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8. The first term of an arithmetical progression is 100, the ratio of decrease 7; to find the nineteenth term.

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9. A man traveled 7 miles the first day, 14 miles the second day, 21 miles the third day, and so, in increasing arithmetical progression; to find how many miles he traveled the tenth day, and the whole distance traveled.

Ans. He traveled 70 miles the tenth day, and the whole distance traveled was 385 miles.

10. A man has traveled 600 miles, of which he traveled. 10 miles the first day and 50 miles the last day; to find how many days he was on his journey, supposing his daily travel increased in arithmetical progression.

Ans. If n stands for the required number of days, we get 10 + 50

n x

2

=

= 30n 600; and dividing these equals by 30,

there results n = 20 the required number of days.

11. Given, the increasing arithmetical progression, 4, 10, 16, 22, 28, 34, etc., whose ratio of increase is 6; to insert two arithmetical means between 4 and 10, also to insert two arithmetical means between 10 and 16, and so on.

Subtracting 4 from 10 we get 6, which is to be divided by the number of means to be inserted (between every two

terms) increased by one, and we have

6

6 = =2, which

2+1 3

is the common difference between the two means, and between them and the terms that correspond to them among the terms between which they are to be inserted. Hence, if we add 2 to 4, and 2 to the sum, we get the first two means; and in the same way by adding 2 to 10, and 2 to the sum, we have the two means to be inserted between 10 and 16, and so on.

The means being inserted as required, the answer is 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, etc. Hence, we see if we insert the same number of arithmetical means between every two adjacent terms of an arithmetical progression, that the result will also be an arithmetical progression.

SECTION IX.

OF GEOMETRICAL PROPORTIONS AND PROGRESSIONS.

(1.) The quotient of one number divided by another, or of one quantity divided by another of the same kind, is called the geometrical ratio or relation of the number or quantity that is taken for the dividend to the number or quantity that is taken for the divisor; the dividend being called the antecedent, and the divisor the consequent of the ratio, and the antecedent and consequent are called the terms of the ratio. Thus, if A and B stand for two numbers or quantities of the same kind, then the fractional expression represents

A

B

the geometrical ratio of A to B; A being the antecedent and B the consequent of the ratio, and A and B the terms of the ratio.

(2.) If A and B are two numbers or quantities of the same kind, and C and D are two numbers or quantities of the same kind with each other (or with A and B, should the nature of

A

the case require it), then if the ratios and

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C

B D are equal to

(1), which is called a geo

metrical proportion; A being called the first antecedent and B the first consequent, and C is called the second antecedent and D the second consequent.

It is clear that

manner

A

B

is number, and not quantity, and in like

C
is number and not quantity; and (1) signifies that
D

A contains B as often as C contains D, or that A is the same part of B that C is of D.

Thus, if A stands for 4 dollars and B for 2 dollars, and C for 6 bushels of wheat and D for 3 bushels of wheat, we shall 4 dollars 2 dollars

have

lars twice; also

= the number 2, since 4 dollars contain 2 dol

6 bushels
3 bushels

the number 2; so that 4 dollars

contain 2 dollars just as often as 6 bushels of wheat contain 3 bushels of wheat, which is in accordance with the true meaning of (1), which merely signifies an equality of geometrical ratios.

(3.) If A, B, C are three numbers or quantities of the same kind, such that we have (2), then A, B, C are said

A B

B

==

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to be in continued geometrical proportion; in which it will be observed that the first consequent equals the second antecedent.

(4.) It is customary to write (1) and (2) in the forms A: B :: C: D, (3), and A : B :: B: C, (4), or in the forms A: BC: D, (5), and A: BB: C, (6).

Because A and D occupy the extreme places in (3) and (5), we call them the extremes of the proportion, and since B and C are between A and D, we call them the means of the proportion; in like manner A and C are called the extremes in (4) or (6), and B is called the geometrical mean between A and C. Also A, B, C, D in (3) or (5) are called geometrical proportionals, A and D being the extremes and B and C the means, and D is called a fourth proportional to A, B, and C ; and A, B, C, in (4) or (6), are called continued proportionals, A and C being the extremes and B the mean proportional, and C is called a third proportional to A and B.

(3) or (5) is read by saying A is to B as C is to D, or we say, as A is to B so is C to D; and it is clear that the meaning is the same as to say that A contains B as often as C contains D, or that A is the same part of B that C is of D; (4) and (6) are read in a similar way, by saying A is to B as B is to C, or as A is to B so is B to C.

It may be observed that the colon (:) which is written between A and B in (3), is used to signify that A and B are numbers or quantities of the same kind, and that A which precedes the colon is to be divided by B which follows it; and the use of the colon is to be understood in like manner in all cases.

The double colon (::), which is used in (3) and (4), signifies that the quotient of the division of A by B is equal to the quotient of the division of C by D, these quotients being abstract numbers, and not quantity; and the same thing (if

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we please) may be expressed by =, as in (5) and (6); yet it may perhaps, for the sake of expressing the species of equality intended in geometrical proportion, be advisable to use the double colon (::) instead of =.

It follows clearly, from what has been done, that for

A: B::C: D we may use the equation

versa.

A C
BD'

and vice

(5.) Resuming (1), and multiplying its two members by any whole number m, we shall have the equation

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MA

B

=

It is clear from (2) that the integral part of the quotient

mA

B

mc is the same as that of and that whether A, B, C, D D

are of rational or irrational forms, so that when we have

A C
B

=

D'

the multiple mA of A can not contain B a greater

or less integral number of times than the equimultiple mC of C contains D.

Reversely, if we have A, B, C, and D, such that by taking any equimultiples of A and C by any whole number m, we find that the multiple mA of A can not contain B a greater or less integral number of times than mC, the equimultiple A C of C, contains D, then we shall have the equation

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B D' one of them must be the

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C

A tion B D

+p, (3), where p is evidently a number, since

A and B are supposed to be numbers or quantities of the same kind, and that C and D are either numbers or quantities of the same kind; and it is clear that p is less than unity, and that it may be rational or irrational, according to the nature of the case.

Now, if we take the integer m, such that the inequality m> 1÷p shall have place, then will the inequality pm >1 also have place.

If we multiply the two members of (3) by m, we shall

have

mA
B

==

mC
D

+mp; consequently, since mp = pm is not

less than unity, mA contains B a greater integral number of times than mC contains D, which is against the hypothesis;

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In the same manner it may be shown, that

A

than ; of course we must have

= as was to be

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Α с

B D'

mC

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B proved. Hence, we may use the conditions, that the inte

gral part of is not greater than that of and that

mA
B

the integral part of is not greater than that of

mC
D

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tests of the proportionality of A, B, C, D, supposing m to stand for any positive whole number. This method of testing the proportionality of quantities will often be much more simple than to use (1), especially in the application of the doctrine of proportion to geometrical investigations.

(6.) If we multiply the two members of (1) by the integer Im, and divide the products by the integer n, we shall evimC dently have

mA
nB nD'

=

; or, which is the same, if we have

the proportion A: B:: C: D, then we shall also have the proportion mA: nB:: mC nD. In words, if we multiply the antecedents of any proportion by any integer, and the consequents by any integer, the multiple of the first antecedent will be to the multiple of its consequent as the multiple of the second antecedent is to the multiple of its consequent, mA mC

=

nB nD

it is not

It is easy to see that in the equation necessary to suppose m and n to be positive integers; for

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