(Sec. 2) it is preferable to take the case where [alpha] = [beta].

[Illustration: Fig. 3.]

4. Let us suppose that the instrument passes from the position I to
position III (Fig. 4). Then the ruler C A will come to occupy the
position B A, from the fact that the instrument, continuing to move in
the same direction, will roll around the point B. It is well, then, to
manage so that the system shall have another point of support. For that
reason I prolong C B, take B C' = B C, draw C' I, and describe the
circumference--the geometrical place of the points C'. I take C' D = C'
B and obtain at D the position of the fixed point at which the needle
is inserted. In Fig. 4 are represented different positions of the
instrument; and it may be seen that all the points C C', and the centers
O O', are found upon the circumferences that have their center at I.

[Illustration: Fig. 4.]

5. The manipulation and use of the instrument are of the simplest
character. Being given any two straight converging lines whatever,
[alpha] [beta] and [gamma] [delta] (Fig. 5), in order to trace all the
others I insert a needle at A and arrange the instrument as seen at S. I
draw A B and A B', and from there carry it to S' in such a way that the
ruler being on [gamma] [delta], one of the resting rulers passes through
A. I draw the line C B which meets A B at the point B, the position
sought for the second needle. In order to draw the straight lines which
are under [alpha] [beta], it is only necessary to hold the needle A in
place and to fix one at B', making A B' = A B. In this case S" indicates
one of the positions of the instrument.