Geometrical proportion is that relation of two quantities of the same kind, which arises from considering what part the one is of the other, or how often it is contained in it.

When four quantities are compared together, the first * and third are called the antecedents, and the second and fourth the consequents.

Ratio is the quotient which arises from dividing the antecedent by the consequent, or the consequent by the antecedent.

Four quantities are said to be proportional when the first is the same part or multiple of the second as the third is of the fourth.

Thus 2, 8, 3, 12, and a, ar, b, br are geometrical proportionals.

Direct proportion is when the same relation subsists between the first term and the second, as between the third and the fourth.

Thus 3, 6, 5, 10, and x, ax, y, ay are in direct proportion.

Reciprocal or inverse proportion is when one quantity increases in the same proportion as another diminishes. Thus, 2, 6, 9, 3, and à, ar, br, b are in inverse proportion.

A series of quantities are said to be in geometrical progression when the first has the same ratio to the second as the second to the third, the third to the fourth, &c.

Thus, 2, 4, 8, 16, 32, 64, &c. and a, ar, ar2, ar3, ar1, ars, c. are series in geometrical progression.

G

The most useful part of geometrical proportion is contained in the following theorems.

I. If four quantities be in geometrical proportion, the product of the two means will be equal to that of the two

extremes.

Thus, if 2, 4, 6, 12, and a, ar, b, br, be geometrically proportional, then will 2×12=4×6, and a×br=b×ar.

II. If four quantities be in geometrical proportion, the rectangle of the means, divided by either of the extremes, will give the other extreme.

Thus, if 3, 9, 5, 15, and x, ax, y, ay, are geometrically

9X5
3

proportional, then will =15, and

axxy

ay

= x.

III. In any continued geometrical progression, the product of the two extremes, and that of any other two terms, equally distant from them, will be equal to each other.

Thus, in the series 1, 3, 9, 27, 81, 243, &c. 1 × 243: 3X81=9×27.

=

IV. In any continued geometrical series, the last tèrm is equal to the first multiplied by such a power of the ratio as is denoted by the number of terms less one.

Thus, in the series 2, 6, 18, 54, 162, &c. 2×34=162.

V. The sum of any series in geometrical progression is found by multiplying the last term by the ratio, and dividing the difference of this product and the first term by the ratio less one.

Thus, the sum of 2, 4, 8, 16, 32, 64, 128, 256, 512 is 512×2-2

2-1

=1024-2=1022.

And the sum of n terms of a, ar, ar2, ar3, ar1, &c. to

arn-1xr-a

arn-1, is

r-l

VI. If four quantities, a, b, c, d, or 2, 6, 5, 15, be proportional, then will any of the following forms of those quantities be also proportional.

1. directly, a: b::c: d, or 2: 6:: 5:15.
2. inversely, b: a::d: c, or 6:2::15:5.
3. alternately, a: c::b: d, or 2 : 5 :: 6:15.

4. compoundedly, a: a+b::c:c+d, or 2:8:: 5:20. 5. dividedly, a ; b—a ; ; c : d—c, or 2 : 4 ; : 5:10, 6. mixed, b+a:b—a::d+c: d-c, or 8: 4 :: 20:10. 7. by multiplication, ra: rb :: c : d, or 2.3 : 6.3:: 5:15. 8. by division, a÷r:b÷r::c: d, or 1: 3 :: 5:15. 9. The numbers a, b, c, d are in harmonical proportion, when a d::ab:cond.

EXAMPLES.

1. The first term of a geometrical series is 1, the ratio 2, and the number of terms 10; what is the sum of the series?

First, 1x29=1×512 = last term.

2. The first term of a geometric series is, the ratio

, and the number of terms 5; required the sum of the series.

3. Required the sum of 1, 3, 9, 27, 81, &c. continued

to 12 terms.

Ans. 265720.

4. Required the sum of 1,,,,, &c. continued

to 12 terms.

5. Required the sum of 1, 2, 4, 8, 16, 32, &c. continued to 100 terms.

SIMPLE EQUATIONS.

An equation is, when two equal quantities, differently expressed, are compared together by means of the sign = placed between them.

Thus 12-57 is an equation, expressing the equality of the quantities 12-5 and 7.

A simple equation is that which contains only one unknown quantity, without including its power.

Thus x-a+b=c is a simple equation, containing only the unknown quantity x.

Reduction of equations is the method of finding the value of the unknown quantity, which is shown in the following rules.

RULE I.

Any quantity may be transposed from one side of the equation to the other by changing its sign.

Thus, if x+3=7, then will x=7—3=4.

And, if x-4+6=8, then will x=8+4—6—6.
Also, if x-a+b=c-d, then will x=c-d+a—b.
And, in like manner, if 4x-8=3x+20, then will 4x
-3x=20+8, or x=28.

RULE II.

If the unknown term be multiplied by any quantity, it may be taken away by dividing all the other terms of the equation by it.

Thus, if ax=ab-a, then will x=b—1.

And, if 2x+4=16, then will x+2=8, and x=8-2

= 6.

3c2

In like manner, if ax + 2ba = 3c2, then will x +26=

a

RULE III.

If the unknown term be divided by any quantity, it may be taken away by multiplying all the other terms of the equation by it.

Thus, if —=5+5, then will x=10+6=16.

36

+12, and 2x=18+12+6=36, or x===18.

2

RULE IV.

The unknown quantity in any equation may be made free from surds, by transposing the rest of the terms by Rule I, and then involving each side to such a power as is denoted by the index of the surd.

Thus, if x

x=82=64.

-2=6, then will x = 6 + 2 = 8,

and

And if√4x+16=12, then will 4x+16=144, or 4x =144—16=128; and if both sides of the equation be divided by 4, x will be

: 32.