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Remarks. Since Division is the reverse of Multiplicacation, we may reduce the rule of Division to that of Multiplication, by changing the signs of the exponents of the terms of the divisor, and then proceeding as in Multiplication.

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the same result as found above. Also, to divide

=

21 7 3.7

2

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7-1 3.71-1 3.70; or, since

4 2 22

2-1

22-1

we have

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=

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2

as found above.

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The product of any number of equal fractional quantities

2 2

is called a power of one of the fractions. Thus, X is 3 3

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as a factor, n times, supposing n to be a positive integer. And since the product of an even number of negative quantities is positive, and the product of an odd number of negative quantities is negative, it follows that the n" power of is the same as that of when n is an even b

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number, and that the sign of the n" power of - is -, when n is an odd number.

Reversely, when any fraction, as, is to be resolved into

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the product of n equal fractions, one of these equal fractions

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is called the n' root of %, and is denoted by (2)

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απ
b

means that two equal fractions are to be

2

found, such that their product shall equal; or, which is the

same thing, that 2 is to be resolved into two equal factors, and that 3 is to be resolved into two equal factors; and that we are to use one of the factors of 2, divided by one of the factors of 3.

SECTION VIII.

OF ARITHMETICAL PROPORTIONS AND PROGRESSIONS.

(1.) THE difference of two numbers or quantities of the same kind is called their arithmetical ratio or difference.

(2.) Four numbers or quantities of the same kind are said to be in arithmetical proportion when the difference of the first and second is equal to the difference of the third and fourth; the first and last, terms being called the extremes, and the other terms the means of the proportion.

(3.) A series of numbers or quantities of the same kind that continually increase or decrease by a constant difference are said to be in continued arithmetical proportion, or to constitute an arithmetical progression; and if the successive terms increase, the progression is said to be increasing, but if the successive terms decrease, the progression is said to be decreasing.

(4.) If four numbers or quantities of the same kind are in arithmetical proportion, the sum of the extremes is equal to the sum of the means.

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For, let a, b, c, d denote four numbers or quantities, such that we have a bcd, then if we add b + d to these equals, we shall have a-b+b+d=c−d+b+d, or (since b+b= 0, and d+d=0) we have a + d= bc, as was to be proved. Thus, if we take the numbers. 12, 8, 10, and 6; since 1284 the arithmetical ratio of 12 to 8, and that 10 - 6 4 the arithmetical ratio of 10 to 6, and since these ratios are equal, it follows that 12 — 8 106; and we have 12+68+ 10 = 18, as it ought to be, for 12, 8, 10, and 6 severally correspond to a, b, c, d.

Since a, b, c, d are connected by the equation a+d=b+c, it follows that if any three of them are given, the fourth can be found. For, suppose a, b, c are given to find d, then, since a + d = b+c, if we subtract a from these equals, we a = b + c get a + d a, or d = b + c − a.

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Hence, if we take either of the extremes from the sum of the means of an arithmetical proportion, we shall get the other extreme; and if from the sum of the extremes we take one of the means, we shall get the other mean.

(5.) If three numbers or quantities of the same kind are in arithmetical proportion, the sum of the extremes is equal to twice the mean.

For let a, b, c be such numbers or quantities that we have a-bb-c, then by adding b+c to these equals, we shall have a - b + b + c = b + b − c + c or a + c = 2b, as required. Thus, since 15 - 11=11-7, we get 15 +7= 2 x 11 = 22.

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Since a, b, c are connected by the equation a + c = 26, it will be sufficient to know any two of them in order to find the third. For if the mean b is given, and either extreme, as a, is also given, then subtracting a from the equals a + c and 26, we get the other extreme c = 2b a; that is to say, if we subtract either extreme from twice the mean, the remainder will equal the other extreme.

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Thus, if the mean is 10, and one of the extremes is 7, then we shall have 10 × 2 720-713 for the other extreme; and it is easy to see that 13, 10, and 7 are in arithmetical proportion, as they ought to be.

Again, when the extremes a and c are given, if we divide

the equals 26 and a + c by 2, we shall have b

=

a + c

is, the mean equals half the sum of the extremes.

the extremes are 11 and 17, we have

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2

; that

11 + 17 28

2

Thus, if

= = 14 = 2

the mean; and it is easy to see that 11, 14, 17 are in arithmetical proportion, so that 14 is an arithmetical mean between 11 and 17, as it ought to be.

It may be observed that the common method of finding the average of two numbers or quantities of the same kind (which consists in taking the half sum for the average) is the

same as to take the arithmetical mean of the two numbers or quantities.

Again, resuming the equation a + c = 2b; if we add b to both sides, we get a + b + c = 36, and dividing these equals a + b + c = b.

by 3, we get

3

Hence, when three numbers or quantities of the same kind are in arithmetical proportion, the mean is equal to one third of the sum of the numbers or quantities, and it is clear that the mean is the average of the numbers or quantities.

Hence, we often call one third of the sum of any three numbers or quantities of the same kind their average or mean; and one fourth of the sum of four numbers or quantities of the same kind is called their average or mean; and generally, if n is any positive integer, the nth part of the sum of ʼn numbers or quantities of the same kind is called their average

or mean.

Thus, if 20. 23, 20. 24, 20. 26, 20. 28, represent four values of a certain quantity, as found by observation or experiment; then dividing the sum of these numbers by 4, we get 20.23 20.24 + 20.26 + 20.28 81.01

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= 20.2525 =

the average of all the quantities, supposing the above numbers to have been correctly found to four decimal places; if, however, they have been correctly found only to two decimal places, we must take 20.25 for the average of the quantities. And it is clear that the average ought to be taken as the most probable value of the quantity as determined from the four observations or experiments.

It is plain that we may proceed in a similar way in all cases to find the most probable values of quantities as determined by observation or experiment, supposing equal skill and care to have been used in each instance, and supposing any result that differs much from the others to be rejected.

(6.) If a denotes the first term of an arithmetical progression, and d the difference of any two of its successive terms, then by the definition of an arithmetical progression, if the progression is increasing, we shall have the progression expressed by the series a, a + d, a + 2d, a + 3d, a + 4d,

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