TOL. I.] PHILOSOPHICAL TRANSACTIONS. lOQ 



at one place only to touch, and at another place to cut the same circle, with 

 others of like nature ; he finds it necessary, that these things may not seem 

 absurd, to allow his lines some breadth, (that so, as he speaks, while a straight 

 line with its outside doth at one place touch the circle, it may with its inside at 

 another place cut it, &c.) But I should sooner take this to be a confutation of 

 his quadratures than a demonstration of the breadth of a mathematical line. 



And what he now adds being to this purpose ; that though Euclid's InfxfTov, 

 which we translate a point, be not indeed nomen quanti ; yet cannot this be 

 actually represented by any thing, but what will have some magnitude ; nor can 

 a painter, no not Apelles himself, draw a line so small, but that it will have 

 some breadth ; nor can thread be spun so fine, but that it will have some big- 

 ness, is nothing to the business, for Euclid does not speak either of such points, 

 or of such lines. 



He should rather have considered of his own expedient, that when one of his 

 broad lines, passing through one of his great points, is supposed to cut another 

 line proposed, into two equal parts ; we are to understand the middle of the 

 breadth of that line passing through the middle of that point, to distinguish 

 the line given into two equal parts. And he should then have considered fur- 

 ther, that Euclid by a line means no more than what Mr. Hobbes would call 

 the middle of the breadth of his ; and Euclid's point is but the middle of Mr. 

 Hobbes's. And then, for the same reason that Mr. Hobbes's middle must be 

 said to have no magnitude, (for else not the whole middle, but the middle of 

 the middle will be in the middle, and the whole will not be equal to its two 

 halves, but larger than both by so much as the middle comes to ;) Euclid's 

 lines must as well be said to have no breadth, and his points no bigness. 



In like manner, when Euclid and others do make the term or end of a line, 

 a point ; if this point have parts or greatness, then not the point, but the 

 outer half of this point ends the line, for that the inner half of that point is 

 not at the end is manifest, because the outer half is beyond it : and again, if 

 that outer half have parts also ; not this, but the outer part of it, and again the 

 outer part of that outer part, and so in infinitum. So that as long as any thing 

 of line remains, we are not yet at the end ; and consequently if we must have 

 passed the whole length before we be at the end, then that end (or punctum ter^ 

 minansj has nothing of length ; for when the whole length is past, there is 

 nothing of it left. And if Mr. Hobbes tells us that this end is not punctum, 

 but only signum (which he does allow non esse nomen quanti) even this will 

 serve our turn well enough. Euclid's Ir^jLiiov, which some interpreters render 

 by signum., others have thought fit, with Tully, to call punctum : but if Mr. 

 Hobbes like not that name, we will not contend about it. Let it ht punctum, 



