# Philosophical Transactions/Volume 1/Number 16

Errata

Num. 16.

PHILOSOPHICAL

TRANSACTIONS.

Munday, August 6. 1666.

The Contents.

An Essay of Dr. John Wallis, exhibiting his Hypothesis about the Flux and Reflux of the Sea, taken from the Consideration of the Common Center of Gravity of the Earth and Moon; together with an Appendix of the same, containing an Answer to some Objections, made by several Persons against that Hypothesis. Some Animadversions of the same Author upon Master Hobs's late Book, De Principiis & Ratiocinatione Geometrarum.

An Essay
Of Dr. John Wallis, exhibiting his Hypothesis about the Flux and Reflux of the Sea.

Ow abstruse a subject in Philosophy, the Flux and Reflux of the Sea hath proved hitherto, and how much the same hath in all Ages perplexed the Minds even of the best of Naturalists, when they have attempted to render an Account of the Cause thereof, is needless here to represent. It may perhaps be to more purpose, to take notice, that all the deficiencies, found in the Theories or Hypotheses, formerly invented for that End, have not been able to deterre the Ingenious of this Age from making farther search into that Matter: Among whom that Eminent Mathematician Dr. John Wallis, following his happy Genius for advancing reall Philosophy, hath made it a part of his later Inquiries and Studies, to contrive and deduce a certain Hypothesis concerning that Phænomenon, taken from the Consideration of the Common Center of Gravity of the Earth and Moon, This being by several Learned Men lookt upon, as a very rational Notion, it was thought fit to offer it by the Press to the Publick, that other Intelligent Persons also might the more conveniently and at their leisure examine the Conjecture (the Author, such is his Modesty, presenting it no otherwise) and thereupon give in their sense, and what Difficulties may occur to them about it, that so it may be either confirm'd or laid aside accordingly; As the Proposer himself expresly desires in the Discourse, we now, without any more Preamble, are going to subjoyn, as it was by him addressed, by way of Letter, from Oxford to Mr. Boyle April 25. 1666. and afterwards communicated to the R. Society, as follows:

YOu were earnest with me, when you last went from hence, that I would put in writing somewhat of that, which at divers times, these three or four years last past, I have been discoursing with your self and others concerning the Common Center of Gravity of the Earth and Moon, in order to salving the Phœnomena as well of the Seas Ebbing and Flowing; as of some perplexities in Astronomical Observations of the Places of the Celestial Bodies.

How much the World, and the great Bodies therein, are manag'd according to the Laws of Motion, and Statick Principles, and with how much more of clearness and satisfaction, many of the more abstruse Phænomena have been salved on such Principles, within this last Century of years, than formerly they had been; I need not discourse to you, who are well versed in it. For, since that Galilæo, and (after him) Torricellio, and others, have applied Mechanick Principles to the salving of Philosophical difficulties; Natural Philosophy is well known to have been rendered more intelligible, and to have made a much greater progress in less than an hundred years, than before for many ages.

The Seas Ebbing and Flowing, hath so great a connexion with the Moons motion, that in a manner all Philosophers (whatever other Causes they have joyned with it) have attributed much of its cause to the Moon; which either by some occult quality or particular influence, which it hath on moyst Bodies, or by some Magnetick vertue, drawing the water towards it, (which should therefore make the Water there highest, where the Moon is vertical) or by its gravity and pressure downwards upon the Terraqueous Globe (which should make it lowest, where the Moon is vertical) or by whatever other means (according to the several Conjectures of inquisitive persons,) hath so great an influence on, or at least connexion with, the Sea's Flux and Reflux, that it would seem very unreasonable, to seclude the consideration of the Moons motion from that of the Sea: The Periods of Tides (to say nothing of the greatness of them near the New-moon and Full-moon) so constantly waiting on the Moon's motion, that it may be well presumed, that either the one is governed by the other, or at least both from some common cause.

But the first that I know of, who took in the consideration of the Earth's motion, (Diurnal and Annual) was Galilæo; who in his Systeme of the World, hath a particular discourse on this subject: Which, from the first time that I ever read it, seemed to me so very rational, that I could never be of other opinion, but that the true Account of this great Phænomen was was to be referred to the Earths motion, as the Principal cause of it: Yet that of the Moon (for the reasons above mentioned) not to be excluded, as to the determining the Periods of Tides, and other circumstances concerning them. And though it be manifest enough, that Galilæo, as to some particulars, was mistaken in the account which there he gives of it; yet that may be very well allowed, without any blemish to so deserving a person, or prejudice to the main Hypothesis: For that Discourse is to be looked upon onely as an Essay of the general Hypothesis; which as to particulars was to be afterwards adjusted, from a good General History of Tides, which it's manifest enough that he had not; and which is in a great measure yet wanting. For were the matter of Fact well agreed on, it is not likely, that several Hypotheses should so far differ, as that one should make the Water then and there at the Highest, where and when the other makes it at the Lowest; as when the Moon is Vertical to the place.

And what I say of Galilæo, I must in like manner desire to be understood of what I am now ready to say to you. For I do not profess to be so well skilled in the History of Tides, as that I will undertake presently to accommodate my general Hypothesis to the particular cases; or that I will indeed undertake for the certainty of it, but onely as an Essay propose it to further consideration; to stand or fall, as it shall be found to answer matter of Fact. And truly had not your importunity (which is to me a great Command) required me to do it; I should not so easily have drawn up any thing about it, 'till I had first satisfied my selfe, how well the Hypothesis would answer Observation: Having for divers years neglected to do it, waiting a time when I might be at leisure thoroughly to prosecute this design.

