with c as a centre and a radius equal to d'h, describe arcs crossing the line drawn through e in c, and c,,, and the line drawn through d in d, and d; again, with c as a centre and a radius equal to gm, describe arcs crossing the line drawn through fin f, and f; and so on. Curve lines a, c, &c., a,, c,, &c., drawn as in the figure, will be the required projections of the two curves in which the cylinder intersects the sphere.

The two curve lines are evidently similar to one another; they will touch the circumference of the circle A B in certain points near c, and d, e,, and d,,; and it is evident that, if the eye of a spectator were above the plane Yox, the parts c, b, d, and c,, b, d, would be invisible.

If the axis of the cylinder pass through the centre of the sphere, the curves of intersection will evidently be circles, and their projections on XOY will be straight lines.

236. The following problems relate to the orthogonal projections, on vertical and horizontal planes, of the figures given to the voussoirs, or wrought stones which are employed in the construction of vaults of the most usual forms.

In the formation of voussoirs, it is of the utmost importance, when the mutual pressures of the stones are considerable, that the plane angles composing any solid angle should be either accurately, or as nearly as possible, right angles, or that the edges which are in the directions of the thickness of a vault should be in the positions of normals to the curve surface of the vault; for if the angles which the faces in contact, or joints, of any two voussoirs make with the faces which meet them are unequal, one of them, being less than a right angle, will be weaker than the other, and consequently, at the line of meeting, the stone is in danger of being splintered.

In simple vaults, such as the usual arches of a bridge, if the thickness be considered as equal throughout, the concave and the convex faces of each voussoir are parts of the cylindrical surfaces, circular or elliptical, of the vault; the contact faces or joints are planes, two of them perpendicular to the axes of the cylinder, and the two others normals to its surface, so that if the cylinder were circular, the planes of the last faces would pass through the axis. If a vault have the form of an upright cone, with a circular base, and be of equal thickness throughout, the concave and convex faces of each voussoir are parts of the conical surfaces; two of the joints are planes which, if produced, would pass through the axis of the vault, and each of the two others is a portion of the curve surface of a cone the circumference of whose base is the exterior edge of the horizontal course of voussoirs of

which the one considered is a part, and whose vertex is in the axis, all lines imagined to be drawn on these conical surfaces from the base to the vertex being normals to both surfaces of the vault. Again, if the vault be a spherical dome everywhere of equal thickness, the concave and convex faces of each voussoir will be portions of the spherical surfaces, while the joints will be similar to the corresponding joints of a conical vault, except that those which have the conical form will have the centre of the dome for their common vertex.

PROPOSITION XXII.

237. To describe the projections, on a vertical and a horizontal plane, of the key-voussoir in a hemicylindrical vault, the vertical plane of projection being perpendicular to the axis of the cylinder, which, for simplicity, is supposed to be circular.

a

A

K

B

C

Let HK on Zox be a vertical line meeting the axis of the vault in H; then I will be the centre of the circular arcs CD and AB, on the concave and convex faces of the voussoir. The radii HC, HA are given; therefore the arcs can be described. The chord of half the arc CD or AB being given, when set on both sides of HK, it will determine the points through which the radii HA, HB are to be drawn for the sides of the voussoir. The lines aa', cc', &c. drawn, as in the figure, through A, C, &c. perpendicularly to ox, if cut by the lines ab, a'b' parallel to ox at distances from one another equal to the given depth of the voussoir, will determine the projection of that voussoir on the horizontal plane XOY.

C

If the cylindrical vault were elliptical, and if the arcs CD and AB had the same centre of curvature, the point I would have been the intersection of the produced sides AC and BD, each of which would have been in the direction of a normal to either of those arcs.

PROPOSITION XXIII.

238. To describe the projections, on a vertical and a horizontal plane, of a voussoir forming part of a hemispherical dome.

B

E

Let ABCD be a vertical section through such a voussoir in a plane zox, on which the vertical projection is to be made; let HV be the axis, and H the centre of the dome. Through A, C, B, D draw lines, as in the figure, perpendicular to HV; these lines will represent the intersections

on zox of the planes of the circles between which the horizontal course of voussoirs, of which ABCD is a section, is contained.

d'

a a

D

K

From any convenient point as a centre in the line ví produced, and with radii equal to the perpendicular distances of A, C, B, D from HV, draw the arcs at A', c', &c.; these will be the projections on XOY of parts of the circles just mentioned.* Then a straight line A'F, being made equal to the given chord of the voussoir on the circumference of the circle passing through A' (half the chord being set on each side of VH produced), and lines being drawn from the centre through A' and F, there will be determined the figure FD', which is the projection of the voussoir on the plane XOY.

