He should rather have considered of his own Expedient, pag. 11. 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 further, that Euclide, by a Line, means no more than what Mr. Hobs would call the middle of the breadth of his; and Euclide's Point, is but the Middle of Mr. Hobs's. And then, for the same reason, that Mr. Hobs'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 Bigger than Both, by so much as the Middle comes to:) Euclide's Lines must as well be said to have no Breadth; and his Points no Bigness.
In like manner, When Euclide and others do make the Terme 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 terminans) has nothing of Length; (for, when the whole Length is past, there is nothing of it left.) And if Mr. Hobs tells us (as pag. 3.) 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 Σημεῖον, which some Interpreters render by Signum, others have thought fit (with Tully) to call Punctum: But if Mr. Hobs like not that name, we will not contend about it. Let it be Punctum, 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 Controversie in Grammar, not in Mathematicks: And his Book should have been intitled Contra Grammaticos, not, Contra Geometras. Nor is it Euclide, but Cicero, that is concern'd, in rendring the Greek Σημεῖον by the Latine Punctum, not by Mr. Hobs's Signum. The Mathematician is equally content with either word.
What he saith here, Chap. 8. & 19. (and in his fifth Dial. p. 105. &c.) concerning the Angle of Contact; amounts but to thus much, That, by the Angle of Contact, he doth not mean either what Euclide calls an Angle, or any thing of that kind; (and therefore says nothing to the purpose of what was in controversie 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 hath some magnitude, his meaning is, that the Arch of a Circle hath some crookedness, or, is a crooked line: and that, of equal Arches, That is the more crooked, 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. Hobs loves to call that Chalk, which others call Cheese. Of this see my Hobbius Heauton-timorumenus, from pag. 88. to p. 100.
What he saith here of Rations or Proportions, and their Calculus; for 8. Chapters together, (Chap. 11. &c,) is but the same for substance, what he had formerly said in his 4th. Dialogue, and elsewhere. To which you may see a full Answer, in my Hobbius Heauton-tim. from pag. 49. to p. 88. which I need not here repeat.
Onely (as a Specimen of Mr. Hobs's Candour, in Falsifications) you may by the way observe, how he deals a Demonstration of Mr. Rook's, in confutation of Mr. Hobs's Duplication of the Cube. Which when he had repeated, pag. 43. He doth then (that it might seem absurd) change those words, æquales
quatuor cubis DV; (pag. 43. line 33.) into these (p. 44. l. 5.) æqualia quatuor Lineis, nempe quadruplus Recta DV: And would thence perswade you, that Mr. Rook had assigned a Solide, equal to a Line. But Mr. Rook's Demonstration was clear enough without Mr. Hobse's Comment. Nor do I know any Mathematician (unless you take Mr. Hobs to be one) who thinks that a Line multiplyed by a Number will make a Square; (what ever Mr. Hobs is pleased to teach us.) But, That a Number multiplyed by a Number, may make a Square Number; and, That a Line drawn into a Line may make a Square Figure, Mr. Hobs (if he were, what he would be thought to be) might have known before now. Or, (if he had not before known it) he might have learned, (by what I shew him upon a like occasion, in my Hob. Heaut. pag. 142. 143. 144.) How to understand that language, without an Absurdity.
Just in the same manner he doth, in the next page, deal with Clavius, for having given us his words, pag. 45 l. 3. 4. Dico hanc Lineam Perpendicularem extra circulum cadere (because neither intra Circulum, nor in Peripherea;) He doth, when he would shew an errour, first make one, by falsifying his word, line 15. where instead of Lineam Perpendicularem, he substitutes Punctum A. As if Euclide or Clavius had denyed the Point A. (the utmost point of the Radius,) to be in the Circumference: Or, as if Mr. Hobs, by proving the Point A. to be in the Circumference, had thereby proved, that the Perpendicular Tangent A E had also lyen in the Circumference of the Circle. But this is a Trade, which Mr. Hobs doth drive so often, as if he were as well faulty in his Morals, as in his Mathematicks.
The Quadrature of a Circle, which here he gives us, Chap. 20. 21. 23. is one of those Twelve of his, which in my Hobbius Heauton-timorumenus (from pag. 104. to pag. 119) are already confuted: And is the Ninth in order (as I there rank them) which is particularly considered, pag. 106. 107. 108. 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 consisteth of Two branches, which are Both false; and each overthrow the other. For if the Arch of a Quadrant be equal to the Aggregate of the Semidiameter and of the Tangent of 30. Degrees, (as he would Here have it, in Chap. 20. and There, in the close of Prop. 27;) Then is it not equal to that Line, Whose Square is equal to Ten squares of the Semiradius, (as, There, he would have it, in Prop. 28. and, Here, in Chap. 23.) And if it be equal to This, then not to That. For This, and That, are not equal: As I then demonstrated; and need not now repeat it.
The grand Fault of his Demonstration (Chap. 20.) wherewith he would now New vamp his old false quadrature, lyes in those Words Page 49. line 30, 31. Quod Impossibile est nisi ba transeat per c. which is no impossibility at all. For though he first bid us draw the Line R c, and afterwards the Line R d; Yet, Because he hath no where proved (nor is it true) that these two are the same Line; (that is, that the point d lyes in the Line R c, or that R c passeth through d:) His proving that R d cuts off from ab a Line equal to the Sine of R c, doth not prove, that ab passeth through c: For this it may well do though ab lye under c. (vid. in case d lye beyond the line R c. that is, further from A:) And therefore, unless he first prove (which he cannot do) that A c ( a sixth part of A D) doth just reach to the line R c and no further, he only proves