Chapter 28

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 {293} 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 {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.)

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 Cometæ,_ A. 1665. exorti; una cum _Mantissa Prodromi Cometici_. 2. _Isaacus Vossius de Nili & aliorum Fluminum Origine_. 3. _Le Discernement du Corps & de l'Ame_, par Monsieur de _Cordemoy_._

* * * * *

_Observations made in several places, Of the late _Eclipse of the Sun_, which hapned on the 22 of _June_, 1666._

The Observations that were made at _London_ by Mr. _Willughby_, Dr. _Pope_, Mr. _Hook_, and Mr. _Philips_, are these:

The Eclipse began at 5h. 43' h. ' { 3/11 diam. at 6. 00 | 5 dig. at 7. 06 { 4 digits at 6. 07 | 4 dig. at 7. 13 It was { 5 dig. at 6. 13 | 3 dig. at 7. 20 darkned,{ 6 dig. at 6. 21 | 2 dig. at 7. 26 { 7 dig. at 6. 39½ | 1 dig. at 7. 32 { 6 dig. at 6. 57 | 0 dig. at 7. 37

Its _Duration_ hence appears to have been one hour and 54 m. Its _greatest Obscurity_ somewhat more than 7. digits. About the middle, between the Perpendicular and Westward Horizontal _Radius_ the Sun, viewing it through Mr. _Boyle_'s 60. foot-_Telescope_, there was perceived a little of the Limb of the Moon without the Diske of the Sun: which seemed to some of the Observers to come from some shining _Atmosphere_ about the Body either of the Sun or Moon.

They affirm to have observ'd the _Figure_ of this _Eclipse_, and measured the {296} _Digits_, by casting the _Figure_ through a 5 foot _Telescope_, on an extended paper, fix't at a certain distance from the Eye-glasse, and having a round figure; all whose _Diameters_ were divided, by 6 _Concentrick_ Circles, into 12 _Digits_.

The Observations made at _Madrid_ by a Noble Member of the _Royal Society_, His Excellence the Earle of _Sandwich_, as they were sent to the Right Honourable, the Lord Vice-Count _Brounker_, are these;

The Eclipse _began_ at _Madrid_ about 5 of the Clock in the morning, at 5 h. 15', the Suns _Altitude_ was 6 deg. 55'.

The _Middle_ of it was at 6 h. 2', the Suns _Altitude_, 15. deg. 5'.

The _End_ was exactly at 7 h. 5'; the Suns _Altitude_, 25. deg. 24'.

The _Duration_, 2h. 4'.

37. Parts of the Suns diameter remained light.

63. Parts of the same were darkened.

The Observations made at _Paris_ by Monsieur _Payen_, assisted by several _Astronomers_, as they were printed in _French_, and addressed to Monsieur de _Montmor_, are these;

The _Eclipse_ began there, at 5 h. 44'. 52". _mane_. It ended at 7 h. 43'. 6". So that its _whole Duration_ was 1 h. 58'. 14". The _greatest Obscuration_ they assign to have been 7. dig. 50. m. but they adde, that it seem'd to have been greater by 3 minuts; which M. _Payen_ imputes to a particular motion of _libration_ of the Suns Globe, which entertain'd that Luminary in the same _Phasis_ for the space of 8. _min._ and some _seconds_, as if it had been stopped in the midst of its Course; rather than to a tremulous Motion of the _Atmosphere_, as _Scheiner_ would have it.

They intimate that they took the time of each _Phasis_ from half _digit_ to half _digit_, as well by a _Pendulum_, as by the _Altitudes_ of the _Suns Center_ above the _Horizon_, corrected by the _Verticall Paralaxes_ and _Æstivall Refractions_, by which they judged, that though the Time by the _Pendulum_ may be sufficient for _Mechanicall_ Operations, yet 'tis not exact enough for establishing the _Grounds of true Astronomy_.

They further conceive that the apparent _Diameters_ were almost equal; seeing that in the _Phasis_ of 6. _Digits_, the _Circumference_ of the _Moons disk_ passed through the _Center_ of that of the _Sun_, so as that two Lines drawn through the two _Horns_ of the Sun, made with the _Common Semi-diameter_ two _Equilateral Triangles_.

