Note. The ufe of the preceding tables is fo obvious and natural, even by a bare infpection, that I prefume it is needlefs to fay any thing about that; the following examples being fufficient to teftify the great ufe and excellency of fuch tables, and will at the fame time give the learner à clear knowledge of the ufe of them. EXAMPLE I. What is the decimal part of a pound for 15s. 9d? In table I. you find against { 9 15 Shillings 75 ,0375 97875 E. 4. ,583333 Anfwer What decimal part of an hundred weight is 12 pounds 4 oz ? In table III. you find against { f 12 Pounds 4 Ounces Answer 107143 ,002232 ,109375 E. 5. What decimal part of a mile is 300 yards 2 feet? In table VII. you find against { 300 Yards ,170454 ,0003787 ,1708327 By the preceding tables all the fpecies of money, weight, measure, &c. contained therein, by the above method are immediately turned into decimals, and are then worked with the fame pleasure and facility as whole numbers. CASE 3. To find the value of any decimal fraction, in money, weight, measure, &c. RULE. Multiply the given decimal by the parts of the next infe rior denomination, and cut off towards the right-hand of the product fo many figures as there are places in the given decimal, and thofe on the left will be integers; then multiply the remaining decimals by the next inferior denomination, and cut off for decimals as before; thus proceed till you have brought it into the loweft parts of the integer. A few examples will make this plain to the young practitioner. EXAMPLE I. What is the value of ,725 of a pound fterling? Note. As often as cyphers fall on the right-hand of your work, always drop them, for they are, of no value. E. z. What is the value of,72083 of a crown? Note. If the multiplicand be a compound repetend, and the multiplier only a fingle digit, to the product of the first figure on the right-hand, add as many units as there are tens in the product of the left-hand place of the repetend. Thus in the above example,,72083 being a repetend as above described, I multiply by 5, the fhillings in a crown, faying 5 times 3 is 15, there being only one ten in that product, I fet down 6, which is one more, and then proceed as in common multiplication with the remainder of the multiplicand. Again ;60416 I multiply by 12, the pence in a fhilling, faying 12 times 6 is 72, there being seven tens in that product, I add 7 to the 2 remaining, which makes 9, which I fet down, and proceed as before with the remainder of the multiplicand, continuing thus till the work is finished. Mo. . d. b. m. See Answer 6 3 5 6 39 55+ Thefe examples I think fufficient to fhew the method of reducing deci mals into the known parts of any fpecies of quantity. LI. EXTRACTION OF THE SQUARE ROOT. XTRACTION of the fquare root, is finding fuch a number that being multiplied by itfelf fhall give the refpective power, out of which the root is to be extracted; as if 36 be proposed to be extracted, you will find its root to be 6, for 6×6=36, the given number. To extract the fquare root of any number, obferve the following, RULE. 1. Begin at the units place, and point the given number into periods of two figures each. 2. Find the greatest fquare that is contained in the first period, towards the left-hand; fet the root in the quotient, and fubtract the fquare from the figures of that period. 3. To the remainder bring down the two figures under the next point for a dividend. 4. Double the quotient or root, and place it for a divifor; seek how often the divifor is contained in the dividend (referving the units place) and put the answer in the quotient, and alfo on the right-hand of the divifor; then multiply the divifor by the last figure put in the quotient (as in common divifion) the product fubtract from the dividend, and to the remainder bring down the next period, and proceed thus till all the figures or periods are brought down. Note: If at laft there be no remainder, the quotient will be the true root; but if any thing remain, annex two cyphers, and work as has been taught above, and for every two cyphers thus annexed, there will be one decimal place in the root. Inftead of doubling the quotient every time for a divifor, you may always add the laft quotient figure to the laft divifor, for a new divifor, and proceed as before. EXAMPLE I. Let it be required to extract the fquare root of 393129? 62. 393129(627 Root 36 122)331 1247)8729 EXPLANATION. The number being feparated or pointed into periods of two figures each, then the neareft fquare to 39 the first period, is 36, whofe root 6, I place in the quotient, and fubtract the fquare 36 from 39, the remainder is 3. Then I bring down 31, the next point, and annex it to 3, and the new dividend is 331, then I double the quotient 6 for a divifor, which is 12, and feek how oft 12 in 33? the answer is 2, which I place in the quotient, and alfo after 12; then the divifor becomes 122, which multiplied by 2, the product is 244, which subtracted from 331, the remainder is 87. Lastly, I bring down 29, the next point, and the dividend is 8729; then I double the quotient 62, which is 124, for a new divifor, and feek how oft 124 in 872? the answer is 7 times. Then I multiply 1247 by 7, and fubtract the product 8729 from the last dividend, and there remains nothing; therefore 393129 is found to be a fquare number, and 627 its root. Το prove the work; if you fquare the root, and to that product add the remainder (if any) that fum fhall be equal to the number first given, thus: 627 × 627 393129, the given divifor in the laft example. 2 E 2 E. 2. 438 |