CARPENTRY. 



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PART II. CONSTRUCTIVE CARPENTRY. 



IV. 

 1. 



we enter upon the consideration of this 

 important subject, we must first lay before our 

 readers a few preliminary problems of a geometrical 

 nature, relating to angles, tangents, arcs of circlet, 

 elliptic, parabolic, and hyperbolic curves, circular 

 and elliptic polygons, concentric ellipses, and other 

 subjects which are absolutely necessary to a proper 

 understanding of the art of Carpentry. From these 

 preliminary problems, the reader will be naturally led 

 to the Stereographical Principles of Carpentry, which 

 are also of indispensible use in Architecture, Joinery, 

 and Masonry. 



BOOK I. 



Preliminary Geometrical Problems. 



1. From a given point A, to draw a tangent to 

 the arc of a circle BD, Fig. 1. 



Join the centre C and the point A ; on AC as a 

 diameter, describe a semicircle CBA ; draw BA, 

 which is the tangent required. 



2. From a given point A in an arc ABC, to draw 

 a tangent, Fig. 2. without having recourse to the 

 centre. 



From the point A, with any radius, describe the 

 arc DBE, and with the same radius from B, describe 

 an arc at C ; join AEC ; make BD equal to BE, 

 and draw AD, which is the tangent required. 



3. To describe the segment of a circle, having the 

 chord AB and the versed sine CD given in posi- 

 tion, Fig. 3. 



Produce DC to E, and make the angle AED equal 

 to the angle EDA ; from E, with the radius ED or 

 EA, describe an arc ADB, which completes the seg- 

 ment. 



4k To describe a segment by means of an angle, 

 the chord AB and versed sine CD being given in 

 position, Fig. 4-. 



Fasten two rods DE and DF together at D, so 

 as to make the angle ADB, each rod not being less 

 than the chord AB ; fasten the rod GH to the other 

 two 30 as to keep the angle ADB invariable : Ha- 

 ving put a pin at A and another at B, bring the 

 angle at D to A; then move the apparatus so that 

 the rod DE may slide upon the pin A, and DF up. 

 on the pin B, until the point D arrive at B, and the 

 point or pencil at D will then have traced out the 

 arc ADB. This apparatus is rather cumbersome ; 

 in order, therefore, to perform the operation more 

 conveniently, let the same data be given Fig. 5; 

 join BD, and draw DE parallel to AB; make DE 

 at least equal to DB ; form a triangle BED; put 

 a pin at A and another at D; move the triangle 

 round, keeping the side DE upon A, and the side 

 DB upon D, until the point B arrive at A ; and this 

 will describe one half of the segment, the other half 

 will of course be described in the same manner. In 

 many situations it is very inconvenient, and frequent- 

 ly impossible, to find a centre. These two last me- 

 thods are well adapted for this purpose, particularly 



VOL. V. PA.RT II. 



COMIX a-. 



the last, a it only require! half the duUnce it the j^^" 

 ends of the chord that the former require! ; but tXIY. 

 should there be no distance, or a very small ipace 

 at the ends, the following method, by finding points, 

 will then be most convenient. 



5. The same thing* being given to find a number 



of points, in order to trace the path of the arc, Fig.G. pif. . 



Draw AE parallel to CD, and DE parallel to 

 CA; produce DE to F; join AD, and draw AF 

 perpendicular to AD : divide AC and FD each into 

 the same number of equal parts; from the points of 

 division 1 , 2, 3, in FD, to the points of division 1 , 2, 3, 

 in AC, draw 1 a 1, 2 A 2, 3c3; also divide AE into 

 the same number of equal parts ; from the points of 

 division 1, 2, 3, draw 1 a D, 2 6 D, 3 c D, and trace 

 the curve AaicD, which will be the one half. The 

 other half is fourd in the same manner. 



6. To trace the curve of an ellipse through pointl, 

 the transverse AB, and semiconjugate axis CD, be- 

 ing given, Fig. 7. Fif. r. 



Take a slip of paper, the edge of an ivory scale, 

 or a rod of any convenient length, and mark the dis- 

 tance ge equal to the semitransverse CA or CB, and 

 the distance gj" equal to the semiconjugate CD. 



In DC produced, take any point e, and apply the 

 point e of the slip to the point e ; cause the slip to 

 have an angular motion, until the point f fall upon 

 the axis AB at^; then mark the point e on the 

 plane of description, and the point g will be in the 

 curve of an ellipse: in like manner all other points A,i 

 are to be found. If AB and CD were produced, and 

 f made to move in AB, the point g will describe the 

 curve BD A, which will be a semiellipse. Thi last 

 method is the operation of the trammel. 



7. The same things being given to find the curre, 



by another method by points, Fig. 8. Fir. 8. 



Draw DE parallel to CB, and BE parallel to CD ; 

 divide BC and BE proportionally at g and/; make 

 CI equal to CD; draw Igh and /AD, and the 

 point h will be in the curve. In the same manner 

 may other points m, n be found. Perhaps to find the 

 points in, h, n, it would be easier to some to divide the 

 lines BC and BE each into the same number of equal 

 parts. 



8. The same things being still given to describe 



the representation with a compass, Fig. 9. fig. . 



Take half the conjugate CD, which apply on the 

 semitransverse from A to E ; divide EC into three 

 equal parts, and set one towards A from E to F ; 

 make CG equal to CF; with the distance FG de- 

 scribe an equilateral triangle FHG ; produce HFto 

 I, and HG to K | from H, with the distance HD, 

 describe an arc IK ; from G, with the distance GK, 

 describe an arc KB, and with the radius IF de- 

 scribe an arc I A ; and A ID KB will be the curve of 

 the semiellipse required. 



9. To describe the curve of a parabola, having the 

 double ordinate AB, and abscissa CD given in posi- 

 tion, Fig. 10. fig. ia 



Draw AH parallel to CD, and HD to AC ; di- 

 vide AC and AH in the same proportion at e and/"; 

 ST 



