Philosophical Transactions of the Royal Society Part 30

And then as to _Ga.s.sendus_, in his discourse _De aestu Maris_; I find him, after the relating of many other Opinions concerning the Cause of it, inclining to that of _Galilaeo_, ascribing it to the Acceleration & r.e.t.a.r.dation of the Earths motion, compounded of the Annual and Diurnal; And moreover attempting to give an account of the _Menstrual Periods_ from the Earths carrying the Moon about it self, as _Jupiter_ doth his _Satellites_; which together with them is carryed about by the _Sun_, as one Aggregate; (and that the Earth with its Moon is to be supposed in like manner to be carried about by the Sun, as one Aggregate, cannot be reasonably doubted, by those who entertain the _Copernican Hypothesis_, and do allow the same of _Jupiter_ and his _Satellites_.) But though he would thus have the Earth and Moon looked upon as two parts of the same moved Aggregate, yet he doth still suppose (as _Galilaeo_ had done before him) that the line of the Mean Motion of this Aggregate (or, as he calls, _motus aequabilis et veluti medius_) is described by the _Center_ of the _Earth_ (about which Center he supposeth both its own revolution to be made, and an Epicycle described by the Moons motion;) not by another Point, distinct from the Centers of both, about which, as the {288} common Center of Gravity, as well that of the Earth, as that of the Moon, are to describe several Epicycles. And, for that Reason fails of giving any clear account of this _Menstrual_ Period.

(And in like manner, he proposeth the Consideration as well of the Earths _Aphelium_ and _Perihelium_ as of the _aequinoctial_ and _Solst.i.tial_ Points, in order to the finding a Reason of the _Annual_ Vicissitudes; but doth not fix upon any thing, in which himself can Acquiesce: And therefore leaves it _in medio_ as he found it.)

It had been more agreeable to the Laws of _Staticks_, if he had, (as I do,) so considered the _Earth_ and _Moon_ as two parts of the same movable, (not so, as he doth, _aliam in Centro et sequentem praecise revolutionem axis, aliam remotius ac velut in circ.u.mferentia_, but,) so, as to make neither of them the Center, but both out of it, describing Epicycles about it: Like as, when a long stick thrown in the Air, whose one end is heavyer than the other, is whirled about, so as that the End, which did first fly foremost, becomes hindmost; the proper line of motion of this whole Body is not that, which is described by either End, but that, which is described by a middle point between them; about which point each end, in whirling, describes an Epicycle. And indeed, in the present case, it is not the Epicycle described by the Moon, but that, described by the Earth, which gives the _Menstrual_ Vicissitudes of motion to the Water; which would, as to this, be the same, if the Earth so move, whether there were any Moon to move or not; nor would the Moons Motion, supposing the Earth to hold on its own course, any whit concern the motion of the Water.

[Ill.u.s.tration]

But now, (after all our Physical, or Statical Considerations) the clearest Evidence for this Hypothesis (if it can be had) will be from Celestial Observations. As for instance; (see _Fig._ 5.) Supposing the Sun at S; the Earths place in its Annual Orb at T; and _Mars_ (in opposition to the Sun, or near it) at M: From whence _Mars_ should appear in the Zodiack at [gamma], and will at Full moon be seen there to be; the Moon being at C and the Earth at c; (and the like at the New-moon.) But if the Moon be in the First quarter at A, and the Earth at a: _Mars_ will be seen, not at [gamma], but at [alpha]; too slow: And when the Moon is at B, and the Earth at b, _Mars_ will be seen at [beta]; yet too slow: till at the {289} Full-moon, the Moon at C, the Earth at c, _Mars_ will be seen at [gamma], 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 [gamma], but at [delta], too forward: and yet more, when the Moon (at the last Quarter) is at E, the Earth at e, and _Mars_ seen at [epsilon]. If therefore _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 confirmation of the Hypothesis. (The like may be fitted to _Mars_ in other positions, _mutatis mutandis_; and so for the other Planets.)

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 it self, or about 1-1/3 thereof (as is conjectured, in the _Hypothesis_, from the Magnitudes and Distances of the Earth and Moon compared;) and there having not as yet been observed any discernable _Parallax_ of _Mars_, even in his neerest 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 cannot say.

_ANIMADVERSIONS of Dr. _Wallis_, upon Mr. _Hobs_'s late Book, _De Principiis & Ratiocinatione Geometrarum_._

These were communicated by way of Letter, written in _Oxford_, July 24.

1666. to an Acquaintance of the _Author_, as follows:

Since I saw you last, I have read over Mr. _Hobs_'s Book _Contra Geometras_ (or _De Principiis & Ratiocinatione Geometrarum_) which you then shewed me.

A New Book of _Old_ matter: Containing but a _Repet.i.tion_ 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 {290} matter 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 _Needless_, 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 _needless_; 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; what ever _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_; &, _where it hath been already Answered_.

The chief of what he hath to say, in his first 10 Chapters, against _Euclids_ 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 Lat.i.tude; 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 Out-side doth at one place {291} Touch the Circle, it may with its In-side at another place Cut it_, &c.) But I shou'd sooner take this to be a _Confutation 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. 119.

And what he now Adds, being to this purpose; That though _Euclid_'s [Greek: Semeion], which we translate, _a Point_, be not indeed _Nomen Quanti_; 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 Expedient, _pag._ 11. That, when one of his (_broad_) Lines, pa.s.sing 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, pa.s.sing 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 pa.s.sed 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 {292} _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 [Greek: Semeion], 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 int.i.tled _Contra Grammaticos_, not, _Contra Geometras_. Nor is it _Euclide_, but _Cicero_, that is concern'd, in rendring the Greek [Greek: Semeion] 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, _aequales {293} quatuor cubis_ DV; (_pag._ 43. _line_ 33.) into these (p. 44. l. 5.) _aequalia quatuor Lineis, nempe quadruplus Recta_ DV: And would thence perswade you, that Mr. _Rook_ had a.s.signed 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 subst.i.tutes _Punctum A._ As if _Euclide_ or _Clavius_ had denyed the _Point A._ (the utmost point of the _Radius_,) to be in the Circ.u.mference: Or, as if Mr. _Hobs_, by proving the _Point A._ to be in the Circ.u.mference, had thereby proved, that the _Perpendicular Tangent A E_ had also lyen in the Circ.u.mference 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_ pa.s.seth 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_ pa.s.seth 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_ a.s.sertion, 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. 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) 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 _majorem_ in _page_ 57. _line_ 4 (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 other wise 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, nor 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 _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.

{295}

_Num._ 17.

PHILOSOPHICAL _TRANSACTIONS._

_Munday_, _Septemb._ 9. 1666.

The Contents.

_Observations made in several places (at _London_, _Madrid_ and _Paris_,) of the late _Eclipse of the Sun_, which hapned _June_ 22.

1666. Some Enquiries and Directions, concerning _Tides_, proposed by _Dr. Wallis_. Considerations and Enquiries touching the same Argument, suggested by Sir _Robert Moray_. An Account of several Books lately publish't: Vid. 1. _Johannis Hevelii Descriptio Cometae,_ A. 1665.

exorti; una c.u.m _Mantissa Prodromi Cometici_. 2. _Isaacus Vossius de Nili & aliorum Fluminum Origine_. 3. _Le Discernement du Corps & de l'Ame_, par Monsieur de _Cordemoy_._