possessed of this quality in a very limited degree, for the globe could not be smoothly covered with so few as two, three, or four pieces of the thinnest paper without its puckering up, shewing that some parts of the material are in excess. The gliding property, or that of malleability and ductility, possessed by the metals, is indispensable to adapt the flat plate to the sphere, by stretching the central portion and gathering up the marginal part, an action that admits of some comparison to the extension or compression of the slides of a telescope, except that the metal becomes thicker or thinner, instead of being duplicated on itself.

SECT. II.—WORKS IN SHEET METAL, MADE BY CUTTING, BENDING

AND JOINING.

Every one in early life, has made the first step towards the acquirement of the various arts of working in sheet metal, in the simple process of making a box or tray of card; namely, by doubling up the four margins in succession to an equal width, then cutting out the small squares from the angles, and uniting the four sides of the box, either edge to edge, by paste sealingwax or thread, or in similar manners by lapped or folded joints. A different mode is to make the sides of the box as a long strip, folded at all the angles but one; or lastly, the bottom and sides may be cut out entirely detached, and united in various ways.

In the above, and also in the most complicated vessels and solids, it is necessary to depict on the material the exact shape of every plane superficies of the work, as in the plans and elevations of the architect; and these may be arranged in any clusters which admit of being folded together, so as to make part of the joints by bending the material. Thus, a hexagonal box, fig. 197, can be made by drawing first the hexagon required for the bottom, as in fig. 198, and erecting upon each side of the same, a parallelogram equal to one of the sides, which in this case are all exactly alike; otherwise the group of sides can be drawn in a line, as in fig. 199, and bent upon the joints to the required angle, or 120 degrees. Either mode would be less troublesome than cutting out seven detached pieces and uniting them; the addition of one more hexagon, dotted in fig. 199, would serve to complete the top of the hexagonal prism, by adding a cover or top surface.

The same mode will apply to polygonal figures of all kinds, regular or irregular; thus fig. 200 would be produced when the group of sides in 201 were bent around the irregular octagonal base; or that the sides of 202 were separately turned up.

The cylinder, is sometimes compared with a prism of so many sides, that they melt into each other and become a continuous curve; and if the hexagon in fig. 199 were replaced by a circle, and the group of sides were cut out of equal length with the circumference of that circle, and in width equal to the height of the vessel, any required cylinder could be produced. And in like manner any vessels of elliptical or similar forms, or those with parallel sides and curved ends, and all such combinations, could be made in the manner of fig. 201, (provided the sides were perpendicular,) by cutting out a band equal in length to the collective margin of the figure, as measured by passing a string around it; or the sides might be made of two, or several pieces, if more convenient, or requisite from their magnitude.

All prismatic vessels require parallelograms to be erected on their respective bases; but pyramids require triangles, and frustums of pyramids require trapezoids, as will be explained by figs. 204 and 205, which are the forms in which a single piece of metal must be cut, if required to produce fig. 203. Each of the group of sides, must be individually equal to one of the sides of the pyramid, whether it be regular or irregular, and 203 being an erect and equilateral figure, all the sides in 204 and 205

are required to be alike, and would be drawn from one templet: an irregular pyramid, &c., would call for each of its superficies to be drawn to its absolute form and size.

The cone is sometimes compared with a pyramid with exceedingly numerous sides, (as the cylinder is compared with the prism,) and fig. 206, intended to make a funnel or the frustum of a cone of the same proportions as 203, illustrates this case. The sides of the cone are extended until they meet in the center o, fig. 203, and then with the slant distances o a, and o b, the two arcs a a, and b b, are drawn with the compasses, from the center o; and so much of the arc a a, is required as equals the circumference a, of the cone: the margins a b, a b, are drawn as two radii. When the figure is curled up until the radial sides meet, it will exactly equal the cone, and the similitude between figs. 205 and 206 is most explanatory, as 206 is just equal to the collective group of the sides required to form the pyramid.

