The following Rule is established for the practice of the Courts in the State of New York: RULE:-"The Rule for casting Interest, where partial payments have been made, is to apply the payment, in the first place, to the discharge of the interest then due. If the payment exceeds the interest, the surplus goes towards discharging the principal, and the subsequent interest is to be computed on the balance of principal remaining due. If the payment be lesa than the Int., the surplus of interest must not be taken to augment the principal, but interest continues on the former principal until the period when the payments, taken together, exceed the interest due, and then the surplus is to be applied towards discharging the principal; and interest is to be computed on the balance of principal as aforesaid."—Johnson's Chancery Reports, Vol. I, page 17. 1. Suppose you have a bond against B. for 1000 dollars, dated May 15th, 1821, upon which you find the following Indorsements, viz. Ans. $869: 19'9 y. mo. da. 4 5 What remained due upon the bond May 20, 1829? Interest to be cast at 7 per cent? $1000 Principal. 94:30'5 Int. to the 1st ind't. being 1 1094: 30'5 Amt. due, at the time the 1st pay't. was made. 943: 70'5 Principal remaining due, Sept. 20, 1822. 1082: 24'6 Amt. at the time of the 2d payment. 881: 34'6 Principal remaining due, Oct. 25, 1824. 179:08'3 $105:56'5 Int. ex'ds. the p't. July 11, 1826. 73:51'8 Int. to Sept. 20, 1827, upon same 179:08'3 sum of the Int's. 1060: 42'9 Amount. [principal. 75:20 112:11 [20, 1827. 187:31 Sum of the 3d and 4th pay'ts. deducted Sept. (Carried up.) 873: 119 Principal remaining due, Sept. 20, 1827. 845: 36'5 Principal remaining due, Dec. 25, 1828. $869: 199 Balance due May 20, 1829. A note was given, April 20th, 1825, of $800: 50 cts.; May 25th, 1826, 250 dollars were indorsed; and Dec. 20th, 1828, 300 dollars were indorsed; what was due on the note June 20th, 1829; Interest at 7 per cent? Ans. $436: 82'4 COMPOUND INTEREST. Compound Interest is that which accrues on the amount of the principal and Interest. That is, the interest for the given time is added to the principal, and the amount constitutes a principal for another given time, and so on. The time may be three, six, or twelve months, as the parties may agree. RULE:-1. Find the amount of the given principal, at the given rate and time, as in simple interest, which will form a new principal for another period of time. 2. Subtract the first principal from the last amount, and the remainder will be the interest. Thus : 1. What is the compound interest of $150 for 5 years, at 4 per cent a year? 150x,04-6:00 interest for 1 year, and 150+6:00-$156, amount of principal for 2d year. 156,04-6:24 interest 2d year, and 156+6:24=$162:24 amount of principal for 3d year. 162: 24×,04-6: 48'9 interest 3d year, and 162:24+6: 48′9= $168: 729 amount or principal for 4th year. 168: 729×04-6:74'9 interest 4th year, and 168:729-+6:749 =175: 47'8 amount and principal 5th year. 175:47 8×,04=7:019 int. 5th year, and 175: 47'8+7 : 01′9= $182: 49'7 amount 5th year. Then $182:497-150=32: 49'7 com. int. 5 years. Ans. 2. What is the compound interest of $210: 50 for 3 years, a 6 per cent a year? Ans. $40: 20'8 A concise and easy Method of casting Compound Interest, at 6 per cent, on any sum in Federal Money. RULE:-Multiply the given sum, if -6 years, 141,8519 189,8298 NOTE. Three of the first or highest decimals, in the above numbers, will be sufficiently accurate for most operations; the product, remembering to move the separatrix two figures from its natural place towards the left hand, will then show the amount of principal and compound interest for the given number of years. Subtract the principal from the amount and it will show the compound interest. INVOLUTION. When a number is multiplied into itself, it is said to be in volved, and the process is called Involution. The product which is obtained by multiplying a number into itself is called a Power. The power is often indicated by a figure placed at the right of the number, thus 24 which is called the index or exponent of that power. The number involved is called the Root, or first power. When the root is used as a factor twice, it is called the second power; when three times, the third power, &c. The different powers have other names beside their numbers, viz. The square, 2d power; cube, 3d power; biquadrate, 4th power; sursolid, 5th power; squarecubed, 6th power, &c. Involution is performed by the following RULE:-Multiply the given root, or number by itself, and that product by the same number, and so on to the required power. Thus: 1. What is the 6th power of 2? Ans. 64 2×2=4, the 2d power; 4×2=8, the 3d power; 8×2=16, the 4th power; 16x2=32, the 5th power; and 32x2=64, the 6th power. What is the 4th power of 4? Ans. 256 What is the cube of 6? Ans. 216 What is the square of 14? Ans. 196. What is the 2d power of 64? Ans. 4096 What is the biquadrate of 5? Ans. 625 To involve a number, multiply it into itself, as often as there are units in the exponent, save once. Note. The