Next, they affirm, That there was so great a Variation in the _Parallaxes_, by reason as well of the Refractions of the Air, which environs the Earth, as of the Alteration of the Air, which encompasses the Moon, that the _Horns_ of the Sun, there formed by the Shaddow of the Moon, appeared in all kinds of _Figures_; Sometimes inclined to the _Vertical_, sometimes _Perpendicular_ to the _Horizon_, and at last _Parallel_; the _Convexe_ part respecting the _Heaven_, and the _Concave_, the _Horizon_. By the crossing (_so they go on_) of the {297} _Horns_ with the _Angles of Inclination_, it will be easie to those, that have exactly observed them, and that are skill'd in the higher _Astronomical_ Calculations, to compute the _true Place_ of the _Moon_ in her _Orbite_, that so it may be compared with that of the _Tables_, and with that, which has been observ'd in other places, for the more precise determinating of the _Difference_ of _Meridians_ (that being the way, esteem'd by _Kepler_ the most certain) and for making a good Judgment of the defect or exactnesse of the Celestial _Tables_.

Then they observe, That the _Beginning_ and the _Middle_ of this _Eclipse_ hapned to be in the _North Eastern Hemisphere_, and the _End_, in the _South-Eastern_. The _first Contact_ (as 'twere) of the two Disks was observ'd in the _Superior Limb_ of the _Suns Disk_ in respect to the _Vertical Line_, and in the _Inferior_ in respect to the _Ecliptick_: But the _Middle_, and the _End_ were seen in the _Superior Limb_, in respect both to the _Vertical_ and the _Ecliptick_: And (what to this Author seems extraordinary) both the _Beginning_ and the _End_ of this _Eclipse_ hapned to be in the _Oriental_ part of the Suns Disk.

Lastly, they take notice, that by their Observations it appears, that there is but little exactness in all the _Astronomical Tables_, predicting the _Quantity_, _Beginning_ and _Duration_ of this Eclipse; Those of _Lansbergius_ importing, That the Obscuration should be of 10. dig. 48'; those of _Ricciolo_, of 9. dig. 1'; and those of _Kepler_, of 7. dig. 30'. 16": Again, that the _Duration_ should be of 2h. 2'. Lastly, The _Beginning_ did anticipate the _Ricciolan Tables_ by 5 _minuts_, the _End_ by 23; and the _Middle_, almost by 11. In the mean time the Author notes, that the _Rudolphin Tables_ come nearest to the Truth; and withal assures the _Reader_ of the goodnesse of the _Instruments_ employed in his _Observations_, and of the singular care, he, together with his skilful Assistants, took in making them.

* * * * *

_Some Inquiries and Directions concerning _Tides_, proposed by Dr. _Wallis_, for the proving, or disproving of his lately publish't _Discourse_ concerning them_.

The Inquisitive Dr. _Wallis_, having in his lately printed _Hypothesis_ of Tides intimated, that he had reason to believe, that the _Annual Spring-tides_ happen to be rather about the beginnings of _Febr._ and _Nov._ than the two _Æquinoxes_, doth in a late Letter to the _Publisher_, written from _Oxford_ in _Aug._ last, desire, that some understanding Persons at _London_, or _Greenwich_, but rather nearer the Sea, or upon the Sea-shore, would make _particular_ Observation of all the _Spring-Tides_ (_New-Moon_ and _Full-Moon_) between this and the End of _November_; and take account of the _Hour_, and of the _Perpendicular height_: that we may see, whether those in _September_, or those of _November_ be highest: And it were not amiss, the Low waters were observed too. Which may be easily done by a mark made upon any standing Post in the Water, by any {298} Water-man, or other understanding Person, who dwells by the Water-side.

It would also deserve (thinks he) to be inquired into, whether, when the Tides be highest, the Ebbs be ever lowest, & _contra_; (which is generally affirmed, and almost put out of question) or rather (which sutes best with this _Hypothesis_) whether, when the Tides are highest, both in the _Annual_ and _Menstrual_ Periods, the Low waters be not also highest; and at Neap Tides, the Ebbes also very low.

He adds, that he should expect, that the Spring Tides now coming, and those at the beginning of _September_, should not be so high, as those at the _middle_ of _September_; and then lower again at the _beginning_ of _October_, and after that, higher at the _middle_ of _October_, and higher yet about the _beginning_ of _November_ (at the usual times of _Spring-tides_ after the _New_ and _Full_.)

