Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/208

 But the curve-line ACB, is greater than the two right-lines AC, and CB; therefore, à fortiori, the curve-line ACB, is much greater than the right line AB, which was to be demonstrated.

I do not think that if one should ransack all the Paralogisms 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.

How so?

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

I do not yet very well perceive wherein lyeth the fallacy.

As the two right lines are greater than AB, (as may be known by Euclid) and in as much as the curve line is longer than the two right lines AC and BC, shall it not not be much greater than the sole right line AB?

It shall so.

That the curve-line ACB, is greater than the right line AB, is the conclusion more known than the middle term, which is, that the same curve-line is greater than the two right-lines AC and CB. 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 between 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?

It shall be a part of its superficies.

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

There is no reason why it should be otherwise.

Then the two spheres which touch each other, shall touch with the two 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 AB, which passeth through the contact. It passeth thorow the point C, and