Principia Mathematica (Motte Translation, 1848) — part 31 of 45

dicular thereto. Our countryman, Mr. Norwood, measuring a distance of 905751 feet of London measure between London and York, in 1635, and observing the diiference of latitudes to be 2^ 28' determined the measure of one deo-ree to be 367196 feet of Lmidon measure, that is 57300 Paris toises. M. Picart, measuring an arc of one degree, and 22' 55" of the meridian be- tween Amiens and Malvoisine, found an arc of one degree to be 57060 Paris toises. M. Cassini, the father, measured the distance upon the me- ridian from the town of Collioure in Roussillon to the Observatory of Paris ; and his son added the distance from the Observatory to the Cita- del of Dunkirk. The whole distance was 486156^ toises and the difter- ence of the latitudes of Collioure and Dunkirk was 8 degrees, and 3P

406 THE MATHEMATICAL PRINCIPLES [BoOK III.

llf". Hence an arc of one degree appears to be 57061 Paris toises. And from these measures we conclude that the circumference of the earth is 123249600, and its semi-diameter 19615800 Paris feet, upon the sup- position that the earth is of a spherical figure.

In the latitude of Paris a heavy body falling in a second of time de- scribes 15 Paris feet, 1 inch, 1^ line, as above, that is, 2173 lines |. The weight of the body is diminished by the weight of the ambient air. Let us suppose the ^Yeight lost thereby to be tt¥o o P^^* of t^^ whole weight ; then that heavy body falling in vacuo will describe a height of 2174 lines in one second of time.

A body in every sidereal day of 23^ 56' 4" uniformly revolving in a circle at the distance of 19615S00 feet from the centre, in one second of time describes an arc of 1433,46 feet ; the versed sine of which is 0,05236561 feet, or 7,54064 lines. And therefore the force with which bodies descend in the latitude of Paris is to the centrifugal force of bodies in the equator arising from the diurnal motion of the earth as 2174 to 7,54064.

The centrifugal force of bodies in the equator is to the centrifugal force with which bodies recede directly from the earth in the latitude of Paris 48° 50' 10" in the duplicate proportion of the radius to the cosine of the latitude, that is, as 7,54064 to 3,267. Add this force to the force with which bodies descend by their weight in tlie latitude of Paris, and a body, in the latitude of Paris, falling by its whole undiminished force of gravity, in the time of one second, will describe 2177,267 lines, or 15 Paris feet, 1 inch, and 5,267 lines. And the total force of gravity in that latitude will be to the centrifugal force of bodies in the equator of the earth as 2177,267 to 7,54064, or as 289 to 1.

j^:j2^ Wherefore if APBQ, represent the figure of the

earth, now no longer spherical, but generated by the rotation of an ellipsis about its lesser axis PQ,; and iQ ACQiqca a canal full of water, reaching from the pole Q,q to the centre Cc, and thence rising to the equator Art ; the weight of the water in the leg of the canal "b ACcrt will be to the weight of water in the other leg

QSucq as 289 to 288, because the centrifugal force arising from the circu- lar motion sustains and takes off one of the 289 parts of the weight (in the one leg), and the weight of 288 in the other sustains the rest. But by computation (from Cor. 2, Prop. XCI, Book I) I find, that, if the matter of the earth was ail uniform, and without any motion, and its axis PQ, were to the diameter AB as 100 to 101, the force of gravity in the place Q, towards the earth would be to the force of gravity in the same place Q. towards a sphere described about the centre C with the radius PC, or QO, as 126 to 125. And, by the same argument, the force of gravity in the place A towards the spheroid generated by the rotation of

