AN IMPROVED SYSTEM FOR OPERATING THE QUADRATURE OF CIRCLE
Year: 1873

page_text: PROVISIONAL SPECIFICATION left by Antonio Giovanni Lanzirotti at
the Office of the Commissioners of Patents, with his Petition, on
the 10th May 1873.
I, ANTONIO GIOVANNI LANZIROTTI, of Boulevard Strasbourg, No. 23,
Paris, Sculptor, do hereby declare the nature of the said Invention for
"AN IMPROVED SYSTEM FOR OPERATING THE QUADRATURE OF CIRCLE," to be as
follows: :-
It is often necessary in the industrial arts to rapidly obtain the
conversion of a circular surface into an equivalent square onc, or vice
versa, or in other words to solve practically the problem of "squaring
the circle.
Hitherto in order to produce this transformation one has been com-
pelled to have recourse to an algebraic operation as imperfect and tedious
as the proportions of the circumference to the diameter is an incom-
mensurable number. It is in order to dispense with this calculation
that I have arranged the instrument which I call "the angle for
squaring the circle," and which permits by an immediate reading to
obtain the side of the square equivalent to a given circle, or the radius

page_text: '
of a circle equivalent to a given square. This instrument is founded
upon the use of a movable reglet or alhidado forming one of the sides
of the right angle of a rectangular triangle, of which the opposite acute
angle is the constant angle of the quadrature. The determination of
this angle of the quadrature can be once for all effected by making use 5
of the formula C³= R³, in which one calculates C in function of R, or
R in function of C, by making use of logarithms. .
I have obtained a very nearly correct result by taking 16 for the side
of the square, which gives 9.03 for the corresponding radius.
Another determination has been made empirically by me in con- 1
sequence of experimental proofs, and I proceed to explain it with
reference to the annexed Drawing.
Figure 1 shows a hollow or solid metal cylinder of any desired
diameter.
Figure 2 the height of the cylinder.
Figure 3 a hollow or solid metal square of a surface equal to that of
the circle Figure 1.
1
Figure 4 shows the height of the cube, the base of which is the
square, Figuro 3, this height being the same as that of the cylinder,
Figure 2. These being obtained any liquid which exactly fills the 2
cylinder (Figures 1 and 2 will also exactly fill the cube (Figures 3
and 4), and since these two bodies are of the same height the circular
form, Figure 1, has taken the square from Figure 3.
If the cylinder and cube are of solid metal, and they are plunged
successively into any liquid, the displacement of the liquid will be the 2
same with both bodies, and since they are of the same height the
surface of the circle is therefore cqual to that of the square, for if it
were not so two equal displacements would not take place; the square
(Figure 3) is therefore the square of the circle (Figure 1).
Figure 5 shows a circle and a square of an equal surface. In order 3
to find graphically the square equal to the circle it is necessary to
multiply the measure of the radius by 6, and to divide the product by 4.
This fourth carried, starting from the centre ○ of the circle upon a
prolonged radius O, R, will extend beyond the circumference for a
certain distance R, R'. One-half of this distance added to the radius 3

page_text: gives the semi-diagonal O, A, of the square. Then by carrying it upon
two perpendicular diameters (O A, O B, O C, O D,) and joining the
four points A, B, C, D, we have the square, which is equivalent to tho
circle.
The square and the circle shown in Figure 6 have also an equivalent
surface. In order to find the circle equal to the square it is necessary
to divide the side of the square by 16, and 9.03 sixteenths give the
radius of the circle required.
Figure 7 shows the instrument which solves the problem; it is in the
form of a right-angled triangle, one side of whose right angle represents
the radius of the circle, and the other side the side of the square. I
call this instrument "angle for the squaring of the circle; it permits
the square equivalent to a given circle, or vice versa, to be found
immediately.
A, B, C, is a right-angled triangle of any suitable material, the
base A, B, of which is graduated in any desired manner into millemetres,
inches, &c. D, E, is a movable alhidade of the same material graduated
in the same way; it moves horizontally in a slide at the base A, B, and
carries a pressure screw V.
In order to immediately find the square equivalent to a circle it is
only necessary to notice on the alhidade the measure of the radius, and
to move this alhidado until the measure of the radius upon it touches
the hypothenuse A, C; and then upon the graduation on the base of the
triangle we read off the measure of the side of the square equivalent
to the circle.
It is of course unnecessary to describe the inverse
operation.
In Figure 8 I have shown an improved arrangement of instrument
based upon the same principle of the transformation of the circle intɔ
an equivalent square, which is represented algebraically by the formula,
R²=0³, whence follows that C=√R=R√ or universely R³=- or
7
0⁹
75
R=V CO. In this improved arrangement of instrument the
movable graduated alhidade D, E (of wood, bonc, ivory, or any metal)
slides in a frame M, which encloses it, and in the interior of which is
engraved the triangle A, B, C, the base A, B, of which is also graduated.

page_text: The use of this instrument is the same as the other, shown in Figure 7;
the little hook a serves to enable the hand to move the alhidade D, E,
to bring it into the position necessary for the solution of the problem.
The opening 6 serves to suspend the instrument to the wall when
necessary, like squares and rules.