ALGEBRA. 323 



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, (-f-), is read plus ; 

 and shows that the quantity placed after it, is to be added to the pre- 

 ceding. 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 writ- 

 ten each in a parenthesis ; or if letters, with simply a point, or with- 

 out any sign, between them. The sign of division, (-r-), 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 3 denotes the 

 square of a; and 3 , its third power, instead of aaa. If a denote 

 5, a 3 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 3 will be 3x125, or 375. Like quantities, are those which con- 

 sist of the same letters, raised to the same powers ; as 6 a 3 b, and 

 12 a 2 b ; which are added or subtracted, simply by adding or sub- 

 tracting 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 con- 

 tain 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 a 3 b, by 

 7 a b* c, is 42 a 4 b*c. 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 



