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PREFAT. DEFINING How far Mathematical Prop: are TRUE
It is to be observed that of the 3 lines lines of a
Triangle as such there is no one in particular to which
the name of Base more properly belongs
than to another. [+] The source from
which That name is given taken only
from the accidental circumstance of the situation
of the figure with respect to the reader.+ And it may be
justly employd
since it is easy
to conceive, that
whatever station
a reader finds thinks proper convenient
to view
the figure from,
that figure itself
must in every respect be still the
same
The purpose for which it is taken is, to
distinguish some one of the 3 that one means
to speak of from the two others that at the instant one
does not mean to speak of, After this distinction two which, two others after this distinction,
others are still (by Euclid) combined under
on the common appellation of “the sides.
Meantime, for whatever purpose and by
whatever reason [& in whatever manner] one way is one justified in
distinguishing any one from the 2 remaining ones for the same
purpose & by the same reason & in the same manner is one justified
in distinguishing those two from one another,
let them be so distinguished, & call one
of them the right the other of them the left.
it - confined
[X extended] aspect
So of the Angles call the one the angle to the right the other
the angle to the left left. Of that which is opposite to the base side,
Some properties however it is necessary should be supposed
to be attributable to certain portions
of quantity at pleasure: & this
be supposed: never we are naturally able to
give to these portions of quantity those propertiesbesets upon
with a truth sufficient for all the purposes of
[common] life. practise.
True it is, that these fundamental properties
we cannot give to portions of quantity to such
a nicety, i:e: to such a degree of proximity
to truth as that whereto can conceive them
to be given in speculation.
I say proximity to truth: for natural truth
is incontestably without above our reach: we
can always perceive some deviations from it
which our organs are unable to do away correct removedefects
defections
& which yet no mind can conceive to be done corrected removed
away: offer a answer analogous to that in
which it had conceived other defects existing
which our bodily organs have
succeeded in doing away correcting
True alas it is therefore, that the ulterior properties
founded upon these do not exert a greater
degree of proximity: but they exist in an
equal one: & that was sufficient for all
purposes of life.
A right lined triangle is not equilateral only
the lines which it consists are right, that is
straight ones: now the truth is that we cannot
draw that is a straight one. That is but
has inequalities incompatible with the idea
of & which we distant, we can
conceive also as to act to non existent, therefore strictly
speaking, the directions of Euclid are insufficient
to enable us to draw an equilateral triangle: but
they are sufficient to enable us to draw a
triangle, that shall come as to this
an equilateral one, as the sides by which […].
When these operations and additions come to be performed,
it is necessary to be able to show that
they are what they are announced for: that is
that they have really possess the properties attributed to them they are announced
for [in the course of the demonstration] and hence it is, that
by these propositions under the name of Problems, the
of comes every here & there to be
interspersed [interrupted.]
The order necessary to be assumed for the
purpose of demonstrating the several propositions
is one thing: the other which would be most convenient &
methodical for the purpose of teaching them
were the only to apprehend & retain
them would be another: the order in which this
occurred to the inventor, probably was a third:
certainly it was different from the first.
We leave therefore 1 three 3 different orders + (unless 2
the 2 latter coincide). 1r The order of
Demonstration. 2dly the order of announcement enunciation, ## of authoritative
instruction. or
3dly the order of invention.
The 1st and only state in which Geometry
has been transmitted to us by the anecedents is
in state four advanced towards perfection;
I mean relatively recording to the order of demonstration
we see it not this science or its first rude essays as we do
those for example of Natural History &
Legislation
Instance the point of a ; & a
CLASSIFICATION
is of one kind
+ Order of
1r Invention
2o Demonstration
3o Systematical
mind of who
first thought of
them.
species of order
It will appear
very plain that
Euclid could never
have first thought
of his propositions
in the order in which
he has placed them
It is not with this
Science, as with
which we may behold
in their
in their to
remove
it
It is necessary to state
that you
what is required: hence
of
comes every here &
there to be interrupted
by Problems
Demonstration is the
establishing the existence of propositions certain
to the
or auditor
are , from
their relations to certain others
that are known to him
The serving
to call to mind
the subject matter
concerning which
the new properties
are predicated, must
have certain
performed upon it
& additions made to
it to call to mind
such the old ones from when
those ones are
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prefat. defining how far mathematical props are true |
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sir samuel bentham |
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