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1824 May 5
Posology — Rudiments
1
The contrivance in
Euclids first proposition.
— namely a Problem — how to construct
an equilateral triangle.
Mode of designating and
exhibiting it.
2
Here it follows expressed
in the asi
and historical inferential or conjectural
method.
3
Take any portion of
matter exhibiting a rectilinear figure
for instance a small
twig of a tree
the straitest you can
find: consider it as if it were
perfectly strait.
4
Describe two circles with
it.
5
Circle the first is described
by keeping the twig fixt
at one of the its ends as
points: which with the
exception of the
point at that end, the rest
of the twig is moved round
till it comes back into the
a position exactly the same
as that which it occupied
before it began to move.
6
If the twig has any
coloring matter on
that part of its surface
which touches the ground
and this matter is sufficiently
copious and
adheres to leave a
mark over the whole of
the surface to which it
has applied itself, the
sort of figure
called a circle will
be the result
7
The line by which the
figure is bounded at all
points over which the moving
end of the twig has passed is the circumferential line: — in one word, the circumference.
8
One mark having thus
been made with the twig,
employ now this same
twig in the formation of
another circle
9
But in the formation
of this new circle the
whole twig must not be
moved from the position
it had when employed
in the formation of the
first circle
10
The end which in the
formation of the first circle
was the first end, must
in the formation of the second
circle be the moving
end: and conversely,
the end which in the formation
of the first circle
was the moving end must
new be the fixt end
11
The position which
the twig occupied the instant
before the formation
of the first circle was commenced
should have
been noted: if supposing
the coloring matter
previously applied to it
on the raising of the the whole
twig from its position
it will have been seen
to have left on the ground
the figure of a line.
12
The second circle
having been formed as
above, the result will
be two circular figures
cutting one another at
two points.
13
If The position of the twig at the
commencement of the second
circle was such
as to be a continuation
of that at which it
occupied at the formation
of the first in
such sort that no angle
or corner can be seen
to be formed between the
two, the whole line is composed
will thus be seen composed
of three distinguishable
parts: 1. a middle
part which belongs equally
to both circles
2. a part peculiar to
the first circle: and being
either on the left or
the right of the middle
line — say on the left.
3. and a part peculiar
to the second circle
and being on the right
of the middle line.
Place now the whole compound
figure with the compound line before
you in such this a horizontal
position
14
Place yourself now in
such a situation as
that nearest to you
shall be the bottom lower
parts of the two the circles
thus connected circles.
furthest from you consequently
the upper top
parts.
15
From the right hand
end of the middle line
trace a line to that one
of the two intersecting parts
of the two circles which
is
is furthest from you: this
you may do by laying that
same twig in that position
Turn the left hand end
of the same middle line that
trace another line to that
same furthest point; in
that same manner
16
You will this have another
line which being
traced by the same twig
can not be different in
its length — from — can not be
otherwise than exactly
equal to the first.
17
Thus then you have
two lines equal to one
another
18
In that same manner
From the right hand
end of that same middle
line with that same twig trace now another
line to that same furthest
point. By this means
you have obtained a
third line exactly equal
to both each of the two
others which you have
already seen to be exactly
equal the one to the other
Thus have you obtained
a figure compleat plain
figure — a surface
bounded on all sides
by three lines all of
them of the same length
a trilateral equilateral
figure.
19
These three lines will
you will see meeting at
three places: and at each
of these places, you will
see two of the lines forming
between them one angle or
corner: on consideration
of this circumstance the
figure is called a triangular
figure and in consideration
of the equality of the all three bounding
lines an equilateral
triangle. M. T. Col. 2
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Identifier: | JB/135/299/001 "JB/" can not be assigned to a declared number type with value 135.
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1824-05-05 |
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135 |
posology |
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299 |
posology rudiments |
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rudiments sheet (brouillon) |
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jeremy bentham |
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