Now if, parallel to VH, lines be drawn through A', C', B', D' and the corresponding points on the opposite side of VH produced till they meet the corresponding straight lines drawn through A, C, B, D as above; they will determine the points a, c, b, d and a', c', b', d' in the projection on zox; but the normals BA, DC, &c. tending to the centre of the dome, it is evident that, in the projection on zox, ba, b' a', dc, d'c' will be straight lines all tending to H. The circular arcs AC, BD, &c. will, in the projection on zox be evidently portions of ellipses, since they are orthogonal projections of parts of great circles of the sphere on a plane inclined to their own planes; but any number of points in those lines may be determined in the following manner: - Take a point as E in BD, and draw Ee' perpendicular to VH; with a radius equal to the perpendicular distance of E, from VH describe an arc E'F' concentric with A'F; then half the chord of that arc

being set on Ee', on each side of VH, the points e, e' will be in the elliptical curve passing through b and d, b' and d'. The figure abde a'b'd'e' is the required projection of the voussoir on the plane zox.

The projection of a voussoir for a conical vault would differ from that which has been described only by the lines ba and de, b'a' and d'c', which are perpendicular to the surface of the cone, being respectively parallel to one another, and by the lines corresponding to ac, bd, a'c', b'd', both in the vault and in the projection being straight.

PROPOSITION XXIV.

239. To describe the projections, on a vertical and a horizontal plane, of a voussoir at an angle of a groined vault, or one formed by the intersection of two hemicylindrical vaults.

T

H

d

Ꭶ

e

b

a

b

X

Let AV, AU be part of two sides of a square formed on a horizontal plane XOY when two hemicylindrical vaults of equal magnitudes intersect one another at right angles; XOY being the plane in which are the axes of the cylinders. For simplicity let the cylinders be circular, and let A'N, supposed to be in a plane pèrpendicular to the paper and cutting it in A'P' parallel to AV, represent part of the arch line in a semicircular section taken vertically over Av; also, let MR, NT indicate the positions of two voussoir joints which are normals to the hemicylinder of which A'N is a section.

R

R

T

P

N

M

Let T'A produced be the direction of a diagonal of the square on AV, AU; and let the produced part represent the direction, on the plane XOY, of the curve line (ellipse in the present case) in which the hemicylinders intersect one another. Let CABD, C'EB'D be horizontal projections of the lower and upper surfaces of the voussoir which rests immediately on XOY, GE being in a line drawn through M parallel to AU, and EB' being parallel to AV. A section

through this voussoir in a vertical plane passing through a line parallel to AV is represented by MA'P'R. Again, let GEFS, G'E'F's be a horizontal projection of the voussoir next above the other, G'E' being in a line drawn through N parallel to AU, and E'F' being parallel to Av. A section through this voussoir in a vertical plane passing through av is represented by the figure NMH.

Let the vertical plane zox, on which the groined voussoirs are to be represented, be perpendicular to the line AT' in YOX; and on a T', produced to cut ox, let the point a be the representation of A. On a t, in the direction of A T' produced, make a e, a e' equal, respectively, to the perpendicular distances of M and N from P'A' produced; and through e and e draw indefinite lines parallel to ox; these will be the projections on zox of the horizontal lines passing through points corresponding to м and N parallel to AB and AC on the intersecting concave faces of the vault.

Making ab and ac, also eb' and ec' respectively equal to the perpendicular distances of B and C, B' and c' from AT′; the points b and c, b' and c' will be representations of B and C. B' and c' on the lower and upper joints of the voussoir A'M RP', The curves bb, cc' are representations of the lines in which the vertical joints of that voussoir meet the faces of the vaults; these being representations of portions of circles projected on a plane, which is oblique to their planes, are portions of ellipses; and any number of points in them may be found by means of points, as K, taken in a’'M, as b' and c' were found by means of the point M.

To determine the projection, on zox, of the upper face of the voussoir MA'P'R; on at make ap equal to P'R; then the straight line b'p will be the projection of the edge vertically over B'P; draw pd parallel to ox, and make it equal to the perpendicular distance of D from AT'; this will be the projection of the (horizontal) edge vertically over PD; and the straight line c'd will be the projection of the edge vertically above c'D.

In the upper surface of this voussoir, the plane, of which b'ep is the projection, is one which if produced would pass through the axis of the hemicylinder vertically over the space UAB, that axis being parallel to VAB; and the plane of which epde' is the projection, is one which if produced would pass through the axis of the hemicylinder vertically over VAC, that axis being parallel to UAC.

The figure gqsf is the projection, on zox, of the inferior surface of the voussoir, of which NMH is a section; it is determined by first making ef and eg respectively equal to the

P