CARPENTRY. 



515 





23. 



otutnic- ordinates or sines to AC { make eiffk, pi, HI, re- 

 spectively equal to the sines. In the same manner 

 _ iy the half figure QRS be found, by taking the 

 ordinates of the ellipse for the ordinates of the stmi- 

 circle. 



This Figure is very frequently required in archi- 

 tecture. The coverings ot cylinders, and domes of 

 every species, are of this form. It may be drawn to 

 any given proportion, by dividing a quadrantal arc 

 into any number of equal parts, and the half base into 

 the same number of equal parts, arid taking the sines 

 as ordinates in the Figure. 



1 H. To construct a trapezium equal and similar to 

 a given trapezium. Fig. 23. 



Let No. 1. be the given trapezium, which divide 

 into two triangles ABC, ACD by a diagonal AC ; 

 make the triangle ABC, No. 2. equal to the triangle 

 ABC, No. 1. (Euclid, book i. prop. 23.) ; and the 

 triangle ACD, No. 2. equal to the triangle ACD, 

 No. 1 ; then will the trapezium ABCD, No. 2. be 

 equal to the trapezium ABCD, No. 1. 



In this manner may any rectilineal figure be made 

 equal and similar to any other given rectilineal figure, 

 by resolving the given figure into triangles, and con- 

 structing these one after the other in the successive 

 order of their contiguity, until all the triangles of the 

 original figure have been constructed in the required 

 figure, and the result will be a polygon equal and si- 

 milar to the given figure. 



20. To construct a rectilineal figure similar to a 

 given rectilineal figure ABCDEFGHA, Fig. 24-. 

 having one of the sides of the required figure corre- 

 sponding to a side of the given figure. 



Make A b, No. 1. equal to the side of the required 

 figure corresponding to AB, the side of the given 

 figure ; draw the diagonals AC, AD, AE, AF, and 

 AG ; make bc> cd, ae, ef, fg, and gh, respectively 

 parallel to BC, CD, DE, EF, FG, and GH, from 

 one diagonal to the other. By the last problem con- 

 struct the rectilineal figure abcdefgha, No. 1. simi- 

 lar and equal to the rectilineal figure AbcdefghA, 

 No. 1.; and abcdefg/in, No. 2. will be equal and 

 similar to ABCDEFGHA, No. 1. 



By means of these problems, it will be easy to 

 adapt a piece of framing to any place required. 



21. Given two lines AB, CD, inclining towards 

 each other, to draw a line from a given point A in 

 one of them, so as to make equal angles with both 

 lines, Fig. 25. 



Through A draw EF parallel to CD ; bisect the 

 angle BAF by AC, and the angles BAG and DCA 

 will be equal. 



BOOK II. 



Pg. 24. 



"gra- 

 phical prin' 

 ciplcs of 

 carpentry. 



Stereographical Principles of Carpentry. 



STEREOGRAPHY is that branch of knowledge which 

 demonstrates the properties, and teaches the whole 

 doctrine of regularly defined solids. It explains the 

 rules for constructing the superficies in piano, so as 

 to form the entire solid, or to cover its surface. It 

 also bhcwi how to form any section thereof, or to 

 had any angles or inclinations relating to two or 



more of its surfaces ; or any angles upon iny one of 

 iti surfaces, formed by a sectional line and adjacent . . 

 side of the surface, by having tin- proper data given l 

 according to what may bt- required. 



Mr Hamilton has denominated the principles of 

 perspective by the term ttercography, contrary to the 

 usage of other authors. Perspective is only a branch 

 of the doctrine of solids ; and all that this branch 

 teaches, is only the methods for finding the section* 

 of pyramids and cones, the eye being considered at 

 the vertex, the original object the base of the pyra- 

 mid or cone, and the picture to be drawn a section 

 thereof; the term is therefore of too general appli- 

 cation, perspective being only a branch of stereogra- 



The eleventh and twelfth books of the Element* of 

 Euclid belong to stereography : these may be looked 

 upon as the theory of the doctrine of solids, and to 

 them we shall refer our readers for the original pro- 

 perties ; but for ttieir practical applications to useful 

 properties in life, it is rather singular that to little 

 has been done in this respect. The present article 

 is entirely new. It is of the greatest importance in 

 the various mechanical departments of architecture. 

 The geometrical principles in masonry, carpentry, 

 joinery, and the other useful branches of the build- 

 ing art, are entirely dependent upon it : in short, the 

 cutting of individual pieces of timber in the art of 

 carpentry, and the formation of separate stones in 

 masonry, is only the application of stereography to 

 practice. 



To the acquirement of these arts this branch of 



feometry is therefore indispensible ; and, as it is a 

 ey to the whole, no farther apology for its intro- 

 duction is necessary. 



DEFINITIONS OF SOLIDS, AND THEIR PROPERTIES. 



A prism is a solid, the ends of which are similar Prim. 

 and equal parallel plane figures, and the sides paral- 

 lelograms ; and if the ends of the prism are perpendi- 

 cular to the sides, the prism is called a right prism, 

 but if otherwise, it is termed oblique : If the sides 

 and ends are equal squares, the prism is called a 

 cube ; and if the base or ends are parallelograms, 

 the prism is called a parallelepiped : If all the planes 

 of the parallelepiped are at right angles to each 

 other, then the prism is called a rectangular prism : 

 If the ends of the prism are circles, the prism is call- 

 ed a cylinder ; but if they are ellipses, it is called a 

 cylindroid. 



AH the sections of a cylinder, or cylindroid, are 

 either circles, ellipses, or parallelograms, excepting 

 when cut partly through the sides, and partly through 

 the ends, and then they are portions of circles or el- 

 lipses. 



All parallel sections of a prism are equal and simi- 

 lar polygons. 



All sections whatever, except partly through the 

 base, and partly through the sides of a parallelepi- 

 ped, are parallelograms. 



All sections of a cylinder, or cylindroid, through 

 the curved surface, are either ellipses or circles ; and 

 all sections parallel to the axis are parallelograms. 



