DEMONSTRATION OF THE RULE. According to the rule, we resolve the dividend into parts, and find, by trial, the number of times the divisor is contained in each of those parts: consequently, to illustrate, let 8686, divided by 43, be separated and arranged as follows: OPERATION. Divisor. Dividend. Quotient. 40+3 8000 8000+600+80+6200+2=202,true quotient. 8000+600 80+6 EXPLANATION.-I find first, that, 40 is contained in 8000, 200; then I multiply the whole divisor, (40+3) by 200, which makes 8000+600. These I put under the first two terms of the dividend, and subtract, and nothing remains. I then bring down the other two terms and proceed in the same manner. Here it will be seen that the 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: consequently, all the quotient figures, taken in order, is the true quotient of the whole dividend by the divisor. NOTE.-Division being the converse of Multiplication, the same difficulty attending the proving of Multiplication, "by casting out the nines," is also liable to Division. VII. PROPERTIES OF NUMBERS. 1. The product of an even, and an odd number, or of two even numbers, is even. 2. The product of any two odd numbers is an odd number. 3. If an odd number measure an even number, it will also measure the half of it. 4. The difference between an integral cube and its root, is always divisible by 6, 5. The product arising from two different prime numbers cannot be a square. 6. A prime number is that which can only be measured by unity. 7. The product of no two different numbers, prime to each other, can make a square, unless each of those numbers be a square. 8. Every prime number above 2, is either 1 greater or 1 less than some multiple of 4. 9. Every prime number above 3, is either one greater or 1 less than some multiple of 6. 10. The number of prime numbers is unlimited. The first ten are, 1, 2, 3, 5, 7, 11, 13, 17, 19, 23. 11. One number is prime to another, when unity is the only number by which both can be measured. 12. If equal quantities be added to, subtracted from, multiplied or divided by equal quantities, the wholes, remainders, products and quotients will be respectively equal. 13. Two quantities respectively equal to a third, are equal to each other. 14. The equal powers or roots of equal quantities are equal. 15. A perfect number is equal to the sum of all its aliquot parts. Thus, 6=3×2×1, here 3,2 or 1, or all, will divide 6. VIII. TABLES OF WEIGHTS AND MEASURES, TROY WEIGHT. Grains. Pennyweight, 1= 24 Pound, 1=12=240=5760 APOTHECARIes' weight, Grains, Scruple, 1 20 Dram, 1= 3= 60 Ounce, 1= 8= 24= 480 Pound,1-12-96-288-5760 NOTE.-175 oz. Troy is 192 oz. Avoirdupois. REFINERS' WEIGHT. Blanks, Perrot, 1 20 Mite, 1= 20— 480 Grain, 1=20=400=9600 NOTE. The Carat is a 24th part of gold or silver, Ounce, 1= 16 Pound, 1= 16= 258 7168 Quarter, 1= 28= 448= 60= 3600 Hour, 1= Day, 1= 24= 1440- 86400 Week,17 168 10080- 604800 Month, 1=4=28— 672— 40320— 2419200 Julian Y'r,1=52w. 1d. 6h.=365=8766=525960=31557600 Periodical Y'r,1=142s. 9m. 365=8766–525969–31558154 Tropical Y'r,1=57s. 48m. 365-8765-525948-31556937 NOTE. The twelve calendar months has each, the following number of days, viz. The fourth, eleventh, ninth and sixth Have thirty days to each affixed; Till leap-year gives it twenty-nine. 594 Rod, 1=5= 161= 198= Furlong, 1= 40= 220= 660= 7920= 23760 NOTE. 2721= 39204 Rood, 1 40 1210 10890 1568160 Acre, 1 4= 160 4840 43560— 6272640 Mile, 1=640=2560-102400-3097600-27878400-4014489600 Inches. Foot, 1 1728 Yard, 1-27-46656 SOLID MEASURE. NOTE 1.-16 cubic feet is 1 foot of wood, and 8 ft. of wood is 1 cord. NOTE 2.-All carpenters in the United States, allow 40 cubic feet of hewn timber to make a ton; consequently, 50 cubic feet of round timber is a ton; yet in every Arithmetic we find it thus: 50 cubic feet of hewn, &c. make a ton. This error, without doubt, was originally, a typographical one, but faithfully copied by every subsequent author; and I hardly know which to wonder at the most,-why the same error should pass uncorrected through so many able hands, or that the copiers should blunder so prodigiously over one another. WINE MEASURE. Cubic Inches. Pint, 1- 287 2= 57 Quart, 1= Gallon, 1= 4= 8- 231 Tierce, 142–168— 336— 9702 DRY MEASURE. Gallons. Cubic Inches. Peck, 1= 2= 5379 Bushel, 14= 8= 2150 Last, 1-80-320=640=172032 |