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to divide each of them, and add the quotients. It remains to be seen what method of separation is niost convenient.
To determine this, it is to be considered that our object is to find the number of units, tens, &c. in the quotient. Now no denomination of figures can arise from the division of a lower denomination, but may arise from the division of one higher. Therefore separating the dividend into units, tens, &c. and commencing the division with the highest denomination in the dividend, we shall certainly determine the true number of that denomination in the quotient (Cor. Prop. 13.) Let then the division be commenced by dividing the number of the highest denomination in the dividend ; if this number contain the divisor, i.e. be not less than the divisor, the number of exact times the divisor is contained may be written as the number of this denomination in the quotient. If there be any remainder, (which is ascertained by subtracting the product of divisor and the quotient from the figures divided,) or if the number do not contain the divisor, then there cannot arise from the division of this remainder, or number, any number of this denomination in the quotient, but there may arise a number of the next denomination, and also from the division of the next number in the dividend ; therefore let the remainder be converted into the next lower denomination, and let the number of this denomination in the dividend be added to it, which is done by affixing this number to the remainder. Now if this number be divided, the exact part of the quotient will be the number of this denomination in the whole quotient. In the same manner the numbers of the other denominations may be found.
Thus the dividend will at length have been separated into several parts, each of which contains the divisor a number of single times, tens of times, &c. The quotients of these parts being written after one another as they occur are in fact added (Cor. Prop. 1), and are equal to the whole quotient.
It being evident that there can be no denomination of figures in the quotient, of which there is not in the dividend a number greater than the divisor, therefore in commencing the division, we may at once take as our first dividend the least number of figures on the left, which compose a number not less than the divisor, and the denomination of the last of them will be the highest denomination in the quotient.
The foregoing explanation will be better understood by an example.Divide 87444 by 347.
Here we see at once that the highest possible denomination of figures in the quotient is hundreds. Therefore let the dividend be separated as below;
874 hundreds + 4 tens + 4 units. We may now commence the division with the 874 hundreds; and we find that 347 is contained twice in 874 with 180 over. Therefore there are 2 hundreds in the quotient. We have next to ascertain the number of tens, which may arise from the division of the 180 hundreds in the remainder just obtained, and also from the 4 tens in the dividend. As the quotients of parts of a dividend are equal to the quotient of the whole, therefore the quotients of 180 hundreds, and 4 tens are equal to the quotient of their sum, i.e. of 1804 tens. Dividing this number by 347, we find 5 as the number of tens in the quotient with 69 tens over. In the same way the number of units in the quotient is found to be 2. Therefore the whole quotient contains 2 hundreds, 5 tens, 2 units, or is equal to 252. The process of Division may be written thus:
divisor. hundreds. tens. units, hundreds.
180 = 1800 tens
add 4 tens
347)1804(5 tens 5 X 347 = 1735
69 = 690 units
347)694(2 units 2 X 347 = 694
The ordinary method will be seen to be merely an abbreviation of this.
Cor. In precisely the same manner the Rule for Compound Division may be explained by merely substituting the denominations of the quantities for the units, tens, &c. in the Proposition.
Prop. 15.- To prove that the quotient obtained by successive
division by several numbers is the same as from the division by their product; and to prove the Rule for the formation of the total remainder.
Let the dividend be represented by counters, equal in number to the units in the dividend. Let them be divided into equal heaps (which call A), the number of heaps being equal to the Ist divisor. The number of counters in each heap will represent the 1st quotient, and the number of those that are over will be the 1st remainder. Now let each of the heaps (A) be divided into equal heaps (which call B), the nnmber of heaps being equal to the 2nd divisor. Then there will altogether be a number of heaps (B) equal to the product of the 1st and 2nd divisors. And the number of counters in each heap will represent the quotient of the 1st quotient by
the 2nd divisor, or the 2nd quotient. Also there will remain over from the division of each heap (A) the same number or 2nd remainder; therefore from the division of the whole of the counters there will remain a number equal to the product of the 2nd remainder and 1st divisor together with the 1st remainder. Now assuming the two remainders to be the greatest possible, viz. less by one than the respective divisors, the product of the 2nd remainder and Ist divisor will be less by the 1st divisor than the product of the two divisors; therefore adding the 1st remainder, we find the total remainder to be less by one at least than the product of the divisors. Hence the number of counters in each heap (B) is the greatest possible number that can be obtained by dividing the whole number into equal heaps, in number equal to the product of the two divisors. In other words the number of counters in each heap (B) represents the quotient from the division by the product of the two divisors. Hence to divide by a number composed of two factors, we may divide by each factor in succession, and the total remainder will be found by adding the 1st remainder to the product of the 1st divisor and the 2nd remainder.
If the divisor be composed of more than two factors, they may be reduced to two, and the quotient obtained by successive division by each of these. But the division by each of these may be performed in like manner by dividing by each of two factors, of which it may be composed ; and the same may be said of any divisor, which is composite. Hence to divide by any composite number, we may divide by each factor in succession.
