432 Prof. Watuace on the Invention of the Pantograph, 
point is fixed at T, the extremity of one arm, and a black lead 
pencil, which serves as a copying point, at P, the extremity of the 
other arm on the opposite side of the beam. The distances of 
these points from the centres of the wheels are in every case 
equal to the distances of the latter from the centre of the beam, 
whatever proportion these have to each other. By this disposi- 
tion of the members of the instrument, a straight line joining the 
tracing and copying points passes through the axis of motion of 
the beam, and is there divided into two parts, which have to 
each other the proportion of the distances between the centres 
of the wheels and the centre of the beam. Thus the conditions 
required by the mathematical theory are satisfied. 
Having described the instrument generally, I shall now no- 
tice its parts, particularly, in detail. 
Tue BEAM @ a, for the sake of lightness and stiffness, is hol- 
low ; it is about thirty inches long, but may be of any required 
length ; its cross section is a square, about nine-sixteenths of an 
inch in the side. There is a scatx of 200 equal parts engraved 
on one of its vertical faces. 'The length of the scale is the exact 
distance between the centres of the axes of the wheels, and each 
division is 74,th part of this length. The division at the middle 
of the beam is 0 (zero), and the scale is numbered both ways ; 
the extreme divisions would be 100, if they were numbered so 
far, but they go only to about 70, no more being ever required. 
Tue socket ¢¢, in which the beam slides, is four inches long ; 
it has an opening on one side, through which the divisions on the 
scale appear, ard there is an index engraved on it, against which 
the zero division is set, when the middle of the beam is exactly 
over the centre of the vertical axis on which the socket turns. 
Fig. 4. is a vertical cross section, through the centre of the beam 
socket, of half the actual size. The opening for viewing the 
scale is at v; c is a steel conical axis, fixed into a strong plate p, 
screwed into the bottom of the socket, and turning in the bored 
tube s, which is screwed to dd, the base of the instrument. 
