ON DRAWING, WRITING, AND MEASURING. 79 



accuracy ; but where the distances of the lines are great, the obliquity of 

 this motion is a considerable inconvenience. The ruler or square of the 

 drawing board affords us lines parallel to each other, in a certain position ; 

 and if it be made with a joint, or as the workmen call it, bevilled, it may 

 be employed for the same purpose in all other directions. The systems of 

 lines, on which music is written, are drawn at one stroke by a pen with 

 five orifices, usually made of brass. It was long since proposed to rule a 

 whole page at once, with a more complicated pen of the same kind, and 

 the greatest part of the paper on which music is written in this country, 

 is actually ruled by such a machine, for which a patent has been taken 

 out. (Plate VI. Fig. 87, 88.) 



The pantograph is used for copying figures, and at the same time 

 reducing or enlarging them ; it consists of four rulers, two of them 

 united by a joint at the extremities, and receiving at the middle the other 

 two, which are but half as long, and are also united together so as to form 

 with the others a jointed parallelogram, of which two of the sides are 

 produced beyond the angles ; if holes be made in these, and in one of 

 the shorter rulers, so situated as to be in the same right line in any 

 position of the instrument, they will remain in a right line in any other 

 position, and they will always divide this line in the same proportion : so 

 that if one of the holes be placed on a fixed axis or pin, a tracing point 

 inserted in another, and a pencil in the third, any figure delineated by 

 the pencil will be similar to that which is described by the tracing 

 point. And instead of holes in the rulers, they may be furnished with 

 sliding sockets, to receive the axis, the point, and the pencil. (Plate VI. 

 Fig. 89.)* 



Proportional compasses are also of great use in reducing lines and 

 figures to a different scale.t This instrument consists of two legs, pointed 

 at each end, and turning on a centre which slides in a groove common to 

 both legs, and is furnished with an index. The divisions of the scale are 

 so laid down that the centre may divide the length of the legs from 

 point to point in a given proportion ; hence by the properties of similar 

 triangles, when the legs are opened to any extent, the intervals between 

 each pair of points must be to each other in the same ratio as the por- 

 tions of the legs. Sometimes a screw is added, for the sake of adjusting 

 the centre with greater accuracy ; and it is usual to lay down scales for 

 dividing the circumference of a circle into a given number of parts, and 

 for some other purposes ; but the instrument might be much improved 

 by inserting, in the common scale, fractional or decimal divisions between 

 the whole numbers, so that the legs might be divided, for example, in 

 the ratio of 2 to 3, 3 to 4, or 4 to 5, or of 10 to 11, 12 or 13, at pleasure. 

 (Plate VI. Fig. 90.) 



The use of the sector depends also on the properties of similar triangles. 



* Leup. Th. Art. t. 26. Langlois's Pantograph : Machines Approuves par 1'Ac. 

 7 vols. 4to, 1735-1777, vii. 207. Sike's Pantograph, Mem. Par. 1778, invented by 

 Sclieiner, who describes it in his Pantographice. An improved instrument for the 

 same purpose is described by Prof. Wallace, in the Trans, of the Roy. Soc. of Edin. 

 vol. xiii. and termed by him the Eidograph. 



f Leon, da Vinci MSS. Leup. Th. Ar. 121. 



