voyage to America, determined that in the island of Cayenne and Granadathe length of the pendulum vibrating in seconds was a small matter lessthan 3 feet and 6| lines ; that in the island of St. Christophers it was3 feet and 6f lines;and in the island of St. Domingo 3 feet and 7lines.And in the year 1704, P. Feuille, at Puerto Bello in America, foundthat the length of the pendulum vibrating in seconds was 3 Paris feet,and only 5--^ lines, that is, almost 3 lines shorter than at Paris ; but theobservation was faulty. For afterward, going to the island of Martinico.he found the length of the isochronal pendulum there 3 Paris feet and5 。 | lines.Now the latitude of Paraiba is 6 38 south ; that of Puerto Bello 933 north ; and the latitudes of the islands Cayenne, Goree, Gaudaloupe}Martinico, Granada, St. Christophers, and St. Domingo, are respectively4C 55 , 14 40", 15 00 , 14 44 , 12 06 , 17 19 , and 19 48 , north. An*J412 THE MATHEMATICAL PRINCIPLES [BOOK IIIthe excesses of the length of the pendulum at Paris above the lengths ofthe isochronal pendulums observed in those latitudes are a little greaterthan by the table of the lengths of the pendulum before computed. Andtherefore the earth is a little higher under the equator than by the preceding calculus, and a little denser at the centre than in mines near the surface, unless, perhaps, the heats of the torrid zone have a little extended thelength of the pendulums.For M. Picart has observed, that a rod of iron, which in frosty weatherin the winter season was one foot long, when heated by lire, was lengthenedinto one foot and -]-line. Afterward M. de la Hire found that a rod ofiron, which in the like winter season was 6 feet long, when exposed to theheat of the summer sun, was extended into 6 feet and f line. In the formercase the heat was greater than in the latter; but in the latter it was greaterthan the heat of the external parts of a human body ; for metals exposedto the summer sun acquire a very considerable degree of heat. But the rodof a pendulum clock is never exposed to the heat of the summer sun, norever acquires a heat equal to that of the external parts of a human body ;and, therefore, though the 3 feet rod of a pendulum clock will indeed be alittle longer in the summer than in the winter season, yet the difference willscarcely amount to 。 line. Therefore the total difference of the lengths ofisochronal pendulums in different climates cannot be ascribed to the difference of heat ; nor indeed to the mistakes of the French astronomers. Foralthough there is not a perfect agreement betwixt their observations, yetthe errors are so small that they may be neglected ; and in this they allagree, that isochronal pendulums are shorter under the equator thanat the Royal Observatory of Paris, by a difference not less than 1{ line,nor greater than 2| lines. By the observations of M. Richer, in the islandof Cayenne, the difference was 1| line. That difference being corrected bythose of M. des Hayes, becomes 。。 line or l line. By the less accurateobservations of others, the same was made about two lines. And this disagreement might arise partly from the errors of the observations, partlyfrom the dissimilitude of the internal parts of the earth, and the height ofmountains ; partly from the different heats of the air.I take an iron rod of 3 feet long to be shorter by a sixth part of one linein winter time with us here in England than in the summer. Because ofthe great heats under the equator, subduct this quantity from the differenceof one line and a quarter observed by M. Richer, and there will remain oneline TV, which agrees very well with l T -oo ^ne collected, by the theory alittle before. M. Richer repeated his observations, made in the island ofCayenne, every week for ten months together, and compared the lengths ofthe pendulum which he had there noted in the iron rods with the lengthsthereof which he observed in Prance. This diligence and care seems tohave been wanting to the other observers. If this gentleman s observationsBOOK I1I.J OF NATURAL PHILOSOPHY. 413are to be depended on, the earth is higher under the equator than at thepoles, and that by an excess of about 17 miles; as appeared above by thetheory.PROPOSITION XXI. THEOREM XVII.That the equinoctial points go backward, and that the axis of the earth,by a nutation in, every annual revolution, twice vibrates towards theecliptic, and as often returns to its former position,.The proposition appears from Cor. 20, Prop. LXVI, Book I; butthat motion of nutation must be very small, and, indeed, scarcely perceptible.PROPOSITION XXII. THEOREM XVIII.That all the motions of the ?noon, and all the inequalities of those motions,follow from the principles which we have laid down.That the greater planets, while they are carried about the sun, may inthe mean time carry other lesser planets, revolving about them ; and thatthose lesser planets must move in ellipses which have their foci in the centres of the greater, appears from Prop. LXV, Book I. But then their motions will be several ways disturbed by the action of the sun, and they willsuffer such inequalities as are observed in our moon. Thus our moon (byCor. 