Thus, 15 and 16, 2 and 3, and a+b and a+b, 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 20ab2 of 5ab. The reciprocal of any quantity, is that quantity inverted, or unity divided by it. Thus, the reciprocal of a, or a —; is 1; and the A function of one or more quantities, is an expression into which those quantites enter, in any manner whatever, either mixed, or not, with known quantities (b). Thus, a-2x, ax + 3x2, 2x − a(a2 — x2)*, - · axTM, a, &c., are functions of x; and axy+bx2, 1y + x(ax2 -- by3)*, &c. are functions of x and y. A vinculum, is a bar, or parenthesis (), made ase of to collect several quantities into one. Thus, a+b× c, or (a + b)c, denotes that the compound quantity a + b is to be multiplied by the simple quantity c; and ab+c, or (ab+c2), is the square root of the compound quantity ab + c2. (b) The term function, which is scarcely to be found in any of our algebraical treatises, was used by the early analysts, to signify the different powers or roots of the same quantity; so' that, supposing m to be any whole or fractional number, aTM and m would have been called by them functions of a and x; but the phrase is now generally understood in the extended sense above mentioned. Practical Examples for computing the numeral Values of various Algebraic Expressions, or Combinations of Letters. Supposing a=6, b=5, c=4, d=1, and e=0. Then a+2ab-c+d=36+60-4+1=93. And 2a-3a2b+c3432-540+64-44. And a2 x a+b-2abc=36 × 11-240=156. And 3a√2ac+c2, or 3a(2ac+c2)=18√64 = 144. And 2a√b2-ac + √2ac + c2 = 12 x 1+8=20. And √2a2 -√2ac+c2=√72−√64 = √72-8=√64=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. ADDITION. (B) 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 commonly divided into the three following cases (c). CASE I. When the quantities are like, and have like signs. RULE. 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. (c) It may here be observed, that the term Addition, which is applied to this rule, in conformity to custom, is too scanty to express the nature of the operations to be performed, which are sometimes those of addition and sometimes subtraction, according as the quantities are positive or negative. It should, therefore, be called by some name signifying incorporation, union, ́or striking a balance, in which case, the incongruity that now subsists, would be removed. When the quantities are like, but have unlike signs. RULE. Add all the affirmative coefficients into one sum, and those that are negative into another, when there are several of the same kind; then subtract the least of these sums from the greatest, and to the difference prefix the sign of the greater, annexing the common letter or letters, as before. The principles upon which the above operations are founded, as well as the demonstrations of all the following rules, are given in the second part of the work; to which the reader, who wishes to obtain a theoretical knowledge of the science here treated of, is referred; where he will find the several corresponding articles designated by the same letters, A, B, C, &c. as in the first, or present part. |