But there be two reasons, by which you have prevailed with me, at least to do something. First, because it is the common Fate of the English, that out of a modesty, they forbear to publish their Discoveries, till prosecuted to some good degree of certainty and perfection; yet are not so wary, but that they discourse of them freely enough to one another, and even to Strangers upon occasion; whereby others, who are more hasty and venturous, comming to hear of the notion, presently publish something of it, and would be reputed thereupon, to be the first Inventers thereof: though even that little, which they can then say of it, be perhaps much less, and more imperfect, than what the true Authors could have published long before, and what they had really made known (publikely enough, though not in print) to many others. As is well known amongst us as to the business of the Lymphatick Vessels in Anatomy: the Injection of Liquors into the veins of Living animals; the Exhibiting of a straight line equal to a crooked; the Spot in Jupiter, whence his motion about his own Axis may be demonstrated; and many other the like considerable Inventions.

The other Reason (which, with me, is more really of weight, though even the former be not contemptible) is, because, as I have been already for at least three or four years last past diverted from prosecuting the inquiry or perfecting the Hypothesis, as I had thoughts to do; so I do not know, but like Emergencies may divert me longer; and whether I shall ever so do it as to bring it to perfection, I cannot determine. And therefore, if as to my self any thing should humanitus accidere; yet possibly the notion may prove worth the preserving to be prosecuted by others, if I do it not. And therefore I shall, at least to your self, give some general account of my present imperfect and undigested thoughts.

I consider therefore, that in the Tides, or the Flux and Reflux of the Sea, besides extraordinary Extravagancies, or Irregularities, whence great Inundations or strangly high Tides do follow, (which yet perhaps may prove not to be so meerly accidental as they have been thought to be, but might from the regular Laws of Motion, if well considered, be both well accounted for, and even foretold;) There are these three notorious Observations made of the Reciprocation of Tides. First, the Diurnal Reciprocation; whereby twice in somewhat more than 24. hours, we have a Floud and an Ebbe; or a High-water and Low-water. Secondly, the Menstrual; whereby in one Synodical period of the Moon, suppose from Full-moon to Full-moon, the Time of those Diurnal Vicissitudes doth move round through the whole compass of the Νυχθήμερον, or Natural day of twenty four hours; As for instance, if at the Full-moon the full Sea be at such or such a place just at Noon, it shall be the next day (at the same place) somewhat before One of the clock; the day following, between One and Two; and so onward, till at the New-moon it shall be at midnight; (the other Tide, which in the Full-moon was at midnight, now at the New-moon coming to be at noon;) And so forward till at the next Full-moon, the Full-sea shall (at the same place) come to be at Noon again: Again, That of the Spring-tides and Neap-tides (as they are called;) about the Full-moon and New-moon the Tides are at the Highest, at the Quadratures the Tides are at the Lowest: And at the times intermediate, proportionably. Thirdly, the Annual; whereby it is observed, that at sometimes of the year, the Spring-tides are yet much higher than the Spring-tides at other times of the year: Which Times are usually taken to be at the Spring and Autumne; or the two Æquinoxes; but I have reason to believe (as well from my own Observations, for many years, as of others who have been much concerned to heed it, whereof more will be said by and by;) that we should rather assign the beginnings of February and November, than the two Æquinoxes.

Now in order to the giving account of these three Periods, according to the Laws of Motion and Mechanick Principles; We shall first take for granted, what is now adayes pretty commonly entertained by those, who treat of such matters; That a Body in motion is apt to continue its motion, and that in the same degree of celerity, unless hindred by some contrary Impediment; (like as a Body at rest, to continue so, unless by some sufficient mover, put into motion:) And accordingly (which daily experience testifies) if on a Board or Table, some loose incumbent weight, be for some time moved, & have thereby contracted an Impetus to motion at such a rate; if that Board or Table chance by some external obstacle, or otherwise, to be stopped or considerably retarded in its motion, the incumbent loose Body will shoot forward upon it: And contrarywise, in case that Board or Table chance to be accelerated or put forward with a considerably greater speed than before, the loose incumbent Body, (not having yet obtained an equal Impetus with it) will be left behind, or seem to fly backward upon it. Or, (which is Galilæo's instance) if a broad Vessel of Water, for some time evenly carried forward with the Water in it, chance to meet with a stop, or to slack its motion, the Water will dash forward and rise higher at the fore part of the Vessel: And, contrarywise, if the Vessel be suddenly put forward faster than before; the Water will dash backwards, and rise at the hinder part of the Vessel. So that an Acceleration or Retardation of the Vessel, which carries it, will cause a rising of the Water in one part, and a falling in another: (which yet, by its own weight, will again be reduced to a Level as it was before.) And consequently, supposing the Sea to be but as a loose Body, carried about with the Earth, but not so united to it, as necessarily to receive the same degree of Impetus with it, as its fixed parts do; The acceleration or retardation in the motion of this or that part of the Earth, will cause (more or less, according to the proportion of it) such a dashing of the Water, or rising at one part, with a falling at another, as is that, which we call the Flux and Reflux of the Sea;

Now this premised, We are next, with him, to suppose the Earth carried about with a double motion; The one Annual, as (Fig. 1.) in B E C the great Orb, in which the Center of the Earth B, is supposed to move about the Sun A.

The other Diurnal, whereby the whole moves upon its own Axis, and each point in its surface describes a Circle, as D E F G.

