nute of refusal on the bill itself, and afterwards the instrument is drawn out and attested under his hand and seal. The want of a protest, can in no case be supplied by noting, which is a mere preparatory minute, of which the law takes no cognizance as distinguished from a protest. If there be no notary resident at or near the place, the bill must, when payable, be protested by some substantial resident in the presence of two or more witnesses, and should in general be made at the place where payment is refused; but when a bill is drawn abroad, directed to the drawee at Southampton or London, or any other place, requesting him to pay the payee in London, the protest for non-acceptance of such bill, may be made either at Southampton or London. Notice in case of foreign bills when to be given. Notice should be given on the day of refusal to accept, if any post or ordinary conveyance set out on the day, and if not by the next earliest couveyance. An usance is generally understood to mean only a month. Instead of an express limitation by months or days, we continually find the bills drawn or payable at Amsterdam, Rotterdam, Hamburgh, Altona, Paris, or any other place in France, Cadiz, Madrid, Bilboa, Leghorn, Genoa or Venice, limited by the usance, that is the usage between those places and this country. An usance between this kingdom and Amsterdam, Rotterdam, Hamburgh, Altona, Paris, or any place in France, is one calendar month from the date of the bill; an usance between us and Cadiz, Madrid, or Bilboa, two; and an usance, between us and Leghorn, Genoa, or Venice, three. A double usance is double the accustomed time; an half usance, half. Upon an half usance, if it be necessary to divide a month, the division, notwithstanding the difference of the length of months, shall contain fifteen days. ALGEBRA LGEBRA is a method of calculating quantities in general by means of signs or characters. It has also been called specious arithmetic, on account of the species or letters made use of in it; and by Sir Isaac Newton universal arithmetic, from its performing all arithmetical questions by indeterminate quantities. The word algebra is from the Arabic, in which language al-gjabr Wal-mokabala means the art of resolution and equation. Definitions. 1. Known quantities are generally represented by the first let ters of the alphabet, as a, b, c, &c. Unknown by the last; as x, y, 2. 2. The sign (plus or more) is the mark of addition. Thus, ab means that the quantities represented by a and b are to be added together. When no sign is prefixed, + is understood. 3. The sign (minus or less) denotes subtraction; as a-b, that is, the number represented by b, is to be subtracted from that represented by a. - 4. Quantities with the sign prefixed, are called positive or affirmative and those with the sign negative quantities. 5. The sign x (or by) denotes multiplication: as 5×4 means that 5 is to be multiplied by 4. 6. is the mark of division; thus, a÷b means that a is di— vided by b. 7 A number prefixed to a letter is called a numeral coefficient. When no number is expressed, 1 is understood. 8. A simple quantity consists of one part or term, as +a, -abc; a compound quantity of more than one, connected by the signs or; as, a+b, a−b+c, are compound quantities. If there are two terms, it is called a binomial, if three a trinomial, &c. 9. Like quantities consist of the same letters repeated: thus, +ab, -5ab, are like quantities; +ab, and +aab are unlike, OF ADDITION. Case 1. To add quantities that are alike and have like signs. Rule. Add the coefficients together, to their sum join the quantities, and prefix the common sign. Case 2. To add quantities that are like but have unlike signs. Rule. Subtract the latter coefficient from the greater, prefix the sign of the greater to the remainder, and subjoin the common letter or letters. Rule. Set them all down one after another with their signs and coefficients. General Rule.-Change all the signs of the quantities to be subtracted into their contrary signs, and add them to the others by the preceding rules; which will give the difference or remainder. From 5a Rem. 5a-3a or 2a Example. Tab-16bc 4ab-16bc-mb The reason of this rule is that tó subtract a negative quantity is the same as adding its positive value. MULTIPLICATION. General Rule for the Signs. If the signs of the two terms are like, that of the product is +, but if unlike, it is -. Case 1. To multiply two terms. Rule. Find the sign of the product by the general rule; then place after it in the product of the numeral coefficients, and the letters one after another. Case 2. To multiply compound quantities. Rule.-Multiply each term of the multiplicand, by the terms of the multiplier, and collect the products into one sum. When several quantities are multiplied together, any of them is termed a factor of the product. Thus aa, The product arising from the continual multiplication of the same quantity are named its powers, and is the root. aaa, &c. are powers of the root a. Powers are sometimes expressed by placing a figure above the root to express the numbers of the factors. As, power of the 3d 4th The figures denoting the number of the factors are called exponents, or indices. Instead of multiplying compound quantities it is usual to set them down with a line over each of the compound factors; thus, a+bxa-b which means the product of a+b multiplied by a--b. The line over the factors is called a vinculum, OF DIVISION. Rule for the Signs. If the signs of the divisor and dividend are the same, that of the quotient is +, if unlike, it is The general rule in division, is to place the dividend above a line, and the divisor under it, expunging those letters which are in all the quantities of the dividend and divisor, and dividing the coefficients of all the terms by a common measure. Thus in dividing 10ab+15ac by 20ad, expunge a out of all the terms, and divide the coefficients by 5, which gives the quotient 2b+3c Powers of the same root are divided by subtracting their exponents. Thus if a5 is divided by a the quotient is a5-2 or a3. In dividing compound quantities the parts are to be ranged according to the powers of some one of its letters: thus if a2+2ab+b2 is the dividend, and a+b the divisor, they are to be ranged according to the powers of a. Then divide the first term ALGEBRA. of the dividend by the first term of the divisor; multiply the quotient by the whole divisor, and subtract the product from the dividend. If nothing remain, the operation is finished; but if there is a remainder, it becomes a new dividend. Thus a divided by a, gives a which is the first quotient and the product of this divided by the whole divisor, a+b, viz. a+ab being subtracted from the dividend there remains ab+b2 Divide the first term of this dividend by the first term of the divisor, and join the quotient with its proper sign; then multiply the whole divisor by this part of the quotient and subtract the product from the new dividend, and continue the same operation till there be no remainder. When the operation may be continued without end, the quo tient is termed au infinite series. OF FRACTIONS. In a fraction, the quantity above the line is termed the numerator, and that below the denominator. If the numerator and denominator be multiplied or divided by the same quantity the value of the fraction is the same. ma Thus: lete, then c; for by the last rule if the quotientc, be multiplied by the divisor b, the product must be a Hence a fraction may be reduced to a more simple one of the |