Algebra[4] is not, as are Arithmetic and Geometry, indispensable to every one. It should be very sparingly introduced into the general education of youth, and we would there willingly dispense with it entirely, excepting logarithms, if this would benefit the study of arithmetic and geometry. The programme of it which we are now to give, considers it purely in view of its utility to engineers, and we will carefully eliminate every thing not necessary for them. [Footnote 4: The true distinction between ALGEBRA and ARITHMETIC is so commonly overlooked that it maybe well to present it here, in the words of Comte. “The complete solution of every question of calculation is necessarily composed of two successive parts, which have essentially distinct natures. In the first, the object is to _transform_ the proposed equations, so as to make apparent the manner in which the unknown quantities are formed by the known ones; it is this which constitutes the _Algebraic_ question. In the second, our object is _to find the value_ of the formulas thus obtained; that is, to determine directly the values of the numbers sought, which are already represented by certain explicit functions of given numbers; this is the _Arithmetical_ question. Thus the stopping-point of the algebraic part of the solution becomes the starting-point of the arithmetical part. “ALGEBRA may therefore be defined as having for its object the _resolution of equations_; taking this expression in its full logical meaning, which signifies the transformation of _implicit_ functions into equivalent _explicit_ ones. In the same way ARITHMETIC may be defined as intended for _the determination of the values of functions_. Henceforth, therefore, we may call ALGEBRA _the Calculus of Functions_, and ARITHMETIC _the Calculus of Values_.”]