2, 3, 4, and 5, Prop. LXVI, Book I) moves faster, and, by a radiusdrawn to the earth, describes an area greater for the time, and has its orbitless curved, and therefore approaches nearer to the earth in the syzygiesthan in the quadratures, excepting in so far as these effects are hindered bythe motion of eccentricity ;for (by Cor. 9, Prop. LXVI, Book I) the eccentricity is greatest when the apogeon of the moon is in the syzygies, andleast when the same is in the quadratures ; and upon this account the perigeonmoon is swifter, and nearer to us, but the apogeon moon slower,arid farther from us, in the syzygies than in the quadratures. Moreover,the apogee goes forward, and the nodes backward ; and this is done not witha regular but an unequal motion. For (by Cor. 7 and 8, Prop. LXVI,Book I) the apogee goes more swiftly forward in its syzygies, more slowlybackward in its quadratures; and, by the excess of its progress above itsregress, advances yearly in consequentia. But, contrariwise, the nodes (byCor. 11, Prop. LXVI, Book I) are quiescent in their syzygies, and go fastestback in their quadratures. Farther, the greatest latitude of the moon (byCor. 10, Prop. LXVI, Book I) is greater in the quadratures of the moonthan in its syzygies. And (by Cor. 6, Prop. LXVI, Book I) the mean motion of the moon is slower in the perihelion of the earth than in its aphelion.And these are the principal inequalities (of the moon) taken notice of byastronomers.414 THE MATHEMATICAL PRINCIPLES [BOOK IIIBut there are yet other inequalities not observed by former astronomers,by which the motions of the moon are so disturbed, that to this day wehave not been able to bring them under any certain rule. For the velocities or horary motions of the apogee and nodes of the moon, and theirequations, as well as the difference betwixt the greatest eccentricity in thesyzygics, and the least eccentricity in the quadratures, and that inequalitywhich we call the variation, are (by Cor. 14, Prop. LXVI, Book I) in thecourse of the year augmented and diminished in the triplicate proportionof the sun s apparent diameter. And besides (by Cor. 1 and 2, Lem. 10,and Cor. 16, Prop. LXVI, Book I) the variation is augmented anddiminished nearly in the duplicate proportion of the time betweenthe quadratures. But in astronomical calculations, this inequalityis commonly thrown into and confounded with the equation of the moon scentre.PROPOSITION XXI1L PROBLEM V.To derive the unequal motions of the satellites of Jupiter and Saturnfrom the motions of our moon.From the motions of our moon we deduce the corresponding motions ofthe moons or satellites of Jupiter in this manner, by Cor. 16, Prop. LXVI,Book I. The mean motion of the nodes of the outmost satellite of Jupiteris to the mean motion of the nodes of our moon in a proportion compounded of the duplicate proportion of the periodic times of the earth about thesun to the periodic times of Jupiter about the sun, and the simple proportion of the periodic time of the satellite about Jupiter to the periodic timeof our moon about the earth ; and, therefore, those nodes, in the space ofa hundred years, are carried 8 24 backward, or in antecedentia. Themean motions of the nodes of the inner satellites are to the mean motion ofthe nodes of the outmost as their periodic times to the periodic time of theformer, by the same Corollary, and are thence given. And the motion ofthe apsis of every satellite in consequential is to the motion of its nodes inantecedentia as the motion of the apogee of our moon to the motion of itsnodes (by the same Corollary), and is thence given. But the motions ofthe apsides thus found must be diminished in the proportion of 5 to 9, orof about 1 to 2, on account of a cause which I cannot here descend to explain. The greatest equations of the nodes, and of the apsis of every satellite, are to the greatest equations of the nodes, and apogee of our moon respectively, as the motions of the nodes and apsides of the satellites, in thetime of one revolution of the former equations, to the motions of the nodesand apogee of our moon, in the time of one revolution of the latter equations. The variation of a satellite seen from Jupiter is to the variation ofour moon in tne same proportion as the whole motions of their node?BOOK IIIJ OF NATURAL PHILOSOPHY. 