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tens, and so on in order; for by so doing we shall certainly find the whole number of units, tens, &c. in the difference.
Let the units of the subtrahend be subtracted; this is done (Prop. 3) by subtracting the units' figure of the subtrahend from that of the diminuend, if possible; or from that of the diminuend, increased by 10, the next higher figure of the diminuend being diminished by 1. The difference so obtained is the number of units in the whole difference : and the diminuend so altered is the first partial difference. We have now to subtract the tens of the subtrahend, which is done (Prop. 3) by subtracting the tens' figure from that of the partial difference, if possible; or (if not) from it when increased by 10, the next higher figure of the diminueud being diminished by I. The difference so obtained is the number of tens in the whole difference; and the first partial difference so altered, is the second partial difference. In like manner the number of hundreds, &c. in the difference will be obtained.
Hence the Rule for Subtraction might be given thus :~When the upper figure is less than the lower, increase it by 10, and diminish the next higher figure of diminuend by 1. But whether we diminish the diminuend, or increase the subtrahend by any number, the final difference is the same, since in the one case we are subtracting in two parts, in the other in one. Hence (and because it is perhaps more convenient) the Rule is given to increase the next higher figure in the subtrahend by 1, whenever any figure of the diminuend is increased by 10.
Note. The increase of the figure of the diminuend by 10, and of the subtrahend by 1, may also be explained by considering that by so doing we are in fact increasing both by the same number. And if two numbers be both increased by the same number, their difference remains unaltered.
Cor. The Rule for Compound Subtraction may be proved and explained in precisely the same manner by substituting for units, tens, &c. the denominations of the given quantities.
Prop. 6.-The product of two numbers is the same, whichever
be the multiplier. Place in a line 12 dots: draw a line after every third, and every fourth dot. It is thus seen that the 12 dots are composed of four groups of 3 dots,
1:1:::::....1 or of 3 groups of four dots. Hence 3 times 4 are equal to 4 times 3. The same may be proved of any numbers whatever, the multiplicand being abstract or concrete.
The same Prop. exhibited algebraically:--If a and 6 be the two numbers, then a Xb=bX a.
Prop. 7.- To prove that the product of one number by another
is equal to the sum of the products of each part of the mul
tiplicand and multiplier. Since Multiplication is only a short process of Addition, and since in adding numbers we are permitted to separate them into parts, and grouping the parts to add each group in succession, therefore in Multiplication we may do the same. The only difference between this process and Addition is, that the parts of each group being the same, the number in each is found by the aid of the Multiplication Table, instead of by direct Addition, and is called the product instead of the sum. In other respects the processes of Multiplication and Addition are precisely similar. The same Prop. exhibited algebraically;-If A be the multiplicand, and
A = a + b + c + &c. and m be the multiplier ; then
Prop. 8.—To prove that the product of two numbers is equal
to the sum of the products of the multiplicand by each part
of the multiplier. Since the product of two numbers is the same, whichever be the multiplier; therefore let the multiplicand be supposed to be the multiplier. Then the product of the two is equal to the sum of the products of the several parts of the present multiplicand by the present multiplier, i. e. is equal to the sum of the products of the several parts of the real multiplier by the real multiplicand; or is equal to the sum of the products of the multiplicand by the several parts of the multiplier.
The same Prop. exhibited algebraically :-If A be the multiplicand, M the multiplier, and M=a+b+c+ &c. then
A XM=M XA (a+b+c+&c.) XA (Prop. 6.)
=aX A +6 XA +cXA + &c. (Prop. 7.) = A Xa+AXb+ AXC + &c.
Prop. 9.–To prove the Rule for Multiplication by a com
The product of two numbers is equal to the sum of the products of the multiplicand by the parts of the multiplier. If all these parts, and therefore the products, be equal, the sum may be formed by the short process of Multiplication by their number. Now a composite number (supposed to consist of two factors) is composed by the sum of one number repeated as often as there are units in another. Therefore the product by a composite number is equal to the sum of as many products by one number as there are units in another, and may therefore be found by multiplying one product by the number of products. That is, to multiply by a number composed of two factors, we may multiply by one first, and this product by the other.
If the multiplier be composed of more than two factors, they may be reduced to two, and the product formed by multiplying by each of these. But the products by each of these two may be formed in the same manner by multiplying in succession by each of two factors of which it may be composed; and the same may be said of every multiplier, which is composite. So that generally to multiply by any composite number, we may multiply by each factor in succession.
