An Account of a Book

moon, the Moon at C, the Earth at c, Mars will be seen at γ, its true place, as if the Earth were at T. But then, after the Full, the Moon at D, the Earth at d; Mars will be seen, not at γ, but at δ; too forward: and yet more, when the Moon (at the last Quarter) is at E, the Earth at e, and Mars seen at ζ. If there- fore Mars (when in opposition to the Sun) be found (all other allowances being made) somewhat too backward before the Full moon, and somewhat too forward after the Full-moon, (and most of all, at the Quadratures:) it will be the best con- firmation of the Hypothesis. (The like may be fitted to Mars in other positions, mutatis mutandis; and so for the other Pla- nets.) But this proof, is of like nature as that of the Parallaxis of the Earths Annual Orb to prove the Copernican Hypothesis. If it can be observed, it proves the Affirmative; but if it cannot be observed, it doth not convince the Negative, but only proves that the Semidiameter of the Earths Epicycle is so small as not to make any discernable Parallax. And indeed, I doubt, that will be the issue. For the Semidiameter of this Epicycle, being little more than the Semidiameter of the Earth itself, or about 1/3 thereof (as is conjectured, in the Hypothesis, from the Mag- nitudes and Distances of the Earth and Moon compared;) and there having not as yet been observed any discernable Parallax of Mars, even in his neatest position to the Earth; it is very suspicious, that here it may prove so too. And whether any of the other Planets will be more favourable in this point, I can- not say. ANIMADVERSIIONS Of Dr. Wallis, upon Mr. Hobbs's late Book, De Principiis & Ratiocinatione Geometrarum. These were communicated by way of Letter, written in Ox- ford, July 24. 1666. to an Acquaintance of the Author, as fol- lows: Since I saw you last, I have read over Mr. Hobbs's Book Contra Geometras (or De Principiis & Ratiocinatione Geometrarum) which you then shewed me. A New Book of old matter: Containing but a Repetition of what he had before told us, more than once; and which hath been Answered long agoe. In which, though there be Faults enough to offer ample mat- ter for a large Confutation: yet I am scarce inclined to believe, that any will bestow so much pains upon it. For, if that be true, which (in his Preface) he saith of himself, *Aut solus insanio Ego, aut solus non insanio*: it would either be Needleless, or to no Purpose. For, by his own confession, All others, if they be not mad themselves, ought to think Him so: And therefore, as to Them, a Confutation would be needleless; who, its like, are well enough satisfied already: at least out of danger of being seduced. And, as to himself, it would be to no purpose. For, if He be the Mad man, it is not to be hoped that he will be convinced by Reason: Or, if All We be so; we are in no capacity to attempt it. But there is yet another Reason, why I think it not to need a Confutation. Because what is in it, hath been sufficiently confuted already; (and, so Effectually; as that he professeth himself not to Hope, that This Age is like to give sentence for him; whatever Nondum imbuta Posteritas may do.) Nor doth there appear any Reason, why he should again Repeat it, unless he can hope, That, what was at first False, may by oft Repeating, become True. I shall therefore, instead of a large Answer, onely give you a brief Account, what is in it; &c., where it hath been already Answered. The chief of what he hath to say, in his first 10 Chapters, against Euclid's Definitions, amounts but to this, That he thinks, Euclide ought to have allowed his Point some Bigness; his Line, some Breadth; and his Surface, some Thickness. But where in his Dialogues, pag. 151, 152, he solemnly undertakes to Demonstrate it; (for it is there, his 41th Proposition;) his Demonstration amounts to no more but this; That, unless a Line be allowed some Latitude; it is not possible that his Quadratures can be True. For finding himself reduced to these inconveniences: 1. That his Geometrical Constructions, would not consist with Arithmetical calculations, nor with what Archimedes and others have long since demonstrated: 2. That the Arch of a Circle must be allowed to be sometimes Shorter than its Chord, and sometimes longer than its Tangent: 3. That the same Straight Line must be allowed, at one place onely to Touch, and at another place to Cut the same Circle: (with others of like nature;) He findes 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 Touch the Circle, it may with its Inside at another place Cut it, &c.) But I should sooner take this to be a Consultation of His Quadratures, than a Demonstration of the Breadth of a Mathematical Line. Of which, see my Hobbius Heauton-timorumenus, from pag. 114. to p. 19. And what he now Adds, being to this purpose; That though Euclid's Σημεῖον, which we translate, a Point, be not indeed Nomen Quantitatis; 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 Bigness; (pag. 2, 3, 19, 21.) is nothing to the Business; For Euclide doth not speak either of such Points, or of such Lines. He should rather have considered of his own Expediteness. 