side of the square produced; again contract the compasses to the side of the square as before, and describe another quadrant, until it cuts the perpendicular side of the square produced from that centre; thus one revolution being described, proceed with the second and third square in the same manner until the last quadrant is described, which will cut the eye on the upper side.

PROBLEM VI.

To describe a spiral to any given height, and to terminate at the end of any given number of revolutions in the circumference of a circle, whose diameter is given.

Divide the height, made less by the diameter of the given circle, in two more parts than four times the number of revolutions that the spiral is intended to have; that is, if the spiral is to have one revolution, it must be divided into four times one, and two more, that is, into six equal parts; and if two revolutions, it will be divided into four times two, and two, which are ten equal parts; or if three revolutions, it will be divided into 4×3, and two, which is fourteen equal parts; then take half the number of these parts, and one more, together with half the diameter of the eye, and set it from the top of the spiral downwards to give the centre of the spiral; or take half the number of those parts, made less by one, together with half the diameter of the eye, and set it on the perpendicular line from the bottom upwards, which will give the centre of the spiral as before; then construct a square, whose centre is the centre of the spiral, having two of its sides parallel to the perpendicular, and the other two consequently at right angles to it; each side of the square being equal to one of the equal parts before mentioned; draw the diagonals of the square, which will cut each other in the centre of the spiral; divide each diagonal into twice as many parts as there are to be revolutions, which will give the centres of the spiral; and proceed as in the last Problem.

PROBLEM VII. PLATE 75.

The centre c of the spiral being given, the perpendicular height CA above the centre, and semi-diameter CR of the eye, to describe the spiral.

On the centre c, with one half of BD, describe a circle, cutting AC at R; through c draw HI perpendicular to AC; divide AR into two equal parts at u; divide UR into one more part

than the number of revolutions, then set half of one of these parts from the centre upon each of the lines AC and HI: that is, from c to K, from c to I, from c to L, and from c to н; and through these points complete the square DEFG, whose sides, DE, FG, are parallel to HI; and DG, EF are parallel to AC: draw the diagonals DF and GE, then divide CD, CE, CF, and CG, each into as many equal parts as there are to be revolutions, and through these points draw squares, whose sides are parallel to the sides of the square DEFG; through a draw AB parallel to HI, produce the side of the square DG, cutting AB in B ; then on D as a centre, with the distance DB, describe the quadrant BM, cutting DE produced at м; on E, with EM, describe the quadrant MN, cutting EF produced at N; on F, with the distance FN, describe the quadrant No, cutting FG at o; on G, with the distance Go, describe the quadrant or, which will make one revolution. Proceed in the same manner with all the other revolutions, the centres always falling on the angles of the next square.

PROBLEM VIII. PLATE 75.* To describe a spiral line in the manner of Goldman's volute, to any given height.

Divide the height Iw of the volute into eight equal parts, 1 being at the top; then on the fourth part from the bottom call AB, or the fifth from the top, or BA as a diameter, describe a circle, whose centre call c; divide BA into four equal parts, and from the centre set one of these parts on the perpendicular from c to downwards, and from c to E upwards, and construct a square EFGH, whose side is FG on the right or left side of HE, according as it is intended to be a right or lefthanded volute; join CG and CF; divide CH, CE, CG, and CF each into three equal parts respectively at q, m; n, i; p, l ; 0, k ; then will E, F, G, H, i, k, l, m, n, o, p, q, be the centres for describing each quadrant. On E, with the distance E1, describe the quadrant IK, cutting EF produced at K; on F as a centre, with the distance FK, describe the quadrant KL, meeting FG produced at L; on G, with GL, describe the quadrant LM, cutting GH produced at м; on H, with the distance Hм, describe the quadrant MN, cutting the cathetus at N. In this manner proceed with the centres i, k, l, m, for the second revolution; and n, o, p, q, for the third, every revolution mecting upon the cathetus CI.

To describe a spiral in the manner of Goldman's volutes, having a given height, and to touch a given circle to any number of revolutions.

