Improvements in the Method of and Apparatus for Teaching Composition of Number and Numeration.
Year: 1896

page_text: Date of Application, 8th Sept., 1896
Complete Specification Left, 6th July, 1897-Accepted, 28th Aug., 1897
Improvements in the Method of and Apparatus for Teaching
Composition of Number and Numeration.
I, WILLIAM RADCLIFFE, Schoolhouse, Andreas, Isle of Man, Schoolmaster, do hereby.
declare the nature of this invention to be as follows:-
This invention consists in improvements in the method of and apparatus for
teaching composition of number and numeration. It is intended to take the place
of the well known ball frame, as well as to be a means of instantaneously forming
symmetrical outlines composed of discs representing units, whereby a very large
number of pictures of numbers can be formed under the notice of a class, for teaching
purposes.
I will describe my invention with reference to the accompanying drawings.
Fig. 1 represents the elevation of my apparatus, which, being in outline an
equilateral triangle, and its purpose such as I have described, I intend naming it
"Radcliffe's Triangle of Number." The apparatus consists, first, of a framework
composed of 3 suitable pieces of wood or other material fastened together in some
well known manner to form the outline of an equilateral triangle. The size of this
frame is determined by the fact that the most convenient size of apparatus for
hanging on a blackboard or easel, or for suspending it in some other way before a
class, will be with sides 1 ft. to 2 ft. in length. Inside this large triangle, is fixed
a figure consisting approximately of 6 equilateral triangles, of equal size, represented
on the drawing by the letters u v w x y z. This figure is formed of wood or other
suitable material, but if lightness is required, more or less of the middle of each of
these 6 solid triangles may be cut away, thus leaving said triangles in skeleton forin.
Their purpose is to divide the large triangle into 10 other empty spaces of equal size
which approximate as nearly to equilateral triangles as the methods of fixing together
of divisional framework into the outer frame work, will permit, or as keeping the
6 triangles in one piece or fastening them into one piece will permit. These 10 empty
triangles are each to be filled with 10 discs of equal size, cach disc representing a
unit of number, so that the whole triangle will contain 10 small triangles of 10 units
each. These discs are arranged so that there are 4 in the bottom line, 3 in the
second, 2 in the 3rd and one in the top of each triangle, and in this particular of
arrangement they resemble the 10 small triangles themselves, which are similarly
arranged in the large triangle. Each disc has 2 faces which are stained, painted,
enamelled or otherwise coloured in such a manner that one face shall be easily
distinguishable from the other. This is best effected by using on one side a
colour in striking contrast to that on the opposite side. The discs may be of
wood or other suitable material and must be of such thickness as to allow of
their free revolution independently of each other on the spindles in such manner
as to show either face upright,-a horizontal row of dises in any triangle having
a common spindle. I have in the drawing represented the spindles as passing
horizontally parallel to each other across the small triangles and through the
dividing framework (which must be sufficiently thick for the purpose) right on from

page_text: side B to side A. This will be found a more convenient way than cutting the
spindles into the short pieces necessary for each small triangle. Having placed the
discs on the spindles, and having the front faces of said discs all one colour, and
the back faces of said discs all another colour, contrasting with the former; and the
discs having been made of such diameter that, though as nearly as possible filling
up each triangle, they yet have room to revolve independently of each other; and
having made the outside triangular frame A B C sufficiently thick in a backward
direction, to allow of the discs revolving clear of the blackboard, or wall, or other
background against which it is placed; they should readily revolve, with a slight
push from the finger tip, so as to bring to the front the opposite and contrasting
faces of the discs. It will be well, in choosing colours for the discs, to avoid
black, inasmuch as the apparatus will be most conveniently hung upon a
blackboard for teaching purposes, and the black discs would not show well against
the black background, which latter would be noticeable between the discs. By
means of bringing to the front view the faces of discs contrasting in colour with those
surrounding them in their respective triangles, it is possible to form pictures or
diagrams of numbers which will give to a child or other learner a pleasing concrete
view of an abstract idea, or enable the teacher to educe from the children the
abstract ideas embodied in the pictures or diagrams.
Thus Fig. 2 gives typical pictures of the numbers 3, 4, 5, 6, 7, 8, 9, 10 which may
be exhibited to view at one time on my apparatus by the process indicated.
Figure 3 gives various pictures of the number 8 as an illustration of the same
number in different aspects, all of which may be exhibited at the same time. I will
give a few of the abstract ideas that may be educed from said pictures of number 8.
