Gross weight is the weight of both goods and packages. Tare is an allowance granted to the buyer for the weight of the barrel, &c. containing the goods, and is deducted from the gross weight. Tret is an allowance of 4lbs. on 104lbs. or on goods liable to waste, and is deducted after the tare. Cloff is an allowance of 2lbs. on 3cwt., or retailers for the turn of the scale, and is deducted tret. given to after the After subtracting the tare from the gross weight, the remainder is called the tare suttle; and after subtracting the tret, the remainder is called the tret suttle; and what remains after all the deductions are made is called neat or net weight. Barter is the method of exchanging goods without profit to either party. Exchange is the method of finding how much money of one country, is equal in value to any proposed sum of money of another country. A Bill of Exchange is the instrument in writing, by means of which the transaction is carried out. When an exchange is mentioned between two places, one place gives a determined price, to receive an undetermined one. Thus London gives a Pound sterling, which is a certain price, to receive from Paris a certain number of francs, more or less to be received there, as the exchange may be. The determined price is called certain, and the undetermined uncertain, because it is always subject to variation. The par of exchange is the intrinsic value of the money of one nation, compared with that of another. This is estimated by the quantity of pure gold or silver. The course of exchange is the current value allowed for the money of one country when reduced to the money of any other country. This is continually fluctuating above or below the par of exchange, according as the demand for bills is greater or less. The demand for bills depends upon what is called the balance of trade, which is for or against a country, according as more or less goods are exported or imported by that country in comparison of some other. Thus, if London ships to Calcutta goods to the amount of £500,000, and Calcutta at the same time sends to London goods only to the amount of £400,000, the balance of trade is said to be in favour of London; and as Calcutta can discharge only to the amount of £400,000, by bills of exchange in the way of trade, i. e. to the amount of the value of goods sent to London, there is a balance against her of £100,000, which she must pay by bills of exchange procured elsewhere, and for these she must pay a premium. If in this case Calcutta pays £1. per cent for bills, she will, to liquidate the debt, have to pay £1000. premium; and in this way the balance of trade affects the fluctuation of exchanges. It is evident, however, that the course of exchange can never exceed the expense of sending gold or silver bullion to the place upon which the bill is drawn. The Arbitration or Comparison of Exchanges, is the determining what rate of exchange called the Par of Arbitration, between any number of places corresponds with any assigned rates between each of them and another place. Arbitration is styled simple or compound, according as three or more places are concerned. Agio is the difference between bank and current money, and also between the intrinsic and circulating value of foreign coin. Usance is the usual time allowed by merchants and bankers to pay bills of exchange. Involution is the method of finding the power of a given number. A Power is the product arising from the multiplication of the given number by itself the proposed number of times. The number itself is called the root. The number of times any factor is repeated is called the index or exponent. Thus 53 denotes 5 x5x5 or 125: where 3 is the index, 5 the root, and 125 the power. Evolution is the method of finding the root of a given number. The root of any power is such a number as, being multiplied into itself a certain number of times, will produce that power. Those roots which can be found exactly are called rational roots. Some numbers have no exact root and can only be found approximately: these roots are called Surds or fractional quantities. An Incommensurable Quantity is that from which no exact root can be extracted. Roots are sometimes denoted by the character or radical sign before the power, with the index of the root against it. Thus the 3rd root of 10 is expressed by 3/10 and the square root of 20 by 20, the index 2 being generally omitted. Roots are often denoted like powers, with fractional indices, thus 8 is written 8, 3 10 as 10. In general when any quantity is represented in the form of a surd by means of a fractional index, it is always understood that the numerator of the index denotes the power to which the number is intended to be raised, and that the denominator expresses the root to be afterwards extracted, thus 27 means the cube root of the square of 27. ALGEBRA. Algebra is a general method of reasoning respecting the relation of abstract quantities, by means of letters and symbols. The fundamental principles and operations of Algebra are the same as those of common Arithmetic: but, in consequence of the use of symbols instead of particular numbers, its reasoning is more general, and its application far more extensive. Quantities whose values are known or determined are usually expressed by the first letters of the alphabet, as a, b, c, d, &c.; and unknown or undetermined quantities by the last, as x, y, z, &c. The characters +, -, ×, ÷, and =, have the same meaning as in Arithmetic. Thus ab means either a -b or b—a, according as a or b is the greater. > denotes greater and < less. .. stands for then or therefore, because or since. All quantities before which + stands are called positive or affirmative, and if no sign stand before a quantity + is understood. Quantities having before them are called negative. The number, whether positive or negative, prefixed to any algebraical quantity is called its coefficient: thus 5 is the coefficient of 5a. If no number is expressed, the coefficient 1 is understood, thus a means once a. When a number is used as a coefficient, it is sometimes called the numerical; when a letter, the literal coefficient. The sign is often left out, and sometimes a full-point is used instead of it: thus a b× c, a. b. c. and abc all signify the same thing. When any quantity is multiplied by itself any number of times, the product is called a power of the quantity, and is briefly expressed by writing down the quantity, with a small figure above it to the right denoting the number of times it is repeated. Thus axaxa is written a3. The small figure in any case is called the index or exponent of the corresponding power. Thus a1 is called the first power of a, a2 the second or square of α, a3 the third power or cube of a, &c. power, The square, cube, fourth (or biquadrate), &c. root of a quantity is that quantity whose square, cube, fourth, &c. power is equal to the given quantity. The symbol used to denote a root is (a corruption of r, the first letter of the word radix), which with the proper index on the left side of it, is set before the quantity whose root is expressed. Thus or simply denotes the square root; 3/ the cube root, and the fourth root, &c. The reciprocal of a fraction is the fraction inverted. Thus - is the reciprocal of; and is the reciprocal of or a. b a α b α a 1 Brackets (), { }, [ ], are employed to show that all the quantities within them are to be taken collectively, as if they formed but one quantity. Sometimes, instead of brackets, a line is drawn above the quantities that are connected, which is called a vinculum. Thus a+b is the same as (a+b). The line which separates the numerator and denominator of a fraction may be considered as a species of vinculum. In taking any quantities out of the bracket, if + precede it, |