of the ratios is called a subquadruplicate ratio; and if a ratio is resolved into five equal ratios, either of the equal ratios is called a subquintuplicate ratio; and if it is resolved into six equal ratios, either of the equal ratios is called a subsextuplicate ratio; and generally, if a ratio is resolved into n equal ratios, we may say that either of the equal ratios is the th ratio of the given ratio, which is plainly the same

n

as the nth root of the given ratio.

Thus, if equals the product of the n equal ratios

A
B

etc., then either of the equal ratios may be called

If a ratio that is compounded of three equal ratios is resolved into two equal ratios, either of the two equal ratios is

X X × etc., for the compound ratio. B C D

Because A, B, C, etc., are (numbers or) quantities of the same sort, it is clear that we may divide A and B by C, so

the dividend and divisor in the fractional product. In like

Hence, if A, B, C, D, etc., represent any (numbers or) quantities of the same sort, we say that the ratio of the first to the last is compounded of the ratio of the first to the second, and of the ratio of the second to the third, and of the ratio of the third to the fourth, and so on to the last; see Def. A in R. Simson's Euclid, Book V.

(30.) Ratios that are compounded of equal ratios are equal to each other.

For, let the equal ratios be expressed by the equations A CE G K M B ̄D'F=HL N'

=

and so on; then, since the products

of equal numbers must be equal, we have

BX

A E K

FXIX etc. C N = X X × etc.; that is, ratios which are compounded DH M

of equal ratios are equal, as was to be shown.

Reciprocally, equal ratios divided by equal ratios or by the products of equal ratios, must give equal ratios for the quotients.

For, let

A C
B D

denote the equal ratios to be divided,

whether they are simple or compound ratios, and

E G
FH

the

equal ratios by which they are to be divided, whether they are simple or compound.

Then, since equal numbers divided by equal numbers must give equal numbers for the quotients, we shall have A E C G

or, since to divide by the numbers

clearly the same as to multiply by their reciprocals

EG

is

F' II

FH

E' G

A

we shall have

B

F C H

X E-DXG ; that is, equal ratios divided

by equal ratios give equal ratios for the quotients, as required.

(31.) We will now show how, from knowing any three terms of a proportion, to find the remaining term.

Suppose that we have the proportion A: B:: C : D, or, which is the same, the equation (1). Then, if we

A C
B D'

A

know A, B, and D, we of course know the quotient and

C

B'

since expresses the number of times that C contains D, if

D

which gives the value of C, since D and the quotient

known.

Hence, we see that the third term of a proportion equals the fourth term multiplied by the ratio of the first to the second term.

B

Again, by (12), we may take the equation

(1); hence (as before), if we know A, B, and C, we shall have

Consequently, if we multiply the third term of a propor tion by the ratio of the second term to the first term, the product will equal the fourth term.

In a similar way, it appears that the first term of a proportion equals the product of the second term by the ratio of the third to the fourth term; and the second term equals the first term multiplied by the ratio of the fourth to the third term.

(32.) We will now adapt what has been done to a continued proportion which consists of three terms. Thus, if in the proportion A: B:: C: D we suppose that CB, we shall have A: B :: B : D, which is a continued proportion of three

terms.

Instead of the proportion we may use the equation

(2), or, by (12), we may use the equation

A B
=
B D'

Since, from equation (1), we have D = C ×

(4) shows that the third term of a continued proportion equals the mean term multiplied by the ratio of the mean to the first term. And it may be shown, in a similar way, that the first term equals the mean term multiplied by the ratio of the mean to the third term.

To get the mean term, we compound the ratio of the first to the mean term with the ratio of the mean to the third

(5) shows that the mean term equals the third term multiplied by the square root of the ratio of the first to the third

term.

It may be shown, in like manner, that the mean term equals the first term multiplied by the square root of the ratio of the third to the first term.

B

B

Since B A × and that D=Bx it follows that

A'

Α'

B

the terms A, B, D of our proportion will become A, A× Ã'

It is manifest, from what has been done, that any one of the three terms of the proportion can easily be found from the other two terms.

(33.)

OF GEOMETRICAL PROGRESSIONS.

Numbers or quantities which are such that the first is to the second as the second to the third, and the second to the third as the third to the fourth, and the third to the fourth as the fourth to the fifth, etc., are said to be in continued geometrical proportion, or in geometrical progression.

If the terms increase from the first term, the proportion (or progression) is said to be increasing, but if the terms decrease from the first term, the progression is said to be decreasing. The ratio of the second term of the progression to the first term is called the ratio of increase or decrease of the progression. When the ratio is greater than unity, the progression is clearly increasing, and when it is less than unity, the progression is decreasing; and it is evident if the ratio is equal to unity, that the progression will be neutral, or that its terms will neither increase nor decrease, or that they will equal each other.

If A, B, C, D, E, etc., represent the successive terms of the continued proportion (or progression) whose first term is A, we shall signify that they are in continued proportion by writing them as follows, viz., A: B:: B: C :: C :D :: D: E: E F etc., (1), which is in accordance with the usual manner of writing the terms of a continued geometrical proportion.

(34.) For A, B, C, D, E, F, etc., we may write A, A ×

B

Α'

; supposing n to de

For, since A, B, C are in continued geometrical proportion, we shall have, from (32), A, B, C expressed by A,

Ах

A x

And because B, C, D are in contin

ued geometrical proportion, we shall have D=Cx

manner, because C, D, E are in continued geometrical pro