(a) 448, or two 4's are 8; or 8 ÷ 4 = 2, or 84 = 4; also 5 + 3 = 8;
(b) 7+ 1
17; (c) 17 = 8; (d) 6+2 8; (e) 6+2 8.
It will be noticed that each picture number is also always shown in its relation to
the number 10. Thus the number 3 in Fig. 2 shows 7 white discs, these being its
complement or the number required to make 10..
=
8; 8
-
=
=
Figure 4 shows how the apparatus may be used for teaching "the device of place"
in numeration. It will be noted that there are 3 complete small triangles of 10 units
each, and from this can be educed the fact that the number 3 in 38 represents 30,
and similarly for all other numbers from 11 to 99.
It is evident that 10 of the larger triangles placed together in a manner similar to
the arrangement of the small triangles would form a triangle representing a thousand
units.
Dated this 4th day of September 1896.
WILLIAM RADCLIFFE,
Applicant.
Improvements in the Method of and Apparatus for Teaching
Composition of Number and Numeration.
I, WILLIAM RADCLIFFE, Schoolhouse, Andreas, Isle of Man, Schoolmaster, 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:-
This invention consists in improvements in the method of, and apparatus for,
teaching composition of number and numeration. It is intended to do the work of
the well known abacus or ball frame, as well as to be a means of making and changing

page_text: instantaneously a very large variety of pictures of numbers such as at present are
published in the form of diagrams on wall sheets. It will also do the work of several
forms of arithmetical demonstrators, and it is pre-eminently fitted to show the
principles underlying the decimal system of notation and numeration. Specifically,
it is intended as an aid in the teaching of numeration, addition, subtraction,
multiplication, division, vulgar fractions, decinal fractions, metric system, proportion,
percentages, and square root.
I will describe my invention with reference to the accompanying drawings,
giving first its primary form and the essential features in the mathematics
underlying its construction. Afterwards I will show the method and materials of
its construction; the mode of operating the apparatus and the manner of applying.
it to the purposes intended by the invention.
The outline of the apparatus being an equilateral triangle, and its use being"
connected with the teaching of number, I propose that the said apparatus shall be,
called "Radcliffe's Triangle of Number."
Fig. 1 shows the mathematical form underlying its construction, and consists
essentially of an equilateral triangle subdivided by the figure u v w x y z into.
10 smaller equilateral triangles, which again are subdivided into 10 circular spaces in,
the manner represented in Figure 1, so that the whole triangle shall represent"
100 circular spaces, the arrangement of said circular spaces in the smaller triangles
being similar to the arrangement of the smaller triangles in the whole or large,
triangle; that is to say, that as there are in the large triangle, 4 smaller triangles in
the base, 3 in the next series, 2 in the next and 1 at the apex, so in each small triangle
there are 4 circular spaces in the base, 3 in the next series, 2 in the next and 1 at the
apex.
I will now describe the method and materials of construction.
There is, first, a framework composed of three suitable pieces of material (preferably
picture frame wood moulding) mitred or otherwise fastened together at their ends, so
as to form the outline of an equilateral triangle.
In one form of the apparatus this frame-work is not absolutely necessary, but I
prefer to retain it in this and all forms of the apparatus as adding to the strength,
and beauty of the contrivance. This frame-work may be of any convenient size, but
perhaps the most convenient size for tenching purposes (the apparatus being hung
upon a wall or blackboard or easel; or mounted on a stand, or otherwise exhibited
before a class) will he with sides about 2 ft. in length. The circular spaces
represented in Figure 1 are embodied in the apparatus either in the form of discs or
balls (each preferably made of wood) and the balls or discs may each be connected
with the apparatus in two ways, thus originating two distinct forms of my invention
which are, nevertheless, essentially the same in principle; i.e., the relation of the
balls or discs to the remainder of the apparatus may be
(1) Removable or
(2) Revolving.
The mode of construction of the apparatus depends chiefly upon whether the balls
or discs are "removable " or " revolving" using these terms in the particular sense
hereafter described.
Figure 2 shows the elevation of the apparatus, using removable balls or discs.
This is that form of the apparatus in which the triangular frame-work is not
absolutely necessary though preferably retained. The frame-work is planted upon
and around a wooden equilateral triangle, flat on both sides, cut out of one piece if
possible (though this triangle may be formed of pieces of boards fastened together)
and said triangular piece of wood must be of sufficient thickness to admit of pegs or
headless nails or things similar, being driven into its front surface vertically, and with
such grip, and in such positions, and in such a manner, as will allow of the balls or,
discs being freely slipped in upon them against the surface of the triangle (as shown
in Figures 2 and 2) and as readily removed again.
