8 XXXI. 1. At $40.00 a hogshead, how many hogsheads of molasses can I buy for $320.00 ? 40 is a composite number, made up of the factors 4 and 10. First divide by 10 according to the rule ; thus, 1,0)32,0 8 hogsheads. 2. Paid 80 laborers $570.00 distributing it equally. How many dollars had each ? 80=8X10 Divide by 10, 1,0)57,0 7+1 Rem. This remainder, of course, is one 10, or 10 dollars by the rule in ♡ XXIX. 3. The paymaster of a garrison, distributed 5,845 dollars equally among 700 men. How many dollars did each receive ? 700=7X100 Divide by 100, 1,00)58,45 RIGHT OF THE DIVIDEND. 8+2 Rem. The 2, remainder, is of course, 2 hundreds=200, to be added to the 45, first remainder. The true remainder, is therefore 245, and is found by writing the last remainder 2, before the first remainder 45; since this brings the 2 to the hundreds' place where it belongs. 4. In an army of 63,474 men, how many regiments of 4,800 ? 4,800=6X8X100. A. 137574. Hence, when there are cyphers at the right of the divisor, II. DIVIDE THE REMAINING FIGURES OF THE DIVIDEND, BY THE RĖ. MAINING FIGURES OF THE DIVISOR. III. PREFIX THE REMAINDER FOUND BY THIS DIVISION, TO THE FIGURES REMOVED FROM THE DIVIDEND, FOR THE TRUE REMAINDER. EXAMPLES FOR PRACTICE. 5. Divide 7,861 by 30. A. 262 . 6. Divide 21,564 by 20. A. 1,07847 7. Divide 31,943 by 300. A. 106442. 8. Divide 1,151 by 20; 2,873 by 30; 9,999 by 90; 2,864 by 80 ; 71,843 by 200; 59,995 by 500; 77,724 by 9,000 ; 325,963 by 70,000 ; 2,541,861 by 80,000. $ XXXII. But there are many cases of Division, for which the preceding rules are insufficient. For instance, 23 1. If 23 yards of cloth cost $322, what cost 1 yard? Our divisor, 23, consists of two figures. Of 23)322/14 course, it cannot be contained in the first figure of the dividend. We must therefore, take the first two figures. Thus, we must 92 find how often 23 is contained in 32. The 92 remainder must be prefixed to the next lower order, as in the preceding examples, Then the only difference between dividing by a single figure, and by several, consists in this ; that in order to obtain the first quotient figure, we must take as many figures of the dividend, as there are places in the divisor. Or, if the divisor be larger than the same number of figures in the dividend, we must take one more figure in the dividend. 2. If a man's income be 1,248 dollars a year what is that per week, allowing 52 weeks to the year? A. $24. 3. A privateer took a prize of $7,735. It was equally divided among 65 men. , What amount had each ? A. $119. 4. Å man bought 529 head of cattle for $15,341. What did he give a head ? Ans. $29. 5. If a man's income be $49,640 a year, what is that a day al. lowing 365 days to the year ? Ans. $136. 6. For $36.56 how many books can I buy at $4.57 each? Ans. 8. It will be observed, that, we make a separate division for each quotient figure. _The numbers, thus successively divided, are some. times called the PARTIAL DIVIDENDS. 