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# JB/135/076/002

Completed

Arithmetic 11. Dec 1794

Square root means
the root of the
square: not a figure root
of a square form,
but the root of a
figure of a square
form.

The ambiguity arises
from the double
signification of
the word square,
having either a substantive
Then In
the above expression
square root it appears
to be used
shews at first sight
(that being its
original signification)
whereas it
is intended to be
taken substantively
The square root,
just as
meaning the root
of the square, just
as we say the Parish
Priest, meaning
the Priest of
the Parish.

By means of this
ambiguity the
same sort of perplexity
is produced
as would be produced
if meaning to stile
a particular man
the Man of the Green,
we were to speak of
him by the name of
the green man.

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So the Cube Root ,
means the root of
the Cube.

A Square number
is so called from
this circumstance,
from this property
belonging to it, that
supposing the units
denoted by it to
be squares, they
would, taken together,
be capable
of being arranged
in the form of a
square: which is
not the case of with
any number that
is not a square.
Thus 4 is a square
number: for 4 equal
squares are capable
of being arranged in
the form of a square.
For the same reason
so is 9: so isare 16,
25 &c. Whereas
at the same time
that no numberthat property
in any of the intermediate
numbers.
You can not make
a square out of two
squares: nor out of
3 squares: nor out
of 5 squares; nor
out of 6,7, 8, 10,
11,12,13,14,15, or
17 squares, and so
on.

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Four being a square
number, 2 is the
square root of that
number: that is
4 being such a
number that a
set of squares
composed of such
a number of squares are
capable of being
placed in form
of a square, when
that form is produced
[+], two of those [+} the figure being
so situated as
that 2 of its sides
shall be parallel
to the horizon

component squares
will form the seat
bottom, base, or
root as it may The truth of the above
explanation this may be seen
exemplified in a
common Draught
Board composed as
it is of 64 squares
On a line of squares
composed of two in
length, as in a base or root you may observe
a squ compound
square standing
composed of
4 elementary squares
on a line of 3, a
square composed of
9: on a line of
4, a square composed
of 16: on a
line of 5, a square
composed of 25:
on a line of 6, a
square composed
of 36: and on a
line of 7, a square
composed of 49:
all these over and
above the whole square
of 64 composed of
the whole line of 8

be called of such
square.

A square number
may accordingly therefore
be defined, any
number of which
the component units
if squares, are capable
of being arranged
in form of
a square without
any interstices, in
the form of a square.

In like manner
a cube number
may be defined
any number of
which the component
units, if
cubes, are capable
of being arranged
without any interstices
in the figure
of a cube.

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Identifier: | JB/135/076/002
"JB/" can not be assigned to a declared number type with value 135.

1794-12-11

135

076

arithmetic

002

rudiments sheet (brouillon)

2

recto

jeremy bentham

46194

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