Lectures on the Principles of Demonstrative Mathematics (Paperback)

This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1843. Excerpt: ... LECTURE II. ON THE PRINCIPLES OF DEMONSTRATIVE GEOMETRY. Geometry, the mainspring of sound education--subject--the properties of space--how called up in the mind--externals suggestive--their agency neither to be confounded with the assistance they afford, nor with the application which is made of geometry to them--space defined--termination--extension--composition--comparison--elements of demonstration--definitions of two kinds--requisites for a good definition--postulates--axioms of two kinds--principle of comparison by superposition--the superiority of equality to congruity asserted--Proclus defended against Barrow and Simson--equality the general of which congruity is the particular mode of comparison in geometry. When first Discipline of the Mind commended itself to men, it assumed the form of Mathematical Demonstration. Whatever might have been known of the rudiments of grammar, of rhetoric, and of logic, in early times, it is exceedingly improbable that they formed any part of the ground-work of education. It is certain, indeed, that very little was known of these sciences in the age which preceded Plato; and perhaps we shall not greatly err if we descend a step lower, and refer their origin in a systematic form, as we do their perfection, to Aristotle.f On the other hand, we have the testimony of the most illustrious philosophers, to the effect that, in the best days of Grecian intelligence, the mathematical sciences formed the basis of instruction. They are styled by Plato " the primary education," " the steps to knowledge;" by Xenocrates, " the handles of philosophy;"t by Timasus, " the way to education." And truly, in reflecting on the blaze of intellectual light at the time we refer to, its gradual extinction, its protracted dawn at the break o...