It is then manifest, that if we suppose, that the Earth moved but by any one of these motions, and that regularly, (with an equal swiftness;) the Water, having once attained an equal Impetus thereunto, would still hold equal space with it: there being no occasion, from the Quickening or Slackening of the Earths motion, (in that part where the Water lyeth) for the Water thereon either to be cast Forward or fall Backward, and thereby to accumulate on the other parts of the Water: But the true motion of each part of the Earths surface being compounded of those two motions, the Annual and Diurnal; (the Annual in B E C being, as Galilæo there supposeth, about three times as fast as a diurnal motion in a great Circle, as D E F;) while a Point in the Earths surface moves about its Center B. from G. to D. and E. and at the same time, its Center B. be carried forwards to C; the true motion of that Point forwards, is made up of both those motions; to wit, of B to C, and of G to E; but while G moves by D to E, B moves backward by F to G, contrary to the motion of B to C; so that the true motion of E, is but the difference of B C, and E G; (for, beside the motion of B. above the Center; G is also[errata 1] put forward as much as from G to E; and E put backward as much as from E to G:) so that the diurnal motion, in that part of the Earth, which is next the Sun, as E F G, doth abate the progress of the Annual, (and most of all at F;) and in the other part, which is from the Sun, as G D E, it doth increase it, (and most of all at D.) that is, in the day time there is abated, in the night time is added to the Annual motion, about as much as is G E, the Earths Diameter. Which would afford us a Cause of two Tides in twenty four hours; the One upon the greatest Acceleration of motion, the Other upon its greatest Retardation.

And thus far Galilæo's Discourse holds well enough; But then in this it comes short; that as it gives an Account of two Tides; so those two Tides are alwayes to be at F and D; that is, at Noon and Midnight whereas Experience tells us, that the Time of Tides, moves in a moneths space through all the 24. hours. Of which he gives us no account. For though he do take notice of a Menstrual Period; yet he doth it onely as to the Quantity of the Tides; greater or less; not as to the Time of the Tides, sooner or later.

To help this, there is one (Vid. * Jo. Baptista Balianus)* Vid. Riccioli Almagest. novum, Tom. 1. lib. 4. cap. 10. n. 111. pag. 216 2. who make the Earth to be but a secondary Planet; and to move, not directly about the Sun, but about the Moon, the Moon meanwhile moving about the Sun; in like manner as we suppose the Earth to move about the Sun, and the Moon about it.

Instead of this, that Surmise of mine, for I dare not yet, with confidence give it any better name,) of what I have spoken to you heretofore, (and which hath occasioned this present account which I am now giving you,) is to this purpose.

The Earth and Moon being known to be Bodies of so great connexion (whether by any Magnetick, or what other Tye, I will not determine; nor need I, as to this purpose;) as that the motion of the one follows that of the other; (The Moon observing the Earth as the Center of its periodick motion:) may well enough be looked upon as one Body, or rather one Aggregate of Bodies, which have one common center of Gravity; which Center (according to the known Laws of Staticks) is in a streight Line connecting their respective Centers, so divided as that its parts be in reciprocal proportion to the Gravities of the two Bodies. As for Example Suppose the Magnitude (and therefore, probably, the Gravity) of the Moon to be about an one and fourtieth part of that of the Earth; (and thereabouts Hevelius in his Selenography page 203. doth out of Tycho, estimate the proportion; and an exact certainty is not necessary to our present businesse.) And the distance of the Moons Center from the Center of the Earth, to be about fifty six semidiameters of the Earth, (as thereabouts he doth there estimate it, in its middle distance; and we need not be now very accurate in determining the numbers; wherein Astronomers are not yet very well agreed.) The distance of the Common Center of Gravity of the two Bodies, will be from that of the Earth, about a two and fourtieth part of fifty six Semidiameters; that is, about ${\displaystyle \scriptstyle {\tfrac {56}{42}}}$ or ${\displaystyle \scriptstyle {\tfrac {4}{3}}}$ of a Semidiameter; that is about ${\displaystyle \scriptstyle {\tfrac {1}{3}}}$ of a Semidiameter of the Earth, above its surface, in the Air, directly between the Earth and Moon.

Now supposing the Earth and Moon, joyntly as one Body, carried about by the Sun in the great Orb of the Annual motion; this motion is to be estimated, (according to the Laws of Staticks, in other cases,) by the motion of the common Center of Gravity of both Bodies. For we use in Staticks, to estimate a Body, or Aggregate of Bodies, to be moved upwards, downwards, or otherwise, so much as its Common Center of Gravity is so moved, howsoever the parts may change places amongst themselves.

And accordingly, the Line of the Annual motion, (whether Circular or Elliptical; of which I am not here to dispute,) will be described, not by the Center of the Earth (as we commonly estimate it, making the Earth a Primary and the Moon a Secondary Planet,) nor by the Center of the Moon, (as they would do, who make the Moon the Primary and the Earth a Secondary Planet, against which we were before disputing:) But by the Common Center of Gravity of the Bodies, Earth and Moon, as one Aggregate.