415respectively during the times in which the satellite and our moon (afterparting from) are revolved (again) to the sun, by the same Corollary ; andtherefore in the outmost satellite the variation does not exceed 5" 12 ".PROPOSITION XXIV. THEOREM XIX.That the flax and reflux of the sea arise from the actions oj the sunand moon.By Cor. 19 and 20, Prop. LXVI, Book I, it appears that the waters ofthe sea ought twice to rise and twice to fall every day. as well lunar as solar ;and that the greatest height of the waters in the open and deep seas oughtto follow the appulse of the luminaries to the meridian of the place by aless interval than 6 hours ;as happens in all that eastern tract of the Atlanticand jEthinpic seas between France and the Cape of Good Hope ; and onthe coasts of Chili and Pern, in the Smith Sea ; in all which shores theilo >d falls out about the second, third, or fourth hour, unless where themotion propagated from the deep ocean is by the shallowness of the chaiirnels, through which it passes to some particular places, retarded to thefifth, sixth, or seventh hour, and even later. The hours I reckon from theappulse of each luminary to the meridian of the place, as well under asabove the horizon ; and by the hours of the lunar day I understand the24th parts uf that time which the moon, by its apparent diurnal motion,employs to come about again to the meridian of the place which it left theday before. The force of the sun or moon in raising the sea is greatest inthe appulse of the luminary to the meridian of the place; but the forceimpressed upon the sea at that time continues a little while after the impression, and is afterwards increased by a new though less force still acting upon it. This makes the sea rise higher and higher, till this new forcebecoming too weak to raise it any more, the sea rises to its greatest height.And this will come to pass, perhaps, in one or two hours, but more frequently near the shores in about three hours, or even more, where the seais shallow.The two luminaries excite two motions, wrhich will not appear distinctly,but between them will arise one mixed motion compounded out of both.In the conjunction or opposition of the luminaries their forces will be conjoined, and bring on the greatest flood and ebb. In the quadratures thesun will raise the waters which the moon depresses, and depress the waterswhich the moon raises, and from the difference of their forces the smallestof all tides will follow. And because (as experience tells us) the force ofthe moon is greater than that of the sun, the greatest height of the waterswill happen about the third lunar hour. Out of the syzygies and quadratures, the greatest tide, which by the single force of the moon oujjht to fallout at the third lunar hour, and by the single force of the sun at the thirdsolar hour, by the compounded forces of both must fall out in an interme416 THE MATHEMATICAL PRINCIPLES [BOOK indiate time that aproaches nearer to the third hour of the moon than tcthat of the sun. And, therefore, while the moon is passing from the syzygies to the quadratures, during which time the 3d hour of the sun precedesthe 3d hour of the moon, the greatest height of the waters will also precedethe 3d hour of the moon, and that, by the greatest interval, a little afterthe octants of the moon; and, by like intervals, the greatest tide will follow the 3d lunar hour, while the moon is passing from the quadratures tothe syzygies. Thus it happens in the open sea : for in the mouths ofrivers the ogreater tides come liter to their heiiOrht.But the effects of the luminaries depend upon their distances from theearth ;for when they are less distant, their effects are greater, and whenmore distant, their effects are less, and that in the triplicate proportion oftheir apparent diameter. Therefore it is that the sun, in the winter time,being then in its perigee, has a greater effect, and makes the tides in thesyzygies something greater, and those in the quadratures something lessthan in the summer season ; and every month the moon, while in the perigee, raises greater tides than at the distance of 15 days before or after,when it is in its apogee. Whence it comes to pass that two highesttides do not follow one the other in two immediately succeeding syzygies.The effect of either luminary doth likewise depend upon its declinationor distance from the equator ; for if the luminary was placed at the pole,it would constantly attract all the parts of the waters without any intension or remission of its action, and could cause no reciprocation of motion.And, therefore, as the luminaries decline from the equator towards eitherpole, they will, by degrees, lose their force, and on this account will excitelesser tides in the solstitial than in the equinoctial syzygies. But in thesolstitial quadratures they will raise greater tides than in the quadraturesabout the equinoxes ; because the force of the moon, then situated in theequator, most exceeds the force of the sun. Therefore the greatest tidesfall out in those syzygies, and the least in those quadratures, which happenabout the time of both equinoxes : and the greatest tide in the syzygies is always succeeded by the least tide in the quadratures, as we findby experience. But, because the sun is less distant from the earth inwinter than in summer, it comes to pass that the greatest and least tidesmore frequently appear before than after the vernal equinox, and morefrequently after than before the autumnal.Moreover, the effects of the luminaries depend upon the latitudes ofplaces. Let AjoEP represent theearth covered with deep waters ; Cits centre; P, p its poles; AE theequator ; F any place without theequator ; F/ the parallel of the place ;/F~ M ^ Drl the correspondent parallel on theK 1STBOOK III.] OF NATURAL PHILOSOPHY. 