Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/209

 another point in the contact being taken as D, conjoyn the two right lines AD and BD, so as that they make the triangle ADB; of which the two sides AD and DB shall be equal to the other one ACB, both those and this containing two semidiameters, which by the definition of the sphere are all equal: and thus the right line AB, drawn between the two centres A and B, shall not be the shortest of all, the two lines AD and DB being equal to it: which by your own concession is absurd.

This demonstration holdeth in the abstracted, but not in the material spheres.

Instance then wherein the fallacy of my argument consisteth, if as you say it is not concluding in the material spheres, but holdeth good in the immaterial and abstracted.

The material spheres are subject to many accidents, which the immaterial are free from. And because it cannot be, that a sphere of metal passing along a plane, its own weight should not so depress it, as that the plain should yield somewhat, or that the sphere it self should not in the contact admit of some impression. Moreover, it is very hard for that plane to be perfect, if for nothing else, yet at least for that its matter is porous: and perhaps it will be no less difficult to find a sphere so perfect, as that it hath all the lines from the centre to the superficies, exactly equal.

I very readily grant you all this that you have said; but it is very much beside our purpose: for whilst you go about to shew me that a material sphere toucheth not a material plane in one point alone, you make use of a sphere that is not a sphere, and of a plane that is not a plane; for that, according to what you say, either these things cannot be found in the world, or if they may be found, they are spoiled in applying them to work the effect. It had been therefore a less evil, for you to have granted the conclusion, but conditionally, to wit, that if there could be made of matter a sphere and a plane that were and could continue perfect, they would touch in one sole point, and then to have denied that any such could be made.

I believe that the proposition of Philosophers is to be understood in this sense; for it is not to be doubted, but that the imperfection of the matter, maketh the matters taken in concrete, to disagree with those taken in abstract.

What, do they not agree? Why, that which you your self say at this instant, proveth that they punctually agree.

How can that be?

Do you not say, that through the imperfection of the matter, that body which ought to be perfectly spherical, and that plane which ought to be perfectly level, do not prove to be the