An Improved Apparatus for the Graphic Demonstration of Trigonometry.
Year: 1898

page_text: Date of Application, 28th Mar., 1898
Complete Specification Left, 23th Jan., 1899-Accepted, 25th Mar., 1899.
PROVISIONAL SPECIFICATION.
An Improved Apparatus for the Graphic Demonstration of
Trigonometry.
GEORGE WILLIAM USILL, 145, Cannon Street, London, E.C., Civil Engineer,
FREDERICK JOHN BROWNE, 145, Cannon Street, London, E.C., do hereby declare
the nature of this invention to be as follows;----
An appliance, appliances, diagram or diagrams actuated mechanically or other-
wise, to demonstrate mathematical science with special reference to trigonometry
and geometry. Such appliance or appliances, diagram or diagrams may include
figures represented by the circle, semi-circle, quadrant or such divisions of the
circumference, as we may deem necessary or by straight lines, triangles
or any other figure which may be delineated or represented graphically or otherwise
upon a plane surface. In carrying out our invention we cause an arm or arms,
triangle or triangle or any plane figure or figures to revolve about a given centre
or Series of centres or to move in such directions as we may determine and by
being actuated by mechanical motion or otherwise may form figures varying in
shape and size from the original figure or figures delineated upon the plane surface
or base upon which the moveable parts rotate or move. Particularly we draw
on the surface of a base or on a card or any suitable material covering a base a
diagram representing the trigonometrical canon and from the centre by suitable
mechanism or by hand without mechanism we cause an arm or arms to rotate
about that centre. Attached to the arm or arms may be a right-angled piece made
of any suitable material and by mechanism or otherwise we cause such right-angled
piece to travel synchronously with the arm or arms so that one leg of the right
angled piece is vertical and the other horizontal or by any suitable means we
cause a triangle or other plane figure of suitable material to rotate about a centre
on a plane surface upon which we draw a diagram representing the position of
such rotating triangle or other plane figure in any one or more positions of its
rotation or by the combination of all or any plane figures rotating about a centre
or a series of centres so as to graphically demonstrate the varying trigonometrical
functions.
GEORGE WM. USILL.
FRE. J. BROWNE.

page_text: 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:-
An appliance, appliances, diagram or diagrams actuated mechanically or other-
wise, to demonstrate mathematical science with special reference to trigonometry
and geometry. Such appliance or appliances, diagram or diagrams may include
figures represented by the circle, semi-circle, quadrant or such divisions of the
circumference of the circle as we may deem necessary or by straight lines, triangles
or any other figure which may be delineated or represented graphically or otherwise
upon a plane surface. In carrying out our invention we cause an arm or arms,
triangle or triangles or any plane figure or figures to revolve about a given centre
or series of centres or to move in such directions as we may determine and by
being actuated by. mechanical motion or otherwise may form figures varying in
shape and size from the original figure or figures delineated upon the plane surface
or base upon which the moveable parts rotate or move. Particularly we draw
on the surface of a base or on a card or any suitable material covering a base a
diagram representing the trigonometrical canon and from the centre by suitable
mechanism or by hand without mechanism we cause an arm or arms to rotate
about that centre. Attached to the arm or arms may be a right-angled piece made
of any suitable material and by mechanism or otherwise we cause such right-angled
piece to travel synchronously with the arm or arms so that one leg of the right-
angled piece is vertical and the other horizontal or by any suitable means we
cause a triangle or other plane figure of suitable material to rotate about a centre
on a plane surface upon which we draw a diagram representing the position of
such rotating triangle or other plane figure in any one or more positions of its
rotation or by the combination of all or any plane figures rotating about a centre
or a series of centres so as to graphically demonstrate the varying trigonometrical
functions.
In order to explain the manner in which we carry out our invention we refer
to Sheet I of the accompanying drawings. Fig. I on that sheet shows the front
of the base or board upon which is mounted various shaped pieces. Fig. II
Sheet I shews the back of the board, A & B being two toothed wheels of equal
diameters suitably mounted upon the board & held in place by the pieces M & N.
D) is a toothed rack gearing with A & B & sliding in the channel S, being
held in position by the pieces P & P C is another toothed wheel gearing with
the wheel B & held in position by the piece R. The spindle or shaft of the
wheel C being so formed that a key or handle may be applied to it so that by
turning C motion is given to the wheel B and through the rack 1) to the wheel A.
In Fig. I a part of the board or cover of the base is cut away shewing the piece G¹
which is mounted upon the spindle or shaft of the wheel B." Upon the wheel A, a
similar piece G is mounted. The object of these pieces is to obtain a greater
radius than can be obtained directly from the wheels without unduly increasing
their diameter. Fig. I A & B are respectively pins or centres placed at equal
distances from the centre of the wheels A & B upon the cranks G&G', E E' is
a right angled piece made of any suitable material preferably of transparent
celluloid upon which suitable provision is made to attach it to the centres A¹ & B'.
Fis a travelling arm made of any suitable material preferably transparent
celluloid & of any required length according to the size of the base or board.
Suitable provision is made for attaching this travelling arm to the centre O &
also to the pin or centre A¹. K & K¹ are suitable coffins sunk into the base or
board beneath the front card in which the cranks G & G can move through a
quadrant of the circle. H is a slot cut in the card so that the centre A¹ may move
completely through the quadrant. To follow out the exact action, by turning the
wheel C from left to right we cause the wheel B to revolve from right to left, thus
causing the rack D to rise which causes the wheel A to turn from right to left and
to carry with it the crank G. By the dotted lines we have shown the position of

page_text: the various parts when the cranks G & G¹ have travelled over 3 of the quadrant.
It will there be seen that the right angled piece E E¹ has moved in towards the
centre O in a horizontal direction & vertically upward away from the centre O. The
top of the arm E¹ being the line of the graphic co-sine of the angle of 60° &
the vertical outside edge of the arm E being the line of the sine of the angle 60°.
The travelling arm F shews the tangent & co-tangent, secant & co-secant
graphically by the point at which it cuts the tangential line X & the dotted
co-tangential line Y.
On Sheet II Fig. I & II we show a means of graphically demonstrating
trigonometry in book or card form. Fig. I Sheet II is a card upon which we
draw a quadrant suitably divided as at 30° & 60°. From the centre O a vertical
line is drawn forming a right angle with the travelling arm F which rotates
about the centre O. We have shewn in dotted lines two positions at 30° & 60°
which the travelling arm may take & as will be seen from the points at which the
travelling arm cuts the tangent & co-tangent lines X & Y the graphic lengths of
the tangent & co-tangent, secant & co-secant are graphically demonstrated. In
Fig. II Sheet II we show upon another card a graphic method demonstrating
that the functions of the angle A are equal to the co-functions of the complemen-
tary angle B. In order to do this we mount a triangle A B C made of celluloid
or other suitable material pivoted at C so that it may be revolved to take up a
position as shewn by the dotted triangle A' B₁ C. From this it will be seen that
as the sine of the angle A is equal to the perpendicular B C divided by the
hypotenuse A B, so the cosine of the complementary angle B is the base B C
divided by the hypotenuse A¹ B' & so each of the other functions of the angle A
may be shown to be equal to the co-functions of the complementary angle B.
Having now particularly described and ascertained the nature of our said
invention and in what manner the same is to be performed we declare that what
we claim is:-
:-
The combination of moveable figures, upon a base, actuated mechanically or by
other suitable means in connection with the graphic demonstration of trigono-
metry.
M
Ja of Tanum 1900
GEORGE WILLIAM USILL.
FRE. J. BROWNE.