proportions; by means of which it may always be resolved; without requiring any special rules. § 5. Of the strictly Mercantile Rules of Arithmetic, most of which depend upon the principles of proportion, we have only room to speak very briefly. Tare and Trett, is a rule for making allowances in selling goods by weight. Tare, is an allowance for the weight of the box, bag, or other recipient; and trett, or draft, is a per centage deduction for refuse, waste, or loss. These being taken from the gross weight, leave the net weight for the remainder. Interest, is an allowance made for the use of money; and is generally reckoned at a certain rate per cent., per annum: that is a certain number of dollars paid for the use of one hundred dollars for one year. Insurance, paid for risks; Brokerage or Commissions, paid for exchanges or sales; and Discount, allowed for the payment of money before it becomes due; are also usually reckoned at a certain rate per cent. Loss and Gain, is the rule by which merchants discover their total profit or loss, in buying or selling certain quantities at fixed rates and Fellowship relates to the division of profit or loss among partners. Alligation, is the rule for finding the price of mixtures, or for making mixtures of a given value. § 6. The Powers, of any number, are the successive products of that number by itself, and of this product by the same number again: the exponent of the power denoting how many times the same number is taken as a factor. One multiplication produces the square, or second power, of which two is the exponent; and a second multiplication produces the cube, or third power, the exponent of which is three. Thus 25 is the square, and 125 is the cube, of the number 5; and its fourth power, or biquadrate, is 725. The Root, of any number, is another number, which, multiplied by itself a certain number of times, will produce the given number. Thus 5 is the square root of 25; and it is the cube root of 125. Involution is the process of finding powers; and the name Evolution is given to that of finding roots. A Progression, is a series of numbers in continued proportion. In an Arithmetical series or progression, each term is found by adding or subtracting the common difference to or from the preceding; according as the series is increasing or decreasing. In either case, the sum of the series is equal to the sum of the two extreme terms multiplied by half the number of terms. A series is in Geometrical progression, when each term is either the product or the quotient of the preceding term by a common ratio. In such a series, any four consecutive terms form a geometrical proportion. The Rules of Position, for which those of Algebra are a far preferable substitute, we have no room to describe. 41 CHAPTER II. ALGEBRA. ALGEBRA is that branch of Mathematics in which the relations of quantities are expressed, and problems resolved, by means of letters and other symbols. The name is derived from the Arabic phrase, Al gebr u al mocabela, signifying the reduction of equations: and from the generality of its results, it has also been called Universal Arithmetic. It presupposes a knowledge of Arithmetic, or at least of the elementary rules, on the general principles of which it also depends; but in representing unknown or variable quantities by letters, and expressing their relations by means of other symbols, it reaches a wide range of useful and curious problems, and theorems, which common Arithmetic could never grasp. The first germs of Algebra are found in the writings of Diophantus of Alexandria; who flourished A. D. 350, and is the reputed inventor of the indeterminate analysis. His works, however, are merely a collection of difficult questions concerning squares and cubes, and the general properties of numbers. Here ends the history of Algebra among the ancients: and, accordingly, its invention is ascribed by some writers to the Hindoos; and by others to the Arabians; to whom we are indebted, as has already been mentioned, for its introduction into Europe. The earliest mentioned Hindoo writer on Algebra, is said to have been the astronomer, Aryabhatta, probably as early as the fifth century of our era. Some of the Arabians admit that they received their Algebra from India; but others attribute its invention to their countryman, Mahomed Ben Musa, about A. D. 800; and, in either case, it was doubtless improved by their mathematical knowledge derived from Greek authors. The first printed treatise on Algebra, entitled Summa de Arithmetica, was published in Italy, in 1494, by Lucas Paccioli de Borgo; but it only extended to quadratic equations. The first resolution of cubic equations, is claimed by Tartaglia, (or Tartalea), about 1535; and that of biquadratic equations is ascribed to Ferrari, by Cardan of Pavia, in his book De Arte Magna, published in 1545. Cardan used letters to represent unknown quantities: but Vieta of France, who died in 1603, first applied them to known quantities; and thus generalized the solutions. Vieta also improved the modes of resolving equations; particularly