Also for the formation of the remainder, we have to consider that the result of any number of divisions is (as has been proved) the same as the result from division by a number equal to the product of the divisors, hence the final result is the same, as that from division by the last divisor, and by the product of all the rest. Therefore (by the first part of the Prop.) the remainder after 3 divisions is equal to the product of the 3rd remainder and the first 2 divisiors together with the remainder after two divisions. Similarly the remainder after 4 divisions is equal to the product of the 4th remainder and the first 3 divisors together with the remainder after 3 divisions. Thus the remainder after any number of divisions is evidently to be obtained by multiplying each remainder into the product of all the previous divisors, adding the products and the first remainder, which is the rule.
The same exhibited algebraically :-Let a1, ll2, ag, &c. be the successive divisors; ru, ru, r3, &c. the successive remainders from each division; R1, R2, Rz, &c. the total remainders after one, two, three, &c. divisions: then Rq=r; R, = r2 X an, tri
R3 =r3 X 22 X ai + R
= rX a, X a, tra X a, tri &c. = &c.
Prop. 16.-—To prove the Rule for Division by any number
having ciphers on the right. Since units, tens, &c. on multiplication by 10, become respectively tens, hundreds, &c. therefore conversely tens, hundreds, &c. when divided by 10, become respectively units, tens, &c. Now if we take away the figure on the right of any number, so as to make the tens' figure occupy the units' place, &c. we thus convert the tens, hundreds, &c. into units, tens, &c. i.e. we effect the same result as by the division by 10. Hence to divide by 10 we have only to take away the figure on the right; the figures that then remain will form the quotient, and that taken away will be the remainder. In a similar manner it may be shewn that the division by 100, 1000, &c. may be performed by taking away from the right hand as many figures as there are O's in the divisor. Hence, and by the previous Prop. the division by any number having ciphers on the right may be performed by taking away (or marking off) as many figures on the right as there are O's in the diviser, and dividing the remaining figures of the dividend by the remaining figures of the divisor, after throwing away the ciphers. Thus to divide by 25000, mark off 3 figures from the right of the dividend (which is in fact dividing by 1000), and divide the remaining figures by 25. The remainder will then be obtained by affixing the ciphers taken away to the remainder from the division, and adding the figures cut off from the dividend. Prop. 17.—Every factor of a number is a measure of the
same, and every measure is a factor. Since factors are numbers, which by their product compose a given number, therefore every factor, or the product of any number of co-factors, will divide the given number without remainder, the quotient being the product of the other co-factors. Therefore every factor of a number is also a measure of the same. Conversely, every measure of a number is a factor of the same. For the measure multiplied by the number of times it is contained in the given number becomes equal to the given number. Therefore the measure is one of those numbers, which by their product compose the given number, and is consequently a factor.
The same Prop. exhibited algebraically ;-Let N be the given number, a, b, c. &c. its factors. Then N=a X 6 XCX &c. = a X (6 XCX &c.) = (a Xb) X (CX &c.) .:. N contains a a number of times equal to (6 XCX &c.) and
N contains a Xb a number of times equal to (c X &c.); &c. &c. .. a, a X 6, &c. are measures of N. Again let n measure N, or be contained in N, a times,
then NEn Xa .. n is a factor of N.
Prop. 18.--If one number measure another, the factors of
the first are factors also of the second, The second may be obtained by multiplying by the first the number of times it is contained in the second. But if the first be a composite number, this Multiplication may be effected by multiplying by each of its factors in succession. Therefore each of these factors being one of the numbers, which by their product form the second, is a factor of the second.
The same Prop. exhibited algebraically:-Let the number n measure N; or let N contain n, a times; let n = b XCX &c.: then
N=aXn=a Xi6x0X &c.) = axbXcX &c. .. b, c, &c. are numbers which, together with a, by their product form N. .. b, c, &c. are factors of N.
Prop. 19.-If one number measure another, the first mea
sures also every multiple of the second. Since the second number is a factor of its multiple, it is therefore a measure of the same (Prop. 17); consequently every factor of the second is a factor of its multiple (Prop. 18); but every measure of the second is a factor of it (Prop. 17), therefore every measure of the second is a factor and measure of every multiple of the second.
The same Prop. exhibited algebraically :-Let n measure or be contained in N, a times; so that N =n Xa; then
N Xm=(n X a) Xm=n X (a X m).
Prop. 20.--If one number measure each of two others, it
measures also their sum or difference. Since the quotient of a sum may be obtained by dividing each part and adding the quotients, therefore if we divide each of two numbers by another, and add the quotients, we shall obtain the quotient of their sum. Now the quotient of each of the two given numbers divided by their measure is an exact quotient, therefore the sum of these quotients, i.e. the quotient of the sum of the two numbers, is also an exact quotient. Hence the given measure of the two numbers measures also their sum.
Again: since the larger of two numbers is equal to the sum of the smaller and their difference, therefore the quotient of the larger divided by any number, is equal to the sum of the quotients of the smaller, and their difference, divided by the same number. But if the divisor be the given measure the first and second of these quotients are exact, therefore the last