Book III.] of natural philosophy. 407

the ellipsis APBQ, about the axis AB is to tLe force of gravity in the same place A, towards the sphere described about the centre C with the radius AC, as 125 to 126. But the force of gravity in the place A to- wards the earth is a mean proportional betwixt the forces of gravity to- wards the spheroid and this sphere ; because the sphere, by having its di- ameter PQ. diminished in the proportion of 101 to 100, is transformed into the figure of the earth ; and this figure, by having a third diameter per- pendicular to the two diameters AB and PQ, diminished in the same pro- portion, is converted into the said spheroid ; and the force of gravity in A, in either case, is diminished nearly in the same proportion. Therefore the force of gravity in A towards the sphere described about the centre C with the radius AC, is to the force of gravity in A towards the earth as 126 to 125|. And the force of gravity in the place Q, towards the sphere de- scribed about the centre C with the radius Q,C, is to the force of gravity in the place A towards the sphere described about the centre 0, with the radius AC, in the proportion of the diameters (by Prop. LXXII, Book I), that is, as 100 to 101. If, therefore, we compound those three proportions 126 to 125, 126 to 125^, and 100 to 101, into one, the force of gravity in the place Q, towards the earth will be to the force of gravity in the place A towards the earth as 126 X 126 X 100 to 125 X 125^ X 101 ; or as 501 to 500.

Now^ since (by Cor. 3, Prop. XCI, Book I) the force of gravity in either leg of the canal ACca, or QCcq, is as the distance of the places from the centre of the earth, if those legs are conceived to be divided by transverse, parallel, and equidistant surfaces, into parts proportional to the wholes, the weights of any number of parts in the one leg ACca will be to the weights of the same number of parts in the other leg as their magnitudes and the accelerative forces of their gravity conjunctly, that is, as 101 to 100, and 500 to 501, or as 505 to 501. And therefore if the centrifugal force of every part in the leg ACca, arising from the diurnal motion, was to the weight of the same part as 4 to 505, so that from the weight of every part, conceived to be divided into 505 parts, the centrifugal force might take off four of those parts, the weights would remain Cv^ual in each leg, and therefore the fluid would rest in an equilibrium. But the centri- fugal force of every part is to the weight of the same part as 1 to 289 ; that is, the centrifugal force, which should be ^ ^^ parts of the weight, is only 2I9 part thereof. And, therefore, I say, by the rule of proportion, that if the centrifugal force j^j make the height of the water in the leg ACca to exceed the height of the water in the leg GiCcq by one yi^- part of its whole height, the centrifugal force ^}g will make the excess of the height in the leg ACca only 2 19 part of the height of the water in the other leg Q,Cc^ ; and therefore the diameter of the earth at the equator, is to its diameter from pole to pole as 230 to 229. And since the mean semi-

408

THE MATHEMATICAL PRINCIPLES

[Book IIL

diameter of the earth, according to Picarfs mensuratioiij is 19615S00 Paris feet, or 3923,16 miles (reckoning 5000 feet to a mile), the earth will be higher at the equator than at the poles by 85472 feet, or 17y'^ miles. And its height at the equator will be about 19658600 feet, and at the poles 19573000 feet.

If, the density and periodic time of the diurnal revolution remaining the same, the planet was greater or less than the earth, the proportion of the centrifugal force to that of gravity, and therefore also of the diameter be- twixt the poles to the diameter at the equator, would likewise remain the same. But if the diurnal motion was accelerated or retarded in any pro- portion, the centrifugal force would be augmented or diminished nearly in the same duplicate proportion ; and therefore the difference of the diame- ters will be increased or diminished in the same duplicate ratio very nearly. And if the density of the planet was augmented or diminished in any pro- portion, the force of gravity tending towards it would also be augmented or diminished in the same proportion : and the difference of the diameters contrariwise would be diminished in proportion as the force of gravity is augmented, and augmented in proportion as the force of gravity is dimin- ished. Wherefore, since the earth, in respect of the fixed stars, revolves in 23^ 56', but Jupiter in 9*". 56', and the squares of their periodic times are as 29 to 5, and tlieir densities as 400 to 94|, the difference of the diameters

of Jupiter will be to its lesser diameter as -^ X fttt X ^^^ to 1, or as 1 to ^ o 94i 229

9}, nearly. Therefore the diameter of Jupiter from east to west is to its

diameter from pole to pole nearly as 10^ to 9|-. Therefore since its

greatest diameter is 37", its lesser diameter lying between the poles will

be 33" 25'". Add thereto about 3" for the irregular refraction of light,

and the apparent diameters of this planet will become 40" and 36" 25'" •

which are to each other as 11 1 to 10 J-, very nearl}^ These things are so

upon the supposition that the body of Jupiter is uniformly dense. But

now if its body be denser towards the plane of the equator than towards

the poles, its diameters may be to each other as 12 to 11, or 13 to 12, or

perhaps as 14 to 13.

