Conic Sections And Analytical Geometry: Theoretically And Practically Illustrated (1862)

PREFACE. In the preparation of the following work the object has been to bring within tho compass of one volumo of con cnien s t ize an elementary treatise on both Conic Sections and Analytical Geometry. In the first part, the properties of the curves known as the Conio Scctions are demonstratcd, principally by geometrical methods that is, in the investigations, the curves and parts connected with them are conshntIy kept before the mind by thcir graphic representations, and we reason directly upon them. In the purcly Analytical Geometry the process is quite different, Here the geome trial magni tudes, themselves, or those having cer tain relations to thcm, arc represented by algebraic symbols, and we seek to express properties and imposed conditions by means of theso symbols. The mind is thus relieved, in a great measure, of thc nc ccssity of holding in view the often-timcs complcx figures required in the intermediate steps of the first method. It is, mainly, at tho beginning and end of our investigations that Fe ham to dcal with concrete quantity. That is, after we have expressed known and imposed conditions, analyticaIIy, our reasoning is independent of the kind of quantity involved, unt, il thc conclusion is rcachcd in the form of an algebreic expression, which must then receive its geometrical interpretation. Much of the value of Analytical Geometry, as a disciplinary study, will be derired from a careful consideration, in each case, of thia process of passing from the concrete to the abstract and the converse, and both teacher and student are earnestly recommended to-give it a large ahare of their at. tention...