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Of Ratios and Proportions.

Let us take the proportions

A : B :: C D, and A: B :: E : F.

which give

AxD-BX C and

AXF BX E.

By subtracting these equalities, we have

A×(D–F)=B×(C—E};

and by Th. II, we obtain

A: B :: C-E : D-F,

in which the antecedent and consequent, C and D, are dimin ished by E and F, which have the same ratio.

Sch. The proposition may be verified by the proportion, 9 : 18 :: 20 : 40,

for, by diminishing the antecedent and consequent by 15 and 30, we have

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If we have several sets of proportions, having the same ratio, any antecedent will be to its consequent, as the sum of the ante cedents to the sum of the consequents.

If we have the several proportions,

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We shall then have, by addition,

AX(D+F+H)=Bx (C+E+G);

and consequently, by Th II.

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

:

Of Ratios and Proportions.

Sch. The proposition may be verified by the following proportions: viz.

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If four quantities are in proportion, their squares or cubes will also be proportional.

If we have the proportion

it gives

A: B :: C : D,

B D

A C

Then, if we square both members, we have

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and then, changing these equalities into a proportion, we have

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Sch. We may verify the proposition by the proportion,

2 : 4 :: 6 : 12,

and by squaring each term we have,

4 16 :. 36 :

144

Of Ratios and Proportions.

numbers which are still proportional, and in which the ratio

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If we have two sets of proportional quantities, the products of the corresponding terms will be proportional.

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and this by Th. II, gives

AXE B× F :: CX G : DXH.

Sch. The proposition may be verified by the following proportions:

8 : 12 :: 10 : 15,

and

3 : 4 :: 6 :

8;

we shall then have

24 : 48 :: 60 : 120

which are proportional, the ratio being 2.

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1 Similar figures, are those which have the angles of the one equal to the angles of the other, each to each, and the sides about the equal angles proportional.

2. Any two sides, or any two angles, which are like placed in the two similar figures, are called homologous sides or angles.

3. A polygon which has all its angles equal, each to each, and all its sides equal, each to each, is called a regular polygon. A regular polygon is both equiangular and equilateral.

4. If the length of a line be computed in feet, one foot is the unit of the line, and is called the linear unit. If the length of a line be computed in yards, one yard is the linear unit

5. If we describe a square on the unit of length, such square is called the unit of surface. Thus, if the linear unit is one foot, one square foot will be the unit of surface, or superficial unit.

1 foot.

unit

Of Parallelograms.

6. If the linear unit is one yard, one square yard will be the unit of surface; and this square yard contains nine square feet.

1 yd. =3 feet.

7. The area of a figure is the measure of its surface. The unit of the number which expresses the area, is a square, the side of which is the unit of length.

8. Figures have equal areas, when they contain the same measuring unit an equal number of times.

9. Figures which have equal areas are called equivalent. The term equal, when applied to figures, implies an equality in all respects. The term equivalent, implies an equality in one respect only: viz. an equality in their areas. The sign , denotes equivalency, and is read, is equivalent to.

+

THEOREM J.

Parallelograms which have equal bases and equal altitudes, are

equivalent.

Place the base of one parallel

E D

B

ogram on that of the other, so that AB shall be the common base of the two parallelograms ABCD and ABEF. Now, since the parallelograms have the same altitude, their upper bases, DC and FE, will fall on the same line FEDC, parallel to AB. Since the opposite sides of a parallelogram are equal to each other (Bk. I. Th. xxiii), AD is equal to BC. Also, DC and FE are ench equal to AB: and consequently, they are equal to each

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