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A For, because

B

adding 1 to both members,

D

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A С A+B C+D
+1

+1, or
B D

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

BD
Also (IV. 8),

whence, as before,
A c'
A+B: A :: C+D : C.

Proposition 10. Theorem.-

If four quantities be in proportion, they are in proportion by division.

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If

A: B::C:D, then

A-B:B:: C-D :D, and

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

А с For, because

subtracting 1 from both members, BD A c A - B C-D

1, or B D

B D or,

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

BD
Also, because

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

we have

Proposition 11. Theorem.- If four quantities be in proportion, the sum of the first and second is to their difference, as the sum of the third and fourth is to their difference. If

A:B::C:D, then A+B: A-B :: C+D: C-D.

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Dividing one equation by the other, and striking out (IV.3) the common terms, B and D, and we have

A + B C +D
A-BC-D'

or,

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

Proposition 12. Theorem.The products of the corresponding terms of two numerical proportions are in proportion. If

A:B::C:D, and

E: F:: G : H, then A.E: B.F :: C.G: D.H.

A С E G For, because

and

B D multiplying the equations together,

A.EC.G

B.F D.H'
A.E: B.F :: C.G : D.H.

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H'

or,

Corollary 1.The continued products of the corresponding terms of any number of proportions are in proportion.

Corollary 2.-If four quantities be in proportion, like powers of them are in proportion.

For, if in last corollary all the corresponding terms be the same, the products would be like powers of the terms.

Proposition 13. Theorem. If two proportions have in each ratio an antecedent of one the same as a consequent of the other, the other terms are in proportion, antecedent remaining antecedent, and consequent, consequent. If

A : B :: С :D, and

B:E:: F: C, then

A : E :: F: D. For multiplying term by term, (IV. 12), A.B: B.E:: C.F: D.C.

Cutting out the common factors from each ratio, (IV. 3, Cor.), A: E:: F: D.

Corollary.-Hence, also, if

A : B :: C:D, and

B E F G, then

A : E:: C.F: D. G.

:

Proposition 14. Theorem.-Equal quantities have the same ratio to the same quantity, and quantities which have the same ratio to the same quantity, are equal to each other.

Let A and B be equal magnitudes, and C another.
Then

A:C::B:C.
A

B
Let

m; then, because A =B,
с

С
А B
Hence

or A :C:: B C.
C C

= m.

:

B

let A = m C, then (IV. 6) b =mC. Hence

A Again, if

С A = B.

С

Proposition 15.
Theorem.-If two proportions have one ratio in each the
sume, the remaining terms are in proportion.
Let

A: B::C:D, and

E:F:: 0 :D; then

A B

::E : F. А С E С

A E For, because

and

therefore
B D F D

B
A : B :: E: F.

:

or

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Proposition 16.
Theorem. If any number of quantities be in proportion,
any antecedent is to its consequent, as the sum of all the ante-
cedents is to the sum of all the consequents.

If A : B::C:D :: E. F, etc.,
then А B :: A+ C+E, etc. : B+D+F, etc.
Let A = mB, then (IV. 6) C=mD and E=mF, etc.
Adding these, we have

A+C+E, etc. = m(B+D+F), etc.,
A+ C + E, etc.

А whence

But

= m; B+D+F, etc.

B therefore A:B :: A+ C+E+etc. : B+D+F+etc.

:

+

= m.

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EXERCISES. 1. If A :B :: B: C, prove A : C :: A’ : B?. 2. If A :C >B : C, prove A >B. 3. If A+B : A :: C+D : C, prove A : B :: C : D. 4. If A : B::C: D, prove A: B :: A C:B-D.

y 5. If

1, what is the ratio of x to ý? y

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BOOK V.

SIMILAR POLYGONS.-MEASUREMENT OF

POLYGONS.

DEFINITIONS.

1. Similar rectilineal figures are those which have their several angles equal, each to each, and the sides about the equal angles proportional.

2. A straight line is cut in extreme and mean ratio when the whole is to the greater -segment, as the greater segment is to the less.

3. The altitude of a triangle is the straight line drawn from its vertex perpendicular to the base, or the base produced.

As any side of a triangle may be considered the base, a triangle may have three altitudes. The altitude of a parallelogram is the perpendicular distance between either pair of parallel sides. 4. The homologous sides of similar rectilineal figures are

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