Improvements in or relating to Drawing "Squares for Calculating and other Purposes.
Year: 1899

page_text: Date of Application, 8th Mar., 1899-Accepted, 13th May, 1899
Improvements in or relating to Drawing "Squares
for Calculating and other Purposes.
" or Triangles
A communication from Madame PLACIDIA FARINI, of Rome, in the Kingdom of
Italy.
I, ALFRED JULIUS BOULT, of 111, Hatton Garden, in the County of Middlesex,
Chartered Patent Agent, do hereby declare the nature of this invention and in what
manner the same is to be performed, to be particularly described and ascertained in
and by the following statement :-
The present invention relates to a triangular scale for drawing purposes which
provides a means of graphically solving several geometrical problems of practical
importance to engineers, architects, mathematicians, designers, students and others.
The scale essentially consists of a scalene right-angled triangle, the three sides of
which are to each other in a certain ratio, which ratio is characterized by the fact
that the projections, of the two catheti or sides containing the right angle, on the
hypothenuse taken as base, and the height of the triangle relatively to the hypo-
thenuse are to each other, as the three quantities which represent the diameter of a
circle (2), the fourth part of the circumference of the same circle (②---
2
π)
πT -) and
the side of a square the area of which is equal to the area of the same circle (r
taking for practical purposes the value of to be equal to 3.14159. This relation forms
the principle of the invention, and having thus established a relation of measure
between rectilineal elements (height of triangle, projection of the greater cathetus)
with circular elements rectified, (fourth part of a circumference or quadrant) one can
now easily, by means of this scale, transform, graphically, given rectilineal segments
into curvilineal equivalents, and vice versa.
2 7,
In the schematic view represented by Figure 1 of the accompanying draw-
ings A B C is a right-angled triangle with the right angle at C, in which A D =
i.e. the diameter; the height C D = √, or 3.14159
(taking to be equal to 3.14159), from which it follows that B D is equal to
i.e. to the fourth part of a circle the radius of which is r, since A D x BD
or, in this case: 2 x BD (" ✓ 2, and consequently :
=
πT
1772453
-
1" T
2
C D²,
In this triangle ABC, if an arbitrary value is given to any of the rectilineal
elements AB, CB, DB, AD, AC, CD, the value of each of the remaining elements is
Ofixed, and it is evident that, in consequence of the known geometrical and trigono-
metrical relation existing in triangles of this construction, one can express in a
great variety of ways the relations between the elements of this special triangle.
For instance.

page_text: The form of the calculating scale being similar to a special scalene triangle, in
which, if we call a the angle contained by the hypothenuse and the greater cathetus
π
7T
TT
Or again, the sides containing the right angle are to each other, as 2: ✓
or, bearing in mind that the value of has been fixed at 3.14159, as 2: 1-772453.
Having now explained the theoretical principle of the scale, some of the uses to
which it can be applied will now be explained.
The calculating scale in its simplest form is a triangle like an ordinary drawing
triangle A B C Figure 2, made of wood, metal, celluloid, glass, or any other con-
venient material, the central portion of which is cut out so as to form another,
similar but smaller triangle DE F. This scale may be made use of as an ordinary
drawing triangle, but it may also be used as a calculating instrument_for_the-rectifica-
tion-of-curves-to express in squares the magnitude of plane surfaces limited by
curves, and also that of curved surfaces of the first degree, so far as they are functions
of a circle; and for use in the construction of special ornamental curves, amongst
others in that of the volute in the Ionic capital; all the operations being carried out
graphically and with a rapidity much greater than can be attained by ordinary
methods. The accuracy too, of the work is very high, seeing that it depends on the
value of , which has been so fixed, as to render it, practically exact, as compared
with other means employed: in fact the error amounts to rather less than to
one-millionth part.
The method of solving several problems of importance, to which the scale is readily
adapted will now be described.
Method of squaring a circle of any radius r, (see Figure 3). Let A B be the
diameter of the given circle. Place a rule or straight edge along the line or
diameter A B and at B draw a line B H perpendicular to A B. Still keeping the rule
along the diameter A B, place the scale with its greater cathetus against the rule and
slide it along the latter, until the hypothenuse of the scale triangle reaches the
point A. Mark the point C, in which the hypothenuse, in this position, intersects
the perpendicular; then BC will represent the side of a square the area of which
will be found to be equal to the area of the given circle up to the desired limit of
accuracy. For, since the two catheti are to each other as 2: J if
π one, as is the
case in this problem, is of the length of A B 2, the other must be ✓
of which the square is, which, is precisely the area of a circle, the diameter of
which is 27.
=
TT
To rectify or find the length of a quadrant of a circumference (see Figure 5).
Let M B be the quarter of a circle which is to be rectified. Place a rule or
straight edge along the diameter A B; at B erect a line B H perpendicular to A B.
Now place the calculator scale with its hypothenuse against the 'rule still held against
. the line A B, so that the larger cathetus is to the left, and slide it along the rule,