And Cassini observed in the year 1691, that the diameter of Jupiter

reaching from east to west is greater by about a fifteenth part than the

other diameter. Mr. Pound with his 123 feet telescope, and an excellent

micrometer, measured the diameters of Jupiter in the year 1719, and found

them as follow.

The Times.

Greatest diam.

Lesser diam.

The diam. to each other.

Day. Hours.

Parts.

Parts.

January 28

6

13,40

12,28

As 12 to 11

March 6

7

13,12

12.20

13| to 12|

March 9

7

13,12

12,08

121 to 111

April 9

9

12,32

11,48

Uh to 13i

Book III.] of natural philosophy. 409

So that the theory agrees with the phsenomena ; for the planets are more heated by the sun's rays towards their equators, and therefore are a little more condensed by that heat than towards their poles.

Moreover, that there is a diminution of gravity occasioned by the diur- nal rotation of the earth, and therefore the earth rises higher there than it does at the poles (supposing that its matter is uniformly dense), will ap- pear by the experiments of pendulums related under the following Propo- sition.

PROPOSITION XX. PROBLEM IV.

To find and compare together the weights of bodies in the different re- gions of our earth.

Because the weights of the unequal legs of the canal ^^i^

of water ACQqca are equal: and the weights of the parts proportional to the whole legs, and alike situated in them, are one to anotlier as the weights of the Pj wholes, and therefore equal betwixt themselves ; the weights of equal parts, and alike situated in the legs, will be reciprocally as the legs, that is, reciprocally as ^S"

230 to 229. And the case is the same in all homogeneous equal bodies alike situated in the legs of the canal. Their weights are reciprocally as the legs, that is, reciprocally as the distances of the bodies from the centre of the earth. Therefore if the bodies are situated in the uppermost parts of the canals, or on the surface of the earth, their weights will be one to another reciprocally as their distances from the centre. And. by the same argument, the weights in all other places round the whole surface of the earth are reciprocally as the distances of the places from the centre ; and, therefore, in the hypothesis of the earth's being a spheroid are given in proportion.

Whence arises this Theorem, that the increase of weight in passing from the' equator to the poles is nearly as the versed sine of double the latitude; or, which comes to the same thing, as the square of the right sine of the latitude; and the arcs of the degrees of latitude in the meridian increase nearly in the same proportion. And, therefore, since the latitude of Paris is 48^ 50', that of places under the equator 00° 00', and that of places under the poles 90° ; and the versed sines of double those arcs are 11334,00000 and 20000, the radius being 10000 ; and the force of gravity at the pole is to the force of gravity at the equator as 230 to 229 ; and the excess of the force of gravity at the pole to the force of gravity at the equator as 1 to 229 ; the excess of the force of gravity in the latitude of Paris will be to the force of gravity at the equator asl X iUl^to 229, or as 5667 to 2290000. And therefore the whole forces of gravity in those places will be one to the other as 2295667 to 2290000. Wherefore since the lengths of pendulums vibrating in equal times are as the forces of

410

THE MATHEMATICAL PRINCIPLES

[Book III.

gravity, and in the latitude of Paris, the length of a pendulum vibrating seconds is 3 Paris feet, and S\ lines, or rather because of the weight of the air, 8f lines, the length of a pendulum vibrating in the same time under the equator will be shorter by 1,087 lines. And by a like calculus the following table is made.

Latitude of

Length of the

Measure of one degree

the place.

pendulum

in the meridian.