417other side of the equator; L the place of the moon three Lours before;H the place of the earth directly under it; h the opposite place ; K, k theplaces at 90 degrees distance ; CH, Ch, the greatest heights of the seafrom the centre of the earth; and CK, Ck, its least heights: and if withthe axes H//, K/.*, an ellipsis is described, and by the revolution of thatellipsis about its longer axis H/i a spheroid HPKhpk is formed, this spheroid will nearly represent the figure of the sea; and CF, C/, CD, Cd,will represent the heights of the sea in the places F/, Dd. But farther ;in the said revolution of the ellipsis any point N describes the circleNM cutting the parallels F/, Dd, in any places RT, and the equator AEin S : CN will represent the height of the sea in all those places R, S,T, situated in this circle. Wherefore, in the diurnal revolution of anyplace F, the greatest flood will be in F, at the third hour after the appulseof the moon to the meridian above the horizon ; and afterwards the greatest ebb in Q,, at the third hour after the setting of the moon ; and thenthe greatest flood in/, at the third hour after the appulse of the moon tothe meridian under the horizon ; and, lastly, the greatest ebb in Q,, at thethird hour after the rising of the moon ; and the latter flood in / will beless than the preceding flood in F. For the whole sea is divided into twohemispherical floods, one in the hemisphere KH/J on the north side, theother in the opposite hemisphere Khk, which we may therefore call thenorthern and the southern floods. These floods, being always opposite the oneto the other, come by turns to the meridians of all places, after an intervalof 12 lunar hours. And seeing the northern countries partake more ofthe northern flood, and the southern countries more of the southern flood,thence arise tides, alternately greater and less in all places without theequator, in which the luminaries rise and set. But the greatest tide willhappen when the moon declines towards the vertex of the place, about thethird hour after the appulse of the moon to the meridian above the horizon ; and when the moon changes its declination to the other side of theequator, that which was the greater tide will be changed into a lesser.And the greatest difference of the floods will fall out about the times ofthe solstices ; especially if the ascending node of the moon is about theHrst of Aries. So it is found by experience that the morning tides inwinter exceed those of the evening, and the evening tides in summer exceed those of the morning ; at Plymouth by the height of one foot, but atBristol by the height of 15 inches, according to the observations of Colepressand Sturmy.But the motions which we have been describing suffer some alterationfrom that force of reciprocation, which the waters, being once moved, retaina little while by their vis insita. Whence it comes to pass that the tidesmay continue for some time, though the actions of the luminaries should27418 THE MATHEMATICAL PRINCIPLES [BOOK IIIoease. This power of retaining the impressed motion lessens the differenceyf the alternate tides, and makes those tides which immediately succeedafter the syzygies greater, and those which follow next after the quadratures less. And hence it is that the alternate tides at Plymouth andBristol do not differ much more one from the other than by the height ofa foot or 15 inches, and that the greatest tides of all at those ports are notthe first but the third after the syzygies. And, besides, all the motions areretarded in their passage through shallow channels, so that the greatesttides of all, in some straits and mouths of rivers, are the fourth or even thefifth after the syzygies.Farther, it may happen that the tide may be propagated from the oceanthrough different channels towards the same port, and may pass quickerthrough some channels than through others;in which case the same tide,divided into two or more succeeding one another, may compound new motions of different kinds. Let us suppose two equal tides flowing towardsthe same port from different places, the one preceding the other by 6 hours ;and suppose the first tide to happen at the third hour of the appulse of themoon to the meridian of the port. If the moon at the time of the appulseto the meridian was in the equator, every 6 hours alternately there wouldarise equal floods, which, meeting writh as many equal ebbs, would so balance one the other, that for that day, the water would stagnate and remainquiet. If the moon then declined from the equator, the tides in the oceanwould be alternately greater and less, as was said ; and from thence