Page images





Giving Rules for the Delineation and Application of varions
Geometrical Lines, Figures, and Curves.

[merged small][merged small][ocr errors][merged small]





183. e. 30.




THE PRESENT TREATISE is written for the purpose of providing the practical man with some simple rules for the delineation of various geometrical figures and for their application to useful and ornamental purposes. In a work upon 'Practical Geometry' no demonstration is given of the rules laid down, the results only of scientific investigation being explained.

Geometrical figures were undoubtedly made use of for building and other purposes long before any demonstrations or investigations of their properties were attempted; since we find examples in ancient temples of the use of parallels, right angles, oblongs, squares, circles, conic sections, spirals, and other figures, long before any treatises upon geometry were written to investigate their properties. When, however, logical deduction was brought to bear upon geometry, and it was thereby formed into a 'science,' a large number of valuable properties were discovered that were hitherto unknown, and the modern application of analysis to geometry has greatly extended our knowledge of this science.

In the following pages all abstruse formulæ or complicated methods have been avoided, the rules laid down being rendered as simple as possible so as to enable persons with very moderate knowledge of geometry to work out the

various problems and processes with the simplest instruments usually at command.

In the first chapter the various combinations of STRAIGHT LINES are given, the circle being used, however, as a necessary aid to the ready solution of the problems.

The second chapter is devoted entirely to the CIRCLE and the various problems that arise in connection therewith, including its application to architectural purposes as well as to the drawing of polygonal figures.

The curves known as the CONIC SECTIONS are treated in the third chapter; the object being to enable the practical man to apply those curves with the greatest possible facility; and several of their uses in architecture are also suggested. The curve formed by a rope or chain suspended at two points, called the CATENARY, is introduced at the end of this chapter, although in no way connected with the conic sections.

In the fourth chapter some of the curves known to mathematicians as CURVES OF FLEXURE are discussed, and rules given for drawing and applying them to architectural purposes. These curves have the form of the letter S or of the figure 8, and are frequently employed for the contour of mouldings and arches. It is, however, too often the case, that instead of a true mathematical curve being employed, a rough imitation is substituted by drawing two arcs of circles turned in opposite directions and meeting at the line joining their centres. This method is, however, quite contrary to the principle of curves of flexure in which the curvature becomes flatter and flatter as the point of contrary flexure is approached, and merges into a straight line at the actual bending point. Such a curve cannot

therefore be imitated by arcs of circles, and it is far better

to draw it by hand through points' found upon its contour, unless a method of continuous motion' can be readily employed.

The fifth chapter is devoted to a large and important class of curves known as SPIRALS. Some few of these can be drawn by continuous motion,' but in all cases methods of finding 'points,' and of drawing the curve through those points either by hand or by the compasses are carefully laid down; so that they can be as accurately drawn as is necessary in practice by aid of a drawing board, T square, set squares, and a scale of parts. On account of the frequent use of spirals in decorations, the author has been at considerable pains to make the delineation of them in great variety as easy as possible to those who have but small knowledge of mathematical science.

The sixth chapter explains the methods that may be employed for describing and applying the curves known under the name of CYCLOIDS, some of which may be used in architecture and in the mechanical arts.

There are numerous other curves known to mathematicians which have not been touched upon in the present work, the author having confined himself to those which appeared to him of most practical utility to the class of readers for whom it was intended.

« PreviousContinue »