RULE. Having placed units under units, tens under tens, &c. draw a line underneath, and begin with the units; after adding up every figure in that column, consider how many tens are contained in their sum, and, placing the excess under the units, carry so many as you have tens, to the next column, of tens: Proceed in the same manner through every column, or row, and set down the whole amount of the last row. PROOF. Begin at the top of the sum and reckon the figures downwards, in the same manner as they were added upwards, and, if it be right, this aggregate will be equal to the first. Or, cut off the upper line of figures, and find the amount of the rest; then, if the amount and upper line, when added, be equal to the sum total, the work is supposed to be right. ADDITION for ev * This Rule as well as the method of proof, is founded on the known axion, "the whole is equal to the fum of all its parts." The method of placing the numbers, and carrying for the tens, is evident from the nature of notation; for any other difpofition of the numbers would alter their value; and carrying one, ery ten, from an inferiour to a superiour column, is, evidently, right, because one unit in the latter cafe is equal to the value of ten units in the former. Befides the method of proof, here given, there is another, by casting out the nines; thus: 1. Add the figures in the upper row together, and find how tained in their fum. many nines are con 2. Reject the nines, and set down the remainder, directly even with the figures in the row. 3. Do the fame with each of the given numbers, and fet all the exceffes of nines in a column, and find their fum; then, if the excefs of nines in this fum, found, as before, is equal to the excess of nines in the fum total; the question is supposed to be right. EXAMPLE. 53241 28689 This method depends upon a property of the mumber 9, which, except 3, belongs to no other digit whatever; viz, that any number, divided by 9, will leave the fame remainder, as the fum of its figures, or digits, divided by 9: which may be thus demonstrated. Demonftration. Let there be any number, as 5432; this, feparated into its feveral parts, becomes 5000+400+30+2; but 5000=5×1000=5×999+1=5×999+5. In like manner 400-4X99+4, and 30=3X9+3. Therefore, 5432=5X999+5, +4×99+4, +3x9+3+2=5X99944X99+3×9+5+4+3+2. And 5132 9 = 5X999+4X99+3×9+5+4+3+2 9 ; but 5x999+4×9943x9 is divifible by 9; therefore, 5492, divided by 9, will leave the fame remainder, as 5+1+3+2, divided by 9; and the fame will hold good of any other number what ever. The fame property belongs to the number 3: However, this inconveniency attends this method, that, although the work will always prove right, when it is fo; it will not, always, be right, when it proves fo; I have, therefore, given this dem onftration more for the fake of the curious, than for any real advantage. When you would add two numbers, look one of them in the left hand column and the other atop, and in the common angle of meeting, or, at the right hand of the first, and under the second, you will find the sum-as, 5 and 8 is 13. When you would subtract: Find the number to be subtracted in in the left hand column, run your eye along to the right hand till you find the number from which it is taken, and right over it, atop you will find the difference-as 8, taken from 13, leaves 5. SUBTRACTION TEACHES to take a less number from a greater, to find a third, shewing the inequality, excess or difference between the given numbers; and it is both simple and compound. SIMPLE SUBTRACTION Teaches to find the difference between any two numbers, which are of a like kind. RULE. Place the larger number uppermost, and the less underneath, so that units may stand under units, tens under tens, &c. then, drawing a line underneath, begin with the units, and subtract the lower from the upper figure, and set down the remainder; but if the lower figure be greater than the upper, borrow ten, and subtract the lower figure therefrom: To this difference, add the upper figure, which, being set down, you must add one to the ten's place of the lower line, for that which you borrowed; and thus proceed through the whole.* PROOF. In either simple, or compound Subtraction, add the remainder and the less line together, whose sum, if the work be right, will be equal to the greater line: Or subtract the remainder from the greater line, and the difference will be equal to the less. * Dem. When all the figures of the lefs number are less than their correspondent figures in the greater, the difference of the figures, in the feveral like places, muft, all taken together, make the true difference fought; because, as the fum of the parts is equal to the whole; fo muft the fum of the differences, of all the fimilar parts, be equal to the difference of the whole. 