Prop. 10.- To prove the Rule for Multiplication by
10, 100, &c. Since 10 times 6 (or any other number) are equal to 6 times 10 (Prop. 6), therefore the multiplication of units, tens, &c. by 10, changes them respectively to tens, hundreds, &c. Now if we place a 0 on the right of a number, it has the effect of increasing by 1 the distance of each figure from the right, and therefore of increasing tenfold its signification, i. e. it has the same effect as multiplication by 10. Therefore to multiply by 10 we have only to place a 0 on the right of the number. Similarly to multiply by 100, 10000, &c. we have only to place on the right of the number as many ciphers as there are in the multiplier.
Cor. Hence, and by the previous Proposition, the multiplication by any number, having ciphers on the right, is effected by adding on the right as many ciphers as there are in the multiplier, and multiplying the result by the figures of the multiplier, when the ciphers have been removed. Prop. 11.- T'o explain the Rule for Multiplication by any
number. If the multiplier be not greater than 12, then (Prop. 7) we may find the product by multiplying each part of units, &c. in succession by the multiplier (with the aid of the Multiplication Table) and adding the products, which in practice is done as they are formed.
If the multiplier be greater than 12, since the product of one number by another is equal to the sum of the products by any parts into which the multiplier is divided; therefore we are at liberty to divide the multiplier into any parts we please, to multiply by each part separately, and add the products; and doing so we shall obtain the whole product. Now by the Rule for Mul. tiplication, the multiplier is divided into units, tens, &c. and the successive products formed are the products by these several parts, the first being that
by the units, the second that by the tens, where the 0, which should be in
5765 The same as usually written 5765
40355 = 7 times the multiplicand
40355 51885 46120 23060
28231205 = 4897
Prop. B.-To explain the Rule for Compound Multiplication. If the multiplier be less than 12, the Multiplication is performed precisely on the same principle as Addition, the sums of the several denominations being found by the short process of Multiplication.
If the multiplier be greater than 12, and a composite nuinber, the true product is found by multiplying by the factors in succession. This is evidently true, when the multiplicand is only of one denomination (Props. 6, 7, 8, 9); and when it is of several, it may be reduced to one, and the Multiplication then performed. But since the product of a whole is equal to the sum of the products of the several parts, therefore we may multiply the parts of the multiplicand as they stand, without reduction.
If the multiplier be greater than 12, and not composite, then by Prop. 8 we may separate it into any parts we please, and, multiplying by these separately, may add the products. And this is done by the rule, according to which the multiplier is separated either into units, tens, &c.; or into the sum of a composite number and another less than 12.
Again since the result of the repetition of a quantity a certain number of times may be obtained by repeating it a greater number of times, and by subtracting the result of its repetition a number of times equal to the difference, therefore the product by any number may be found by multiplying by a greater number, and subtracting from the result the product by the difference.
Prop. 12.—To prove that the quotient of one number by
another is equal to the sum of the quotients of parts of
the dividend divided by the divisor. Since the quotient expresses the number of times which the divisor is contained in the dividend ; or, in the case of a concrete dividend, is the part of it denoted by the divisor, therefore the dividend is equal to the product of divisor and quotient. Hence
Divisor X quotient = dividend = sum of parts of dividend but each part of dividend = divisor X its quotient .. Divisor X quotient = sum of products of divisor into each partial quotient
(Prop. 8) = divisor X sum of partial quotients
.. Quotient = sum of partial quotients. The same Prop. exhibited algebraically:-Let D be the dividend, d the divisor; and let D=a+b+c+ &c.; let Q be the whole quotient; 9, r, s, &c. the quotients of a, b, c, &c. divided by d; Then
=dxq+ d xgtdxs + &c.
=dx (a tots + &c.)
.. Q=a +r+s + &c. Prop. 13.— To shew that the quotient of the product of two
or more numbers may be obtained by dividing one of them,
and multiplying the quotient by the rest. Divisor X quotient = dividend = product of the factors of dividend. Putting therefore in place of one factor the product of the divisor into its quotient, which is equal to the factor, we have Divisor X quotient=(divisor X partial quotient) x the other factors
(Prop, 10) = divisor X (partial quotient X the other factors)
.. Quotient = partial quotient X the other factors. The same Prop. exhibited algebraically:-Let D be the dividend, d the divisor, Q the quotient; and let D=a XbXCX &c.; let the quotient of a by d be q: then
d XQ=D= axbXCX &c.
= (d X 9) X 6 XCX &c.
- dx (X6 XCX &c.)
.: Q =9X6 XCX &c. Cor. Hence if we divide an exact number of tens, hundreds, &c. by any number, the quotient obtained by throwing away the ciphers and dividing will be number of tens, hundreds, &c. in the whole quotient: in other words a number of tens, hundreds, &c. when divided, give a number of tens, hundreds, &c. as the quotient. Prop. 14.- To explain the Rule for Division of Numbers.
The quotient of one number divided by another is equal to the sum of the quotients of any parts into which the dividend is separated (Prop. 12.) Hence we are at liberty to separate the dividend into any parts we please,