History of the Plow. 53 



thirty inches or more, and this being clividecl into ten equal parts, 

 the parallels 11, 22, 33, &c., are to be drawn upon the cylindrical 

 surface, and between the points b d & curve has to be described 

 that shall 1)0 tlie line of transit of the slice. After investigating 

 the application of various curves to this purpose, I have found 

 that the circular arc is the only one that can be adopted. It pre- 

 sents the least attainable resistance in the first staffes of the 

 ascent, where the force required to raise the slice is greatest, and 

 in the last stages where the force of raising has vanished, leaving 

 only wdiat is necessaiy to turn the slice over, there the resistance is 

 greatest ; and, above all, the circle being of equal flexure through- 

 out, it is in every w^ay best adapted to the objects here required. 

 To determine the radius of curvature of this arc, we must evolve 

 the cylindrical surface, cbde, and from it construct the diagram. 

 Fig. 124, Plate IV. Draw eh equal to c cZ of Fig. 123; ecZ^qual 

 to the length of the arc, cb or d e, and at right angles to eh; 

 divide e b into ten equal parts, and from the points of division 

 draw the ordinates \f. 2g, 3h, etc., parallel to e d; from b set off 

 ten inches for the length of the share along the line b e, which 

 wall fall one inch beyond the division 7, and at this distance 

 draw the dotted line parallel to 7m; upon this set ofli'tw^o and one- 

 half inches, and through the three points, dmb, describe an arc 

 of a circle, whose radius will be found equal to the circumference 

 of the cylinder, of which a b c, Fig. 123, is a quadrant. The 

 circular arc thus found is to be transferred to the cylindrical 

 surface c b d e. The transfer may be performed by drawing the 

 arc on paper, and the paper then laid over the cylindrical surface 

 in such a manner that the points, b m d, shall be brought to coin- 

 cide wdth the points, b m d, of the cylindrical surface; when the 

 remaining points/*^ h i, or any number more, may be marked on 

 the cylindrical quadrant by pricking through the paper with a 

 pointed instrument at short intervals along the arc; or the lengths 

 of the ordinates, If, 2(/, 3h, of Fig. 124, may be transferred to the 

 corresponding parallels of Fig. 123, when the lengths of the 

 ordinates will cut the parallels in the points fgr h, etc. In either 

 case the curve can now be traced through the points b p n ?7i, etc., 

 on the cylindrical surface. Through the points b p «w?, etc., draw 

 the dotted lines,//', (/ g', hh', parallel to cd or be, and from the 

 centre, a, draw the radii of, ag', a It', etc.; the unequal divi- 

 sions of the arc, c h, will thus show the proportion of angles of 

 ascent of the slice along the transif now found, b p a, etc., foi* 