Figure 2
Figure 3

Now supposing A B C D E to be a part of the See Fig. 2. and 3.great Orb of the Annual motion, described by the Common Center of Gravity, in so long time as from a Full-Moon at A to the next New-Moon at E; (which, though an Arch of a Circle or Ellipse, whose Center we suppose at a due distance below it; yet being but about ${\displaystyle \scriptstyle {\tfrac {1}{25}}}$ of the whole, may well enough be here represented by a streight Line:) the Center of the Earth at T, and that of the Moon at L, must each of them (supposing their common Center of Gravity to keep the Line A E) be supposed to describe a Periphery about that Common Center, as the Moon describes her Line of Menstrual motion. (Of which I have (in the Scheme) onely drawn that of the Earth; as being sufficient to our present purpose; parallel to which, if need be, we may suppose one described by the Moon; whose distance is also to be supposed much greater from T than in the figure is expressed, or was necessary to expresse.) And in like manner E F G H I, from that New-moon at E, to the next Full-moon at I.

From A to E (from Full-moon to New-moon,) T moves (in its own Epicycle) upwards from the Sun: And from E to I, (from New-moon to Full-moon) it moves downwards, toward the Sun. Again, from C to G, (from last quarter to the following first quarter,) it moves forwards according to the Annual motion; But from G forward to C, (from the first Quarter to the ensuing last Quarter,) it moves contrary to the Annual motion.

It is manifest therefore, according to this Hypothesis, that from Last quarter to First quarter (from C to G, while T is above the Line of the Annual motion) its Menstrual motion in its Epicycle adds somewhat of Acceleration to the Annual motion; and most of all at E, the New-moon: And from the first to the last quarter (from G forward to C, while T is below the Line of the Annual motion,) it abates of the Annual motion; and most of all at I, or A the Full-moon.

So that in pursuance of Galilæo's Notion, the Menstrual adding to or detracting from the Annual motion, should either leave behinde, or cast forward the loose waters incumbent on the Earth, (and thereby cause a Tide, or accumulation of Waters;) and most of all at the Full-moon and New-moon, where those Accelerations or Retardations are greatest.

Now this Menstrual motion, if nothing else were superadded to the Annual would give us two Tides in a moneth, and no more; (the one upon the Acceleration, the other on the Retardation;) at New moon and Full-moon; and two Ebbs, at the two Quarters; and in the Intervals, Rising and Falling water.

But the Diurnal motion superadded doth the same to this Menstrual, which Galilæo supposeth it to do to the[errata 2] Annual; that is, doth Add to, or Substract from, the Menstrual Acceleration or Retardation; and so gives us Tide upon Tide.

For in whatsoever part of its Epicycle, we suppose See Fig. 4 T to be; yet because, while by its Menstrual motion the Center moves in the Circle L T N; each point in its surface, by its diurnal motion moves in the Circle L M N: whatever effect (accelerative or tardative) the Menstrual would give, that effect by the Diurnal is increased in the parts L M N (or rather l M n. the Semicircle) and most of all at M: but diminished in the parts N O L (or rather n O l) and most of all at O. So that at M, and O, (that is when the Moon is in the Meridian below or above the Horizon,) we are to have the Diurnal Tide or High water, occasioned by the greatest Acceleration or Retardation, which the Diurnal Arch gives to that of the Menstrual: which seems to be the true cause of the Daily Tides. And withall gives an account, not onely why it should be every day; but likewise, why at such a time of the day; and why this time should in a moneth run through the whole 24. hours; viz. because the Moons coming to the Meridian above and below the Horizon, (or as the Seamen call it, the Moons Southing, and Northing) doth so: As likewise of the Spring tides and Neap-tides. For, when it so happens, that the Menstrual and Diurnal Accelerations or Retardations, be coincident, (as at New-moons and Full-moons they are,) the effect must needs be the greater. And although (which is not to be dissembled) this happen but to one of the two-Tides; that is, the Night-tide at the New-moon (when both motions do most of all Accelerate,) and the Day-tide at Full moon (when both do most Retard the Annual motion;) Yet, this tide being thus raised by two concurrent causes; though the next Tide have not the same cause also, the Impetus contracted will have influence upon the next Tide; Upon a like reason, as a Pendulum let fall from a higher Arch, will (though there be no new cause to occasion it) make the Vibration on the other side (beyond the Perpendicular) to be also greater: Or, of water in a broad Vessel, if it be so jogged, as to be cast forward to a good height above its Levell, will upon its recoyling, by its own gravity, (without any additional cause) mount so much the higher on the hinder part.

But here also we are to take notice, that though all parts of the Earth by its Diurnal motion do turn about its Axis, and describe parallel Circles; yet not equal Circles; but greater neer the Æquinoctial, and lesser near the Poles, which may be a cause why the Tides in some parts may be much greater than in others. But this belongs to the particular considerations, (of which we are not now giving an Account:) not to the general Hypothesis.

Having thus endeavoured to give an account of the Diurnal and Menstrual Periods of Tides; It remains that I endeavour the like as to the Annual. Of which there is, at least, thus much agreed; That, at some times of the year, the Tides are noted to be much higher, than at other times.

But here I have a double task; First, to rectify the Observation; and then, to give an account of it.

As to the first; It having been observed (grosly) that those high Tides have used to happen about the Spring and Autumn; it hath been generally taken for granted (without any more nice observation) that the two Æquinoxes are the proper times, to which these Annual high Tides are to be referred; And such causes sought for, as might best sute with such a Supposition.

But it is now, the best part of twenty years, since I have had frequent occasions to converse with some Inhabitants of Rumney-marsh in Kent; where the Sea being kept out with great Earthen walls, that it do not at high water overflow the Levell; and the Inhabitants livelyhood depending most on grazing, or feeding Sheep; they are (as you may believe they have reason to be) very vigilant and observant, at what times they are most in danger of having their Lands drowned. And I find them generally agreed, by their constant Observations, (and Experience dearly bought) that their times of danger are about the beginning of February and of November: that is, at those Spring Tides which happen near those times; to which they give the names of Candlemass-stream and Allhallond-stream: And if they scape those Spring-titles, they apprehend themselves out of Danger for the rest of the year. And as for March and September (the two Æquinoxes) they are as little solicitous of them, as of any other part of the year.