by approximation. Harriot, of England, who died in 1621, first discovered that every algebraic equation is composed of as many factors of the first degree, as are indicated by the degree of the equation. Descartes first introduced the use of exponents; and explained the nature of the negative roots of an equation and he also made the application of indeterminate coefficients, to resolve equations into their several factors. Newton enriched Algebra, not only by farther discoveries concerning equations, but by the invention of the binomial theorem, for problems of involution and evolution. The later discoveries of Maclaurin, Clairaut, Euler, Lagrange, and others, we have no room to describe. The invention of Logarithms, by Napier, of Scotland, in 1614, with the improvement of Professor Briggs, has particularly facilitated the numerical operations of Algebra, to which science they belong : and the Arithmetical Triangle of Pascal, who died in 1662, by exhibiting the properties of figurate numbers, originated the Calculus of Probabilities; a distinct and interesting application of Algebra. We proceed to treat of Algebra under the heads of 1. Preliminary Rules; 2. Simple Equations; 3. Quadratic Equations; 4. Powers and Roots in general; 5. Equations in general; and 6. Series and Logarithms. §1. The Preliminary Rules of Algebra, relate to its peculiar symbols, and their simple applications. In this science, quantities, or rather numbers, are expressed by letters: and it is the general practice to use the first letters of the alphabet for known quantities, and the last for unknown. The sign of addition, (+), is read plus ; and shows that the quantity placed after it, is to be added to the preceding. The sign of subtraction, (—), is read minus; and is placed before quantities that are subtractive, or to be subtracted. The sign of multiplication, (x), called St. Andrew's Cross, is read into, and placed between quantities that are factors: or they may be written each in a parenthesis; or if letters, with simply a point, or without any sign, between them. The sign of division, (÷), may be read divided by, being placed after the dividend, and before the divisor: but division is more generally indicated by writing these quantities as a fraction; the divisor becoming the denominator; and the value of the fraction being the quotient. The power of a quantity, in Algebra, is expressed by writing its exponent above the quantity, on the right. Thus a denotes the square of a; and a3, its third power, instead of aaa. If a denote 5, a will denote 125. The co-efficient of a quantity, is properly the number written as its first factor: thus 3a denotes three times a, and three is the co-efficient. If a denote 5, then 3a will be 15; and 3a will be 3x125, or 375. Like quantities, are those which consist of the same letters, raised to the same powers; as 6 ab, and 12 ab; which are added or subtracted, simply by adding or subtracting their co-efficients, and appending the literal part. Unlike quantities, do not admit of this reduction; but must all be written with their proper signs. To subtract any quantity, we must change its sign, and append it to the subtrahend; or if no sign be written, plus is understood. A term, in Algebra, is a simple expression, not separated into parts by the signs, plus, or minus. A single term is called a monomial; but a quantity having two terms is called a binomial; and one having more than two terms, a polynomial. In Algebraic multiplication, the product of two terms must contain all the factors of them both; and its sign will be plus, if the terms have like signs, but minus, if their signs are unlike, that is, one positive, and the other negative. Thus the product of 6 a3 b, by 7 ab c, is 42 a' b3c. The product of two polynomials, is the sum of all the products of each term of the multiplicand by each term of the multiplier. Algebraic division of monomials, is the reverse of multiplication; and consists in cancelling from the dividend all the factors which it has in common with the divisor; the remaining factors being the quotient. Division of polynomials is performed in much the same manner as arithmetical division; requiring first that all the terms both of the dividend and divisor should be arranged according to the powers of some one letter; after which the first term of the quotient is found by dividing the first term of the dividend by the first of the divisor. Of algebraic fractions, which are similar to arithmetical, we have no room to speak farther. § 2. An Equation, is an expression denoting the equality of two quantities and a Simple Equation, is one in which no unknown quantity is multiplied either by itself or by any other unknown. quantity. The sign of equality, (=), is read, equal to, and is placed between the two equal qualities which are the first and second members of the equation. Common algebraic problems are most frequently solved by means of equations; or by proportions, from which equations are easily obtained. To form the equation, we usually express the unknown quantity, if there be but