[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

___ ___
___ AD squared CD squared
CD x DI = AD squared. CD = ---- = --------- tang squared[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]