7. Divide 9,391 by 32. Ans. 293 . II. FOR THE FIRST QUOTIENT FIGURE, DIVIDE AS MANY PLACES ON THE LEFT OF THE DIVIDEND AS THERE ARE PLACES IN THE DIVISOR ; OR IF THESE BE NOT SUFFICIENT, TAKE ONE MORE. III. MULTIPLY THE DIVISOR BY THIS QUOTIENT FIGURE, PLACE THE PRODUCT UNDER THE PARTIAL DIVIDEND, FROM WHICH SUBTRACT IT, AND, TO THE REMAINDER, ANNEX THE NEXT FIGURE OF THE DIVIDEND. THIS WILL BE THE SECOND PARTIAL DIVIDEND WHICH DIVIDE AS BEFORE. IV. PROCEED IN THIS MANNER, TILL ALL THE FIGURES OF THE DIVIDEND ARE EMPLOYED, AND IF ANY PARTIAL DIVIDEND BE TOO SMALL TO CONTAIN THE DIVISOR, WRITE A CYPHER IN THE QUOTIENT, AND TREAT IT AS A REMAINDER. EXAMPLES FOR PRACTICE. 9. Divide 251,104 by 472. Ans. 532. 10. Divide 863,256 by 736. Ans. 1,17296 11. Divide 1,893,312 by 912. Ans. 2,076: 12. Divide 47,254,149 by 4,674. Ans. 10,110T 121 912314 13. Divide 761,858,465 by 8,465. Ans. 90,001. 14. Divide 119,181,693 by 38,473. A. 3,097 33143. 15. Divide 230,209,122,081 by 912,314. Ans. 307,140 16. Divide 7,328,946,264,418,232 by 814,313,515,623,303. Ans. 9 _13_4_63_38_0 8 5 0 5 8 14 3 1 35 156 23 30 3 As the quotient shows how many times the divisor is contained in the dividend, and the remainder what is left ; it is plain, that io prove Division, we must MULTIPLY THE DIVISOR AND QUOTIENT TOGETHER, AND ADD THE REMAINDER TO THE PRODUCT. THE SUM OUGHT TO BE EQUAL TO THE DIVIDEND. 17. Divide 7,154 by 14 | 24. Divide 29,993 by 25 8,561 1 230,031 1 39 2,881,943 " 46 20. 9,001 » 4,960,002 " 201 13,801,804 » 315 22. 17,020 41 29. 175,990,806 ” 7,864 35,700 » 26 | 30. 27,341,020,003 "91,992 23. pt. qt. gal. ♡ XXXIII. We have given (ŞXV111.) some of the tables of weights, measures, &c. The following are some that remain to be given. The first table contains the denominations of WINE MEASURE. 1 quart 1 gallon ank. run. 31; gallons 1 barrel b!, 42 gallons 1 tierce tier. 63 gallons I hogshead hhd. 81 gallons 1 puncheon pun. 2 hogsheads 1 pipe or butt " 2 pipes 1 tun. Note. This measure is used for wine, brandy, spirits, mead, vinegar, hones; perry, cider oils, &c. EAXMPLES FOR PRACTICE. 1. How many runlets in 925 gals. ? A. 51 run. 7 gals. How many tierces in 824 gals.? A. 19 tier. 26 gals. 66 p. or b. 3. How many puncheons in 976 gallons ? In 1,823 ? A. 11 pun. 52 gals. & 21 pun. 59 gals. 4. How many hhds. gals. qts. pts. and gi. in 11,934 gi. ? A: 5 hhds. 57 gal. 3 qts. 1 pt. 2. gi. Note. Begin by dividing by 4, because 4 gi. make 1 pt. Then divide by 2, because 2 pts. make 1 qt., and so on. 5. How many tior. gal. qts. pts. and gi. in 38,254 gi. ? A. 28 tier. 19 gal. 1 qt. 1 pt. 2 gi. 6. How many run. gal. qts. pts. and gi. in 38,254 gi. ? A. 66 run. 7 gals. 1 qt. 1 pt. 2 gi. 7. How many gal. qts. and pts. in 218,363 pts. ? A. 27,295 gal. 1 qt. 1 pt. 8. Change 3,834,579 gi. to hhds. gal. qts. &c. 9. How many bls. in 3,826 gal ? A. 1213 k. The pupil will probably be at a loss how to divide by 31d. But he can find llow many half gallons there are in 31t, and also how many half-gallons there are in 3,826. Then he can divide the half-gallons in 3,826' by the half-gallons ja 315. 10. How many bl. in 15,835 gal. ? CLOTH MEASURE. qr. 4 quarters 1 yard yd. 3 quarters 1 Ell Flemish E. FI. 4; quarters Si Ell French E. Fr. 5 quarters 1 Ell English E. E. NOTE. This measure is used for cloths, and all goods sold by the yard or ell. 13. How many yds. in 275 nls. ? A. 17 yds. O qr. 3 nls. 14. How many E. Fl. qrs. &c. in 2,753 in. ? A. 101 E. Fl. 2 qrs. 3 nls. 14 in. NOTE. The pupil must reduce the 21, and the 2,753 to quarters of an inch, before dividing, us, in example 9, he reduced his numbers to halves. 