1869.] Practical Carpentry and Joinery. 443 C and P, respectively, for centres, with C I, and P G-, respectively, for radii, describe the arcs A I, and B Gr; thus completing the ellipsis. Having the Chord, to construct the Cen- tering for an Egg-shaped Culvert. Pig. 2. Take the chord A B, and di- viding it into two equal parts, strike a semi-circle from the centre. On the chord, as a base, construct a reversed equilateral triangle A H B, the apes of which will be the centre of the circle H, the radius of which is one-third of a line H C drawn from this centre to the for- mer. Prolong the respective sides of the equilateral triangle ; and taking A, B, alternately, as centres, and the length of the chord as radii, describe the two arcs ; and thus unite the circle with the semi-circle, and complete the figure. Another Method : — With half the length of the chord A B as a radius, D C, describe a semi-circle A D B ; and with the whole chord A B as a radius, describe an arc A P ; and with B A for a radius, describe an arc B I. On the chord A B construct a reversed equi- lateral triangle A H B ; produce the sides A H and B H to F and I respect- ively. Then, taking H P as a radius, connect the arcs A P and B I by the small arc P I. A Third Method: — Divide the chord A B into five equal parts. Apply four of these parts to the line D H ; one of these parts will be the radius H P. The semi-circle segments and small circle will be all struck as before. Fig. 3. — Having the Chord and Height of the Segment of a Circle, to describe the Segment without finding the Centre. Join D B, and draw D H parallel to B A. Apply the trammel, so that its angle may coincide with that of the figure ; previously putting pins at the points A and B, and sliding the tram- mel along them. A pencil at D will describe half the arc ; and by moving the pin out of B and putting it in A the other half will be described in the same manner. Fig. 4. Another method: — Let A B be the chord of the segment, and C D its height. Join D A and D B ; stick a pin at A and another at B. Now take a trammel and fix a pencil point at its apex and apply it on D ; slide the tram- mel along the pins A and B, keeping the sides close up to the pins, and the pencil point will correctly mark out the required segment. The trammel above mentioned is thus formed : Put together two slips of wood, so as to form an angle equal to A D B, each of the sides pro- longed to the length of the chord A B ; secure this angle by a cross-piece, and the trammel is complete. It is formed so that the segment of a circle will be equal to that of Fig. 3 or 4 ; and is made practical when the circle is of great diameter. Either of these tram- mels may be applied in flat segments ; but the latter, by reason of the obtuse- ness of the angle, slides with less fric- tion along the pins, takes up much less room, and can be applied in all cases, whereas in many it is impossible to use the other. If the height of the arch is very small, the instrument may be in one piece. To describe an Octagon within a Square. Fig. 5. Draw the diagonal A D in the given square ; and taking one-half of it for a radius, with A for a centre, describe a segment. Do the same at the other three centres, B and D ; cut off the angles of the square by joining the points ; and the octagon is complete. Having one Centre given for an Un- equal-sided Groin, to form the Other, so that the Intersection thereof shall produce the Angle, or Mitre-Arch, to hang perpendicularly over its Base, and, moreover, to draw the Centre thereof. Fig. 6. Draw the lines A B and B D, and D and C A, each equal to one another, to represent the walls from
