514 



CARPENTRY. 



Construe 



PLATE 

 CXIV. 

 Fig. 1 1. 



Fig. 15. 



Fig. 16. 



Fig. 17. 



Fig. 18. 



- draw eg parallel to CD, and draw/g D, the point g Draw AF parallel to CD, and FD to AB ; divide 

 - r a parabola.^ In^ the same manner AC and AF, the former in/; and the latter in g, in the 



same proportion ; draw/A E and gkD, and the point 



all other points h,i are to be found. 



10. To describe the curve of a hyperbola, having h wilfbe'in the curve. ^In the same manner "ail other 

 a double ordinate AB, and a diameter DE, given in points z, k are to be found, 

 position, Fig 11. 



Demonstration for the ellipse and hyperbola, Fig. 12, 13. 



Let AB be the diameter, CD the ordinate, AC the abscissa, and K a point in the curve. 

 Draw KN and gM parallel to CD, cutting AC in N and M. 



Then because of the similar triangles . i? N ' BCG >' BN : BC * : : NK : CG 



|ANK,AMg-... AN: AM, erg E : : NK : Ms, or CD 

 And by construction gg . DE> or CA : : CG : CD 



CXIV. 



Therefore by multiplication we have BN X NA : BC X CA : : NK* CD 1 



A well known property of the ellipse. 



For the parabola, Fig. 14. 



By similar triangles ANK and AM- AN : AM : : NK : Mg, or CD 



By construction AM : AC : : CG, or NK : CD 



By multiplication 



The property of the parabola. 



11. Upon a given straight line AB., to describe 

 any regular polygon, Fig. 15. 



Produce the side AB to H, on AH, as a diameter, 

 describe the semicircle ACH ; divide the semicircum- 

 ference HCA into as many rqual parts as the poly- 

 gon is to have sides ; draw BC through the second 

 division ; bisect AB at K, and BC at L ; draw KI 

 at right angles to AB, and LI at right angles to 

 BC; from I, with the distance IA, IB, or 1C, de- 

 scribe the circle ABCD, &c. to A, which will con- 

 tain the side AB the number of times required. 



12. To cut off the angles of a square ABCD, so 

 as to form an octagon, Fig. 16. 



Draw the diagonals AC and BD, intersecting each 

 other at E ; through E, draw FG and HI parallel to 

 the sides ; make El, EG, EH, EF, each equal to 

 any half diagonal ED ; join IF, FH, HG, and GI, 

 cutting the sides of the square at P, Q, R, K, L, M, 

 N, O ; join KL, LM, MN, NO, OP, PQ, QR, RK, 

 and KLMNOPQR will be the octagon required. 



13. To inscribe an octagon in a square ABCD, 

 having four of its angles in the middle of the sides, 

 Fig. 17. 



Draw the diagonals AC and BD, cutting each 

 other at E ; through E draw FK and HM parallel 

 to the sides, cutting the sides of the square in the 

 middle at F, H, K, M; make EG, El, EL, EN, 

 each equal to the half side of the square ; join FG, 

 GH, HI, IK, KL, LM, MN, NF, FG, and FG 

 HIKLMN will be the octagon required. 



14. To cut off the angles of an oblong ABCD, 

 so as to form an elliptic octagon, Fig. 18. 



Draw the diagonals AC and BD, cutting each 

 other at E; draw FG and HI parallel to the sides ; 

 make FU and AK each equal to FA ; and join UK 

 and AU ; make UM equal to UF ; draw ML pa- 

 rallel to FK, cutting AD at L ; make FT equal to 

 FL, and GP and GQ each equal to FL ; draw LN 

 and QR parallel to DB ; PO and TS parallel to 

 AC ; then will LNOPQRST be the elliptic octagon 

 required. 



AN : AC : : 



NK 1 



CD- 



15. In a given oblong ABCD to inscribe an el- 

 liptic octagon, Fig. 19. Fig. l<>. 



Draw the diagonals AC and BD, cutting each 

 other at E ; make IK equal to IB, and join KB, 

 cutting FG at L ; make EM equal to LB, and EN 

 equal to KB ; OG, GR, RH, HQ, QF, EP, PI, 

 IO, and OIPFQHRGO will be the elliptic octagon 

 required. 



16- Through a given point K, to draw the circum- 

 ference of an ellipse concentric with a given ellipse 

 ABCDEFGH, Fig. 20. Fig. 20. 



Let I be the centre ; take any number of points A, 

 B, C, D, E, F, G, H, in the circumference of the 

 given ellipse ; draw the semidiameters AI, BI, CI, 

 DI, EI, FI, GI, HI, and the chords AB, BC, CD, 

 DE, EF, FG, GH, HA ; parallel to these draw 

 KL, LM, MN, NO, OP, PQ, QR, meeting the se- 

 midiameters, and through the points KLMNOPQR 

 to K draw a curve, which is the ellipse required. 



17. About a given rectangle ABCD, to circum- 

 scribe an ellipse which shall have its axis parallel to, 

 and in the same proportion as the sides of the rect- 

 angle, Fig. 21. Fig. 21. 



Draw the diagonals AC and DB, cutting each 

 other at N ; through N draw EF and GH parallel 

 to the sides of the rectangle ; produce GH to K, 

 making IK equal to I A, or IB ; join KA and KB ; 

 from K, with the distance KI, describe the arc LI, 

 cutting KA at L ; draw LM parallel to HG, cut- 

 ting AC at M ; join MO and MI ; draw AH paral- 

 lel to MI, and AE parallel to MO ; make NF equal 

 to NE, and NG equal to NH ; then EF is the trans- 

 verse axis, and HG the conjugate, by which the re- 

 quired ellipse may be described. 



18. To find the figure of the sines to any given 

 height, Fig. 22. Fig. 22. 



Describe a semicircle ABC, to a given height; 

 divide the quadrant AB into any number of equal 

 parts by the points 1, 2, 3 ; make De equal to A 1, 

 D/ equal to A 2, Dg equal to A3, and DH equal 

 to AB j draw the ordinates ei, fk, gl, HI, also the 