exponent shows, not how many times we are to multiply, but how many times the root is used as a factor Involve 93. Ans. 729 Involve 65. Ans. 7776 Involve 10. Ans. 10000 Involve 211. Ans. 9393931 OBS. A vulgar fraction is involved by multiplying the numerator by itself, and the denominator by itself. A mixed number must first be reduced to an improper fraction, or a decimal before involving it. A decimal, is involved the same as a whole number-point off the same as in multiplication of decimals. What is the 4th power of? Ans. of ? Ans. What is the cube of 4? to the 4th power. Ans. 1 What is the square of 5? Ans. 30 4? Ans. 18 What is the square Ans. 4 What is the square of What is the square of 30? Ans. 915 What is the square of ,5? Ans.,25 1,2? Ans. 1,44 What is the square of 37,5? Ans. 1406,25 What is the 6th power of 5,03? Ans. 16196,005304479729 Norr. Involving a vulgar or decimal fraction diminishes it in the same proportion, as a whole number becomes increased. EVOLUTION. This is the extracting or finding the roots of any given powers; or it is exactly the reverse of Involution. There, a root was given to find a power. Here, a power is given to find a root. The root of any number or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root, or 2d root of 4, because 22 =2×2=4; and 3 is the cube root, or 3d root of 27, because 33= 3x3x3=27. The power of any given number or root may be found exactly by multiplying the number continually into itself. But there are numbers, of which a proposed root can never be exactly found. Yet, by means of decimals, we may approximate or ap proach towards the root, to any degree of exactness. Those numbers whose roots only approximate towards the true roots, are called surd numbers; but those whose roots can be exactly found, are called rational numbers. The roots are sometimes denoted by writing the character V, before the power, with the index of the root against it. Thus, the square root of 25 is expressed 25, and the cube root of 64 is expressed 164; and the 5th root of 16807, v516807. The index to the square root is always omitted; the character only, being placed before it; thus, v16, the index 2, being omitted. When the power is expressed by several numbers, with the sign +, or, between, a line is drawn from the top of the sign over all the parts of it; thus the square root of 41-5, is √41—5 or thus, V (41-5,) enclosing the numbers in a parenthesis. But all roots are now frequently distinguished by fractional indices; thus, the square root of 8, is 8 the cube root of 64, is 64 and the square root of 41-5, is 41-5 or (41-5)2. It is very necessary for practical purposes, to be able to find the amount of surface there is in any given quantity. The rule for finding the amount of surface, is to multiply the length by the breadth, and this will give the amount of square inches, feet, or yards. It is important for the pupil to learn the distinction between a square quantity, and a certain extent that is in the form of a square. For example, four square inches, and four inches square are dilferent quantities. A four inch square, then, is a square whose sides are four inches Jong, but it makes a square which is on each side, only two inches. Four square inches are four squares that are each an inch on every side. But it contains sixteen square inches. When we wish to find the square contents of any quantity, we seek to know how many square inches, or feet, or yards, there are in the quantity given, and this is always found by multiplying the length by the breadth. When the length and breadth of any quantity are given, we find its square contents, or the amount of surface it will cover, by multiplying the length by the breadth. If any quantity is placed in a square form the length of one side is the square root of the square contents of this figure. The length of the side of a square is the square root, made by the given quantity. If we have one side of a square given, by the process of Involution, we find what are the square contents of the quantity given. If, on the contrary, we have the square contents given, by the process of Evolution, we find what is the length of one side of the square, which can be made by the quantity given. Thus if we have a square whose side is four inches, by Involution we find the surface, or square contents to be 16 square inches. But if we have 16 square inches given, by Evolution we find what is the length of one side of the square made by these 16 inches. EXTRACTION OF THE SQUARE ROOT. Extracting the square root is finding a number, which, multiplied into itself, will produce the given number; or, it is finding the length of one side of a certain quantity, when that quan tity is placed in an exact square. It will be found by trial, that the root always contains just half as many, or one figure more than half as many ngures as are in the given quantity. To ascertain, therefore, the number of figures in the required root, we point off the given number into periods of two figures each, beginning at the right, and |