1. LET A and B be two fixed points and A C and C B two straight lines converging at C and moving in their plane so as to always remain based on this point (Fig. 1). The geometrical place of the positions occupied by C is the circumference of the circle which passes through the three points A, B, and C. Now let C F be a straight line passing through C. On prolonging it, it will meet the circumference A C B I at a point I. If the system of three converging--lines takes a new position A C' F B, it is evident F' B' prolonged will pass through I, because the angles [alpha] and [beta] are invariable for any position whatever of the system. [Illustration: Fig. 1.] 2. In the particular case in which [alpha] = [beta] (Fig. 2), the point I is found at the extremity of the diameter, and, consequently, for a given distance A B, or for a given length C D, such point will be at its maximum distance from C. [Illustration: Fig. 2.] 3. This granted, it is easy to construct an instrument suitable for drawing converging lines which shall prove useful to all those who have to do with practical perspective. For this purpose it is only necessary to take three rulers united at C (Fig. 3), to rest the two A C and C B against two points or needles A and B, and to draw the lines with the ruler C F, in placing the system (§ 1) in all positions possible. The three rulers may be inclined in any way whatever toward each other, but (§ 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.]