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1832 Jany 15


Observations on Euclid


In this same manner may be described parallel lines
in any number: with the same moving line for instrument, all, described by the same operation, at the
same time: having for their common base the inferior line, on
which the moving line moves.

Of these superior lines call that which is at the greatest
distance from the base the supreme line or say parallel: the other
of but one, the intermediate line: if there be more than one
the others call intermediate or intervening lines

In the same manner may be drawn described, also in any number, not only right
lines as above but curve lines: and these, of any mode of curvature which has
place in the instance of the base line.

When in this manner parallel lines are actually
formed, the parallelism will for its accuracy, depend on
the accuracy of the machinery employed in the operation. But, by
this circumstance no objection is opposed to the fitness
of the manner of performing the operation, or say to the accuracy
of the result: for in no other way does this result depend on the machinery for
its accuracy, than the accuracy of the circle, in the case when
a circle is the result

For the purpose of demonstration, in the case of the circle
actual performance is now regarded as necessary: conception is at
times regarded as sufficient: and so in the case of every other
figure: so therefore may it with equal propriety, may in be
in the case of every other sort of figure.

As between a circle, and a pair of parallel lines

In the case of two parallel lines the inferior will be analogous
to the centre of the circle, the superior to the circumference, the
area thus travelled over, to the circle itself

If what is desired be actually to draw a superior parallel line
without drawing any intermediate area between that and the base
the moving line must be fixt so as to occupy a plane
superior to that in which the parallel lines are meant to be described:
it will, in that case form an angle which in relation to a
perpendicular to the inferior plane will can not be greater than
a right angle, but may be in any degree less: if the whole area that which is
desired be the whole area, the moving line will be in the same
place throughout the whole length of it: if, that which is desired be the superior
line above — say the quasi-
circumference a line
must be in the manner of
a pen must be let fall
from the superior extremity
of the moving line upon the
plane, in which it is intended
that the line parallel to the base shall be described.

Identifier: | JB/135/170/001
"JB/" can not be assigned to a declared number type with value 135.



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