10. Second Method:-Let AB (fig. 7, pl. I,) be the length, and DD the breadth, of the ellipsis, and C the centre.

With the radius AC, and centre D, describe arcs, cutting AB in F, f. The points F, f, are called the foci of an ellipsis. Take any point, n, in the length AB, and with the radii, n A and nB, and centres F,ƒ, describe arcs, intersecting one another in M, then M is a point in the curve.

11. Hence, if a thread of the same length as the ellipsis have its ends fastened to two pins in the foci, F, f, and it be stretched to M, by moving a pencil round within the thread, so as to keep it uniformly stretched, the curve may be described.

12. Third Method. To describe an ellipsis by finding points in the curve. Divide AE and AF, (pg. 8,) each into the same number of equal parts, as five for example. Through the points of division, 1, 2, 3, &c., in AE, draw the lines Bh, Bi, Bk, &c.; and through the points of division, 1, 2, 3, &c., in AF, draw the lines 1D, 2D, 3D, &c., intersecting the former lines in the points h, i, k, &c. Through the points AhiklD, draw the curve, and it is one quarter of

the ellipsis required. In the same manner the other parts may be described.

To describe the False Ellipsis, or an Elliptical Figure, by means of Circular Arcs.

13. Let AB be the length, and CD the breadth, (fig. 9, pl. I.) Join BD, and make GD equal to the difference between DE and AE. Through the middle of the line BG draw ab perpendicular to BG, intersecting EB in f, and EC produced in b. From the centres ƒ and b describe the arcs /Bm, and mDn; and complete the curve in the same manner. Blackfriars' bridge has arches nearly of the same

This curve is frequently used for bridges.

figure as would be obtained by this method.

When the length is not above one-third greater than the breadth, the circles meet one another without the change of curvature being strongly marked; but when the length exceeds this proportion, a greater number of centres should be employed. The arch of the bridge of Neuilly was drawn from eleven centres; but it becomes more troublesome to draw a curve of good form by arcs of circles, than to draw it of the true elliptical figure, which is decidedly more beautiful. The arches of the Waterloo-bridge are ellipses.

To describe a Parabola.

14. First Method, by Tangents.-Let AC (fig. 10 or 11,) be the base, and ED the height. Produce ED to B, and make DB equal to DE. Join AB and BC; and divide AB into any even number of equal parts, numbering them from A to B; also divide BC into the same number of equal parts, numbering the parts from B to C. Join 1, 1; 2, 2; 3,3; &c., and the lines so drawn will be tangents to the parabola; and a curve, ADC, drawn to touch these tangents is the parabola required.

This curve is adapted for arches in some cases: and this method of drawing it is much used for rounding off angles, as will be shown in other parts of this Work.

15. Second Method, by Ordinates.-Let AC, (fig. 1, pl. II,) be the width, and ED the height of the arch. Make EC equal to EA, and complete the rectangle AFGC, so that the side FG may pass through D. Divide AE and AF each into the same number of equal parts; and join 1D, 2D, 3D, &c. from the divisions on AF. And from the divisions 1, 2, 3, &c., on AE, draw lines parallel to ED, meeting the former lines in the points h, i, k, &c., which are points in the curve.

The parabola answers very well for a Gothic arch when the line ED is made the springing

line.

An example is shown of its application to the head of a window, in fig. 2; the mode of describing the curve is the same as in fig. 1, and the figures of reference the same; but any of the other methods of describing the parabola will apply to the same purpose.

16. Third Method, by continued Motion.-Let GH (fig. 3,) be the edge of a straight ruler, and KLQ the internal angle of a square, of which the edge is parallel to KL, and coincides with the straight edge, GH. Then, if one end of a string be fastened at F, and the other end to the point Q of the square, and the side of the square be moved along GH, while the parts QM, FM, of the string are kept uniformly stretched by a pencil at M, the pencil will be moved and describe a parabola.

If AC be the breadth, and DE the height of the curve, the point F may be found by drawing a line from D to a, the middle of EC: and make ab perpendicular to a D, intersecting DE produced in b, then make DF equal to Eb. The length of the string must not be less than the line Db; and GH should be parallel to AC, and at any convenient distance from D.

To describe a Hyperbola.

17. In this figure (see fig. 4, pl. II,) the degree of curvature is not fixed by the height and width of the arch, but is capable of every degree of variation between the curvature of the parabola and the straight lines of a triangle. This variation depends on the position of the point B; for the nearer that point is to D, the nearer the figure will be to a triangle; and the more distant the point B is from D, the nearer the curve will be to a parabola.

