An Extract out of Doctor John Wallis his Hypothesis concerning the Ebbing and Flowing of the Sea. In His Letter to Mr. Boyle Dated Aprill 25. 1666. Inserted in the Philosophick Transactions No. 16 page 263. Drawn out by CM. to be added by way of Appendix to our Phisicks.
THE Periods and Magnitudes of Tides doe So constantly attend the moons motion that we may well presume they are either governed one by the other or both of them by some common cause and So there must be a certain connection between them.
The Moons occult quality to influence moist bodies Leaves us in the Dark as to this matter and only confidently assumes that it is So without Shewing us how or why: the magnetick vertue fancied to be in the moon whereby it Should draw the water to it or drive it from it does as Little Satisfy for drawing to would make it always high water where the moon is verticall; and driving from or depression by certain gravitating emanations would make it (on the Contrary) always Lowest water at the Same time of verticity: but both of them are false. The Dr. therefore Layes aside Such Hypotheses and closes rather with Gallileus referring it to the Earths motion according to the generall Laws of motion and the Statick principles; Onely he notes in Gallileus a defect of the true history of tides which is a blemish on So excellent a person or a prejudice to the maine Hypothesis; This defectiveness of the History Still remaining in great part; He therefore proposeth his own Supposall onely as an essay to be farther considered which is this. That the Earth’s motion causes the tides and the Moon’s Phases Determine (or rather point out) their periods. Of Tides Some are extraordinary (Either from Some accidentall cause or the naturall not Sufficiently observed) others are ordinary and constantly taken notice of: Tis of the Latter he offers to give the Reason. These are 3 Diurnall Menstruall and  Annuall.
1. Diurnall Reciprocation is twice in Somewhat more than 24 hours.
2. Menstruall has two considerations of the time and of the magnitude.
1. The Time that the tides remove round the day and night in one Synodicall period of the moon as if at full moon (at any place) it be full Sea at noon this Day tomorrow it will be full there a Little before one the Next near two (Loosing near an hour every day) till at the next full Moon twill be full Sea again just at noon as it was before.
2. The Magnitude is, that about the full and new moons the Tides are highest and the Ebbs Lowest; that at the Quadratures are Least tides and at the times intermediate they rise proportionably: This variety is expressed by Spring and neap tides.
3. Annuall that twice in a year the Springs are Higher than at other times. These times were formerly Said to be near the Equinoxes but indeed by better observation they are found to be in the begginning of February and November. To Solve these Phainomenas he assumes Divers things with Gallileus which either will be readily granted or are elsewhere proved, as
1. That a body in motion is apt to continue its motion in the Same velocity unless hindered by Some other body as a body in Rest will continue in rest unless it be put into Motion.
2. That fluids do not perfectly recieve their change either to Rest or motion from their conteining vessels but upon the vessels change the fluid rises at one or other end above Level for a while and then falls back again into Level by its own gravity.
3. That the Seas are but as water in Large vessels and the vessels of Earth being carried about Swifter or Slower at divers times the water is not So fastned to the Earth but that at the acceleration or retardation of that part of the Earths Surface where the water Lyes it will Swell up for a time at one end before it can recieve the Same degree of motion with its vessel.
4. He assumes also the Earths Double motion. Annuall about the Sun and Diurnall about its own axis or Center. He allows (with Gallileus) the annuall to be about 3 times as Swift as the Diurnall in its æquator or other great circle; and that therefore a Superficiall part of the Earth adds to and takes from the annuall Swiftness of its Center according as that part comes nearer to or goes farther from the Sun. As Suppose (c) the Center of the Earth move equally in the great orb about the Sun (c. x. a. z.) yet the point (b) in the Surface rolling about to (d. m. n.) will alter its pace for in the day time at (d) the part will take from the Swiftness of the centers motion because tis moving the contrary way but in the night time when the  Same part comes to (n) it adds to the annuall Motion of the center because its revolution is the same way and has a compound motion both of the annuall and diurnall: whence it follows that the Superficiall part (b) moves Slower in the day and Swifter in the night and by consequence the Sea must be affected with these changes of velocity So as to make two tides in about 24 hours. And thus far the Dr. agrees with Gallileus. But then (Saith he) all this would give the tides at that place of the Surface onely and always at the Same times (noon and midnight) every day and not make them Shift round in a month as we see they do. To Solve this Johannes Baptista Ba[li]anus2 (vid. Rice alm: novum page 216)3 makes the Earth but a Secondary planet to the moon and to move about it while the moon goes about the Sun contrary to what is commonly Supposed that the Earth goes about the Sun and the Moon as a Secondary about the Earth. this (though it might Serve for a menstruall period of tides) he admitts not because The Earth is bigger than the moon (as Jupiter is than his Satellites and Saturn than his) and therefore twere Less Consonant with the generall System of the World that the Less body Should carry about the greater with it: As the Suns Motion about its axis is (with good reason Saith he) judged to be the Phisicall cause of the primary planets moving about so tis Likely the rotation of Jupiter (discovered Lately by its fixed Spotts) is the cause of its Satellites motion about its body; and So also the Earths dayly revolution does more Likely carry about the Lesser moon which does not So revolve as appears by the Same face always towards us. If it be Said this happens because the moon carries about the Earth in just the Same period in which it turns about its own axis This is contrary to that of the Suns carrying about the other planets or Jupiter carrying about his Satellites for in these the Mover turns his period Sooner than any of those that are moved do accomplish theirs and thus he refells Ba[li]anus.
N.B. I must confess I am not So well Satisfied with his Second reason as to think the Sun has So powerfull a Vortex; or if it have I See not why it Should cause revolution in the planets about their own axis (as tis Said all the primaries do). The Sun might indeed Swim them round about himself; but this would not turn them about themselves also and therefore I rather incline to admitt Gallileus notion of their being created at Some Distance from their place and then dropt trolling into it. But this only by the by.