2. When any figure in the greater number is lefs than its correfpondent figure in the lefs, the ten, which is added by the Rule, is the value of an unit in the next higher place, by the nature of notation; and the one which is added to the next place of the lefs number, is to diminish the corrrefpondent place of the greater, accordingly; which is only taking from one place, and adding as much to another, whereby the total is never changed: And, by this mean, the greater is refolved into such parts, as are, each, greater than, or equal to, the similar part of the Ans. 35177. Ans. 91751. Ans. 345358. Ans. 9744. 9. In .36 12s. 10d. 1qr. how many farthings? 10 In 95 11s. 5d 3qrs. how many farthings? 11 In 719 9s. 11d. how many half pence? 12 In 29 guineas, at 28s. how many pence? 13. In 37 pistoles, at 22s. how many shillings, pence, and farthings? Ans. 814s 9768d.39072qrs. 14. In 49 half johannes, at 48s. how many sixpences ? Ans. 4704. 15. In 473 French crowns, at 6s. 8d. how many threepences? Ans. 12613. 16. In 53 moidores, at 36s. how many shillings, pence and farthings? Ans. 1908s 22896d. 91584qrs. 17. In. 29 how many groats, threepences, pence, and farthings? Ans 1740 groats, 2320 threepences, 6960d. 27840qrs. 18. Reduce 47 guineas and one fourth of a guineà into shillings, sixpences, groats, threepens, twopences, pence and farthings. Ans. 1323 shillings 6 sixpences, 3969 groats, 5292 threepences, 7938 twopences, 15876 pence, and 63504 qrs. REDUCTION ASCENDING. RULE. Divide the lowest denomination given, by so many of that name, as make one of the next higher, and thus continue till you have brought it into that denomination which your question requires Note. From this rule and the note under Case II of Simple Division, it appears, that Federal Money is reduced from lower to higher denominations by cutting off as many places as the given denomination stands to the right of that required; the figures cut off belonging to their respective denominations. EXAMPLES. 1. How many eagles in 32000 mills? 2. In 9175 cents, how many dollars? 3. In 500 dollars how many Eagles ? 4. In 4414 mills, how many dimes? 5. In 9317 milis, how many dollars? 6. How many dollars in 28175 mills? Ans. 3 E. 2 D. Ans. 91 D. 75 c. Ans. 50. 7. In 547325 farthings, how many pence, shillings, and pounds? Farthings in a penny Pence in a shilling = 4)547325 12)136831 1 qr. Shillings in a pound = 20)11402 7d. £.570 2s. 7d. 1 qr. Ans. 136831d. 11402s and 1..570 Note. The remainder is always of the same name as the dividend, 8. Bring 35177 farthings into pounds. 9. Bring 9. Bring 91751 farthings into pence, &c. 10. Bring 345358 half pence into pence, shillings, and pounds. 11 Reduce 9744 pence to guineas, at 28s. per guinea. 12 In 39072 farthings, how many pistoles, at 22s. ? 13. In 4704 sixpences, how many half johannes? 14 In 12613 threepences, how many French crowns, at 6s. 8d. ? 15. In 91584 farthings, how many moidores, at 36s ? 16 In 27840 farthings, how many pence, threepences, groats, shillings and pounds? 17. In 63504 farthings, how many pence, twopences, threepences, groats, sixpences, shillings and guineas? Note. The preceding questions may serve as proofs to those in Reduction descending. REDUCTION DESCENDING AND ASCENDING. 1. MONEY. 1. In £.97 how many pence and English or French crowns, at 6s. Sd.? Ans 23280d. and 291 crowns. 2. In 947 English crowns, at 6s. 8d. how many shillings and English guineas? Ans. 6313s. 4d. and 225 guineas 13s. 4d. 3. In 519 English half crowns, how many pence and pounds? Ans. 20760d. and .86 10s. 4. In 1259 groats, how many farthings, pence, shillings, and guineas ? Ans. 20144qrs. 5036d 419s. 8d. and 14 guin. 27s. 8d. 5. In 75 pistoles, how many pounds? Ans. £.82 10s. 6. In 755 French crowns, how many shillings and French guineas, at 26s. 8d.? Ans. 4900s. and 183 guin. 24s. 7. In 5793 pence, how many farthings, pounds and pistoles? Ans. 23172qrs. £.24 2s. 9d. and 21 pistoles, 20s 9d. 8. In £.99, how many shillings, and half johannes, at 48s. ? Ans. 1980s. and 41 half joes. 12s. Ans. 127 guin 24s. Ans. 191 moid. 24s. 9. In .179, how many guineas? 10. In .345 how many moidores ? 11. In 59 half joes, 37 moidores, 45 guineas, 63 pistoles, 24 English crowns, and 19 doliars; how many pounds, half joes, moidores, guineas, pistoles, English crowns, dollars, shillings, pence and farthings? Ans. 354 4s. 147 half joes, 28s 196 moidores, 28s 253 guineas, 322 pistoles, 1062 English crowns, 4s. 1180 dollars, 4s 7084 shillings, 85008d. and 340032qrs. When it is required to know how many sorts of coin, of different values, and of equal number, are contained in any number of another kind; reduce the several sorts of coin into the lowest denomination mentioned, and add them together for a divisor; then reduce the money given, into the same denomination, for a dividend, and the quotient. arising from the division, will be the number required. Note, Observe the same direction in weights and measures. |