By preference also the pegs or headless nails or similar things should extend to
near the top of the central boring made through said balls or discs (which boring"

page_text: they must all have) to prevent their too readily slipping off. On the other hand, the
pegs or things similar should not project too far through the bore holes of the balls
or discs as this would be unsightly. The balls or discs must be of such a size as just
to fill up the small triangles and yet allow of their removal by being taken off with
one's fingers. It is obvious that the balls will be more easily gripped than the discs,
and I will use the balls by preference in this form of the apparatus.
I propose to attach to the frame-work of the triangle a tray or box or netting or
other convenient receptacle in which to throw the balls or discs when not wanted in
position upon the pegs. Each dise has, of course, two flat circular surfaces or sides,
and, on the peg, either circle can be presented to view. Similarly, a ball is made up
of two opposite hemispheres, and on the peg, either hemisphere can be presented to
the view. I propose to paint, stain, or enamel the opposite sides or surfaces of the
bulls or discs so, that each hemisphere or circle may be presented to view in
contrasting colours. This is not absolutely necessary in this form of the apparatus,
but. I retain it as highly convenient in certain cases of its use.
Figure 1 shows also that the ten smaller triangles are divided from each other.by
means of the outline u v w x y 2, which really consists of six equilateral triangles of
equal size. In the form of the apparatus now being described it is highly desirable
(though not absolutely necessary) that six equal sized equilateral triangles of wood,
and, by preference, of a thickness equal to about half the diameter of the balls,
should be fastened upon the large triangle so as to separate the ten small ball-filled
triangles from each other. When discs fill the small triangles these divisions are
still less necessary, but as they in both cases add to the good appearance of the article
I retain them.
Figure 3 shows the elevation of the apparatus when revolving balls or discs are
used. In this form, spindles pass horizontally parallel to each other across the
breadth of the triangle. This is more convenient than making short spindles to
suit the breadth of the small triangles separately. Either way of fixing the
spindles will, however, be, satisfactory. The spindles inust be of sufficient thickness
to ensure strength. I have found steel rods inch diameter very suitable for use in
a triangle the side of which measures 2 ft.
In the case of the balls, the borehole must be sufficiently wide to allow of them
revolving on the spindle; and in the case of dises being used, they must be
sufficiently thick to allow of a central hole being bored through them for spindle.
See Figures 3 and 3". While as nearly as possible filling up each small triangle,
the balls or discs should yet have room upon the spindle to revolve independently of
each other. The way of causing them to revolve is by a touch with the point of a
finger.
In this form of the apparatus the outer frame-work A B C is absolutely necessary
and will have to be of sufficient thickness in a backward direction to allow the
balls or discs to revolve clear of the back of the apparatus (in case such should
be used) and clear of any article upon which the triangle has been fixed, or from
which it has been suspended, for teaching purposes. It is not essential that a
back should be placed in the apparatus, but at the same time, it enhances the
good appearance of the contrivance.
In using discs I have found it more convenient to use a peculiar kind than
those just described, and illustrated in Figure 3". A section of the latter kind is
shown in 3 and the method of obtaining them, viz., by cutting away 3 sections
from a globular ball, is also shown. Balls can be cut to show more than 3 circular
surfaces but it makes the working of the apparatus more complicated.
In this form of my apparatus it is absolutely necessary that the opposite halves
of the balls, or the successive faces of the discs, shall be stained, painted, or
enamelled in contrasting colours so that in the case of the balls, two different
colours can be brought to view, and in the case of the discs two or more different
colours. In this form of apparatus also, the embodiment of the outline u v w x y z
in Figure 1 is preferably one solid piece (as shown in Figure 3) or the smaller

page_text: triangular pieces may be jointed together, or they may be fixed to the back of the
triangle when such back is used.
Having described the two forms of my apparatus, I now proceed to show the mode
of operating these forms and the application of the apparatus to the purposes intended
by the invention.
In using the removable balls or dies, pictures or diagrams may be built up or
analysed, the various outlines appearing in strong relief against the naked or
unoccupied pegs. As before described, the element of colour can be brought to the
assistance of form for this purpose.