page_text: until the larger cathetus reaches the extremity A of the diameter A B, cutting the
perpendicular B H at a point C; now slide the scale back, while still holding the rule
firmly in its original position, until the lesser cathetus which is to the right passes
through the said point. C, cutting the prolongation of the diameter A B at a point D ;
then B D will be the length of the quarter circle B M; and twice B D will be equal
to the length of half the circle, and four times B D the length of the whole circle.
To find the length of any are (see Figure 6).
Given the angle of an arc M N as 95° and the radius M O. Find the centre
which will be the point of intersection of two perpendiculars drawn from the centre
of any two chords of the are given, and complete the semicircle A M N B. Having
recourse to the solution of the preceding problem, take on the prolongation of the
diameter A B a length B D equal to a length of a quarter of the circle of which the
arc M N forms part; from D draw a line D E in any direction, and make D E equal
to 90 mm., and join E with B. Now, on E D at the point D, add as many millimetres
DF as are necessary to make the length E F equal to the number of degrees in the
arc, in this case 95 (fractions of degrees may safely be estimated or marked by sight
in proportional lengths of a millimeter), which will give the point F. Through F
draw F G parallel to B E, giving the point G. . B G will be the length of the
are M N within the desired limits of accuracy.
With this scale it is also possible, and always with the same considerable approxi-
mation of accuracy, based on the hypothesis that the value of r is = 3.14159 to
square an ellipse; to develop the curve of half an ellipse; to describe au are with a
given radius, the length of which should be equal to the length of a given
straight line; to find the length of an are given the chord it stands on, and its
pitch; to calculate the surface of a sector of a given circle; of a circular elliptical or
parabolic section; to calculate the development of a circle; to trace cycloids, given
the diameter of the generating circle; to trace special spiral curves by successive
changes in the position of the scale; to transform the perimeter of a polygon into a
circle; to trace a co-ordinate curve, given the angle of the tangents and the radius of
curvature; to calculate the lateral surface of a perpendicular cylinder; to transform
the surface of a sphere to that of a square; to calculate graphically the convex
surface of a calotte or segment of a sphere by transforming it to a rectangle; to
calculate the surface of a spherical zone; to calculate the surface of vaults; to trans-
form into each other, the cone, cylinder and sphere; in short, to calculate a number
of analogous graphic operations, by which to represent graphically all trigonometrical
formulæ and expressions.
It will be seen therefore, from the example given, that the scale is of very great
practical utility for assisting the work of geometrical calculations. It may also be
remarked that the inner space as shown in Figure 4 may be cut so as to represent a
protractor or angle calculator or any other like device; and a two-decimeter or other
scale inay be marked, along the hypothenuse side of the triangle. This arrangement
renders it possible to draw lines at an angle of 45°, which would be impossible with
the triangle alone; not being an isosceles triangle, as well as to draw lines at any
other inclination which is of very great help to a draughtsman.
Having now particularly described and ascertained the nature of my said
invention and in what manner the same is to be performed, I declare that what I
claim is:
7'
1. A scale or instrument for use in drawing and graphic calculations having the
form of a right angled scalene triangle, in which the projection of one cathetus of
the triangle on the hypothenuse, is equal to the the diameter 2 r, of a circle, and the
height of the triangle, the hypothenuse being taken as the base, is equal to the side
3.14159, of a square exactly equal to the surface of the circle 3.141592, and
the projection of the other cathetus of the triangle, on the hypothenuse is equal in
length to the fourth part of the saine circle, viz. :
; the scale being of any
size and material and provided preferably on the edges of the triangle with a two
3.14159.
2

page_text: decimeter or other scale, and in the central part with a protractor, or any like
device.
2. The complete drawing and calculating triangle substantially as described and
illustrated in Figure 4 of the accompanying drawings.
Dated this 8th day of March 1899
BOULT & WADE,
Agents for the Applicant.