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Also for every one besides in these places take one farthing, subtracting 1 if the number be greater than 24. By converting the decimal into farthings, which is easily done by multiplying by 1000 — 40, this approximation may be corrected.

3.- To explain the Rules of Practice. The cost of a quantity of goods at a given price is equal to the sum of the costs at other prices, which are together equal to the given price; or is equal to the sum of the costs of parts of the goods. Also the cost at any price, which is an exact part of another price, is the same part of the cost at this latter price, and may therefore be obtained from the latter by Division. And the cost of any quantity of goods, which is an exact part of another quantity, is the same part of the cost of this latter quantity, and may therefore be obtained from the latter by Division.

Hence if the price given be separated into parts, each of which is an exact part of some price, the cost at which is known, the costs at each of these may be found in succession by division, and their sum will be the total cost. Or if the quantity of goods given be separated into parts, each of which is an exact part of some quantity, the cost of which is known, the costs of each of these may be found by division, and their sum will be the total cost. On these principles the Rules for the two cases in Practice are founded.

Note to the observations in page 10 of the · Practice.' 1. It is axiomatical that every even number is divisible by 2.

2. Since any number of hundreds is divisible by 4, therefore any number is divisible by 4, in which the number composed of the last two digits is divisible by 4. Thus 1364 is divisible by 4, because 1300, and 64, are each so divisible.

3. Since any number of thousands is divisible by 8, therefore any number is divisible by 8, in which the number composed of the last 3 digits is divisible by 8. Thus 10512 is divisible by 8, because 10000, and 512, are each so divisible.

4. The truth of Obs. 4 is evident.

5. Any number of tens is divisible by 5, and 5 is divisible by 5, therefore any number ending with 5 is evidently divisible by 5.

6. Since 10 = 9+ 1, 100 = 99 + 1, 1000 = 999 + 1, &c. therefore any number as 6564 may be put into this form :

6 X 999 +6 +5 X 99 +5 +6 x 9 + 6 + 4

or (6 X 999 + 5 X 99 +6 X 9) + (6 +5+6+4) of which the first part is divisible by 3 and by 9; therefore, if the latter part be divisible also by 3 and by 9, the whole number will be so.

7. Since 10 = 11 1; 100 = 99 +1; 1000 = 1001 - 1; 10000 =9999 + 1, &c. therefore any number as 57694 may be put into this form :

S

5 X 9999 + 5+7 X 1001 — 7 + 6 X 99 +6 +9 X 11 - 9+4 or (5 X 9999+7X 1001 +6 x 99+9x11)+{15+6+4) — (7+9)} of which the first part is divisible by 11 ; therefore if the latter part (which is the difference of the sums of the digits in the odd and even places) be also divisible by 11, the whole number will be so.

Note to Prop. 4. It is important to observe, that all conclusions respecting abstract numbers are arrived at by means of reasoning upon concrete numbers. For, abstract numbers being symbols of an operation, viz.-repetition, we have no means of ascertaining the equivalence of two or more operations, except by considering their effect upon some concrete quantity. Thus because we find that 2 articles added to 3 articles produce 5 articles, we conclude, therefore, that the aggregate result of the operations denoted by 2 and by 3 is the same as that of the one operation denoted by 5, and consequently that the two operations 2 and 3 are together equivalent to the one 5. By this means it will be seen, that the Rule for Addition of numbers is obtained. For it being established, that the sum of several numbers of articles is the same as the sum of the smaller collections into which these may be formed, it is hence concluded that the operation, which is equivalent to the aggregate of several operations, is also equivalent to the aggregate of those to which the former are equivalent. The same remarks apply to the Rule of Subtraction.

Note to Prop. 6. A product is defined to be the sum of several, the same numbers, and the operation, by which the product is obtained, is called Multiplication; the number, which is repeated, being the multiplicand, that, which shows how often it is repeated, being the multiplier. Hence it is evident that the multiplier is always an abstract number, but the multiplicand may be abstract or concrete. In the Prop. it is proved, that the repetition of three things 4 times is the same as the repetition of four things 3 times, and hence it is concluded that 4 times 3 are equal to 3 times 4, and that the abstract numbers in the multiplier and multiplicand may be interchanged.

Note to Prop. 9. The Rule for Multiplication by a composite number might be deduced immediately from the nature of the product of two abstract numbers. Such a product is a number, representing an operation equivalent to the repetition of that denoted by the multiplicand, as often as there are units in the multiplier. Now an abstract number is the symbol of the operation of repetition, or multiplication; therefore multiplication by a composite number is equivalent to the repetition of the multiplication by one factor,

as often as there are units in the other; and the result may be obtained by successive multiplications by the factors.

Note to Prop. 12. The first idea of Division is, that it is the operation of finding what is that number or quantity which, repeated a number of times equal to the divisor, will produce the dividend. Hence the divisor must be abstract, the dividend either abstract or concrete. When the dividend is abstract, we may put the definition into another form, and say that the object of Division is to find the symbol of the operation, which, repeated a number of times equal to the divisor, is equivalent to that denoted by the dividend. But since the repetition of the operation denoted by one number, as often as there are units in another, is equivalent to the successive performance of the operations denoted by the two numbers, therefore the object of Division might be defined to be, the finding the symbol of an operation, upon whose result if that denoted by the divisor be performed, a result will be obtained the same as by performing the operation denoted by the dividend. This is the view taken of Division in Prop. 35, 2nd case.

