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.
Fig. 4.
5. The manipulation and use of the instrument are of the simplest character. Being given any two straight converging lines whatever, α β and γ δ (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 γ δ, 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 α β, 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.
Fig. 5.
6. The point A was chosen arbitrarily, but it is evident that that of the needles depends on its distance from the point of convergence. Thus, on taking A' instead of A in the case of Fig. 3, they approach, while the contrary happens on choosing the point A". It is clear that the different positions that a needle A may take are found on a straight line which runs to the point of meeting.
7. If the instrument were jointed or hinged at C, that is to say, so that we could at will modify the angle of the resting ruler, we might make the position of the needles depend on such angle, and conversely.