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Arithmetic
As the truth of the
explanation of the
meaning of the expression
square root may
be exemplified by a
plain figure like that
on a Draught board,
exhibiting sufficient containing a competent
number of squares
so may the truth
of the explanation of
the meaning of the
expression cube root
be exemplified by
a sufficient number
of dice.
Upon a line of 3
dice considered taken as a
base or root you
may so heap up lines
of dice (2 in number,
making 3 lines in
the whole) that of the
figure composed of
them two of the surface
shall exhibit
each of them the
appearance of a square
which done, upon the square thus
formed, taking it for
a root or base, you
heap up similar sets
of dice, 2 in number,
(making, in the whole 3 such sets
of lines of dice, three
dice in a line) in such
manner that the figure
composed of these
525 dice shall itself
be of the form of a die. In
In the same manner
you may make
a cube or die
a compound cube
or die by lines
and sets of 2 dice
of composed of 4
dice, or the square
of the square of 2
dice, of 64 dice,
or the square of the
square of 4 dice,
and so on: but
not of any of num
intermediate number
of dice between 8
27, and 64, and
so on forth:
Surd in its archetypal
meaning is
deaf: analogically, & therefore better ashizous,
rootless.
Second, 3d, 4th power
of a number: the
expression elliptical: 
at length it means
the 2d, 3d, 4th product
of the self
multiplicative power
of the number. The
term power, thus
accompanied is
proper and significative:
since no
other power number
in question is
capable of producing
the product in
question, at least without
the help of some
other number.
Biquadrate used
improperly for the
4th power of a number:
it should rather
be applied to
the 3d power, to the
cube: the cube being
the square of the
square: being to the
square what the
square is to the
line
A circulate is a
compound repetend:
i:e: a repetend
composed of
several figures instead
of one.
At the 3d power or
cube the analogy
between the
figure and number
ceases. By
raising the number
of elementary units
to the 4th power, you
do not get a figure
an object which is to thea cube
what the cube is to
its component square.
It cannot be called
a double cube: for
though were the elementary
line is composed
of two units,
two component cubes,
by involving raising the number
of those units to the
4th power you get
enough and just
enough to make
2 cubes, yet this is
not the case with
the cube of any other
number: by raising
3 to the 4th power
you get units enough
not merely to make
2 cubes each having
for its cube root a
line of 3, but 4,3
such cubes: so by
raising 4 to the 4th
power 4 such cubes,
by raising 5, 5 cubes
and so on.
This explanation
of the word square
in an arithmetical sense
as applied to numbers,
does not agree
precisely with the
notion of the word
square in a geometrical
sense, as applied
to figures: it
does not agree with
the notion of the square
of a line, because it
does not agree with
the geometrical notion
of a line. Add
units to units in
sufficient numbers
you make any
square number: but
you may add geometrical
lines to
geometrical lines
for ever, you will
never make the geometrical
figure of
a square. A geometrical
line, a line
in the geometrical
sense is considered
as having no breadth
but with lines having
no breadth you
will never make a
square figure: you
will never make any
such figure as in
the Books of Geometry
as exhibited under
the appellation of a
square. To make
as in with linesa draught
board a square
of which the side shall
be eight inches in length
Identifier:  JB/135/076/003 "JB/" can not be assigned to a declared number type with value 135.



17941211 

135 

076 
arithmetic 

003 

rudiments sheet (brouillon) 
2 

recto 

jeremy bentham 

46194 
