THE CONTENTS. Involution, or the raising of powers Evolution, or the extraction of roots Of irrational quantities, or surds Of geometrical proportion Of the nature and formation of equations in general 113 Of the resolution of equations by various methods 123 Of the resolution of cubic equations Of the resolution of biquadratic equations To find the roots of equations by approximation and To find the roots of pure powers in numbers To find the roots of exponential equations Of indeterminate or unlimited problems Of the summation and interpolation of infinite series 163 ALGEBRA. DEFINITIONS. ALGEBRA is the art of computing by symbols. 1. Like quantities are those which consist of the same letters. 2. Unlike quantities are those which consist of diffel'ent letters. 3. Given quantities are those whose values are known. 4. Unknown quantities are those whose values are unknown. 5. Simple quantities are those which consist of one term only. 6. Compound quantities are those which consist of several terms. 7. Positive or affirmative quantities are those which are to be added. 8. Negative quantities are those which are to be subtracted. 9. Like signs are all affirmative (+), or all negative (-). 10. Unlike signs are when some are affirmative (+) and others negative (-). 11. The co-efficient of any quantity is the number prefixed to it. B 12. i binomial quantily is one consisting of two terms; a trinomial of three terms; a quadrinomial of four, &c. 13. A residual quantity is a binomial where one of the terms is negative. 14. The power of a quantity is its square, cube, biquadrate, &c. 15. The index or exponent of a quantity is the number which denotes its root or power. 16. A surd or irrational quantity is that which has no exact root. 17. A rational quaniity is that which has no radical sign (V) or index annexed to it. 18. The reciprocal of any quantity is that quantity inverted, or unity divided by it. Thus, a t- b shows that the number represented by 6 is to be added to that represented by a. a-b shows that the number represented by b is to be subtracted from that represented by a. a's b represents the difference of a and b when it is not known which is the greatest. ab, or axb, or a.) denotes the product of the numbers represented by a and b. a 1 1 a = b, or shows that the number represented by a o is to be divided by that represented by b. u :b::6:d denotes tint a is in the same proportion to b as c is to d. x =u--b +cis an equation, showing that x is equal to the difference of a ind og udlikud l the quantity c. Va, or uŹ, is the square root of a; fra, or a”, is the , 1 cube root of a; and " is the nth root of a. 42 is the square of «; 43 vie cube of it; at the fourth power of a ; and am the mib power of u. 6 is the reciprocal of , and the reciprocal of a. 6 a + bxc, or (a + b)c is the product of the compound quantity a + b multiplied by the simple quantity c. ato a + 0 + 0-0, or (a + b)-(---1), or is the 6 quotient of a + b divided by a--- b. Vab teel or (ab + cu) is the square root of the compound quantity ab +cd. a +6-6 -c or (a +1-c)3 is the cube, or third power, of the quantity a + 6 5a denotes that the quantity a is to be taken 5 times, and 7(b + c) is 7 times otoc. It is also to be remarked that the sign + is generally expressed by the word fills, or more, and the sign by minus, or less. And, in the computation of problems, it must be observed, that the firsi letters of the sphabet are usually put for known quantities, and the last for those which are inkuown, |