* * * * *

_Considerations and Enquiries concerning _Tides_, by Sir _Robert Moray_; likewise for a further search into Dr. _Wallis's_ newly publish't _Hypothesis_._

In regard that the High and Low waters are observed to increase, and decrease regularly at several seasons, according to the Moons age, so as, about the _New_ and _Full Moon_, or within two or three daies after, in the Western parts of _Europe_, the _Tides_ are at the _highest_, and about the _Quarter-Moons_, at the _lowest_, (the former call'd _Spring-tides_, the other _Neap-tides_;) and that according to the height and excesses of the _Tides_, the _Ebbes_ in opposition are answerable to them, the heighest Tide having the lowest Ebbe, and the lowest Ebbe, the highest Tide; the Tides from the _Quarter_ to the _highest Spring-tide_ increasing in a certain proportion; and from the _Spring tide_ to the _Quarter-tide_ decreasing in like proportion, as is supposed: And also the _Ebbes_ rising and falling constantly after the same manner: It is wished, that it may be inquired, in what proportion these Increases and Decreases, Risings and Fallings happen to be in regard of one another?

And 'tis supposed, upon some Observations, made in fit places, by the above-mentioned Gentleman, though, (as himself acknowledges) not thoroughly and exactly performed, that the Increase of the Tides is made in the _Proportion_ of _Sines_; the first Increase exceeding the lowest in a small proportion; the next in a greater; the third greater than that; and so on to the mid-most, whereof the excess is greatest, diminishing again from that, to the highest Spring-Tide; so as the proportions, before and after the _Middle_, do greatly answer one another, or seem to do so. And likewise, from the _highest Spring-tide_, to the _lowest Neap-tide_, the _Decreases_ seem to keep the like proportions; the _Ebbes_ rising and falling in like manner and in like proportions. All which is supposed to fall out, when no Wind or other Accident causes an alteration. {299}

And whereas 'tis observed, that upon the main Sea-shore the Current of the Ebbings and Flowings is sometimes swifter, and sometimes slacker, than at others, so as in the beginning of the Floud the Tide moves faster but in a small degree, increasing its swiftness constantly till towards the _Middle_ of the Floud; and then decreasing in velocity again from the _Middle_ till to the top of the High water; it is supposed, that in Equal spaces of Time, the Increase and Decrease of velocity, and consequently the degrees of the Risings and Fallings of the same, in Equal spaces of time, are performed according to the _Proportion_ of _Sines_.

But 'tis withall conceived, that the said _Proportion_ cannot hold _exactly_ and _precisely_, in regard of the _Inequalities_, that fall out in the _Periods_ of the _Tides_, which are commonly observed and believed to follow certain _Positions_ of the _Moon_ in regard of the _Equinox_, which are known not to keep a _precise_ and _constant_ Course: so that, there not intervening equal portions of Time between one New Moon and another, the Moons return to the same _Meridian_, cannot be alwaies perform'd in the same Time; and consequently there must be a like Variation of the Tides in the Velocity, and in the Risings and Fallings of the Tides, as to equal spaces of time. And the Tides from New-moon to New-moon being not alwaies the same in number, as sometimes but 57, sometimes 58, and sometimes 59, (without any certain order of succession) is another evidence of the difficulty of reducing this to any great exactness. Yet, because 'tis worth while, to learn as much of it, as may be, the _Proposer_ and many others do desire, That Observations be constantly made of all these Particulars for some Months, and, if it may be, years together. And because such Observations will be the more easily and exactly made, where the Tides rise highest, it is presumed, that a fit _Apparatus_ being made for the purpose, they may be made about _Bristol_ or _Cheap-stow_, best of any places in _England_, because the Tides are said thereabout to rise to ten or twelve fathoms; as upon the coast of _Britanny_ in _France_, they do to thirteen and fourteen.

In order to which, this following _Apparatus_ is proposed to be made use of. In some convenient place upon a Wall, Rock, or Bridge, &c. let there be an _Observatory_ standing, as neer as may be to the brink of the Sea, or upon some wall; and if it cannot be well placed just where the Low water is, there may be a Channel cut from the Low water to the bottom of the Wall, Rock, &c. The Observatory is to be raised above the High water 18. or 20. foot; and a Pump, of any reasonable dimension, placed perpendicularly by the Wall, reaching above the High water as high as conveniently may be. Upon the top of the Pump a Pulley is to be fastned, for letting down into the Pump a piece of floating wood, which, as the water comes in, may rise and fall with it. And because the rising and falling of the water amounts to 60. or 70. foot, the Counterpoise of the weight, that goes into the Pump, is to hang upon as many Pulleys, as may serve to make it rise & fall within the space, by which the height of the Pump exceeds the height of the Water. And because by {300} this means the Counterpoise will rise and fall slower; and consequently by less proportions, than the weight it self, the first Pulley may have upon it a Wheele or two, to turn _Indexes_ at any proportion required, so as to give the minute parts of the motion, and degrees of risings and fallings. All which is to be observed by _Pendulum-watches_, that have _Minutes_ and _Seconds_, with _Checks_, according to Mr. _Hugens's_ way.