Salusbury, Thomas, Mathematical collections and translations (Tome I), 1667

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1But 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
The
tion of a
tick, to prove the
right line to be the
ſhorteſt of all lines.
The Paralogiſm
of the ſame
tetick, which
veth ignotum per
ignotius.
SALV. I do not think that if one ſhould ranſack all the
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.
SIMP. How ſo?
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
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?
SIMP. It ſhall ſo.
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
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
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