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,t 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 stated, 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 of, shall contain 120 degrees, or the third of a circle; 4 o f, 90 degrees, or the fourth; 5 of, 72, or the fifth; 6 of, 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 the several figures 207,209,211, but with 10 sides; we will suppose the original ordinate to reach from o to x on the radius of, the required measure would be the length of the arc x 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 two pairs of intermediate pieces a a, and b b, which latter would necessarily be parallel. In the like manner a pyramidal vessel built upon the same base, would require equal angles for all 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," etc. by John Lodge Cowley, F.R.S., 1759. The schemes include the five regular solids, and various irregular solids, prisms, pyramids, and frustums thereof; all of which are cut out in the plates, and may be folded up so as to become exact models of the solids.

forms out of flat plates, by the process called raising. this is done with the hammer alone, by stretehing some parts of the metal and contracting others (the drawing and upsetting of the black-smith), a process not required in any of the foregoing figures, the whole of which might be made in pasteboard, a material, that as before observed does not admit of being raised or bulged into figures of double curvature. To conclude this chapter, it only remains to advert in the same general manner to the modes of bending and uniting the edges of the works represented.

The various works having been drawn upon the sheet of metal, the first process is to cut them out; this is almost always done with the shears; * sometimes however for thick metal, the cold-chisel and hammer are used, the work being laid upon the bare anvil or upon a cutting-plate, as in forging: occasionally the metal is fixed in the jaws of the tail-vice, and cut off with the cold-chisel applied in contact with the vice; the edge of the chisel is placed nearly parallel with the jaws, which serve as a guide. In some cases very long vices with a screw at each end, are used in a similar manner, for the thick iron plates employed for boilers;† but the shears are the most generally convenient.

Although the tools used in working the sheet metals are extremely various as regards their sizes and specific forms, they may, with the exception of the shears and soldering-tools, be principally resolved into numerous varieties of hammers, anvils, swage-tools, and punches. Figs. 215 to 229 represent some few of the most common of these tools, which are used alike both in bent and raised works, and their close resemblance to those for ordinary forging in iron and steel will not escape observation. The most remarkable points of difference are in their greater height and length, which enable them to be applied to the interior of large objects, and also in their square shanks, by which they are fixed in holes in the wooden blocks and lunches.

The hammers arc nearly alike at both ends; many of them have circular faces, either Hat or convex; others resemble the straight or cross-panes of ordinary hammers, and arc also either flat or convex; and those used in finishing, are exceedingly

* See the Chapter on Shears, Vol II., especially the figures on page 915. † See Plate 60. Buchanan, on Mill Work. Edited by Rennie. 1841.

CC bright, in order that they may impart their own degree of polish to the work, which process is called planishing.

When thin metal is struck between tools both of which are of metal, it is invariably more or less thinned; and should the blows be given partially, such parts will become stretched or cockled, and will distort the general figure. It is therefore usual, whenever admissible, to employ wooden hammers of the forms described, and also wooden blocks or anvils when metal hammers are used; reserving the employment of tools both of metal, either for the concluding steps, or for those cases, where from the substance of the metal and the nature of the work, the wooden hammers would be ineffective, or a greater definition of form is required than wooden tools could give.