“Ως οἷόν τ' ἄρα, ἦν δ ̓ ἐγώ, μάλιστα προστακτέον, ὅπως οἱ ἐν τῇ καλλιπόλει σοι μηδενὶ τρόπῳ γεωμετρίας ἀφέξονται· πρὸς γὰρ πάσας μαθήσεις, ὥστε καλλίον ἀποδέχεσθαι, ἴσμεν που ὅτι τῷ ὅλῳ καὶ παντὶ διοίσει ἡμμένος . τε γεωμετρίας καὶ μή. τῷ παντὶ μέντοι νὴ Δί ̓, ἔφη.” PLATO, Repub. Bk. VII. 527. This was Biuine Plato his Eudgement, both of the purposed, chief, and perfect vse of Geometrie; and of his second, dependyng and deriuative commodities. And for vs, Christen men, a thousand thousand mo occasions are to haue nede of the helpe of Megethologicall Contemplations; wherby to trayne our Emaginations and Myndes, by litle and litle, to forsake and abandon the grosse and corruptible Objectes of our vtward senses: and to apprehend, by sure Doctrine Demonstrative, Things flathematicall. John Bee his Mathematicall Preface to Euclides Elementes. A.D. 1570. INTRODUCTION. THE Science of Geometry treats of the properties and construction of solids, surfaces, and lines. Plane Geometry treats only of the line and plane or flat surface; and the elements of Plane Geometry include the properties of the straight line and circle only, and of combinations of straight lines and circles. The science of Geometry is called deductive, because certain fundamental truths being assumed as obviously true, the remaining truths of the science are deduced from them by reasoning. Propositions admitted without demonstration are called Axioms. Of the Axioms used in Geometry those are termed General which are applicable to magnitudes of all kinds : the following is a list of the general axioms more frequently used. (a) The whole is greater than its part. (b) The whole is equal to the sum of its parts. (c) Things that are equal to the same thing are equal to one another. (d) If equals are added to equals the sums are equal. (e) If equals are taken from equals the remainders are equal. |