Deg.

Feet. Lines.

Toises.

0

3 . 7,468

56637

5

3 . 7^482

56642

10

3 . 7,526

56659

15

3 . 7.596

56687

20

3 . 7^692

56724

25

3 . 7,812

56769

30

3 . 7,948

56823

35

3 . 8.099

56882

40

3 . 8,261

56945

1

3 . 8,294

56958

2

3 . 8,327

56971

3

3 . 8,361

56984

4

3 . 8,394

56997

45

3 . 8,428

57010

6

3 . 8,461

57022

7

3 . 8,494

57035

8

3 . 8,528

57048

9

3 . 8,561

57061

50

3 . 8,594

57074

55

3 . 8,756

57137

60

3 . 8,907

57196

65

3 . 9,044

57250

70

3 . 9,162

57295

75

3 . 9,258

57332

80

3 . 9,329

57360

85

3 . 9,372

57377

90

3 . 9,387

57382

By this table, therefore, it appears that the inequality of degrees is so small, that the figure of the earth, in geographical matters, may be con- sidered as spherical ; especially if the earth be a little denser towards the plane of the equator than towards the poles.

Now several astronomers, sent into remote countries to make astronomical observations, have found that pendulum clocks do accordingly move slower near the equator than in our climates. And, first of all, in the year 1672, M. Richer took notice of it in the island of Cayenne ; for when, in the month of August, he was observing the transits of the fixed stars over the meridian, he found his clock to go slower than it ought in respect of the mean motion of the sun at the rate of 2' 28" a day. Therefore, fitting up a simple pendulum to vibrate in seconds, which were measured by an ex- cellent clock, he observed the length of that simple pendulum ; and this he did over and over every week for ten months together. And upon his re- turn to France, comparing the length of that pendulum with the length

Book III.] of natural philosophv. 411

of the pendulum at Paris (which was 3 Paris feet and 8| lines), he found it shorter by 1{ line.

Afterwards, our friend Dr. Halley^ about the year 1677, arriving at the island of St. Helena, found his pendulum clock to go slower there than at London, without marking the difference. But he shortened the rod of his clock by more than the | of an inch, or 1^ line ; and to effect this, be- cause the length of the screw at the lower end of the rod was not sufficient, he interposed a wooden ring betwixt the nut and the ball.

Then, in the year 16S2, M. Varin and M. des Haijes found the length of a simple pendulum vibrating in seconds at the Royal Observatory of Paris to be 3 feet and 8f lines. And by the same method in the island of Goree, they found the length of an isochronal pendulum to be 3 feet and 6f lines, differing from the former by two lines. And in the same year, going to the islands of Giiadalowpe and Martinico, they found that the length of an isochronal pendulum in those islands was 3 feet and 6|- lines.

After this, M. Couplet, the son, in the month of July 1697, at the Royal Observatory of Paris, so fitted his pendulum clock to the mean motion of the sun, that for a considerable time too^ether the clock ao^reed with the motion of the sun. In November following, upon his arrival at Lisbon, he found his clock to go slower than before at the rate of 2' 13" in 24 hours. And next March coming to Paraiba, he found his clock to go slower than at Paris, and at the rate 4' 12" in 24 hours ; and he affirms, that the pen- dulum vibrating in seconds was shorter at Lisbon by 2\ lines, and at Pa- raiba by 3| lines, than at Paris. He had done better to have reckoned those differences ^ and 2f ; for these differences correspond to the differ- ences of the times 2' 13" and 4' 12". But this gentleman's observations are so gross, that we cannot confide in them.

In the following years, 1699, and 1700, M, des Hayes, making another voyage to America, determined that in the island of Cayenne and Granada the length of the pendulum vibrating in seconds was a small matter less than 3 feet and 6| lines ; that in the island of St. Christophers it was 3 feet and 6f lines ; and in the island of St. Domingo 3 feet and 7 lines.