This, I confess, I much wondred at, when I first heard it; and suspected it to be but a mistake of him, that first told me, though he were indeed a person not likely so to be mistaken, in a thing wherein he was so much concerned: But I soon found, that it was not onely his, but a general observation of others too; both there, and elsewhere along the Sea coast. And though they did not pretend to know any reason of it, (nor so much as to enquire after it;) Yet none made doubt of it; but would rather laugh at any that should talk of March and September, as being the dangerous times. And since that time, I have my self very frequently observed (both at London and elsewhere, as I have had occasion) that in those months of February and November, (especially November) the Tides have run much higher, than at other times: Though I confess, I have not been so diligent to set down those Observations, as I should have done. Yet this I do particularly very well remember, that in November 1660. (the same year that his Majesty returned) having occasion to go by Coach from the Strand to Westminster, I found the Water so high in the middle of King-street, that it came up, not onely to the Boots, but into the Body of the Coach; and the Pallace-yard (all save a little place near the West-End) overflow'd; as likewise the Market-place; and many other places; and their Cellars generally filled up with Water. And in November last, 1665. it may yet be very well remembred, what very high Tides there were, not onely on the Coasts of England, (where much hurt was done by it) but much more in Holland, were by reason of those Inundations, many Villages and Towns were overflow'd. And though I cannot so particularly name other years, yet I can very safely say, that I very often observed Tides strangely high about those times of the year.

This Observation did for divers years cause me much to wonder, not only because it is so contrary to the received opinion of the two Æquinoxes; but because I could not think of anything signal at those times of the year: as being neither the two Æquinoxes, nor the two Solstices, nor the Sun's, Apogœum and Perigœum; (or Earths Aphelium and Perihelium;) nor indeed, at contrary times of the year, which at least, would seem to be expected, From Alhollandtide to Candlemass being but three months; and from thence to Alhollandtide again nine months.

At length it came into my mind, about four years since, that though there do not about these times happen any single signal Accident, which might cast it on these times, yet there is a compound of two that may do it: Which is the Inequality of the Natural day (I mean that of 24. hours, from noon to noon) arising at least from a double cause; either of which singly would cast it upon other times, but both joyntly on those.

It's commonly thought, how unequal soever the length be of the Artificial dayes as contradistinguished to nights, yet that the natural days[errata 3], reckoning from noon to noon, are all equal: But Astronomers know well, that even these dayes are unequal.

For, this Natural Day is measured not onely by one intire conversion of the Æquinoctial, or 24. Æquinoctial hours, (which is indeed taken to be performed in equal times,) but increases by so much, as answers to that part of the Sun's (or Earths,) Annual motion as is performed in that time. For, when that part of the Æquinoctial, which (with the Sun) was at the Meridian yesterday at noon, is come thither again to day, it is not yet Noon (because the Sun is not now at the place where yesterday he was, but is gone forward about one degree, more or less) but we must stay till that place, where the Sun now is, comes to the Meridian before it be now Noon.

Now this Additament (above the 24 Æquinoctial hours, or intire conversion of the Æquinoctial) is upon a double account unequal unequal. First, because the Sun, by reason of its Apogœum and Perigœum, doth not at all times of the year dispatch in one day an equal Arch of the Ecliptick; but greater Arches neer the Perigœum, which is about the middle of December; and lesser neer the Apogœum, which is about the middle of June; As will appear sufficiently by the Tables of the Sun's Annual motion. Secondly, though the Sun should in the Ecliptick move alwaies at the same rate; yet equal Arches of the Ecliptick do not in all parts of the Zodiack answer to equal Arches of the Æquinoctial, by which we are to estimate time: Because some parts of it, as about the two Solsticial Points, lie nearer to a parallel position to the Æquinoctial, than others, as those about the two Æquinoctial points, where the Ecliptick and Æquinoctial do intersect; whereupon an Arch of the Ecliptick, neer the Solsticial points answers to a greater Arch of the Æquinoctial, than an Arch equal thereunto neer the Æquinoctial points: As doth sufficiently appear by the Tables of the Suns right Ascension.

According to the first of these causes, we should have the longest natural daies in December, and the shortest in June, which if it did operate alone, would give us at those times two Annual High-waters.

According to the second cause, if operating singly, we should have the longest daies at the two Solstices in June and December, and the two shortest at the Æquinoxes in March and September; which would at those times give occasion of four Annual High-waters.

But the true Inequality of the Natural Days, arising from a Complication of those two causes, sometimes crossing and sometimes promoting each other: though we should find some increases or decreases of the Natural daies at all those seasons answerable to the respective causes (and perhaps of Tides proportionably thereunto:) yet the longest and shortest natural daies absolutely of the whole year (arising from this complication of Causes) are about those times of Allhallontide and Candlemas; (or not far from them) about which those Annual High-tides are found to be: As will appear by the Tables of Æquation of Natural daies. And therefore I think, we may with very good reason cast this Annual Period upon that cause, or rather complication of causes. For (as we before shewed in the Menstrual and Diurnal) there will, by this inequality of Natural daies, arise a Physical Acceleration and Retardation of the Earths Mean motion, and accordingly a casting of the Waters backward or forward; either of which, will cause an Accumulation or High-water.