From the height cut off a part equal to the diameter of the eye; divide the remainder of the height into four times as many equal parts, and two more, as the number of revolutions; from the bottom of the cathetus set up half the number of these parts made less by one part, together with half the diameter of the eye; or from the top of the cathetus set downwards half the number of these parts, and one part more, together with half the diameter of the eye; take one of these parts and make a square in such a manner that one of the sides may be in the cathetus, and that side must be bisected by the centre of the volute; the square being made upon the right or left side of the cathetus, according as the volute is to be upon the right or left hand. Proceed in every other respect, in what follows, as in the last Problem: that is, from the centre draw lines to each of the opposite angles of the square; divide each of those lines into as many equal parts as there are intended to be revolutions; through the points draw lines parallel to the sides of the square, cutting the cathetus, forming in all as many squares as there are revolutions, then the outside square contains the centres of the first revolution; the next square, if more than one revolution, contains the centres of the second revolution; and the third square, if more than two revolutions, contains the centres on its angles of the third revolution; and so on for any other number of revolutions above three, if required.

PROBLEM X.

The cathetus CA of the spiral from the centre c being given, to describe it to any number of revolutions to touch a given circle whose centre is c, the centre of the spiral.

On the centre c, with a radius equal to the radius of the eye, describe a circle for the eye, cutting the cathetus CA upwards at R; divide RA into two equal parts at 1, and divide RI into one part more than the number of revolutions: that is, if one revolution, it will be divided into two equal parts; and if two revolutions, it will be divided into three equal parts;

and if three revolutions, it will be divided into four equal parts, and so on; then take one of these parts for the side of the square, and proceed as in the last Problem.

PROBLEM XI.

To draw the spiral of Archimedes with a pair of compasses, having the whole height, and the distance between the spiral space given.

Divide the given distance between the spiral space into four equal parts; then take half the whole height and one of these parts and set it from the top downwards; or take half the whole height diminished by one part, and set it from the bottom upwards, it will give the centre of the spiral, through which draw a line at right angles to the height; then construct a square, whose sides are equal to one of the before-mentioned parts, having the centre of the square in the centre of the spiral, and having two of its sides parallel to the perpendicular of the spiral; then the four angles of the square are the four centres. Setting one foot of the compasses on the upper side of the square in that angle which is upon the same side that the spiral is to be drawn, and extending the other leg to the height, describe a quadrant; the second centre is the other angle of the upper side of the square; the third directly under, and so on, moving round in the same direction, until there are as many revolutions made as required.

PROJECTION.

DEFINITIONS.

1. When straight lines, drawn according to a certain law, from the several parts of any figure or object are cut by a plane, and by that cutting or intersection a figure is described on that plane, the figure so described is called the projection of the other figure or object.

2. The lines taken altogether, which produce the projection of the figure, are called a system of rays.

3. When the rays are all parallel to each other, and are cut by a plane perpendicular to them, the projection on the plane is called the orthography of the figure proposed.

4. When the parallel rays are perpendicular to the horizon, and projected on a plane parallel to the horizon, the projection is then called the ichnography, or plan of the figure proposed.

5. When the rays are parallel to each other and to the horizon, and the projection is made on a plane perpendicular to those rays and to the horizon, it is called the elevation of the figure proposed.

In this kind of projections, the projection of any particular point or line is sometimes called the seat of that point or line on the plane of projection..

6. If a solid be cut by a plane passing quite through it, the figure of that part of the solid which is cut by the plane is called a section.

7. When any solid is projected orthographically upon a plane, the outline or boundary of the projection is called the contour, or profile of the projection.

8. The orthographical projection of a line, which is parallel to the plane of projection, is a line equal and parallel to its original.

Note. Although the term orthography signifies, in general, the projection of any plane which is perpendicular to the projecting rays, without regarding the position of the plane on which the object is projected, yet writers on projection substitute it for elevation, as already defined; by which means it will be impossible to know when we mean that particular position of orthographical projection which is made on a plane perpendicular to the horizon.