And in the year 1704, P. Feuille, at Puerto Bello in America, found that the length of the pendulum vibrating in seconds was 3 Paris feet, and only 5y 2 lines, that is, almost 3 lines shorter than at Paris ; but the observation was faulty. For afterward, going to the island of Martinico, he found the length of the isochronal pendulum there 3 Paris feet and 5 II lines.

Now the latitude of Paraiba is 6° 38' south ; that of Puerto Bello 9° 83' north; and the latitudes of the islands Cayenne, Goree, Gaudaloupe, Martinico, Granada, St. Christophers, and St. Domingo, are respectively 4° ^6', 14° 40", 15° 00', 14° 44', 12° 06', 17° 19', and 19° 48', north. And

412 THE MATHEMATICAL PRINCIPLES [BoOK III.

the excesses of the length of the pendulum at Paris above the lengths of the isochronal pendulums observed in those latitudes are a little greater than by the table of the lengths of the pendulum before computed. And therefore the earth is a little higher under the equator than by the prece- ding calculus, and a little denser at the centre than in mines near the sur- face, unless, perhaps, the heats of the torrid zone have a little extended the length of the pendulums.

For M. Picart has observed, that a rod of iron, which in frosty weather in the winter season was one foot long, when heated by fire, was lengthened into one foot and -} line. Afterward M. de la Hire found that a rod of iron, which in the like winter season was 6 feet long, when exposed to the heat of the summer sun, was extended into 6 feet and | line. In the former case the heat was greater than in the latter ; but in the latter it was greater than the heat of the external parts of a human body ; for metals exposed to the summer sun acquire a very considerable degree of heat. But the rod of a pendulum clock is never exposed to the heat of the summer sun, nor ever 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 a little longer in the summer than in the winter season, yet the difference will scarcely amount to \ line. Therefore the total difference of the lengths of isochronal pendulums in diiferent climates cannot be ascribed to the differ- ence of heat ; nor indeed to the mistakes of the French astronomers. For although there is not a perfect agreement betwixt their observations, yet the errors are so small that they may be neglected ; and in this they all agree, that isochronal pendulums are shorter under the equator than at the Royal Observatory of Paris, by a diiference not less than 1^ line, nor greater than 2f lines. By the observations of M. Richer^ in the island of Cayenne, the difference was 1^ line. That difference being corrected by those of M. des Hayes, becomes 1\ line or If line. By the less accurate observations of others, the same was made about two lines. And this dis- agreement might arise partly from the errors of the observations, partly from the dissimilitude of the internal parts of the earth, and the height of mountains ; 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 line in winter time with us here in England than in the summer. Because of the great heats under the equator, subduct this quantity from the difference of one line and a quarter observed by M. Richer, and there will remain one line yV, which agrees very well with ly-oFo li^^ collected, by the theory a little before. M. Richer repeated his observations, made in the island of Cayenne, every week for ten months together, and compared the lengths of the pendulum which he had there noted in the iron rods with the lengths thereof which he observed in France.^ This diligence and care seems to have been wanting to the other observers. If this gentleman's observations

Book IIL] op natural philosophy. 413

are to be depended on, the earth is higher under the equator than at the poles, and that by an excess of about 17 miles ; as appeared above by the theory.

PROPOSITION XXI. THEOREM XVII.

That the equinoctial points go backioard, and that the axis of the earth, by a nutation in every annual revolution, tioice vibrates towards the

ecliptic, and as often i^eturns to its former position. The proposition appears from Cor. 20, Prop. LXVI, Book I ; but that motion of nutation must be very small, and, indeed, scarcely per- ceptible.

PROPOSITION XXII. THEOREM XVIIl. .

