no PHILOSOPHICAL TRANSACTIONS. [aNNO 1666. 



or let it be signum, or if he please he may call it vexillum. But then he is to 

 remember that this is only a controversy in grammar, not in mathematics ; 

 and his book should have been intituled Contra Grammaticos^ not Contra 

 Geometras. Nor is it Euclid, but Cicero, that is concerned, in rendering the 

 Greek Sn/xsroi/, by the Latin punctum., not by Mr. Hobbes's signum. The 

 mathematician is equally content with either word. 



What he saith concerning the angle of contact amounts but to thus much, 

 that by the angle of contact he does not mean either what Euclid calls an angle, 

 or any thing of that kind ; and therefore says nothing to the purpose of what 

 was in controversy between Clavius and Peletarius, when he says, that an 

 angle of contact hath some magnitude : but that by the angle of contact, he 

 understands the crookedness of the arch ; and in saying the angle of contact 

 has some magnitude, his meaning is, that the arch of a circle has some crook- 

 edness, or is a crooked line : and that of equal arches, that is the more crook- 

 ed whose chord is shortest ; which I think none will deny, for who ever 

 doubted but that a circular arch is crooked ? or that of such arches equal in 

 length, that is the more crooked whose ends by bowing are brought nearest 

 together ? But why the crookedness of an arch should be called an angle of 

 contact, I know no other reason, but because Mr. Hobbes loves to call that 

 chalk which others call cheese. 



What he says here of ratios or proportions, and their calculics for eight 

 chapters together, is but the same for substance, as he had formerly said in his 

 4th dialogue and elsewhere. To which you may see a full answer in my Hob- 

 bius Heauton-tim. from page 49 to p. 88, which I need not here repeat. 



The quadrature of a circle, which here he gives us, is one of those twelve of 

 his, which in my Hobbius Heauton-timorumenus, are already confuted ; and 

 is the ninth in order, as I there rank them. I call it one, because he takes it 

 so to be ; though it might as well be called two. For as there, so here, it 

 consists of two branches, which are both false ; and each overthrows the 

 other. 



His demonstration of chap. 23, where he would prove that the aggregate of 

 the radius and of the tangent of 30 degrees, is equal to a line whose square 

 is equal to 10 squares of the semiradius, is confuted not only by me in the 

 place forecited, where this is proved to be impossible, but by himself also in 

 this same chapter, where he proves sufficiently, and doth confess, that this de- 

 monstration and the 47th proposition of the first of Euclid cannot be both truel 

 But, which is worst of all, whether Euclid's proposition be false or true, his 

 demonstration must needs be false. For he is in this dilemma : if that propo- 

 sition be true, his demonstration is false, for he grants that they cannot be 



