Page:Mathematical collections and translations, in two tomes - Salusbury (1661).djvu/207

 amongst those who know nothing thereof. Now to shew you how great their errour is who say, that a Sphere v. g. of brasse, doth not touch a plain v. g. of steel in one sole point, Tell me what conceipt you would entertain of one that should constantly aver, that the Sphere is not truly a Sphere.

I would esteem him wholly devoid of reason.

He is in the same case who saith that the material Sphere doth not touch a plain, also material, in one onely point; for to say this is the same, as to affirm that the Sphere is not a Sphere. And that this is true, tell me in what it is that you constitute the Sphere to consist, that is, what it is that maketh the Sphere differ from all other solid bodies.

I believe that the essence of a Sphere consisteth in having all the right lines produced from its centre to the circumference, equal.

So that, if those lines should not be equal, that same solidity would be no longer a sphere?

True.

Go to; tell me whether you believe that amongst the many lines that may be drawn between two points, there may be more than one right line onely.

There can be but one.

But yet you understand that this onely right line shall again of necessity be the shortest of them all?

I know it, and also have a demonstration thereof, produced by a great Peripatetick Philosopher, and as I take it, if my memory do not deceive me, he alledgeth it by way of reprehending Archimedes, that supposeth it as known, when it may be demonstrated.

This must needs be a great Mathematician, that knew how to demonstrate that which Archimedes neither did, nor could demonstrate. And if you remember his demonstration, I would gladly hear it: for I remember very well, that Archimedes in his Books, de Sphæra & Cylindro, placeth this Proposition amongst the Postulata; and I verily believe that he thought it demonstrated.

I think I shall remember it, for it is very easie and short.

The disgrace of Archimedes, and the honour of this Philosopher shall be so much the greater.

I will describe the Figure of it. Between the points A and B, [in Fig. 5.] draw the right line AB, and the curve line ACB, of which we will prove the right to be the shorter: and the proof is this; take a point in the curve-line, which let be C, and draw two other lines, AC and CB, which two lines together, are longer than the sole line AB, for so demonstrateth Euclid.