That all the 'motions of the moon, and all the inequalities of those motions^ follow from the principles which lue have laid down. That the greater planets, while they are carried about the sun, may in the mean time carry other lesser planets, revolving about them ; and that those lesser planets must move in ellipses which have their foci in the cen- tres of the greater, appears from Prop. LXV, Book I. But then their mo- tions will be several ways disturbed by the action of the sun, and they will suffer such inequalities as are observed in our moon. Thus our moon (by Cor. 2, 3, 4, and 5, Prop. LXVI, Book I) moves faster, and, by a radius drawn to the earth, describes an area greater for the time, and has its orbit less curved, and therefore approaches nearer to the earth in the syzygies than in the quadratures, excepting in so far as these effects are hindered by the motion of eccentricity ; for (by Cor. 9, Prop. LXVI, Book I) the eccen- tricity is greatest when the apogeon of the moon is in the syzygies, and least when the same is in the quadratures ; and upon this account the pe- rigeou moon is swifter, and nearer to us, but the apogeon moon slower, and farther from us, in the syzygies than in the quadratures. Moreover, the apogee goes forward, and the nodes backward ; and this is done not with a regular but an unequal motion. For (by Cor. 7 and S, Prop. LXVI, Book I) the apogee goes more swiftly forward in its syzygies, more slowly backward in its quadratures ; and, by the excess of its progress above its regress, advances yearly m consequentia. But, contrariwise, the nodes (by Cor. 11, Prop. LXVI, Book I) are quiescent in their syzygies, and go fastest back in their quadratures. Farther, the greatest latitude of the moon (by Cor. 10, Prop. LXVI, Book I) is greater in the quadratures of the moon than in its syzygies. And (by Cor. 6, Prop. LXVI, Book I) the mean mo^ tion 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 by astronomers.

414 THE MATHEMATICAL PRINCIPLES [BoOK III.

But there are yet other inequalities not observed by former astronomers, by which the motions of the moon are so disturbed, that to this day we have not been able to bring them under any certain rule. For the veloc- ities or horary motions of the apogee and nodes of the moon, and their equations, as well as the difference betwixt the greatest eccentricity in the syzygies, and the least eccentricity in the quadratures, and that inequality w^hich we call the variation, are (by Cor. 14, Prop. LXYI, Book I) in the course of the year augmented and diminished in the triplicate proportion of the sun's apparent diameter. And besides (by Cor. 1 and 2, Lem. 10, and Cor. 16, Prop. LXVl, Book I) the variation is augmented and diminished nearly in the duplicate proportion of the time between the quadratures. But in astronomical calculations, this inequality is commonly thrown into and confounded with the equation of the moon's centre.

PROPOSITION XXIII. PROBLEM Y.

To derive tlie unequal motions of the satellites of Jupiter and Saturn from the m,otions of our moo?i. From the motions of our moon we deduce the corresponding motions of the 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 Jupiter is to the mean motion of the nodes of our moon in a proportion compound- ed of the duplicate proportion of the periodic times of the earth about the sun to the periodic times of Jupiter about the sun, and the simple propor- tion of the periodic time of the satellite about Jupiter to the periodic time of our moon about the earth ; and, therefore, those nodes, in the space of a hundred years, are carried 8° 24' backward, or iji antecedentia. The mean motions of the nodes of the inner satellites are to the mean motion of the nodes of the outmost as their periodic times to the periodic time of the former, by the same Corollary, and are thence given. And the motion of the apsis of every satellite in coiisequentia is to the motion of its nodes in antecedentia as the motion of the apogee of our moon to the motion of its nodes (by the same Corollary), and is thence given. But the motions of the apsides thus found must be diminished in the proportion of 5 to 9, or of about 1 to 2, on account of a cause which I cannot here descend to ex- plain. The greatest equations of the nodes, and of the apsis of every satel- lite, are to the greatest equations of the nodes, and apogee of our moon re- spectively, as the motions of the nodes and apsides of the satellites, in the time of one revolution of the former equations, to the motions of the nodes and apogee of our moon, in the time of one revolution of the latter equa- tions. The variation of a satellite seen from Jupiter is to the variation of our moon in the same proportion as the whole motions of their nodes

Book III.] of natural philosophy. 415

respectively during the times in which the satellite and our moon (after parting from) are revolved (again) to the sun, by the same Corollary ; and therefore in the outmost satellite the variation does not exceed 5" 12'".