'Tis true, that these longest and shortest daies, do (according to the Tables, some at least) fall rather before, than after Allhallontide and Candlemas (to wit the ends of October and January;) but so do also (sometimes) those high Tydes: And it is not yet so well agreed amongst Astronomers, what are all the Causes (and in what degrees) of the Inequality of Natural daies; but that there be diversities among them, about the true time: And whether the introducing of this New Motion of the Earth in its Epicycle about this Common Center of Gravity, ought not therein also to be accounted for, I will not now determine: Having already said enough, if not too much, for the explaining of this general Hypothesis, leaving the particularities of it to be adjusted according to the true measures of the motions; if the General Hypothesis be found fit to be admitted.

Yet this I must add, (that I be not mistaken) that whereas I cast the time of the daily Tydes to be at all places, when the Moon is there in the Meridian; it must be understood of open Seas, where the water hath such free scope for its motions, as if the whole Globe of Earth were equally covered with water: Well knowing, that in Bayes and In-land-Channels, the position of the Banks and other like causes must needs make the times to be much different from what we suppose in the open Seas: And likewise, that even in the Open Seas, Islands, and Currents, Gulfs and Shallows, may have some influence, though not comparable to that of Bays and Channels. And moreover, though I think, that Seamen do commonly reckon the time of High-water in the Open Seas; to be then, when the Moon is there in the Meridian (as this Hypothesis would cast it:) Yet I do not take my self to be so well furnished with a History of Tides, as to assure my self of it; much less to accommodate it to particular places and cases.

Having thus dispatched the main of what I had to say concerning the Seas Ebbing and Flowing: Had I not been already too tedious, I should now proceed to give a further reason, why I do introduce this consideration of the Common Center of Gravity in reference to Astronomical Accounts. For indeed, that which may possibly seem at first to be an Objection against it, is with me one reason for it.

It may be thought perhaps, that if the Earth should thus describe an Epicycle about the Common Center of Gravity, it would (by this its change of place) disturbe the Cælestial motions; and make the apparent places of the Planets, especially some of them, different from what they would otherwise be. For though so small a removal of the Earth, as the Epicycle would cause (especially if its Semidiameter should not be above 113 of the Earths Semidiameter) would scarce be sensible (if at all) to the remoter Planets; yet as to the nearer it might.

Now though what Galilæo answers to a like Objection in his Hypothesis; (that its possible there may be some small difference, which Astronomers have not yet been so accurate, as to observe) might here perhaps serve the turn; Yet my answer is much otherwise; to wit, that such difference hath been observed, and hath very much puzzeled Astronomers to give an account of. About which you will find Mr, Horrocks (in some of his Letters, whereof I did formerly, upon the Command of the Royal Society, make an Extract) was very much perplexed; and was fain, for want of other relief, to have recourse to somewhat like Keplers amicable Fibres, which did according to the several positions of the Moon, accelerate or retard the Moon's motion; which amicable Fibres he had no affection to at all (as there appears) if he could any other waies give account of those little inequalities; and would much rather (I doubt not) have embraced this Notion of the Common Center of Gravity, to salve the Phænomenon, had it come to his mind, or been suggested to him. And you find, that other Astronomers have been seen to bring in (some upon one supposition, some upon another) some kind of Menstrual Æquation, to solve the inequalities of the Moon's motion, according to her Synodical Revolution, or different Aspects (of New-moon, Full Moon, &c.) beside what concerns her own Periodical motion.

For which, this consideration of the Common Center of Gravity of the Earth and Moon, is so proper a remedy (especially if it shall be found precisely to answer those Phænomena, which I have not Examined, but am very apt to believe) that it is so far from being, with me, an Objection against it, that it is one of the reasons, which make me inclinable to introduce it.

I must before I leave this, add one Consideration more, That if we shall upon these Considerations think it reasonable, thus to consider the Common Center of Gravity of the Earth and Moon; it may as well be thought reasonable, that the like Consideration should be had of Jupiter and his four Satellites, which according to the Complication of their several motions, will somewhat change the position of Jupiter, as to that Common center of Gravity of all these Bodies; which yet, because of their smallness, may chance to be so little, as that, at this distance, the change of his[errata 4] apparent place may not be discernable. And what is said of Jupiter, is in the like manner to be understood of Saturne and his Satelles, discovered by Hugenius: For all these Satellites are to their Principals, as so many Moons to the Earth. And I do very well remember, in the Letters forecited, Mr. Horrocks expresseth some such little inequalities in Saturnes motion, of which he could not imagine what account to give, as if (to use his Expression) this crabbed Old Saturn had despised his Youth. Which, for ought I know, might well enough have been accounted for, if at that time the Satelles of Saturn had been discovered, and that Mr. Horrocks had thought of such a notion[errata 5] as the Common Center of Gravity of Saturn and his Companion, to be considerable, as to the guiding of his motion.

You have now, in obedience to your Commands, an Account of my thoughts, as to this matter, though yet immature and unpolished; What use you will please to make of them, I shall leave to your prudence, &c.

An APPENDIX, written by way of Letter to the Publisher; Being an Answer to some Objections, made by several Persons, to the precedent Discourse.

