78 LECTURE X. 



the other ; and when great accuracy is required, hair compasses may be 

 employed, having a joint with a spring in one of the legs, which is bent a 

 little by means of a fine screw. Beam compasses* are useful for drawing 

 circles of larger radii : they have also the advantage of being steadier than 

 the common compasses, and of admitting readily the application of a gra- 

 duated scale, so as to indicate the measure of the radius of the circle which 

 is described. Sometimes, for drawing portions of very large circles, two 

 wheels, differing a little in diameter, are fixed on a common axis, and thus 

 made to revolve round a point, which is more or less distant, accordingly 

 as the wheels are set at a greater or less distance on the axis, the surface 

 of the wheels tracing the circles on the paper ; or two rulers joined toge- 

 ther, so as to form an angle, are made to slide against two points, or 

 edges, projecting from a third ruler ; so that the angular point remains 

 always in the arc of a circle. The same effect may be produced, somewhat 

 more commodiously, by means of a thin piece of elastic wood, which 

 is made to assume any required curvature by the action of screws 

 applied to different parts of its concavity : it would, however, be more 

 simple and accurate to employ only one screw, in the middle of the arc, 

 and to make the flexible ruler, or bow, every where of such a thickness as 

 to assume a circular form in its utmost state of flexure : it would then 

 retain the circular form, without a sensible error, in every other position. 

 (Plate VI. Fig. 82... 85.) 



For drawing a line perpendicular to another, we often employ a square ; 

 and if we use a rectangular drawing board, there is an additional conve- 

 nience in making the square to slide on its margin. Rulers also, of various 

 descriptions, are commonly made rectangular, in order to answer occasion- 

 ally the same purpose. 



Triangular compasses are sometimes used for laying down a triangle 

 equal to a given triangle ;t and by repeating the operation, any figure 

 which can be divided into triangles, may be copied without the intersection 

 of arcs ; but the same end is more commonly obtained by pricking off the 

 figure with a steel point. (Plate VI. Fig. 86.) 



Various properties of parallel lines are employed in constructing parallel 

 rulers : a parallelogram with jointed angles is the most commonly used ; 

 two equal rulers being united by equal cross bars placed in an oblique 

 position, and turning on pins fixed in the rulers : the instrument is much 

 improved by adding a third ruler, similarly united to the second, for then 

 the obliquity of one of the two motions may be made to correct that of the 

 other. A simple cylinder, or a round ruler, answers the purpose in a rough 

 manner, and two small rollers, fixed on the same axis, are also sometimes 

 attached to a flat ruler, and cause it to move so as to be always in parallel 

 positions. A very useful instrument for drawing parallel lines, at any 

 given distances, is now generally known by the name of Marquois's scales, 

 although it is by no means of late invention ; by sliding a triangle along 

 a graduated ruler, we read off the divisions on an amplified scale with great 



* Shuckburgh, Ph. T. 1798. 

 f Leupold, Th. Art. t. 28. J Ibid. t. 21, a. 



