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dg, equals the circumference, and its width de or gf, the height of the cylinder ; and if it be conceived to roll on a plane with that part of its surface represented by the line de in No. 2 touching it when starting, it is evident that it will roll over the surface included in the rectangle defg, and at the completion of one revolution, the line de (if drawn on the cylinder's surface) would be found to fall exactly on the line gf, as the meridians 1 to 16 assumed to be drawn on that surface would each fall on its corresponding one drawn on the development. By laying out the surface of any solid of revolution (if developable) in a similar way to that of the cylinder, it will be apparent that the development of any point or line on its surface in any position, or direc- tion, may be soon found by the help of a few meridians. For instance, let xy be two points on the front surface of the cylinder, and it is wished to find their position on its envelope. First let fall projectors from x and y in No. 2, on to the plan of the front face of the cylinder in No. 1 ; they will be found to cut it in xy ; ,the first between points 5 and 6 ; and y between 2 and 3 in the semi- circle 1 8, 4, 0. Then on ef (the developed edge of the base of the cylinder) set off from points 3 and 5 in it, the distances that xy are from the corresponding points in the semi-circle 8, 4, ; and through these draw faint lines parallel to de. Then the points x",y" in No. 3, where these lines are cut by projectors drawn through x and y in No. 2 parallel to ef, are the positions of those points on the envelope of the cylinder. Again, if a line be given in position on the surface of a cylinder, drawn, say between the points x and y, and its development is required, then, as a line is the path of a moving point, take any convenient points