a a 9 9 3 Q3 16 27 A rational quantity, is that which can be expressed in finite terms, or without any radical sign, or fractional index; as a, or ža, or 5a, &c. An irrational Quantity, or Surd, is that of which the value cannot be accurately expressed in numbers, as the square roots of 2, 3, 5. Surds are commonly expressed by means of the radical sign Vi as V2, Va, væs, or a fractional index; as 2 &c A square or cube number, &c., is that which has an exact square or cube root, &c. Thus, 4 and 9a” are square numbers; and 64 and are cube numbers, &c. A measure of any quantity, is that by which it can be divided without leaving a remainder. Thus, 3 is a measure of 6, 7a is a measure of 35a, and 9ab of 27 a-62. Commensurable quantities, are such as can be each divided by the same quantity, without leaving a remainder. Thus, 6 and 8, 2v2 and 3/2, 5a+b and 7ab, are commensurable quantities; the common divisors being 2, V2, and ab. Incommensurable quantities, are such as have no common measure, or divisor, except unity. Thus, 15 and 16, V2 and 13, and a +b and a2 + 62 are incommensurable quantities. A multiple of any quantity, is that which is some exact number of times that quantity. Thus, 12 is a multiple of 4, 15a is a multiple of 3a, and 20a62 of 5ab. The reciprocal of any quantity, is that quantity inverted, or unity divided by it. 1 Thus, the reciprocal of a, or is ; and the reciprocal of 1 6 is a a a a A function of one or more quantities, is an expression into which those quantities enter, in any manner whatever, either combined, or not, with known quantities. * This definition of a Surd, or irrational Quantity, is due to Robert Adrian, LL.D., Professor of Mathematics and Natural Philosophy in the University of Pennsylvania, who had first published it in his edition of Hutton's Course of Mathematics.--ED. Thus, a Axm, ai", a", &c., are functions of x; and axy+bx+, ay+ (ax? - by), &c., are functions of x and y. A vinculum, is a bar - or parentheses ( ), made use of to collect several quantities into one. Thus a + b xc, or (a + b)c, denotes that the compound quantity a + b. is to be multiplied by the simple quantity c; and vab+c?, or (ab + c), is the square root of the compound quantity ab + c. Practical Examples for computing the numeral values of various Algebraic Expressions, or Combinations of Letters. Supposing a = 6, b = 5,0 5,c = 4, d = 1, and e = 0. Then 1. a? + 2ab 36 + 60 4+1= 93. 2. 203 3a2b + c3 = 432 540 + 64 44. 3. as x: (a + b) 2abc = 36 X 11 240 = 156. 4. 2a V (69 -- ac) + V (2ac + c) = 12 x1 +8= 20. I 을 5. 3av (2ac +-c“), or 3a (2ac + c^) 18 v 64 144. 6. V [2a - V (2ac + c)]=v(72 - 764)= V(72 8) =V64 -.8. 2a + 30 4bc 12 + 12 80 24 nagy + 十 + 6d + 4e' ✓ (2ac to c^) 6 to ✓ (48 + 16) 6 80 toa 14. 8 Required the numeral values of the following quantities; supposing a, b, c, d, e, to be 6, 5, 4, 1, and 0, respectively, as above. 1. 2a + 3bc 5d = 127 · + 2ae=1 d ✓ 56 +3votd За a a Хd ADDITION. ADDITION is the connecting of quantities together by meåns of their proper signs, and incorporating such as are like, or that can be united into one sum; the rule for performing which is commonly divided into the three following cases.* CASE I. When the Quantities are like, and have like signs. RÚLE.-Add all the coefficients of the several quantities together, and to their sum annex the letter or letters belonging to each term, prefixing, when necessary, the common sign. * The term Addition, which is generally used to denote this rule, is too scanty to express the nature of the operations that are to be performed in it; which are sometimes those of addition, and sometimes subtraction, according as the quantities are negative or positive. It should, therefore, be called by some name signifying incorporation, or striking a balance; in which case, the incongruity here mentioned would be removed. CASE II. When the Quantities are like, but have unlike signs. RULE.--Add all the affirmative coefficients into one suni, and those that are negative into another, when there aro several of the same kind : then subtract the less of these sums from the greater, and to the difference prefix the sign of the greater, annexing the common letter or letters as before. When the Quantities are unlike; or some like and others unlike. RULE.-Collect all the like quantities together, by taking their sums or differences, as in the foregoing cases, and set down those that are unlike, one after another, with their pro per signs. Ans. a. er. er. 1. Required the sum of (a + b) and (a - b). 2. Add 500 3a + b + 7 and 4a 3x + 26 - 9 togeth Ans. 20 na + 3b – 2. 3. Add 2a + 36 - 40 - 9 and 5a 3b +20 – 10 togeth Ans. 7a - 20 - 19. 4. Add 3a+2b - 5, a +-5b - c, and 6a – 2c +3 together. Ans. 10a +76 - 30 -2. 5. Add 23 + ax + bx + 2 and 20.3 + cx? + dx - 1 together. Ans. 2x3 + (a + c)2 + (6 + d) x + 1. 6. Add 6xy 4x2 + 3xy, 4x2 Зху + 4x2 together. Ans. 4xy - 822 2xy, and |