Again, since 3 times 4 are equal to 4 times 3, therefore the symbol of the operation which, repeated 3 times, is equivalent to that denoted by 12, is the symbol of the number of times, which the operation denoted by 3 must be repeated to be equivalent to the same operation ; i.e. the third part of 12 is the number of times that 3 is contained in 12. Hence has arisen the ordinary definition of Division, as applied to abstract numbers; which is the view taken of it in Prop. 35, 3rd case.

Note to Prop. 26. In the 6th conclusion read “ which 2 units are of 5 units.” The 8th conclusion follows from Prop. 35, 3rd case.

Note to Prop. 27.

а Хc This Prop. may be proved otherwise, thus:

bXc

represents the operations of Multiplication by a X c, and Division by 6 Xc; or of Multiplication by a and by c, and Division by b and by c. Now the order in which these operations are performed is a matter of indifference; therefore having multiplied by a and by c, let us divide by c; we thus of course obtain the same result as by merely multiplying by a. We now have still to divide

a Xc by b: so that the operations denoted by ő Xc are equivalent to those denoted by ñ.

Note to Prop. 29. The sum of a series of fractions, considered abstractedly, is the symbol of some operations, which are equivalent to those represented by the several

- fractions taken together. Now if all the fractions have the same denominator, they indicate that the products of the quantity, upon which the operations are supposed to be performed, by each numerator, are all to be divided by the same number, and added; but the sum of the quotients of several quantities is equal to the quotient of the sum (Prop. 12); therefore the result may be obtained by adding the products, and dividing the sum by the denominator; or-since the sum of the products by several numbers is equal to the product of their sum (Prop. 8)—by multiplying by the sum of the numerators, and dividing by the denominator. Hence the sum of several abstract fractions with the same denominator is equivalent to the fraction, whose numerator is the sum of the numerators, and denominator the common denominator. Therefore if fractions be brought to a common denominator, they may be readily added.

In a similar manner the method of Subtraction of fractions may be explained.

Note to Prop. 41.- To explain the abbreviated method of

Multiplication of decimals. If it be required to find the product of two decimals correct only to a given number of figures, (for instance 6,) evidently there is no need to trouble ourselves with any more than 6 places of decimals in the partial products, if we can find these correctly, provided that we also know what is to be carried from the sum of the figures in the 7th place.

Now a figure in the 6th place of decimals in any product may arise from the multiplication of a number of units, tens, &c. by a number in the 6th, 7th, &c. places respectively, or of a number in the 1st, 2nd, &c. places by a number in the 5th, 4th, &c. places respectively, these products being increased by the numbers carried from previous multiplications. Hence if units, tens, &c. tenths, hundredths, &c. in the multiplicand be placed over figures in the 6th, 7th, &c. 5th, 4th, &c. places in the multiplier, each figure in the multiplier will stand under the first figure of the multiplicand, whose complete product is to form part of the total. But this arrangement exactly reverses the order of the figures of the multiplier, and places its units' figure under the last place of decimals of the multiplicand which is to be preserved, viz. the 6th. Hence the rule prescribes that this be done. This being done, the several products may be formed by multiplying by each figure of the multiplier, the figures of the multiplicand, as far as that which is over the particular multiplier being used. The first figures of these products being all in the 6th place of decimals (or in the last which is to be reserved), must all be ranged under each other for addition. But in order to obtain these first figures at all correctly, it will be necessary to commence each multiplication mentally with the figure to the right of the multiplier, so to find the number to be carried. At the same time an allowance may be made for the carriage from the sum of the figures in the 7th place by carrying not the actual number, which appears due, but the nearest number of tens; i.e. if the product be between 5 and 15 carry 1, if between 15 and 25 carry 2, and so on.

As an example, let the number 999.9 be multiplied by 99.9, according to the above Rule, so as to retain six places of decimals :999.99999999

999999

89999.999999 = 90 times the multd
8999.999999 = 9
899.999999 = .9
89.999999 = .09.
8.999999 = .009
.899999 = .0009.

a number less than .000001

.000001 .000001 .000001 .000001 .000001

99999.899994 = 99.9999

.000009

In this case, (which is an extreme one, all the figures being 9's) it appears that not one out of the six decimal figures is correct; but at the same time it is seen that the product obtained differs from the true product by a number less than .00001; so that even in this extreme case the error cannot be very great; and the error may be lessened by increasing the number of decimal places reserved. Thus if 6 decimal figures are required, and it be attempted to obtain 7, the error will be less than .000001. Of course the larger the quantities involved in a question, which are affected by such products, the greater number of places should be reserved.

Note to Prop. 42.-To explain the abbreviated method of

Division of decimals.

n.

1. Let d be a divisor, containing m + 1 digits, D a dividend : let DEN X 10m to n, n being the number composed of the last m digits in D: then

D N X 10m

+ d d

d and, as n is less than 10m, and d is not less than 10m, is a proper fraction ;

d therefore the integral quotient of D by d is the same as that of N X 10m

n

by d.

2. If a divisor contain a decimal places, and p be required in the quotient, the dividend must be made to have p + q decimal places; and the figures of the quotient will then be found by dividing the dividend, so

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