Similar pictures or diagrams can be also shown in a vast number of cases by the
use of that pattern of the apparatus in which revolving balls or discs are used: the
forms or outlines in this case being made up of similarly coloured discs showing out
from a ground work of contrasting colour. When discs having more than two
coloured faces are used it is possible to analyse the picture or diagram by the aid of
these further colours. In practice, however, I have found the "Movable Ball" type
of the apparatus far more effective than the "Revolving Ball or Disc" type
I now proceed to show the application of various diagrams to the elucidating of the
arithmetical ideas already enumerated by me. In giving these diagrams (which are
only types of what may be effected on my apparatus) I have indicated which pattern
of the apparatus they may be obtained from, and for this purpose I have described.
the "Movable Ball "pattern as "M" and the "Revolving Disc" pattern as "R."
Fig. 4 gives representations of the Nos. 1 to 11, showing in a rational way that
11 means 1 ten plus I unit. Similarly all tens and units up to 99 can be represented
on the apparatus, whilst the whole triangle represents 1 hundred, 0 tens, 0 units.
The number 133 is represented in the graphic notation of this apparatus.
Figure 5 shows how the sum of 8+ 5 = 13. By the "Movable Ball" method
2 of the 5 balls would be placed with the 8, thus completing one small triangle or ten'
with 3 units to spare. On the "Revolving Disc" system 2 units would be brought to
view, of the same colour as the other 8, and to compensate, 2 out of the 5 similar
colours would be changed, thus leaving 3 to go along with the completed ten:
hence 13.
Subtraction (Figure 6) can be demonstrated (M) by taking away 9 units out of
1 ten, leaving a remaining unit to be added to the other 3, whilst I ten remains
untouched: hence 14; (R) by changing 9 units in one of the tens to another
colour, thus leaving of the original colour, I ten and 4 units, i.e. 14.
Figure 7 represents a diagram that cannot be exhibited with much satisfaction by
the "R" method. On the "M" system it only requires the 4 sevens to be
distributed into the small triangles, filling them to their utmost capacity, when it
will be found that there are 2 complete tens with 8 units: hence 28. This process
also serves to fix the reverse truths in the mind, for if a person thoroughly
understands how 4 sevens make 28, he will also understand the statement that 4 goes
into 28 seven times. In Figure 8 the shaded discs represent the fractions to be added,
and the demonstration needs no further explanation. The same may be said of
Figure 9.
In using the triangle to give a representation of a decimal fraction I consider the
whole triangle as unity and represented by the number 1. The small triangles are
similar to the large one. I take one of these and show that while it is 1 just the
same as the larger triangle, yet, not being of the same value, we must find a means
of representing it so as to give a proper conception of its value: hence 1. From
this, by similar reasoning I show that 1 dise in comparison with 1 whole triangle
ought to be represented as 01. For an elementary idea of the metric system I point
to one of the smaller triangles as a unit of length or area, or other measure, and I
show that on this supposition, the whole triangle would represent a multiple, while
1 of the dises would represent a submultiple. To give an illustration of the principle
of proportion being "the Equality of Katios," I use the triangle in two ways, and I
think these will be understood from Figs. 10 & 11. A picture of the meaning of
percentage can readily be obtained by the comparison of any number of dises with

page_text: the large triangle of which they form a part, giving a graphic idea of the value of the,
percentage in question.
Fig. 12 gives a picture of the number 7 contained seven times in an outline which
is still a picture of the number 7 on a larger scale. The roots from all squares from
1 to 100 can be represented in a similar sort of way, forming at any rate a pleasing
and novel picture of a root in relation to its square.
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:
(1) The method of dividing and subdividing the surface of an apparatus having
the outline of an equilateral triangle, substantially as described.
(2) The various modifications of details & the arrangement of the details
composing the surface of said equilateral triangular apparatus, substantially as
described.
(3) The application of said equilateral triangular apparatus to the purpose of
illustrating & explaining the "Device of Place" in numeration.
(4) The application of this apparatus to the making of instantaneous pictures of
numbers and of demonstrating their composition and relationship, substantially as
described.
(5) The methods of forming number pictures by slipping wooden or other kinds
of balls on pegs (whether balls are coloured or uncoloured) and by discs on pegs
(coloured or uncoloured) or by revolving coloured balls & discs, substantially as
described.
(6) The use of a receptacle attached to some part of the triangle for the purpose
of containing unused balls or discs.
Dated this 2nd dow of Tulu 1807
WILLIAM RADCLIFFE,
Applicant.