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Click Here To Edit EUCLID LI Prop. 8
This may be found
better without than
with a diagram.
From the same point are drawn two different right lines one to
one point, another to another
a right line is the shortest that can be drawn
between any two points.
If then any other 2 right lines are once more use drawn from
the first point point in question to the 2 others, they must
be either the same with the two others
or different.
If they are the same they don't come under
the terms of the proposition which is predicated
of such as are different.
They are either must
taken together they must either be greater
than the two first or less or equal.
If taken together they are greater than one
at least if not both, must be greater & so
each is not equal to the corresponding
one of the other: if taken together they
are of less, than one at least if not both
must be less, & so in this case too, each
is not equal.
If taken together they are equal with the two.
corresponding ones taken together, they must coincide
with them.
CLASSIFICATION
This Proposition may serve as an example to
shew prove that the order of demonstration is not the
same with the order of invention.
It seems as if the Author had gotten some
inexplicit notion of the truth of it in the
ways I have been mentioning: but that which he
has used happen'd to be the first method that
occurred to him of proving it against
a gainsayer.
[+] to the several parts that are to be considered
names are given according to the
Ordinary method of Notation, it is included
in 3 boundaries to one of these boundaries is given
the name of A to another B to the remaining one
C. to the whole triangle thus included the
complex name ABC.
Having then The Appellatives Upon precisely
this footing that is the footing of Proper names
stand those Appellatives which in the ordinary
method of notation in use in Geometry
are applied to denote any figure or part
of a figure upon which is drawn the demonstration
is exemplified. The Proposition we will say conceives
Triangles {Triangles we will say are the subject of the
Proposition} a Triangle is drawn[+] This
Triangle the moment it is drawn is
of a particular sort of Triangle it
is either an equilateral triangle an
Isosceles or a scalene not only so it is that an
Individual one triangle which is drawn
it is that Individual triangle which is
drawn. Of what is it then that these
letters are expressive? preparatively the Individual
lines which are its boundaries together. Of the
Individual space included between these
lines either the of those lines be situated
as they are so as to bound that whole individual triangle
or as we may say the whole area of that triangle
or else the Area itself that is so bounded.
---page break---
[+] Sad consequence
would accrue from
the diagrams, but
this seems a difficult
matter to effect
is it not better
to leave the felon
at large with liberty
to spare his figure
himself keeping only
within the bounds
of the genus specified.
o See +
See Geometry
No I
This Individual triangle we will suppose is of the sort
which is called equilateral. and
Let us suppose that the proportion in
question manner that the it has been
demonstrated that its 3 angles are
equal to two right ones. the Scholar
we will supposed perfectly convinced of the
truth of it but at the end of the demonstration
perhaps concludes thus are these words therefore the 3
angles of all triangles are equal to
2 right ones. at this he is staggered
and it costs him much pain to be able
to convince himself of that the proposition
would hold good of all under the Genus
of triangles. Euclid does not take pains
enough to make his Scolar consider the
figure he gives as example only and to
convince them of the truth of the Proof.
with respect to all figures of the Genus
he gives. It may be said that if the
Scholar were sufficiently apprised always
the figure in question as an example only no[+]
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