[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
[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.]
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.]