o Thus, –1 is the index of al., 2 is the index of a”, and 4 of ałorv/a. When a quantity appears without any index, or expo nent, it is always understood to have unity, or 1. , Thus, a is the same as a”, and 2x is the same as 2x" ; the 1, in such cases, being usually omitted. A rational quantity, is that which can be expressed in finite terms, or without any radical sign, or fractional in dex; as a, or +a, or 5a &c. An irrational quantity, or surd, is that which has no exact root, or which can only be expressed by means of the radical sign, or a fractional index; as V2 or 2%, 3/ao or ał, &c. A square or cube number, &c. is that which has an exact square or cube root, &c. A measure of any quantity, is that by which it can & divided without leaving a remainder. Commensurable quantities, are such as can be each divided by the sama quantity, without leaving a remain- . der. Thus, 6 and 8, 2 v2 and 3 v2, 5aab and 7aba, are commensurable quantities ; the common divisors being 2, v.2, and ab. Incommensurable quantities, are such as have no common measure, or divisor, except unity. Thus, 15 and 16, v2 and v3, and a + b and a” + 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 2002 b" of 5ab. . The reciprocal of any quantity, is that quantity invert ed, or unity divided by it. Thus, the reciprocal of a, or +, is # ; and the reci ... b procal of ; is a . A function of one or more quantities, is an expression into which those quantities enter, in any manner what ever, either combined, or not, with known quantities. A vinculum, is a bar , or parenthesis ( ), made use of to collect several quantities into one. Thus, a + b X c, or (a + b) c, denotes that the compound quantity a + b is to be multiplied by the simple quantity c ; and Vab-Hc", or (ab-i-co)}, is the square root of the compound quantity ab-Hco. Practical Eramples for computing the numeral Values of various Algebraic Expressions, or Combinations of Letters, uired the numeral values of the following quantities; supposing a, b, c, d, e, to be 6, 5, 4, 1, and 0, respectively, as above. Addition is the connecting of quantities together by means of their proper signs, and incorporating such as are like, or that can be united, into one sum ; the rule for performing which is commenly divided into the three fol. lowing cases (a): is CASE I. When the Quantities are like, and have like Signs. RULE. Add as 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. (a) 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, -, |