ARITHMETIC.

ARITHMETIC is the science of numbers; and, has five principal rules for its operation, viz. Numeration, Addition, Subtraction, Multiplication and Division. Numeration is the art of numbering. Simple Addition is putting together sev. eral smaller numbers of the same denomination, into one larger equal to the whole or sum total. Simple Subtraction is taking a less number from a greater to show the difference between them. The greater number is called the minuend, the less the subtrahend, and the difference between the num. bers is called the remainder. Multiplication is a concise mode of performing addition, when the numbers to be added are equal; or, multiplication is finding a number equal to a given number of repetitions for some other number. The number to be multiplied, is called the multiplicand. The number to multiply by is called the multiplier. The result obtained from multiplication, is called the product. Whenever two or more numbers are multiplied together, each of them is called a factor, or the multiplicand and multiplier are factors. In performing multiplication, either factor may be made the multiplicand without altering the product. The finding of two or more numbers, whose product shall be equal to a given number, is called resolving the given number into factors. A number which can be resolved into factors, is called a compound or composite number. Rule for multiplication. 1. Place the multiplier under the multiplicand, so that the same orders may stand under each other. 2. Multiply the multiplicand by each significant figure in the multiplier, placing the first figure of each partial product under the order of its multiplier. 3. Add the several pro. ducts as they stand. The amount will be the product required. To prove multiplication. Invert the arrangement of the factors, and multiply again. Simple Division is finding how many times one whole number is contained in ano

ther, and also what remains. There are four principal parts in division. 1. The dividend, or number given to be divided. 2. The divisor, or number given to divide by. 3. The quotient or answer, which shows how many times the divisor is contained in the dividend. 4. The remainder, which is always less than the divisor, and of the same name with the dividend. Rule for division. 1. Place the divisor on the left of the dividend. 2. 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. 3. Multiply the divisor by this quotient, 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. 4. 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. Short division is when the divisor does not exceed twelve.

COMPOUND Numbers. Compound Addition is the adding of several numbers together, having different denominations, but of the same generic kind, as pounds, shillings, pence, &c. Rule. 1. Place the same denominations under each other. 2. Add, first, the lowest denomination. Find how many of the next higher are contained in the sum, which number carry to the next higher, and place the remainder under the denomination added. 3. Proceed thus with all the denominations. Compound Subtraction is finding the difference, inequality, or excess, between any two sums of diverse denominations. Rule. Place the same denominations under each other. Take each denomination in the subtrahend, beginning with the lowest, from the same in the minuend, and to compensate for borrowing from one denomination to another, increase the next higher denomination of the subtrahend, by one. Compound Muliplication is when the multiplicand consists of several denominations. Rule. Place the multiplier under the multiplicand, multiply each denomination separately, beginning with the lowest, and carry as in addition. Compound Division is finding how many times one number is contained in another of different denominations. Rule. 1. Divide each denomination of the dividend separately by the divisor, and the several quotients

will belong to the denominations of their respective dividends. 2. If a remainder be left, in dividing any denomination, reduce it to the next lower denomination, add in all of that denomination, in the given dividend, and then divide as before.

REDUCTION. Reduction is changing numbers from one name to another without altering their value. It is either descending or ascending. Descending is when great names are brought into small, as pounds into shillings, days into hours, &c. This is done by multiplication. Ascending is when small names are brought into great, as shillings into pounds, hours into days, &c. This is performed by di

vision.

FRACTIONS are expressions for any assignable part of a unit or whole number, and are divided into vulgar and deciA vulgar fraction is represented by two numbers placed one above the other, with a line drawn between them, as,, &c. The figure or figures above the line is the numerator, those below the denominator. Every fraction is an instance of division, in which the numerator is the dividend, and the denominator is the divisor. Multiplying the numerator of a fraction multiplies the value, and dividing the numerator divides the value. Multiplying the denominator of a fraction divides the value, and dividing the denominator multiplies the value. Hence multiplying the numerator produces the same effect upon the value, as dividing the denominator, and visa versa. If the numerator and denominator be both multiplied, or divided by the same number, the value of the fraction will not be altered. To reduce several fractions to a common denominator, multiply both terms of each fraction by the denominators of all the other fractions. A proper fraction is when the numerator is less than the denominator, as . An improper fraction is when the numerator is greater than the denominator, as 3. A compound fraction is the fraction of a fraction, as of. A mixed fraction is composed of a whole number and a fraction. Vulgar fractions may be added, subtracted, multiplied or divided, by first reducing them to a common denominator, and then treating them according to the rules in simple numbers. A vulgar fraction may be reduced to a decimal, by annexing cyphers to the numerator and dividing by the denominator.