Ba[li]anus his fancy being rejected the Doctors Surmise (as he is pleased to call it) for the Solving of this Phainomenon is, That the Earth and Moon are connected (whether by Magnetick or other  Tye matters not to the present purpose) So as the Earth is the center of the moons periodicall motion in 27 dayes And that these two together may be accounted as one aggregate of bodys having one common center of Gravity which (according to the Laws of Staticks) is the Line connecting their Respective centers So divided that its parts be reciprocall proportionable to the gravities of the 2 bodies. As Suppose (with the Learned Hevelius in his Selenographie, out of Tycho)4 the magnitude (and therefore probably the gravity) of the Moon 1/41 of that of the Earth; and Suppose (with them) the mean Distance of Centers to [be] 56 Semi Diameters of the Earth; then the Distance of the Common Center of gravity from the Center of the Earth (the bigger body) will be about 1⅓ of its Semidiameter that is a third of a Semidiameter from the Earths Surface in the air Directly between the Earth and moon. This common Center of Gravity is next Supposed to move in the great orb about the Sun (whither a true Circle or an Ellipsis is not now to be enquired) and this in the annuall period; but the Earth meanwhile is to roll round about its own axis dayly and its Center is to go about the common center of gravity (as in an Epicicle) in the time of the Moons period; and this revolution of the Earth about the Common Center of gravity gives the monthly inæqualities as the daily revolution does the Diurnall. (a) The Center of the Earth: (b.c.d.) the revolution of the Earths Surface daily: (C) the Common Center of gravity: (f.g.h.) the revolution of the Earths Center about the Common Center monthly: (i. k. 1.) the Moons Monthly revolution about the Same common Center which though it Serves for the distinctions of Seasons in the month yet it is not concerned in the business of acceleration or retardation of the Earth and So causing Tides only because of its Synchronism it may tell us when the Spring and neap tide will be.
N.B. Both the Distance of the Common center and the great orb about the Sun in which the Center is  Supposed to move is far Less than it Should be if the paper would contain it. Onely this is Sufficient to Shew that when tis new moon and the Earth is beyond the common center of gravity farther from the Sun then is the greatest acceleration because the Earths Center goes the same way with the Common Center in the great orb: but when tis full moon and the Earths Center is at (g) moving towards (h) then is the greatest retardation because then it moves towards (h) the contrary way to the Common Centers motion in the great orb.
Having thus Endeavoured to account for the Diurnall and menstruall Periods of tides he comes to (that most obstruse point of) the Annuall highest Spring which must be from the former causes Somewhat augmented in those times of the year wherein Such great tides are found. The times in the open Sea (as at Rumney marish) are 2 the highest Springs are in the beginning of February and November (Cold Candle-mass Stream and Allhallows Stream) which Seems Strange for that they are none of the remarkable times of the year (as the Equinoxes Solstices Apoge and Perige of the Sun) Nor opposite parts of the year there being but 3 months between them on the one Side and 9 on the other. After Long wonder in this part he at Last (in the year 1664) thought that the inæquality of naturall Dayes (or the 24 hours from noon to noon) might be the cause of this which he thus Explaines. Everyone knows that the artificiall days from Sun to Sun are unæquall and Astronomers know (what is not noted by others) that the naturall dayes of 24 hours from noon to noon are unæquall also for this naturall day is measured not only by one intire conversion of the Equinoctiall (taken to be performed in æquall times) but increases So much as the Sun (or Earth) moves in that time (of 24 Equinoctiall hours) in the Ecliptick according to the annuall motion (which is one degree or one 1/15 of an hour more or Less) for when that part of the Ecliptick which was yesterday with the Sun at our meridian comes thither again today tis not yet noon because the Sun is not now in the Same place of the Ecliptick it was yesterday at noon; but gone Eastwards about a degree, and therefore he hath one 15th of an hour yet  To Spend in going on westward ere he comes to our meridian and So make it noon. This Shifting of place by a planet is most observable in the moon which passes on about 12 degrees Eastward every 24 hours. Now this addittament to the 24 Equinoctiall is unæquall upon a double account.
1. The Sun in Apoge and Perige dispatches not æquall arches of the Ecliptick in the Same time; but greater near the Perige (about the middle of December) and Less near the Apoge (About the middle of June) as appears by the Suns annuall motion.
2. Æquall Arches of the Ecliptick by which time is measured (if the Sun did move æqually in it) do not answer to æquall arches of the Equator because of their obliquity one to the other: for in the Solstices where the Equator and the Ecliptick are near paralel an Arch of the Ecliptick (say 10 Degrees) will be near æquall to an Arch of the Equator of the same number: but near the Æquinoxes (where the intersections of these two arches are) Such an arch of the Ecliptick will not require above 6 or 7 degrees in the Equator to answer it: i.e. a meridian drawn to cut 10 degrees in the Ecliptick continued on will cut but 6 or 7 in the Equator (as in the Table of the Suns right ascension and on the globe is easily Seen). According to the former of these causes (if it did operate alone) we Should have the Longest days in December and the Shortest in June; and it would give us two annuall tides to be at those times. According to the Latter (if also operating alone) the dayes would be Longest in the Solstices (June and December) and Shortest in the Equinoxes (March and September) and So there would be 4 Annual Highwaters. But the true inæquality of Naturall Days is from the complication of both these causes Sometimes crossing Sometimes promoting each other; and this gives that (though we Should find Some increases and decreases of Naturall days at all those mentioned Seasons answerable to their respective causes and perhaps of Tides proportionably yet) the Longest and Shortest naturall days absolutely of the whole year are about those times of Allhallow and Candlemass.