twogreater and two lesser tides wrould be alternately propagated towards thatport. But the two greater floods would make the greatest height of thewaters to fall out in the middle time betwixt both ; and the greater andlesser floods would make the waters to rise to a mean height in the middletime between them, and in the middle time between the two lesser floods thewaters would rise to their least height. Thus in the space of 24 hours thewaters would come, not twice, as commonly, but once only to their greatest, and once only to their least height ; and their greatest height, if themoon declined towards the elevated pole, would happen at the 6th or 30thhour after the appulse of the moon to the meridian ; and when the moonchanged its declination, this flood would be changed into an ebb. An example of all which Dr. Halley has given us, from the observations of seamen in the port of Bntshnm, in the kingdom of Tunqvin, in the latitudeof 20 50 north. In that port, on the day which follows after the passageof the moon over the equator, the waters stagnate: when the moon declinesto the north, they begin to flow and ebb. not twice, as in other ports, butonce only every day : and the flood happens at the setting, and the greatestebb at the rising of the moon. This tide increases with the declination ofthe moon till the ?th or 8th day ; then for the 7 or 8 days following itBOOK III.] OF NATURAL PHILOSOPHY. 419decreases at the same rate as it had increased before, and ceases when themoon changes its declination, crossing over the equator to the south. After which the flood is immediately changed into an ebb; and thenceforththe ebb happens at the setting and the flood at the rising of the moon : tillthe moon, again passing the equator, changes its declination. There aretwo inlets to this port and the neighboring channels, one from the seas ofChina, between the continent and the island of Lenconia ; the other fromthe Indian sea, between the continent and the island of Borneo. Butwhether there be really two tides propagated through the said channels, onefrom the Indian sea in the space of 12 hours, and one from the sea ofCliina in the space of 6 hours, which therefore happening at the 3d and9th lunar hours, by being compounded together, produce those motions : orwhether there be any other circumstances in the state of those seas. I leaveto be determined by observations on the neighbouring shores.Thus I have explained the causes of the motions of the moon and of thesea. Now it is fit to subjoin something concerning the quantity of thosemotions.PROPOSITION XXV. PROBLEM VI.To find the forces with which the sun disturbs the motions of the moon.Let S represent the sun, T theearth, P the moon, CADB themoon s orbit. In SP take SKequal to ST; and let SL be toSK in the duplicate proportionof SK to SP: draw LM parallelto PT ; and if ST or SK is supposedto represent the accelerated force of gravity of the earth towards thesun, SL will represent the accelerative force of gravity of the moon towardsthe sun. But that force is compounded of the parts SM and LM, of whichthe force LM, and that part of SM which is represented by TM, disturbthe motion of the moon, as we have shewn in Prop. LXVI, Book I, andits Corollaries. Forasmuch as the earth and moon are revolved abouttheir common centre of gravity, the motion of the earth about that centrewill be also disturbed by the like forces; but we may consider the sumsboth of the forces and of the motions as in the moon, and represent the sumof the forces by the lines TM and ML, which are analogous to them both.The force ML (in its mean quantity) is to the centripetal force by whichthe moon may be retained in its orbit revolving about the earth at rest, atthe distance P J, in the duplicate proportion of the periodic time of themoon about the earth to the periodic time of the earth about the sun (byCor. 17, Prop. LXVI, Book I) ; that is, in the duplicate proportion of 27 d.7。 43 to 365 1. 6". 9 ; or as 1000 to 178725 ; or as 1 to 178f J. But in theJ23 THE MATHEMATICAL PRINCIPLES [BOOK 1114ih Prop, of this Book we found, that, if both earth and moon were revolvedaoout their common centre of gravity, the mean distance of the one fromthe other would be nearly 60^ mean semi-diameters of the earth : and theforce by which the moon may be kept revolving in its orbit about the earthin rest at the distance PT of 60^ semi-diameters of the earth, is to theforce by which it may be revolved in the same time, at the distance of 60semi-diameters, as 60| to 60 : and this force is to the force of gravity withu,;3 very nearly as I to 60 X 60. Therefore the mean force ML is to theforce of gravity on the surface of our earth as 1 X 60-} to 60 X 60 X 60X l~8f, or as 1 to 638092,6 : whence by the proportion of the lines TM,ML, the force TM is also given; and these are the forces with which thesun disturbs the motions of the moon. Q.E.I.PROPOSITION XXVI. PROBLEM VII.To find the horary increment of the area which the moon, by a radius