To draw the curve, divide AF and AE, each into the same number of equal parts; and from the points of division 1, 2, 3, &c. on AF, draw lines to D. Also, from the points of division 1, 2, 3, &c. on AE, draw lines to the point B, cutting the former lines in the points h, i, k, &c. Through the points A, h, i, k, &c. draw a curve, and it will be the hyperbola required.

To describe the Sections of a Cone by a general Method.

18. The ellipsis, parabola, and hyperbola, are curves formed by cutting a cone in different directions in respect to its sides; hence they are sometimes called conic sections; but as these figures are formed by various operations both of nature and art, it seems improper to name them from any particular ones.

Let FI (figs. 5, 6, or 7,) be a line drawn through the foci of the curve, and A the vertex or top of the curve. Make AI equal to FA; and when the curve has two foci, as in the hyperbola, fig. 7, and the ellipsis, fig. 6, from the focus f, as a centre, with the radius fI, describe an arc QI.

To find any point, M, in the curve, draw fQM, and join QF; also from a, the middle of QF, draw a M perpendicular to QF, and it will meet ƒQM in the point M, a point in the curve.

In the parabola, (fig. 5,) as there is only one focus, draw QI perpendicular to IF; and to find any point M in the curve, draw QM parallel to IF; join FQ, and a M being drawn perpendicular to QF, it will meet QM in M, a point in the curve; and any other point may be found in the same manner.*

In all the cases FM is equal to MQ, and Ma is a tangent to the curve at the point M; also, a line drawn perpendicular to Ma would be the proper direction for a joint at M, in a brick or stone arch, of any of these forms.

* This method of drawing the sections of a cone was ascribed to Mr. Gibson; but his was not complete, and only differed in want of completeness from methods described in Emerson's Conics

To describe Gothic Arches.

19. A Gothic arch is generally composed of a curve, which has different degrees of curvature at different points; and a graceful curve of this kind cannot be produced by circular arcs; neither is it easy to describe them for the flat parts of the arch. To avoid this difficuity, we have used the following simple instrument for several years.

CB (fig. 8, pl. II.) is a bar of wood of equal breadth and thickness, which is straight, as shown by the dotted lines C'B, when the string, Ca, is loose. The bar CB is fixed at one end into a strong piece, AB, of equal thickness to the breadth of the bar CB. The piece AB is provided with a groove on each side to receive a button, a, with a flat head to fix the string to, when the bar CB is to be retained at any degree of curvature. The part b is added to prevent the bar curving below the line AB.

To use the instrument, set the line AB to the springing-line of the arch, with the point B adjusted to the line of the jamb, and bend the bar CB by the string till some point on its upper edge coincides with the height of the arch, and the string should be adjusted so as to be perpendicular to the line AB, by means of the sliding button; then, the upper edge is of the proper form for the arch; and by turning over the instrument, the other half of the arch may be described.

20. A Gothic arch may also be described by points in this manner: divide the base AE into eight equal parts, (see fig. 9,) and on each point of division draw a perpendicular. Then, divide ED, the height, into 100 equal parts, and make 7g equal to 96 of these parts; 6f equal 91 parts. 5e equal 86 parts; 4d equal 79 parts; 3c equal 72 parts; 2b equal 63 parts; and 1a equal 50 parts. Through the points Aabcdefg D draw the curve.

The example shows the head of a Gothic window, the arches of which may be described either by this method or the preceding one.

21. To describe a Gothic Arch by Arcs of Circles.-Let AB (fig. 1, pl. III.) be the springing-line, and EC the height of the arch. Draw BD perpendicular to AB, and make it equal to

two-thirds of the height EC. Join DC, and from C draw CH perpendicular to CD. Make BF and CG each equal to BD. Join FG; and from the middle of FG, draw a H perpendicular to FG, meeting CH in H. Then, F and H are the centres for describing the curve, and the two arcs will meet in the line HFb, which passes through their centres.

When a line drawn from A to C is equal to the width AB, the point H coincides with the point A, and the arch is drawn from one centre in A. Also, when the height is in any proportion greater than that which makes the line AC equal to AB, the arch may be described from one centre in the line AB continued.

If this rule be compared with the remains of the best examples of Gothic architecture in this country, it will be found to nearly agree with them.

TRANSFERRING CURVES.

22. It often happens in making drawings, that a complex curve is to be transferred from one drawing to another. A very convenient instrument for making such transfers, has lately been invented by Mr. WARCUP.* Figs. 4, 5, 6, and 7, (pl. III,) represent this instrument and its parts. AA, (fig. 4,) is a plain slip of whalebone, forming the ruler; BB, a series of graduated

"Transactions of the Society of Arts." Vol. XXXV, p. 109.

C