PROPOSITION XXIV. THEOREM XIX.

That the flux and reflux of the sea arise from the actions of the sun

and moon.

By Cor. 19 and 20, Prop. LXVI, Book I, it appears that the waters of the 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 ought to follow the appulse of the luminaries to the meridian of the place by a less interval than 6 hours ; as happens in all that eastern tract of the Atlantic and jEthiopic seas between France and the Cape of Good Hope ; and on the coasts of Chili and Peru in the South Sea ; in all which shores the flood falls out about the second, third, or fourth hour, unless where the motion propagated from the deep ocean is by the shallowness of the chan- nels, through which it passes to some particular places, retarded to the fifth, sixth, or seventh hour, and even later. The hours I reckon from the appulse of each luminary to the meridian of the place, as well under as above the horizon ; and by the hours of the lunar day I understand the 24th parts of that time which the moon, by its apparent diurnal motion, employs to come about again to the meridian of the place which it left the day before. The force of the sun or moon in raising the sea is greatest in the appulse of the luminary to the meridian of the place ; but the force impressed upon the sea at that time continues a little while after the im- pression, and is afterwards increased by a new though less force still act- ing upon it. This makes tlie sea rise higher and higher, till this new force becoming 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 fre- quently near the shores in about three hours, or even more, where the sea is shallow.

The two luminaries excite two motions, which 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 con- joined, and bring on the greatest flood and ebb. In the quadratures the sun will raise the waters which the moon depresses, and depress the waters which the moon raises, and from the diflerence of their forces the smallest of all tides will follow. And because (as experience tells us) the force of the moon is greater than that of the sun, the greatest height of the waters will happen about the third lunar hour. Out of the syzygies and quadra- tures, the greatest tide, which by the single force of the moon ought to fall out at the third lunar hour, and by the single force of the sun at the third solar hour, by the compounded forces of both must fall' out in an interme-

416

THE MATHEMATICAL PRINCIPLES

[Book III.

diate time that aproaclies nearer to the third hour of the moon than to that of the sun. And, therefore, while the moon is passing from the syzy- gies to the quadratures, during which time the 3d hour of the sun precedes the 3d hour of the moon, the greatest height of the waters will also precede the 3d hour of the moon, and that, by the greatest interval, a little after the octants of the moon ; and, by like intervals, the greatest tide will fol- low the 3d lunar hour, while the moon is passing from the quadratures to the syzygies. Thus it happens in the open sea ; for in the mouths of rivers the o-reater tides come later to their heio;ht.

But the effects of the luminaries depend upon their distances from the earth ; for when they are less distant, their effects are greater, and when more distant, their effects are less, and that in the triplicate proportion oi their 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 the syzygies something greater, and those in the quadratures something less than in the summer season ; and every month the moon, while in the peri- gee, 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 highest tides do not follow one the other in two immediately succeeding syzygies.

The effect of either luminary doth likewise depend upon its declination or 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 inten- sion or remission of its action, and could cause no reciprocation of motion. And, therefore, as the luminaries decline from the equator towards either pole, they will, by degrees, lose their force, and on this account will excite lesser tides in the solstitial than in the equinoctial syzygies. But in the solstitial quadratures they will raise greater tides than in the quadratures about the equinoxes ; because the force of the moon, then situated in the equator, most exceeds the force of the sun. Therefore the greatest tides fall out in those syzygies, and the least in those quadratures, which hap- pen about the time of both equinoxes : and the greatest tide in the syzy- gies is always succeeded by the least tide in the quadratures, as we find by experience. But, because the sun is less distant from the earth in winter than in summer, it comes to pass that- the greatest and least tides more frequently appear before than after the vernal equinox, and more frequently after than before the autumnal.

K N

Moreover, the effects of the lumi- naries depend upon the latitudes of places. Let A^EP represent the earth covered with deep waters ; C its centre; P, jo its poles; AE the equator ; F any place without the equator ; F/ the parallel of the place ; Bd the correspondent parallel on the

Book III.] of natural philosophy. 417