be multiplied, or divided, by any quantity, as also the third and fourth, the resulting quantities will be in proportion.

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Multiply both terms of the first fraction by m, and both terms of the second by n.

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293. Sch. Either m or n may be unity.

In a similar manner it may be shown that if the first and third terms be multiplied, or divided, by any quantity, and also the second and fourth, the resulting quantities will be in proportion.

a 294. Cor. Since

7 7

a

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That is, equimultiples of two quantities are in the same ratio as the quantities themselves.

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