Irrational Functions, strictly speaking, are functions whose numerical values cannot be expressed by finite fractions. In the present connexion the term irrational has a wider signification. It is used to denote any function whose numerical value cannot, in general, be expressed exactly within the limits as to decimal places, to which we restrict ourselves in the table under formation. As here employed therefore the term designates not only transcendental functions, (as exponential, logarithmic, circular, &c.,) and algebraical irrational functions, (which are such as contain fractional powers of the variable in either numerator or denominator, or both,) but also algebraical fractional functions, which are such as contain integer powers of the variable in both numerator and denominator, or in the latter only. And it is to the formation of tables of the values of functions such as these that our attention is now to be directed.