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. [Illustration: 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. 8. Being given the length C I (Fig. 6), to establish the position of the needles so that all the lines outside of the sheet shall converge at I. To do this, it is well to determine C D, and then to draw the straight line A D B perpendicular to C I, so as to have at A and B the points at which the needles must be placed. [Illustration: Fig. 6.] Then ___ ___ ___ AD² CD² CD x DI = AD². CD = ---- = --------- tang²[alpha], DI CI - CD [TEX: CD \times DI = \overline{AD^2}.\ CD = \frac{\overline{AD^2}}{DI} = \frac{\overline{CD^2}}{CI-CD} \tan^2 \alpha] whence CI CD = ------------------ or CD = CI cos²[alpha]. (1) I + tang²[alpha] [TEX: CD = \frac{CI}{I + \tan^2 \alpha}\ \text{or}\ CD = CI \cos^2 \alpha.] 9. If the instrument is jointed, the absolute values being _____________ / AD = \ / CD(CI - CD) , (2) \/ [TEX: AD = \sqrt{CD(CI - CD)}] it suffices to take for CD a suitable value and to calculate AD. If, for example, the value of C D is represented by C D', the instrument takes the position A' C B', and the needles will be inserted at A' and B' on the line A' D' B', which is perpendicular to C I. 10. If the position of the instrument, and consequently that of the needles, has been established, and we wish to know the distance C I, we will have CD CI = ------------ ; (3) cos²[alpha] [TEX: CI = \frac{CD}{\cos^2 \alpha}] or, again, ___ AC² CI = ----- (4) CD'