[Footnote: Paper by Prof. Fr. Smigaglia, read at the reunion of the Engineers and Architects of Rome.]

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 α and β are invariable for any position whatever of the system.

Instrument For Drawing Converging Straight Lines 363 4a

Fig. 1.

2. In the particular case in which α = β (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.

Instrument For Drawing Converging Straight Lines 363 4b

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 α = β.

Instrument For Drawing Converging Straight Lines 363 4c

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.

Instrument For Drawing Converging Straight Lines 363 4d

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.

Instrument For Drawing Converging Straight Lines 363 4e

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.

Instrument For Drawing Converging Straight Lines 363 4f

Fig. 6.

Then Instrument For Drawing Converging Straight Lines 363 tex1

whence CD = \frac{CI}{I + \tan^2 \alpha}\ \text{or}\ CD = CI \cos^2 \alpha. (1)

9. If the instrument is jointed, the absolute values being

AD = \sqrt{CD(CI   CD)} (2)

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

CI = \frac{CD}{\cos^2 \alpha} (3)

or, again,

CI = \frac{\overline{AC^2}}{CD'} (4)

11. In order to avoid all calculation, we may proceed thus: If I wish to arrange the instrument so that C I represents a given quantity (§ 8), I take (Fig. 7) the length Ci = CI/n, where n is any entire number whatever.

Instrument For Drawing Converging Straight Lines 363 4g

Fig. 7.

In other terms, Ci is the reduction to the scale of CI.

I describe the circumference C b i a, and arrange the instrument as seen in the figure, and measure the length C b.

It is visible that

 C i 1 C b C d

----- = --- = ----- = ------; then C B = n.C b (5)

C I n C B C D 
 CD = n.C d; (6) 

and, consequently, the position of the needles which are found at A and B are determined.

12. The question treated in § 10, then, is simply solved. In fact, on describing the circumference C b i a with any radius whatever, I shall have

 C B

n = -----; (7)

c b 

and, consequently,

 C I = n.C i (8) 

13. As may be seen, the instrument composed of three firmly united rulers is the simplest of all and easy to use. Any one can construct it for himself with a piece of cardboard, and give the angle 2 α the value that he thinks most suitable for each application. The greater 2 α is, the shorter is the distance at which we should put the needles for a given point of meeting.

14 The jointed instrument may be constructed as shown in Figs 8, 9, and 10. The three pieces, A. B, and C, united by a pivot, O, in which there is a small hole, are of brass or other metal. Rulers may be easily procured of any length whatever. The instrument is Y-shaped. In the particular case in which α = 180° it becomes T-shaped, and serves to draw parallel lines.

Fig. 8, Fig. 9, Fig. 10

Fig. 8, Fig. 9, Fig. 10

15. The instrument may be used likewise, as we have seen, to draw arcs of circles of the diameter C I or of the radius A O = r, whose center o falls outside the paper. The pencil will be rested on C. We may operate as follows (Fig. 2): Being given the direction of the radii A O and B O, or, what amounts to the same thing, the tangents to the curve at the given points, A and B to be united, we draw the line A D and raise at its center the perpendicular D C, which, prolonged, passes necessarily through the center. It is necessary to calculate the length C D.

We shall have

CD (2r   CD) = \overline{AD^2}.\overline{CD^2}   2r.CD + \overline{AD^2} = o.

CD = r  \sqrt{r^2   \overline{AD^2}}.

It is evident that the lower sign alone suits our case, for d < r; consequently,

CD = r   \sqrt{r^2   \overline{AD^2}}. (9)

Having obtained C, we put the instrument in the direction A B C. Then each point of C F describes a circumference of the same center o.

16. If the distance of the points A and B were too great, then it would be easy to determine a series of points belonging to the arc of circumference sought (Fig. 4).

Being given C, the direction C I, and C I = R, on C I I lay off C E = d, draw A E B perpendicularly, and calculate C A or A E. I shall have

d = (R   d) = \overline{AE^2};

or, as absolute value,

AE = \sqrt{d (R d)} (10)

The instrument being arranged according to A C B, I prolong C B and take B C' = B C, when C' will be one of the points sought. It will be readily understood how, by repeating the above operations, but by varying the value of d, we obtain the other intermediate points, and how we may continue the operation to the right of C' with the process pointed out.

17. If the three rulers were three arcs of a large circle of a sphere, the instrument might serve for drawing the meridians on such sphere.

18. If we imagine, instead of three axes placed in one plane and converging at one point, a system of four axes also converging in one point, but situated in any manner whatever in space, and if we rest three of them against three fixed points, we shall be able to solve in space problems analogous to those that have just been solved in a plane. If we had, for example, to draw a spherical vault whose center was inaccessible, we might adopt the same method.--Le Génie Civil.