Archimedes Texts

183
But the curve-line A C B, is greater than the two right-lines A C,
and C B; therefore, à fortiori, the curve-line A C B, is much
greater than the right line A B, which was to be demonstrated.

The demonstra
tion of a Peripate
tick, to prove the
right line to be the
shortest of all lines.

SALV. I do not think that if one should ransack all the Para
logisms of the world, there could be found one more commodious
than this, to give an example of the most solemn fallacy of all
fallacies, namely, than that which proveth ignotum per ignotius.

SALV. Do you ask me how so? The unknown conclusion
which you desire to prove, is it not, that the curved line A C B, is
longer than the right line A B; the middle term which is taken
for known, is that the curve-line A C B, is greater than the two
lines A C and C B, the which are known to be greater than A B;
And if it be unknown whether the curve-line be greater than the
single right-line A B, shall it not be much more unknown whether
it be greater than the two right lines A C & C B, which are known
to be greater than the sole line A B, & yet you assume it as known?

SIMP. I do not yet very well perceive wherein lyeth the fal
lacy.

SALV. As the two right lines are greater than A B, (as may be
known by Euclid) and in as much as the curve line is longer than
the two right lines A C and B C, shall it not not be much greater
than the sole right line A B?

SALV. That the curve-line A C B, is greater than the right
line A B, is the conclusion more known than the middle term,
which is, that the same curve-line is greater than the two right
lines A C and C B. Now when the middle term is less known
than the conclusion, it is called a proving ignotum per ignotius.
But to return to our purpose, it is sufficient that you know the
right line to be the shortest of all the lines that can be drawn be
tween two points. And as to the principal conclusion, you say,
that the material sphere doth not touch the sphere in one sole
point. What then is its contact?

SIMP. It shall be a part of its superficies.

SALV. And the contact likewise of another sphere equal to the
first, shall be also a like particle of its superficies?

SIMP. There is no reason vvhy it should be othervvise.

SALV. Then the tvvo spheres vvhich touch each other, shall
touch vvith the tvvo same particles of a superficies, for each of them
agreeing to one and the same plane, they must of necessity agree
in like manner to each other. Imagine now that the two spheres

[in Fig. 6.] whose centres are A and B, do touch one another:
and let their centres be conjoyned by the right line A B, which
passeth through the contact. It passeth thorow the point C, and