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Each to Each
The expression in Euclid is "each to each".
This is inadequate: for the import he annexes
to it, is not conformable to any import the
wor it can be shewn to bear to bear
on any other occasion.
The two sides of the one triangle, says
Euclid, I suppose to be equal to two sides
of the other triangle each to each. The real
each meaning is, each to it's correspondent one:
namely that one a side of the one triangle
is equal to that side of the other triangle
that corresponds with it in some particular and the arrangement
he has for the purpose given the parts sides
of the each triangle two triangles among themselves: given
I say in his own mind, tho' not announced
which he should have done to have made
his meaning clear.
Correspondent one to what? the answer is in situations:
for there is no other particular in which
the two sides of the same triangle are
have any such arbitrary difference assigned
them as can serve to characterize them.
To ascertain by a general description beyond in a
clear and indubitable manner between which of
a doubt the four objects he means the equality
should be supposed, we must give the two
figures an arbitrary situation
correspondent to it that is on some assignable
particular of resemblance; tho' none has he
assigned. In length magnitude for example that is, in length: the only
property particular those except situations {in} which
{they have} they can either differ or again {in}. That is If there
is any difference in length the longest side have
in one Triangle, equal to the longest in the
other: the shortest to the shortest. If there is
no difference in length between those of
the same triangle: then if any one of the two side in
one triangle is equal to one in the other
be that one which it may, all four sides are
equal according to by the . . . . . an axiom.
Identifier:  JB/135/027/002 "JB/" can not be assigned to a declared number type with value 135.



not numbered 

135 

027 

002 

copy/fair copy sheet 
2 

recto 

sir samuel bentham 
[[watermarks::[gr with crown] [pro patria motif]]] 

46145 