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. Hobbs would call the middle of the breadth of his; and Euclide's Point, is but the Middle of Mr. Hobbs's. And then, for the same reason, that Mr. Hobbs'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. Hobbs tells us (as pag. 3.) that this End is not Punctum, but only Signum (which he does allow non esse nomen Quantitatis 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. Hobbs 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 Euclid, but Cicero, that is concern'd, in rendering the Greek Σημεῖον, by the Latine Punctum, not by Mr. Hobbs's Signum. The Mathematician is equally content with either word. What he saith here, Chap.8. &c. 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 Euclid 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. Hobbs loves to call that Chalk, which others call Cheese. Of this see my Hobbius Heauton-timorenum, 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. Hobbs's Candour, in Falsifications) you may by the way observe, how he deals with a Demonstration of Mr. Rook's, in confutation of Mr. Hobbs's Duplication of the Cube: Which when he had repeated, pag. 43. He doth then (that it might seem absurd) change those words, aequales quatuor Quatuor cubis DV; (pag. 43, line 33.) into these (p. 44. l. 5.) aqua quater Linis, nempe quadruplum Recl. DV: And would thence persuade you, that Mr. Rook had assigned a Solide, equal to a Line. But Mr. Rook's Demonstration was clear enough for Mr. Hobbe's Comment. Nor do I know any Mathematician (unless you take Mr. Hobs to be one) who thinks that a Line multiplied by a Number will make a Square; (whatever Mr. Hobs is pleased to teach us.) But, That a Number multiplied 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 Heb. Heart. pag. 142. 143. 144.) How to understand that I 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. 7. 4. Dico hanc Linam Perpendiculariorem extra circulum cadere (because neither intra Circulum, nor in Peripheria;) He doth, when he would shew an error, first make one, by falsifying his words, line 15. where instead of Linam Perpendiculariorem, he substitutes Punctum A. As if Euclide or Clavius had denied 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 been 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. 27. is one of those Twelve of his, which in my Hobbiius Heauton-timorumenus (from pag. 104. to pag. 119.) are already confused: 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 36 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 Semitradius, (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; lies 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 Rc, and afterwards the Line Rd: Yet, Because he hath no where proved (nor is it true) that these two are the same Line; (that is, that the point d lyeth in the Line Rc, or that Rc passeth through d:) His proving that Rd cuts off from ab a Line equal to the line of Be, doth not prove, that ab passeth through c: For this it may well do, though ab lyeth under c. (vid. in case d lyeth beyond the line Rc, that is, further from A:) or though it lyeth above c, (vid. in case d be nearer, than Rc, to the point A.) And therefore, unless he first prove (which he cannot do,) that Ad (a sixth part of AD) doth just reach to the line Rc and no further; he only proves that that a sixth part of \( ab \) is equal to the Line of \( BC \). But, whether it lyse above it, or below it, or (as Mr. Hobbs would have it) just upon it; this argument doth not conclude. (And therefore Hagenius's assertion, which Mr. Hobbs, 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 consuted 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 confess, 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. For 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) lies in those words, page 56. line 26. Erit ergo \( MO \) minus quam \( MR \). 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 Falsam 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 Hobbins Heanton 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 \( Lf \) incidit in \( I \); is not proved; nor doth it follow from his Quoniam igitur. In the second Demonstration; page 68. line 34 35. Relta \( Lf \) incidit in \( x \); is not proved; nor doth it follow from his Quare. In his third Demonstration; page 71: line 7. Producta \( YP \) transit 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. I doth still stand fast (which he tells us, page 59, 61, 78. must have Fallen, if his Demonstrations had stood;) And so, Geometry and Arithmetick 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.