15. How many E. E. qrs. &c. in 7,286 in. ? A. 161 E. E. 4 qrs. 2 nls. i in. 16. How many yds. qrs. &c. in 27,854 in. ? 17. How many aunes in 754 qrs. ? 18. How many aunes, qrs. &c. in 5,876 in. ? 19. How many aunės, qrs. &c. in 47,854 in.? 20. How many yds. qrs. &c. in 123,456,789 in. ? CIRCULAR MEASURE, OR MOTION. 1 degree, 1 sign s. 12 signs or 360°, The whole circle of the Zodiac: or aune 66 NOTE. By Circular Motiox, is meant the motion of the earth and planeta round the sun. Circular MEASURE is used for reckoning latitudes and longi. tudes on the earth. Every circle, whether great or small, is divided into 360 de. grees; those degrees into 60 minutes each ; these minutes into 60 seconds. Seconds are sometimes subdivided into thirds, thirds into fourths, &c. The sun secms to pass entirely round the earth in a circle, once in 24 hours. Of course, his apparent motion is 360 degrees, from noon to noon. One minute on a great circle of the earth, is a GEOGRAPHICAL MILE. 21. How many degrees in 3,600 minutes? A. 60°. 22. How many degrees in 3,600 seconds ? A. 1o. 23. How many S. degrees, &c. in 217,554" ? A. 2 S 0° 25' 54". 24. How many degrees does the sun pass over in an hour ? A. 15° 25. What part of a degree does the sun pass over in a minute ? How many minutes motion has the sun in one minute of time? Last Ans. 15'. 26. Boston is about 75° west of LONDON. As the sun seems to come from the east to the west, it will be past noon at LONDON, when it is noon at Boston. How many hours difference of time between the two places ? and what time will it be at London when it is noon at BOSTON? A. 5 hours dif.—5 o'clock, P. M. at LONDON 27. A gentleman in England and his friend in America agreed to look at a bright star every evening, at the same time. 9 o'clock was the hour fixed on, but as they were 75° apart, one was probably asleep, while the other was looking at the star. What was the dif. of time, and which was probably asleep? Note. From these examples, we see that time is later in all places east of us, and earlier in all west; and that the difference is one hour for every 15° of longitude, one minute for every 15' of lon. gitude, and one second for every 15" of longitude. 28. How many hours difference of time between us and the part of the earth exactly opposite to us, there being 180 degrees dif. of lon. ? A. 12. NOTE. Of course, when it is noon to us, it is midnight at that place ; and when it is midnight here, it is noon there. 29. Suppose a meteor so high in the heavens, as to be visible, at the same moment, at Boston, longitude 71° 3' W.; at WashingTon, longitude 77° 43' W.; and at the Sandwich Islands, longitude 1550 W.: and suppose the time of appearance at WASHINGTON to be 5 minutes past 10, P.M. What time is it by the clocks at the other places ? 65 LONG MEASURE, 3 barley corns, b. c. make 1 inch, inarked in, 12 inches 1 foot, ft. 3 feet 1. yard, yd. 5 yards, or 16 feet « i rod, perch, or pole, rd.orp. 40 rods 1 furlong fur. 8 furlongs 1 statute mile, m). 3 miles I league, 1. |