Of Ratios and Proportions.

THEOREM X.

If four quantities are proportional, and one antecedent and its consequent be augmented by quantities which have the same ratio us the antecedent and consequent, the four quantities will still he in proportion

in which the antecedent C and its consequent D, are augmented by the quantities E and F, which have the same ratio.

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

in which the ratio is 2.

If we augment the antecedent and its consequent by 15 and 30, which have the same ratio, we have

If four quantities are proportional, and one antecedent and its consequent be diminished by quantities which have the same ratio as the antecedent and consequent, the four quantities will still be in proportion

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

AX(D-F)=Bx(C-E);

and by Th. II, we obtain

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

in which the antecedent and consequent, C and D, are diminished 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

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

If we have the several proportions,

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.

2. 4 :: 6 : 12 and

Then,

that is,

1 : 2 :: 3

:

6

in which the ratio is still 2.

THEOREM XIII.

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

If we have the proportion

A: B :: C: D,

it gives

B D

A C

Then, if we square both members, we have

and then, changing these equalitics into a proportion, we have for the first,

Soh. We may verify the proposition by the proportion,

Of Ratios and Proportions.

numbers which are still proportional, and in which the ratio

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

and this by Th. II, gives

AXE : BxF :: CXG DXH.

:

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

24 : 48 :: 60 : 120

which are proportional, the ratio being 2.

1 Similar figures, are those which have the angles of the onc 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