Publisher's Synopsis
Excerpt from A General Geometry and Calculus, Vol. 1: Including, of the General Geometry, Treating of Loci in a Plane; And an Elementary Course in the Differential and Integral Calculus
In the treatment of the Calculus I have used the Infinitesi mal method instead of the method of Limits, on account of its greater simplicity, as well as because it is the only conception which enables us to apply the Calculus to practical problems with any degree of facility. The general use of the method of limits in our text books has done not a little to prevent the com mon study of this elegant and useful branch of mathematics. This method is not only exceedingly cumbrous, but it has the misfortune that its element, a differential coefficient, is a ratio. The abstract nature of a ratio, and the fact that it is a com pound concept, peculiarly unfit it for elementary purposes. The beginner will never use it with satisfaction, for it does not give him. Simple, direct and clearly defined conceptions. But while I have adopted the infinitesimal theory, I have felt free to intro duce the doctrine of limits, and to illustrate and apply it. The metaphysical objections to this method, if not rebutted by equal difficulties of a similar character encountered in the method of limits, are immensely overborne by its practical advantages; for, let it be remembered that no writer adheres to the Newton ian method throughout, but glides into the other in the Integral Calculus, and adopts it exclusively in most geometrical and physical applications.
The sources from which the material has been drawn will be readily perceived by the mathematician, and need not be enum crated here. That the treatise is sufficiently different from others of a similar purpose to justify its existence, the author feels more sure than that these differences will commend them selves to his fellow laborers in the work of mathematical train ing. One thing, however, is certain, nothing in matter, arrange ment, or manner of treatment, has been introduced without careful reference to the capabilities and wants of such students as I have been accustomed to meet in the class room for more than twenty years; and few things will be found in the volume but what have been put to the test of class room use many times over.
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