CONSTRUCTIVE CA RPENTRY.
we enter upon the consideration of this impor tant subject, we must lirst lay before our readers a few preliminary problems of a geometrical nature, relating to angles, tangents, arcs of circles, elliptic, parabolic, and hyperbolic curves, circular and elliptic polygons, concentric ellipses, and other subjects which are abso lutely necessary to a proper understanding of the art of Carpentry. From these preliminary problems, the reader will be led to the Stercographiczd Prin ciples of Carpentry, which are also of indispensible use in Architecture, Joinery, and Ma:onry.
Preliminary Geometrical Problems 1. From a given point A, to draw a tangent to the arc of a circle BD, Plate CX IV. Fig. I.
Join the centre C and the point A ; on AC as a dia meter, 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 131) equal to BE, and draw AD, which is the tangent required.
3. To desclibe the segment of a circle, having the chord AB and the versed sine CD given in position, Fig. 3.
Produce DC to E, and make the angle AED equal to the angle EDA ; from E, with the radius El) or EA, describe an arc ADB, which completes the segment.
4. 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 GII to the other two, so as to keep the angle ADB invariable : Ilaving pia a pin at A and another at B, bring the angle at I) to A ; then move the apparatus so that the rod DE may slide upon the pin A, and DF upon the pin B, until the point I) ar rive at B, and the point or pencil at D will then have traced out the arc ADB. This apparatus is rather cum bersome ; in order, therefore, to perform the operation more conveniently, let the same data be given Fig. 5 ; join BD, and (haw DE parallel to AB; make DE at least equal to 1)11; form a triangle 13E1); put a pin at A and another at D ; move the triangle round, keeping the side DE upon A, and the side 1)13 upon 1), 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 Cry II,
COIWellient, and frequently impossible, to find a centre. These two last methods are well adapted for this pur pose, particularly the last, as it only requires half the distance at the ends of the chord that the former re quires ; but should there be no distance, or a very small space at the ends, the following method, by finding points, will then be most convenient.
5. The same things being given to find a number of points, in order to trace the path of the arc, Fig. 6.
Draw AE parallel to CI), and DE parallel to CA ; produce 1)E to F; join AI), and draw Al-' perpendicu lar to AD: divide AC and FD each into the same num ber of equal parts; from the points of division I, 2, 3, in F1), to the points of division 1, 2, 3, in AC, draw I a 1. 2 b 2, 3 c 3 ; also divide AE into the same number of equal parts; from the points of division 1, 2, 3, draw 1 a D, 2 b I), 3 c D, and trace the curve A a b c D, which will be the one half. The other half is found in the same manner.
6. To trace the curve of an ellipse through points, the transverse AB, and the semiconjugate axis CD, being given, Fig. 7.
Take a slip of paper, the edge of an ivory scale, or a rod of any convenient length, and mark the distance g c equal to the scmitransvcrse CA or C13, and the distance f equal to the semiconjugate CD.
In DC produced, take any point c, and apply the point c of the slip to the point e ; cause the slip to hale: an angular motion, until the point f fall upon the axis AB at f; then mark the point g on the plane of desciiption, and the point g will be in ute cut ye of an ellipse : in like manner all other points h, i arc to be found. If AB and Cl) were produced, and f made to move in AB, the point g will describe the curve BDA, which will be a semiellipse. This last method is the operation of the trammel.