It will now be easily seen that mixed polygonal figures, such as figs. 207, 209, and 211, may be produced in a similar manner, provided their sides are radiated from the square, the hexagonal or other bases, in the manner of figs. 208, 210, 212, but the sides. of the rays not being straight, it is no longer possible to group them by their edges, as in figs. 199, 201, and 205. The object with plane surfaces, fig. 207, is only the meeting of two pyramids, at the ends of a prism, and when unfolded, as in fig. 208, the center a, is equal to the base a, of the object; the sides b, radiate and expand from the hexagon at the angle of the faces

PLANE AND CURVED SURFACES.

383 of the inverted or lower pyramid b, and their vertical height in the sheet is equal to the slant height in the vessel; the superficies c, are those of a prism, therefore they continue parallel, and have the vertical height of the part c, of the figure; lastly, the sides d, again contract as in the original, and at the same angle as the sides of the six upper faces; in a word, the faces b, c, d, are identical in the vase and in the radiated scheme.

Should the vessels have surfaces of single curvature, instead of planes, as in figs. 209 and 211, the method is nearly as simple. The object is drawn on paper, and around its margin are marked several distances, either equal or unequal, and horizontal lines or ordinates are drawn from each to the central line. The radiating pieces for constructing the polygonal vases are represented in figs. 210 and 212, in which the dotted lines are parallel with the sides of the hexagons or the bases, and at distances equal to those of the steps 1, 2, 3, to 8, around the curve of the intended vases; the lengths of these lines, or ordinates, 1, 1, 2, 2, 3, 3, are in regular hexagonal vessels exactly the same in the radiated plans as in the respective elevations, because the side of the hexagon and the radius of its circumscribing circle are alike.

In all other regular polygonal vessels, the new ordinates will be reduced for figures of 8, 10, 12 sides, in the same proportions as the sides of these respective polygons bear to the radii of their

circumscribing circles, and the ordinates for 3, 4, and 5 sided figures will be similarly increased*.

It would have been easy to have extended these particulars to numerous other figures, such as the regular geometrical solids, oblique solids, and many others †, but enough has been advanced to explain the cases of ordinary occurrence, and in the delineations of which, the tinman, coppersmith, and others are very expert. Much of that which has been given, as it will eventually appear, has been partly advanced in elucidation of the next chapter, on the less apparent methods practised in making similar

All the above cases could be accurately provided for without any calculation, by the employment of a very simple scale represented in fig. 213, in which the angle 3 o f, shall contain 120 degrees, or the third of a circle ; 4 o ƒ, 90 degrees, or the fourth; 5 o f, 72, or the fifth ; 6 o f, 60, or the sixth ; and 8, 10, and 12, respectively the 8th, 10th, and 12th parts of a circle. The circular arcs are struck from the center o, and may be the 6th, 8th, 10th of an inch, or any small distance apart.

To learn the altered value of any ordinate, as for constructing a vase like each of the figures 207, 209, 211, but with 10 sides; we will suppose the original ordinate to reach from o to × on the radius, the required measure would be the length of the arc X, where intersected by the line 10, or that for a decagon; but it would be more convenient to make the angle half the size, as then the new ordinate would be at once bisected, ready for being set off on each side the central line of the radiated plan. When one side had been carefully formed, a curved templet or gage would be made to the shape, by which all the other sides could be drawn. For polygonal vessels with unequal sides, such as fig. 214, the curvatures of the edges of the rays will be identical, notwithstanding the differences of the sides. For example, the octagon drawn in the one corner shows that the figure resembles the regular octagon as far as the angles are considered; and that the regular octagon may be considered to be cut into four quarters and to be removed to the four corners, by the insertion of the pair of intermediate pieces a a, and b b, which latter would necessarily be parallel. In the like manner a pyramidical vessel built upon the same base, would require equal angles for each of its sides.

† See "An Appendix to the Elements of Euclid, in Seven Books, containing Forty-two moveable Schemes for forming the various kinds of Solids and their Sections," &c. by John Lodge Cowley, F.R.S., 1759.

The schemes include the five regular solids, and various irregular solids, prisms, pyramids, and frustrums thereof; all of which are cut out in the plates, and may be folded up so as to become exact models of the solids.