Schools of Design and Practical Art*. Let me urge upon the managers of such Schools the expediency of beginning their work at the right end. Let principles be taught before rules. Let Geometry as an Art be systematically preceded by Geometry as a Science. Then, but not till then, we may hope to see the desired result in the improved taste and skill of our designers, and to be saved the continuance of that sense of humiliation which every Englishman must experience on reading the statement here subjoined. T. L. MORTON RECTORY, ALFRETON, April 20, 1854. On a late public occasion, at the inauguration of one of these Schools, the Duke of Argyll remarked that "a very large proportion of the works of art preparing for the Crystal Palace are being executed almost entirely by foreign artists, and that our manufacturers also have been obliged to send abroad for designs; and, as he was convinced that there was no natural disqualification in our population for such work, he trusted that the defect would be remedied by the adoption of a more complete system of education." By the same Author, 1. WOOD'S ALGEBRA, much enlarged and improved, with numerous Examples and Easy Exercises. Fourteenth Edition, 12s. 6d. boards. 2. COMPANION to WOOD'S ALGEBRA, containing SOLUTIONS of all the difficult Questions and Problems in the former work. Second Edition, 6s. boards. 3. A SHORT AND EASY COURSE OF ALGEBRA, with Easy Exercises. Second Edition, 3s. 6d. boards. 4. KEY to the EXERCISES in the Short and Easy Course of Algebra, 5s. 5. The NECESSITY of a STUDIOUS and LEARNED CLERGY. A Visitation Sermon, preached at the primary visitation of John, Lord Bishop of Lichfield, ls. ELEMENTS OF GEOMETRY AND MENSURATION. PART I. GEOMETRY TREATED AS AN EXACT SCIENCE. DEFINITIONS AND FIRST PRINCIPLES. 1. GEOMETRY* is the science which treats of the form and extent of bodies, and also of the relations which different bodies, or different parts of the same body, bear to each other in respect of magnitude, or position. Hence it is necessarily much concerned with the terms 'length', 'breadth', 'height', 'depth', 'thickness', 'area', 'content', or 'volume'; and that which does not possess some one or more of these properties is not a subject for Geometry. For instance, to determine how many quarts of water there are in a vessel which we can measure is within the province of Geometry; but to determine what will become of the water, if a fire be placed under the vessel, is beside the subject. So likewise the size of a certain carpet is a subject for Geometry, but not the colour of the carpet, or the material of which it is made. 2. It is clear then, that much of our business, as Geometricians, is with the boundaries of bodies, (that is, with whatever bounds their extent) for upon these the magnitude of every body manifestly depends, as well as its form and other properties relating thereto. We shall have to do with more than boundaries in certain cases; but whatever be the subject of our inquiry, it will always have reference to magnitude and position, one or both, to the exclusion of all dissimilar properties, such as colour, softness, hardness, &c. The word Geometry strictly signifies 'land-measuring'; and is supposed to have had its origin in the necessity of re-arranging the fences or other land-marks destroyed by the periodical overflowings of the river Nile. It is clear also that, having to treat of bodies, or parts of bodies, in respect of magnitude and position, we have to provide for taking measurements of various kinds; and hence is required a sort of geometrical language in the first onset, which must be learnt from the following Definitions: 3. We measure a distance by a 'line'; so that a line will represent any one of the dimensions length, breadth, height, girth, depth, or thickness. We do not inquire as to the thickness of the line, when used for this purpose of measurement. Hence the common DEFINITION. A LINE is length without breadth or thickness. It is not meant that any line we can actually use or make is without breadth or thickness; but that for Geometrical purposes, that is, as a measure of length, the length only of a line is considered. Thus, for illustration, if the length of a room be in question, we regard not the fact of its being measured by a broad tape or a narrow tape-even the finest thread we can use will serve our purpose, if it be inextensible, -we expect the same result in each case, because it is length only we are concerned with. In the case here supposed, the broad tape is not inferior to the finest thread; but, as there are numberless other cases in which this is not so, (as will appear hereafter), the Definition of a 'line' above given is the only one which can insure general accuracy of measurement. 4. Another term in common use in Geometry is 'point', by which is meant generally no more than a place to start from, or to stop at, in drawing or measuring a line. A point hath position only, and is nothing for us to measure; and hence the common tude. DEFINITION. A POINT hath no parts and no magni It is true we cannot exhibit such a point, (because that which hath no magnitude cannot be visible to the human eye); but the more nearly the points we use in practice approach the strictness of this Definition, the more accurate, it is obvious, will be the measurements which begin or end at those points. |