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 {294} that a sixth part of _ab_ is _equal_ to the Sine of _B c_. But, whether it _lye above it_, or _below_ it, or (as Mr. _Hobs_ would have it) just _upon_ it; this argument doth not conclude. (And therefore _Hugenius's_ assertion, which Mr. _Hobs_, _Chap._ 21. would have give way to this Demonstration, doth, notwithstanding this, remain safe enough.)