A SERIES OF PROBLEMS, INTENDED 10 SAMILIARIZE THE PUPIL WITH GEOMETRIOAL OONCEPTIONS, AND TO EXERCISE HIS INVENTIVE FACULTY. BY WILLIAM GEORGE SPENCER. WITH A PREFATORY NOTE BY HERBERT SPENOER. NEW YORK ::: CINCINNATI ::: CHICAGO: PRIMER SERIES. SCIENCE PRIMERS. HYGIENE. UNITED STATES. HISTORY PRIMERS. LITERATURE PRIMERS. MAR EXERCISES. SPENCER-INV. GEOM. COPYRIGHT, 1876, BY D. APPLETON & CO. W. P. 13 INTRODUOTION When it is considered that by geometry the architect constructs our buildings, the civil engineer our railways; that by a higher kind of geometry, the surveyor makes a map of a a county or of a kingdom; that a geometry still higher is the foundation of the noble science of the astronomer, who by it not only determines the diameter of the globe he lives upon, but as well the sizes of the sun, moon, and planets, and their distances from us and from each other; when it is considered, also, that by this higher kind of geometry, with the assistance of a chart and a mariner's compass, the sailor navi. gates the ocean with success, and thus brings all nations into amicable intercourse-it will surely be allowed that its elements should be as acces sible as possible. Geometry may be divided into two parts-practical and theoretical: the practical bearing a similar relation to the theoretical that arithmetic does to algebra. And just as arithmetic is made to precede algebra, should practical ge ometry be made to precede theoretical geome. try. Arithmetic is not undervalued because it is inferior to algebra, nor ought practical geometry to be despised because theoretical geometry is the nobler of the two. However excellent arithmetic may be as an instrument for strengthening the intellectual powers, geometry is far more so; for as it is easier to see the relation of surface to surface and of line to line, than of one number to another, so it is easier to induce a habit of reasoning by means of geometry than it is by means of arithmetic. If taught judiciously, the collateral advantages of practical geometry are not inconsiderable. Besides introducing to our notice, in their proper order, many of the terms of the physical sciences, it offers the most favorable means of comprehending those terms, and impressing them upon the memory. It educates the hand to dexterity and neatness, the eye to accuracy of perception, and the judgment to the appreciation of beautiful forms. These advantages alone claim for it a place in the education of all, not excepting that of women. Had practical geometry been taught as arithmetic is taught, its value would scarcely have required insisting on. But the didactic method hitherto used in teaching it does not exhibit its powers to advantage. Any true geometrician who will teach practical geometry by definitions and questions there on, will find that he can thus create a far greater interest in the science than he can by the usual course; and, on adhering to the plan, he will perceive that it brings into earlier activity that highly-valuable but much-neglected power, the power to invent. It is this fact that has induced the author to choose as a suitable name for it, the inventional method of teaching prac tical geometry. He has diligently watched its effects on both nexes, and his experience enables him to say |