IReceived yours; and am very well contented, that objections be made against my Hypothesis concerning Tydes: being

proposed but as a conjecture to be examined; and, upon that Examination, rectified, if there be occasion; or rejected, if it will not hold water,

1.To the first objection of those you mention; That it appears not how two Bodies, that have no tye can have one common Center of Gravity: that is (for so I understand the intendment of the objection) can act or be acted in the same manner, as if they were connected: I shall onely answer, that it is harder to shew How they have, than That they have it. That the Load-stone and Iron have somewhat equivalent to a Tye; though we see it not, yet by the effects we know. And it would be easy to shew, that two Load-stones, at once applyed, in different positions, to the same Needle, at some convenient distance; will draw it, not to point directly to either of them, but to some point between both; which point is, as to those two, the common Center of Attraction; and it is the same, as if some one Load-stone were in that point. Yet have these two Load-stones no connexion or tye, though a Common Center of Virtue according to which they joyntly act. And as to the present case, How the Earth and Moon are connected; I will not now undertake to shew (nor is it necessary to my purpose;) but, That there is somewhat, that doth connect them, (as much as what connects the Load-stone, and the Iron, which it draws,) is past doubt to those, who allow them to be carryed about by the Sun, as one Aggregate or Body, whose parts keep a respective position to one another: Like as Jupiter with his four Satellites, and Saturn with his one. Some Tye there is, that makes those Satellites attend their Lords, and move in a Body; though we do not See that Tye, nor Hear the Words of Command. And so here.

2.To the second objection; That, at Chatham and in the Thames, the Annual Spring-tydes, happen about the Æquinoxes, not (as this Hypothesis doth suppose elswhere to have been observed) about the begining of February and November. If their meaning be, that Annual High Tydes, do then happen, and then onely: If this prove true, it will ease me of half my work. For it is then easily answered, that it depends upon the Obliquity of the Zodiack; the parts of the Æquinoctial answering to equal parts of the Zodiack, being neer the Solstitial points greatest, and near the Æquinoctial points least of all. But beside this Annual Vicissitude of the Æquinoxes, not to say, of the 4. Cardinal Points (which my Hypothesis doth allowance assert;) I believe it will be found, that there is another Annual Vicissitude answering to the Suns Apogœum and perigœum. And that the greatest Tydes of all, will be found to be upon a result of these two causes Cooperating; which (as doth the Inequality of Natural dayes, depending on these same causes) will light nearer the times, I mention. To what is said to be observed at Chatham and in the Thames, contrary to that I allege as observed in Rumney marsh: I must at present ἀπέχειν, and refer to a melius iniquirendum. If those who object this contrary observation, shall, after this notice, find, upon new Observations heedfully taken, that the Spring-tydes in February and November, are not so high, as those in March and September; I shall then think the objection very considerable. But I do very well remember, that I have seen in November, very high Tydes at London, as well as in Rumney marsh. And, the time is not yet so far past, but that it may be remembered (by your self or others then in London) whether in November last when the Tydes were so high at Dover, at Deal, at Margate, and all along the Coast from thence to Rumney Marsh, as to do in some of those places much hurt, (and, in Holland, much more;) whether, I say, there were not also at the same time, at London, (upon the Thames) very high Tydes. But a good Diary of the Height and time both of High-water, and Low-water; for a year or two together, even at Chatham, or Greenwich; but rather at some place in the open Sea, or at the Lands end in Cornwal, or on the West parts of Ireland; or at St. Hellens, or the Bermudas, &c. would do more to the resolving of this point, than any verbal discourse without it.

3. To the third Objection, That supposing the Earth and Moon to move about a Common center of Gravity; if that the highest Tydes be at the New-moon, when the moon being nearest to the Sun, the Earth is farthest from it, and compound motion at the swiftest; and that the Tydes abate as the Earth approacheth nearer, till it comes into the supposed Circle of her Annual motion: It may be demanded, why do they not still abate as the Earth comes yet nearer to the Sun and the swiftnesse of its compound still slackens? And so, why have we not Spring tides at the New Moon (when the motion is the swiftest) and Neap-tides at Full Moon (when the motion is slowest) but Spring tides at both? The answer (if observed) is already given in my Hypothesis it self. Because the effect is indifferently to follow, either upon a suddain Acceleration, or a suddain Retardation. (Like as a loose thing, lying on a moving body; if the body be thrust suddainly forward, that loose thing is cast back, or rather left behind, not having yet obtained an equal impetus with that of the body, on which it lyes; but if stopped, or notably retarded, that loose incumbent is thrown forward, by its formerly contracted impetus not yet qualified or accommodated to the slowness of the Body, on which it lyes.) Now both of these happening, the one at the New Moon, the other at the Full Moon, do cause high Tides at both.

4.To the fourth Objection, That the highest Tydes are not at all places, about the New Moon and Full Moon; and particularly, that, in some places of the East Indies, the Highest Tydes are at the Quadratures: I must first answer in general; That as to the particular varieties of Tydes in several parts of the World, I cannot pretend to give a satisfactory account, for want of a competent History of Tydes, &c. Because (as is intimated in what I wrote in the general) the various positions of Chanels, Bays, Promontories, Gulfs, Shallows, Currents, Trade-winds, &c. must needs make an innumerable variety of Accidents in particular places, of which no satisfactory account is to be given from the general Hypothesis (though never so true) without a due consideration of all those. Which is a task too great for me to undertake, being so ill furnished with materials for it. And then as to the particular instance of some places in the East Indies, where the highest Tydes are at the Quadratures: I suppose, it may be chiefly intended of those about Cambaia, and Pegu. At which places, beside that they are situate at the inmost parts of Vast Bayes, or Gulfs (as they are called) they have also vast In-draughts of some hundred Miles, within Land; which when the Tydes are out, do lye (in a manner) quite dry: And may therefore very well be supposed to participate the effect of the Menstrual Tydes many dayes after the cause of them happens in the open Sea, upon a like ground as in Straights and narrow Channels the Diurnall Tydes happen some hours later than in the Ocean. And a like account must be given of particular accidents in other places, from the particular situation of those places, as Bays, Chanels, Currents, &c.