one, by the letter æ; and with this we form an expression which, by the conditions, is equal to some other expression or formula; after which it only remains to find the value of x from the equation thus formed. Thus, to find a certain number, twice which, being added to 76, and the sum divided by 4, the quotient will be equal to 10 times the same number, 2x+76 we write the equation 4 : 10x; as the first operation. If we multiply each member of the above equation by 4, it will form another equation, free from denominators, and without changing the value of x; viz. 2 x + 76 = 40 x. The next step, is, to bring all the terms containing the unknown quantity to stand by themselves, in one member, usually the first member of the equation. In the present example, to transpose the term 2 x, to the second member of the equation, we cancel it, where it stands, which is really subtracting it from the first member and hence we must also subtract it from the second member; and write 76-40 x-2x; or by reduction, 76=38 x. If, now, we divide both sides of the equation by 38, the co-efficient of the unknown quantity, we shall have 2 = ~ ; or x = 2. When the problem involves two distinct unknown quantities, say x and y, there must be two distinct equations; from one of which we find the value of x, in terms containing y; and then substitute this value of x, wherever a occurs, in the other equation: which will then contain only one unknown quantity, y. § 3. Quadratic Equations, are those which contain the square or second power of the unknown quantity; but no higher power. Το resolve them, we first transpose, if necessary, so as to bring all the terms containing a to stand first in order; those containing to stand next; and all the known terms, that is, those which do not contain x, form the other member of the equation. We then divide both members by the co-efficient of x2, which reduces the equation to the regular form, x2 = a, for pure quadratics, and x2 + ax = b, for those which are affected, or complex: a and b here simply denoting any known quantities. A pure quadratic, is then resolved, = simply by extracting the square root of both of its members. Thus, from the equation 10 1082 x2, we obtain 2 108-10; or 298; or x2 = 49; or x 7. = In this case we may have x=7, or x=-7; since a negative quantity multiplied by itself produces a positive square. As the square root of any quantity is denoted by the radical sign, (✔), we might have written above, x=49= = ±7. The square root of a monomial, is also a monomial: but if we multiply x + a by x+a, we shall have (x + a)2 = x2 + 2 ax + a2 ; that is, the square of a binomial, is made up of the square of the first term, plus twice the product of the two terms, plus the square of the last term. This suggests the rule for extracting the square root of a polynomial; which we have no room here to present. Hence, to resolve a complex quadratic equation, when reduced to the regular form, x2 + ax = b, we must consider ax as twice the product of the two terms of a binomial root; and ≈ being one of them, a will necessarily be the other. We must therefore add the square of a to each member of the equation; making x2 + ax + § a2 = b + ¦ a2 ; and the first member will then become a trinomial and perfect square; while the second member will contain only known quantities. Then, extracting the square root of each member, we have x + a = ± ✔b + a2; from which, as a simple equation, the value of x may readily be found. For example, if we have x+6x=27, then is x2+6x+9=27 +936; and x + 3 = = 6, or x = ±6—3 -9. = 3, or § 4. The theory of Powers and Roots in general, comes next in order, as a preparation for the more general study of equations. If we form the successive powers of the binomial a + b, we shall have (a + b)2 = a3 + 2 a b + b2. (a + b)3 = a3 + 3 a2 b + 3 a b2 + b3. (a + b)* = a* + 4 a3 b + 6 a2 b2 + 4 a b3 + ba. In the formation of these powers, we observe certain remarkable laws, which have been generalized by Newton, in the binomial theorem. We see that the number of terms in the power, is one greater than its exponent. The exponents of the leading factor, a, go on diminishing by unity from term to term; while those of the succeeding factor, b, go on increasing, according to the same law. And to form the coefficient of any term, we multiply the co-efficient of the preceding term by the first exponent in that term, and divide the product by the number denoting the place of that term, counting from the first. By these same rules, we may develope the powers of any other binomial. Thus, to develope (2 x + y), we write 2 x instead of a, and y instead of b; and the result becomes, (2x)3 + 3 (2 x)3 y + 3 (2 x) y3 + y3 : or by reduction we have (2 x + y) =8 x3 + 12 x2 y +6 x y3 + y3. Roots, in general, are denoted by the radical sign, (v ), with the index of the root written above and on the left; except the square root, whose index is understood, but not written. As we multiply the exponent, in raising to a power, so we may divide the exponent, to extract the root; thus forming a fractional exponent. |