( 294 )
that a sixth part of ab is equal to the sine of[errata 1] 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.)
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 Chap. pag. 59 (where he proves sufficiently and doth confesse, that this demonstration, and the 47. Prop. of the first of Euclide, cannot be both true,) But, (which is worst of all,) whether Euclid's Proposition be False or True, his demonstration must needs be False. I or he is in this Dilemma; If that Proposition be True, his demonstration is False, for he grants that they cannot be both True, page 59 line 21. 22. And again, if that Proposition be False, his Demonstration is so too; for This depends upon That, page 55. line 22 and therefore must fall with it.
But the Fault is obvious in His Demonstration (not in Euclid's Proposition:) The grand Fault of it (though there are more) lyes in those words, page 56. line 26. Erit ergo M O minus quam M R. Where, instead of minus, he should have said majus. And when he hath mended that Error; he will find, that the major in page 56. line penult, will very well agree with majorum in page 57. line 1 (where the Printer hath already mended the Fault to his hand) and then the Falsum ergo will vanish.
His Section of an Angle in ratione data, Chap. 22. hath no other foundation, than his supposed Quadrature of Chap. 20. And therefore, that being false; this must fall with it. It is just the same with that of his 6. Dialogue. Prop. 46. which (besides that it wants a foundation) how absurd it is, I have already shewed; in my Hobbius Heauton timor. page 119. 120.
His Appendix, wherein he undertakes to shew a Method of finding any number of mean Proportionals, between two Lines given: Depends upon the supposed Truth of his 22. Chapter; about Dividing an Arch in any proportion given: (As himself professeth: and as is evident by the Construction, which supposeth such a Section.) And therefore, that failing, this falls with it.
And yet this is otherwise faulty, though that should be supposed True. For, in the first Demonstration; page 67. line 12. Producta L f incidet in I; is not proved, not doth it follow from his Quoniam igitur.
In the second Demonstration; page 68 line 34. 35. Recta L f incidit in x; is not proved, nor doth it follow from his Quare.
In his third Demonstration, page 71: line 7. Producta Y P transibit per M; is said gratis; nor is any proof offered for it. And so this whole structure falls to the ground. And withall, the Prop. 47. El. 1 doth still stand fast (which he tells us, page 59, 61, 78. must have Fallen, if his Demonstrations had stood:) And so, Geometry and Arithemetick do still agree, which (he tells us, page 78: line 10.) had otherwise been at odds.
And this (though much more might have been said,) is as much as need to be said against that Piece.
Printed with Licence for John Martyn and James Allestry, Printers to the Royal Society.