Having thus given you some Answers to the Objections you signifie to have been made by several persons to my Hypothesis, and that in the same order your Paper presents them to me: I shall next give you some account of the two Books, which you advised me to consult; so far as seems necessary to this business: Which, upon your intimation, I have since perused, though before I had not.

And first, as to that of Isaac Vossius, De motu Marium & Ventorum; Though I do not concur with him in his Hypothesis; That all the Great motions of the Seas, &c. should arise onely from so small a warming of the water as to raise it (where most of all) not a Foot in perpendicular, (as in his 12th Chapter;) Or that there is no other connexion between the Moons motion, and the Tydes menstrual period, than a casual Synchronism (which seems to be the doctrine of his 16th and 18th Chapters;) Beside many other things in his Philosophy, which I cannot allow: Yet I am well enough pleased with what is Historical in it, of the matter of Fact: Especially if I may be secure, that he is therein accurate and candid, not wresting the Phænomena to his own purpose. But I find nothing in it, which doth induce me to vary from my Hypothesis. For, granting his Historicals to be all true; the account of the constant Current of the Sea Westward, and of the constant Eastern Blasts, &c. within the Tropicks, is much more plausibly, and (I suppose) truly rendered by Galilæo long since, from the Earths Diurnal motion: (which, neare the Æquator, describing a greater Circle, than nearer the Poles, makes the Current to be there more conspicuous and swift, and, consequently, the Eddy, or recurrent motion, nearer the Poles, where this is, more remiss:) than can easily be rendered by so small a Tumor, as he supposeth. Not to adde; that his account of the Progressive motion, which he fansieth to follow upon this Tumefaction, and by Acceleration to grow to so great a height near the Shoar (as in Chap. 13. and 14.) is a Notion, which seems to me too extravagant to be salved by any laws of Staticks. And that of the Moons motion onely Synchronizing with the Tydes, casually, without any Physical connexion; I can very hardly assent to. For it can hardly be imagined, that any such constant Synchronism should be in Nature; but where, either the one is the cause of the other, or both depend upon some Common cause. And where we see so fair a foundation for a Physical connection. I am not prone to ascribe it to an Independent Synchronism. In sum; His history doth well enough agree with my Hypothesis; and I think, the Phænomena are much better salved by mine, than his.

And then as to Gassendus, in his discourse De Æstu Maris, I find him, after the relating of many other Opinions concerning the Cause of it, inclining to that of Galilæo ascribing it to the Acceleration & Retardation 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 Galilæo had done before him) that the line of the Mean Motion of this Aggregate (or, as he calls, motus æquabilis 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 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 Æquinoctial and Solstitial 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 sequantem præcise revolutionem axis, aliam remotius av velut in circumferentia, 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.

Figure 5

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 γ, 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 γ, but at α; too slow: And when the Moon is at B; and the Earth at b. Mars will be seen at β; yet too slow: till at the Fullmoon, the Moon at C, the Earth at c, Mars will he 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 ς. 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 Parallax is 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 123 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.

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. Hob's Book Contra Geometras (or De Principiis & Ratiocinatione Geometrarum) which you then shewed me. A New Book of Old master: 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 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 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 Out-side doth at one place Touch the Circle, it may with in In-side at another place Cut it, &c.) But I should 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 Σημογ, 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 Expedience. 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. Hobs would call the middle of the breadth of his; and Euclide's Point, is but the Middle of Mr. Hob's. And then, for the same reason, that Mr. Hobbs 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. Hobs tells us (as pag. 3.) that this End is not Punctum, but only Signum (which he does allow non esse nonem Quanti) even this will serve our turn well enough, Euclid's Σημογ, 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 intituled Contra Grammaticos, not, Contra Geometras. Nor is it Euclide, but Cicero, that is concern'd, in rendring the Greek Σημογ, 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, has 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 Heuton-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 with 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 quator cubis dv; (pag. 43 line 33.) into these (p. 44. l. 5.) aqualia quator 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[errata 6] 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 hans Lineam Perpendicularem extra circulum cadere (because neither intra Circulum nor in Peripheria;) He doth, when he would shew an errour, first make one, by falsifying his words, line 15. where instead of Lineam Perpendicularum, 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. 30.) wherewith he would now New-vamp his old False quadrature; lyes in those words Page 40. line 30, 31. Quod Impossible 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[errata 7] B 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:) or though it lye above c, (vid. in case d be neerer, than R c, to the point A.) And therefore. unless he first prove (which he cannot do) that A d (a sixth part of A D) doth just reach to the line R a and no further; he onely proves that a sixth part of ab is equal to the sine of[errata 8] 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.
1. Original: motion of B the Center G is also was amended to motion of B. above the Center; G is also: detail
2. Original: is to do to that was amended to it to do to the: detail
3. Original: Natural Day was amended to natural days: detail
4. Original: of this was amended to of his: detail
5. Original: a motion was amended to a notion: detail
6. Original: enough for was amended to enough without: detail
7. Original: to the line of was amended to to the sine of: detail
8. Original: to the line of was amended to to the sine of: detail