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A decimal fraction is that whose denominator is an unit, with a cypher or cyphers aunexed to it.

CENTAGE is rate by the cent. or hundred. Insurance is an allowance made to a company who agree to make up loss occasioned by fire, storms, &c. Commission is an allowance made to a broker or factor, or an agent or correspondent, for assisting in procuring and disposing of goods. Duty is a tax required by government to be paid on the importation, exportation or consumption of goods. Interest is an allow ance made for the use of borrowed money, or for debts of any kind remaining unpaid after they are due. Discount is an allowance made for the payment of money before it is due. An allowance of a certain rate per cent. on any given sum, may be calculated by multiplying the same by the rate per cent. expressed decimally. In interest, the sum lent is called the principal. The sum per cent. agreed on is called the rate. The principal and interest added together is called the amount. GENERAL RULE FOR SIMPLE INTEREST. 1. Find the interest of the given sum for one year. 2. Multiply the interest of one year by the given number of years, and the product will be the answer for that time. 3. If there be parts of a year, as months and days, work for the months by the aliquot parts of a year, and for the days by the rule of three direct, or by allowing 30 days to the month and taking aliquot parts of the same. Compound interest is when the interest is added to the principal at the end of the year, and on that amount the interest cast for another year, and added again, and so on; this is called interest upon interest. Rule. 1. Find the amount for the first year as in simple interest, and make it the principal for the second; on this, find the amount for the second, and make it the principal for the third, and so on, up to the number of years given. The last amount is the amount at compound interest. 2. Subtract the principal from the amount, and the remainder will be the interest. Common rule for calculating interest on notes when there are endorsements. 1. Find the amount of the principal of the note for the whole time. 2. Find the amount of each payment, from the time it was paid, till the time of settlement. 3. Add the amount of the payments, and deduct the sum from the amount of the principal. Massachusetts rule. Find the amount of the principal to the first time when a payment was made, which either

alone, or together with the preceding payments, (if any,) exceeds the interest then due. From this, subtract the pay. ment, or the sum of the payments, made within the time for which interest was computed, and the remainder will be a new principal, with which, proceed as before; and so on, to the time of settlement. Connecticut rule. 1. Find the amount of the principal to the time of the first payment, (if it be a year or more after the interest commenced,) and sub. tract this payment from it. 2. The remainder will be a new principal; on which calculate the amount to the next payment, and subtract; and so on, from payment to payment, until they are all employed, provided a year or more intervenes, between each two payments. 3. But if the intervening time be less than a year, find the amount of the last principal for a year, and of the payment up to the same time, and subtract the latter from the former. If, however, a year overruns the time of settlement, find the amount up to that time, instead of for a year. 4. If any remainder, after subtraction, be greater than the preceding principal, then the preceding principal is to be continued as the principal for the succeeding time, instead of the remainder, and the difference to be regarded as so much unpaid interest.

RATIO AND PROPORTION. Ratio is a measure of the comparative size of two numbers or quantities. The ratio between two numbers, or quantities, may be expressed by the quotient of one divided by the other, or by the numerator and denominator of a fraction. The denominator is called the antecedent, and the numerator the consequent of the ratio; and both taken together are called a couplet. Proportion, in arithmetical language, is synonymous with rule of three. Direct proportion is when one ratio increases as another increases, or diminishes as another diminishes. Inverse proportion is when one ratio increases as another diminishes, or diminishes as another increases. In both direct and inverse proportion, there are three terms given to find a fourth. In the case of direct proportion the fourth term is found by dividing the product of the second and third by the first. In inverse proportion the fourth term is found by dividing the product of the first two by the third. Com. pound proportion involves two or more statings in simple proportion. In this rule there are five terms given to find a sixth. When this is direct, multiply the first two terms for