3. If the remainder fhould be fo fmall, that when the figure of the dividend joined with it make a fum less than the divifor, then a cypher is to be placed in the quotient, and another figure brought down; for every figure brought down, a cypher or figure must be placed in the quotient. This is called LONG DIVISION. EXAMPLE 1. What is the quotient of 14122 divided by 46 ? 138 To work this example, fay how often can I have 4 in one (no times) then, how often 4 in 14, which is 3 times, then place 3 in the quotient, and multiply 46 by it, fetting the product 138 under 141, and fubtract it therefrom, and there remains 3. Then bring down the next figure 2 from the dividend, and annex it to 3, which makes 32; then enquire how often 4 in 3, the answer is o, which I place in the quotient, and bring down the next figure 2, the dividual is then 322; then feek how often 4 in 32, the answer would be 8; but 46 multiplied by 8-would exceed 322; therefore I place 7 in the quotient, by which I multiply 46, and the product is 322; that, fubtracted from 322, leaves nothing, therefore 307 is the quotient. SCHOLIUM. There are various ways of proving divifion, and for the exercife of the learner I fhall prove it by three different ways; first, by multiplying the quotient by the divifor; fecondly, by cafting away the nines; and laftly, by addition, thus: Take the quotient of the last example, and multiply it by the divisor. 2 X Take this example, and caft away the nines in the divifor and quotient, which put on each fide of the crofs, and caft away the nines out of the dividend; put the remainder at the top of the crofs; then multiply the fide figures thereof into each other, and caft the nines out of the product, and if the work be right, the remainder to be written at the bottom of the crofs will be the fame as the top, as appears by this example. Ꭰ E. 3. To prove this example, add up all the lines marked thus*; and as there is nothing but a cypher to be added to 9 in the remainder, put down 9, and for the fame reason put down 8: then fay, 2 and 3 is 5, and 7 is 12, and 5 is 17; fet down 7 and carry 1. Then I and 1 is 2, and 4 is 6; fet down 6, and fay 2 and 3 is 5, and 9 is 14; fet down 4 and carry 1; 1 and 6 is 7, which fet down, and the fum is the fame as the dividend, which proves the work to be right. Note. If there be a remainder when you prove by the cross, it must be added to the product on the fides of the cross, and the nines thrown out as before. 205413 E. 7. 476085)988390547(2076 952170 3622054 3332595 2894597 2856510 38087 CASE 3. When the divifor has cyphers on the right hand; Strike them off, and likewife ftrike off as many places of the dividend on the right hand; and perform the divifion by the re maining maining figures. And when the divifion is finished, annex the figures cut off to the remainder. When the dividend has the fame number of cyphers on the right hand, as the divifor, ftrike them off from each, and the remainder will be fo many of what you divide by, without annexing the cyphers that were cut off. CASE 4. When the divifor is fuch a number, that any two figures (in. the multiplication table) multiplied together, will produce it. RULE. Divide the given number by thofe numbers or component parts, which is much eafier than dividing by all the divifor at once; fee the following examples worked at full length. NOTE. If there be a remainder in the last divifion, it will be fo many times the first divifor, which added to the first remainder (if any) will give the true remainder fought. E. 1. 845876912306 by 32. E. 2. To prove by multiplication all examples of this kind, you must add, or take in feperately the two remainders, when you multiply by their refpective divifors that produced them. D 2 E. 3. Those who are well acquainted with the nature of divifion, may subtract each figure of the product as it is produced, and only write down the remainders; this will shorten the work, and is commonly called Italian divifion; to perform which the following examples will fufficiently explain : TE 1253 Remainder VI. REDUCTION. EACHETH to reduce all great names into small, by multiplying continually the given number with fo many of the next lower name, as makes one of the higher, keeping them equivalent in value; this is called Reduction Defcending. On the contrary, where the quantity is to be reduced to a higher denomination, divide continually the given number by fo many of the leffer name as makes one of the greater; this is termed Reduction Afcending. TABLES |