3d payment April 1, 1801, 12,00 4th payment May 1, 1801, 30,00 How much remains due on said note the 16th of Sep tu nber, 1801 ? $ cts. Principal dated. February 1, 1800, 500,00 Interest to May 1, 1800, (3 mo.) 7,50 Amount 507 50 Paid May 1, 1800, a sum exceeding the interest 40,00 New principal, May 1, 1800, 467,50 Interest to May 1, 1801, (1 year,) 28,05 Amount 495,55 Paid Nov. 4, 1800, a sum less than the interest then due, 8,00 L’aid April 1, 1801, do. do. 12,00 l'aid May 1, 1801, a sum greater, 30,00 50,00 New principal May 1, 1801, 445,55 Interest to Sept. 16, 1801, (4; mo.) 10,92 Balance due on the note, Sept. 16, 1801, $455,57 The payments being applied according to this Rule, keep down the interest, and no part of the interest ever forms a part of the principal carrying interest. COMPOUND INTEREST BY DECIMALS. Rule.-Multiply the given principal continually by the amount of one pound, or one dollar, for one year, at the rate per cent. given, until the number of multiplications are equal to the given number of years, and the product will be the amount required. Or, In Table I, Appendix, find the amount of one dollar, or ope pound, for the given number of years, which multiply by the given principal, and it will give the amount as before. EXAMPLES. 1. What will 4001. amount to in 4 years, at 6 per cent per annum, compound interest ? 400 X 1,06 * 1,06 X 1,06 x 1,06=£504,99+ [£504 19s. 9d. 2,75 grs. + Ans. Whole amount=£504,98800 2. Required the amount of 425 dols. 75 cts. for 3 years, at 6 per cent. compound interest ? Ans. $507,74 cts. + 3. What is the compound interest of 555 dols. for 14 per cent. ? By Table I. Ans. 543,86 cts. + 4. What will 50 dollars amount to in 20 years, at 6 pet cent. compound interest? Ans. $160, 35 cts. 64m. years at 5 INVOLUTION, IS the multiplying any number with itself, and that pro. duct by the former multiplier; and so on; and the several products which arise are called powers. The number denoting the height of the power, is calles the index or exponent of that power. EXAMPLES 64 =2d power, or square. 512 =3d power, or cube. 8 4096 =4th power, or biquadrate. 8 32768 -5th power, or sursolid. Ans What is the squarn, of 17,1 ? Ans. 292,41 Ans. 20736 1 EVOLUTION, OR EXTRACTION OF ROOTS. WHEN the root of any power is required, the business of finding it is called the Extraction of the Root. The root is that number, which by a continued multipli cation into itself, produces the given power. Although there is no number but what will produce a perfect power by involution, yet there are many numbers of which precise roots can never be determined. But, by the help of decimals, we can approximate towards the root ta uy assigned degree of exactness. The roots which approximate are called surd roots, and hose which are perfectly accurate are called rational roots. A Table of the Squares and Cubes of the nine digits. Roots. 1 | 2 | 3 | 4 | 51 61 71 81 91 Siqares. 11/41 9 | 16 | 25 | 36 | 49 | 64 81 Cubes. 1118 | 27 | 64 | 125 | 216 | 343 | 512 729) EXTRACTION OF THE SQUARE ROOT. Any number multiplied into itself produces a square. To extract the square root, is only to find a number, which being multiplied into itself shall produce the given tumber. Rule.-). Distinguish the given number into periods of lwo figures each, by putting a point over the place of units, another over the place of hundreds, and so on; and if there are decimals, point them in the same manner, from units towards the right hand; which points show the number of figures the root will consist of. 2. Find the greatest square number in the first, or left nand period, place the root of it at the right hand of the given number, (after the manner of a quotient in divisiun, for the first figure of the root, and the Sejuure number on der the period, and subtract it therefrom, and to the re mainder bring down the next period, for a dividend. 3. Place the double of the 100t, already found, on the left hand of the dividend, for a divisor. 4. Place such a figure at the right hand of che divisor, and also the same figure in the root, as when niultiplied into the whole (increased divisor) the product srail le equal to, or the next less than the dividend, and it will be the second figure in the root. 5. Subtract the product from the dividendi, and to the remainder join the next period for a new dividend. 6. Double the figures already found n the root, for a new divisor, and from these find the next figure in the root as last directed, and continue the operation in the same manner till you have brought down ali ine periods. Or, to facilitate the foregoing Rule, when you have brought down a period, and formed a dividend in order 10 find a new figure in the root, you may divide said dividend (omitting the right hand figure thereof) by double the root already found, and the quotient will commonly be the figures sought, or being made less one or two, will generally give the next figure in the quotient. EXAMPLES 1. Required the square root of 141225,64. 141225,64(375,8 the root exactly without a remainder; 9 but when the periods belonging to any given number are exhausted, and still 67)512 leave a remainder, the operation may 469 be continued at pleasure, by annexing periods of ciphers, &c. 745)4325 3725 7508)60064 60064 0. remaine. 2 What is the square root of 1296 ? 3 Of 56044? 4. Of 54990:25 ? 5. Of 36372961 ? 6 Of 184,2? 7. Of 9712,693209? 8. Of 0,45369? 9. Of ,002916 ? 10. Of 45? ANSWERS. 36 23,8 2345 6031 13,57 + 98,553 ,6734 ,055 6,708+ TO EXTRACT THE SQUARE ROOT OF VUL GAR FRACTIONS. RULE. Reduce the fraction to its lowest terms for this and all otser roots; then 1. Extract the root of the numerator for a new numera. sov, and the root of the denoininator, for a new denominator. 2. If the fraction be a surd, reduce it to a decimal, and Artruct its root. EXAMPLES. 1. What is the square root of money ? ANSWERS. 3 2. What is the square root of putt? 3. What is the square root of 14%? 4. What is the square root of 20; ? 45 5. What is the square root of 248 to ? 157 SURDS. 6. What is the square root of 11? 9128+ 7. What is the square root of 45? ,7745+ 8. Required the square root of 361 ? 6,0207 + APPLICATION AND USE OF THE SQUARE ROOT. PROBLEM I.-A certain general has an army of 5184 apen; how many must he place in rank and file, to form them into a muare ? |