Elements of Plane and Solid Free-Hand Geometrical Drawing, with Lettering; And Some Elements of Geometrical Ornamental Design, Including the Principals of Harmonic Angular Ratios, Etc (Paperback)

Purchase of this book includes free trial access to www.million-books.com where you can read more than a million books for free. This is an OCR edition with typos. Excerpt from book: CHAPTER II. HLEMENTARY AND PRACTICAL EXERCISES ON RIGHT ANGL38. Principles. Beauty of form, considered as residing in certain geometrical properties of regular figures, results from certain proportions between their parts. These proportions may be regarded as arising from the relati/oe lengths of the distinguishing lines of the objects; or from the relative sises of their angles. In moving, whether to walk, or to merely draw a line, we must begin each movement at a given point. The direction of our movement is first in our thoughts, rather than its extent. "We first, if not of tenest, think, or ask, " which way " than " how far." Direction is therefore a more primary idea than length. An angle, however, is merely difference between directions from a certain point. Hence angular proportions, or the proportion between the angles of a figure, are more elementary than linear proportions, or those between the lengths of the lines of the figure, and will be first considered. In doing this it will be convenient to find first some angle as a natural standard of comparison for all others, and this we now proceed to do. When, then, two lines are so situated that, in moving on one of them, we do not at all move in the direction of the other, their directions are said to be independent.. C f 9 Thus, in these figures, by going from a to b, we find ourselves at the distance ac to the right of a. So by moving from d to e we go a distance equal to df in the direction of the line df. But, when the two angles formed by the meeting lines are equal, as at mgh and kgh, we do not, in moving to any distance on gh, progress at all to the right or left of gh. Hence thedirections of gh and mk are independent, and the angle included between them is the natural standard with which to com...