TANGENTS TO DEVELOPABLE SURFACES. 257
are cut away, the more will the prism approach to therigorous geometrical form of a cylinder.
Planes, tangents to the cone. —If we draw an edgeSABC on the cone, fig. 12, all the tangents in A, B, C,to the parallel sections A a, B b, Cc, are parallel to oneanother. The whole of these tangents form the planePQMN, tangent to the cone through the whole length ofthe edge SABC .
This property of the cone permits us, by circumscribingits base with a polygon, to construct a pyramid, the faces ofwhich shall be tangents to the cone through their wholelength. By cutting away successively with any properinstrument the edges of this pyramid, we form new tan-gent planes, multiply more and more the edges, and thusform a surface, which represents a cone with any requireddegree of precision. (See Tenth Lesson.)
Planes, tangents to developable surfaces. —The propertywhich the same tangent plane has of touching the cylinderor cone through the whole length of an edge, belongs alsoto other species of developable surfaces. Such surfacesmay at all times be considered as formed of a great num-ber of extremely small conical faces, having, like thoseof the cone, the same plane a tangent through the wholelength of each edge.
We can make a developable surface pass through two given curves,by circumscribing polygons about these curves, so that the sameplane will pass at the same time through one side of each polygon;this plane will be the tangent to the developable surface. Cuttingaway the edges formed by the intersection of these planes, we maymultiply the sides of the two circumscribing polygons, and, of course,the small planes which are tangents to the developable surfaces re-quired to be produced.
Cylinders, tangents to each other in the direction of oneof their edges. —Placing two right lined circular cylindersA BCD, BCEF, fig. 10, close to each other, so that theiraxes shall be parallel, and distant from each other by aquantity equal to the sum of the radii of the bases, weshall find that the two cylinders will touch each other in
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