2. Find how many times the divisor may be had in as many figures of the dividend, as are just necessary, and write the number in the quotient. 3. Multiply the divisor by the quotient figure, and set the product under that part of the dividend used. 4. Subtract value of which it is taken in the operation; according as there are 1, 2, or 3, &c. figures standing before it; and consequently the true value of the quotient figure, belonging to that part of the dividend, is also 10, 100, or 1000, &c. times its simple value. But the true value of the quotient figure, belonging to that part of the dividend, found by the rule, is also 10, 100, or 1000, &c. times its simple value for there are as many figures set before it, as the number of remaining figures in the dividend. Therefore this first quotient figure, taken in its complete value, from the place it stands in, is the true quotient of the divisor in the complete value of the first part of the dividend. For the same reason, all the rest of the figures of the quotient, taken according to their places, are each the true quotient of the divisor, in the complete value of the several parts of the dividend, belonging to each; because, as the first figure on the right hand of each succeeding part of the dividend has a less number of figures, by one standing before it, so ought their quotients to have; and so they are actually ordered : consequently, taking all the quotient figures in order as they are placed by the rule, they make one number, which is equal to the sum of the true quotients of all the several parts of the dividend ; and is, therefore, the true quotient of the whole dividend by the divisor. Q. E. D. To leave no obscurity in this demonstration, I shall illustrate it by an example. EXAMPLE. 4. Subtract the last found product from that part of the dividend, under which it stands, and to the right hand of the remainder bring down the next figure of EXPLANATION. It is evident, that the dividend is resolved into these parts, 85000+600+00+9: for the first part of the dividend is considered only as 85, but yet it is truly 85000; and therefore its quotient, instead of 2, is 2000, and the remainder 13000; and so of the rest, as may be seen in the operation. When there is no remainder to a division, the quotient is the absolute and perfect answer to the question; but where there is a remainder, it may be observed, that it goes so much toward another time, as it approaches to the divisor: thus, if the remainder be a fourth part of the divisor, it will go one fourth of a time more; if balf the divisor, it will go half of a time more; and so on, the dividend; which number divide as before; and so op, till the whole is finished. Method on. In order, therefore, to complete the quotient, put the last remainder at the end of it, above & small line, and the divisor be low it. be It is sometimes difficult to find how often the divisor may had in the numbers of the several steps of the operation; the best way will be to find how often the first figure of the divisor may be had in the first, or two first, figures of the dividend, and the answer made less by one or two is generally the figure wanted: beside, if after subtracting the product of the divisor and quotient from the dividend, the remainder be equal to, or exceed the divisor, the quotient figure must be increased accordingly. If, when you have brought down a figure to the remainder, it is still less than the divisor, a cypher must be put in the quotient, and another figure brought down, and then proceed as before. The reason of the method of proof is plain: for since the quotient is the number of times the dividend contains the divisor, the product of the quotient and divisor must evidently be equal to the dividend. There are several other methods made use of to prove division: the best and most useful are these following. RULE I. Subtract the remainder from the dividend, and divide this number by the quotient, and the quotient found by this divis ion will be equal to the former divisor, when the work is right. The reason of this rule is plain from what has been observed above. Mr. MALCOLM, in his Arithmetic, has been drawn into a mistake concerning this method of proof, by making use of particular numbers, instead of a general demonstration. He says, the dividend being divided by the integral quotient, the quotient of this division will be equal to the former divisor, with the same remainder.This is true in some particular cases; but it will not hold, when the Method of PROOF. Multiply the quotient by the divisor, and this product, added to the remainder, will be equal to the dividend, when the work is right. the remainder is greater than the quotient, as may be easily demonstrated; but one instance will be sufficient; thus 17, divided by 6, gives the integral quotient 2, and remainder 5; but 17, divided by 2, gives the integral quotient 8, and remainder 1. This shews how cautious we ought to be in deducing general rules from particular examples. RULE II. Add the remainder, and all the products of the several quotient figures, by the divisor, together, according to the order, in which they stand in the work, and the sum will be equal to the dividend, when the work is right. The 3. Divide 3756789275474 by 2., Ans. 18783946377374. Divide 12345678900 by 7. Ans. 1763668414. 5. Divide The reason of this rule is extremely obvious: for the numbers, that are to be added, are the products of the divisor by every figure of the quotient separately, and each possesses, by its place, its complete value; therefore, the sum of the parts, together with the remainder, must be equal to the whole. RULE III. Subtract the remainder from the dividend, and what remains will be equal to the product of the divisor and quotient; which may be proved by casting out the nines, as was done in multiplication. This rule has been already demonstrated in multiplication. To avoid obscurity, I shall give an example, proved according to all the different methods. 123456789 Proof by Addition. For illustration, we need only refer to the example; except for the proof by addition; where it may be remarked, that the asterisms shew the numbers to be added, and the dotted lines their order. |