KWIC Index

lest the systems of the fixed stars should, by their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another.

This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκράτωρ, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without

their gravity, fall on each other mutually, he hath placed those systems at immense distances one from another.

This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκράτωρ, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without

distances one from another.

This Being governs all things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκράτωρ, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel,

things, not as the soul of the world, but as Lord over all; and on account of his dominion he is wont to be called Lord God παντοκράτωρ, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say,

on account of his dominion he is wont to be called Lord God παντοκράτωρ, or Universal Ruler; for God is a relative word, and has a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of

a respect to servants; and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually

and Deity is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but

is the dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is

dominion of God not over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a

over his own body, as those imagine who fancy God to be the soul of the world, but over servants. The Supreme God is a Being eternal, infinite, absolutely perfect; but a being, however perfect, without dominion, cannot be said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a God. It is the

said to be Lord God; for we say, my God, your God, the God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful

God of Israel, the God of Gods, and Lord of Lords; but we do not say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is

say, my Eternal, your Eternal, the Eternal of Israel, the Eternal of Gods; we do not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and

not say, my Infinite, or my Perfect: these are titles which have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence

have no respect to servants. The word God[1] usually signifies Lord; but every lord is not a God. It is the dominion of a spiritual being which constitutes a God: a true, supreme, or imaginary dominion makes a true, supreme, or imaginary God. And from his true dominion it follows that the true God is a living, intelligent, and powerful Being; and, from his other perfections, that he is supreme, or most perfect. He is eternal and infinite, omnipotent and omniscient; that is, his duration reaches from eternity to eternity; his presence from infinity to infinity; he governs all things, and

in different organs of sense and motion, still the same indivisible person. There are given successive parts in duration, co-existent parts in space, but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In

but neither the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him[2] are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance

the one nor the other in the person of a man, or his thinking principle; and much less can they be found in the thinking substance of God. Every man, so far as he is a thing that has perception, is one and the same man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him[2] are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the

man during his whole life, in all and each of his organs of sense. God is the same God, always and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him[2] are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he exists always and every where. Whence also he is all similar, all eye, all ear, all brain, all arm, all

and every where. He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without substance. In him[2] are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he exists always and every where. Whence also he is all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all

only, but also substantially; for virtue cannot subsist without substance. In him[2] are all things contained and moved; yet neither affects the other: God suffers nothing from the motion of bodies; bodies find no resistance from the omnipresence of God. It is allowed by all that the Supreme God exists necessarily; and by the same necessity he exists always and every where. Whence also he is all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly unknown to

all similar, all eye, all ear, all brain, all arm, all power to perceive, to understand, and to act; but in a manner not at all human, in a manner not at all corporeal, in a manner utterly unknown to us. As a blind man has no idea of colours, so have we no idea of the manner by which the all-wise God perceives and understands all things. He is utterly void of all body and bodily figure, and can therefore neither be seen, nor heard, nor touched; nor ought he to be worshipped under the representation of any corporeal thing. We have ideas of his attributes, but what the real substance of any

their figures and colours, we hear only the sounds, we touch only their outward surfaces, we smell only the smells, and taste the savours; but their inward substances are not to be known either by our senses, or by any reflex act of our minds: much less, then, have we any idea of the substance of God. We know him only by his most wise and excellent contrivances of things, and final causes: we admire him for his perfections; but we reverence and adore him on account of his dominion: for we adore him as his servants; and a god without dominion, providence, and final causes, is nothing else but

necessity, which is certainly the same always and every where, could produce no variety of things. All that diversity of natural things which we find suited to different times and places could arise from nothing but the ideas and will of a Being necessarily existing. But, by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And

and places could arise from nothing but the ideas and will of a Being necessarily existing. But, by way of allegory, God is said to see, to speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy.

Hitherto we have explained the phenomena of the

speak, to laugh, to love, to hate, to desire, to give, to receive, to rejoice, to be angry, to fight, to frame, to work, to build; for all our notions of God are taken from the ways of mankind by a certain similitude, which, though not perfect, has some likeness, however. And thus much concerning God; to discourse of whom from the appearances of things, does certainly belong to Natural Philosophy.

Hitherto we have explained the phenomena of the heavens and of our sea by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause

2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped; but erroneously.





THE SYSTEM OF THE WORLD.





* * *





It was the ancient opinion of not a few, in the earliest ages of philosophy, that the fixed stars stood immoveable in the highest parts of the world; that under the fixed stars the planets were carried

and those heavy fluids, quick silver and the spirit of vitriol, gently evaporate, as I have tried by the thermometer; and therefore there can be no fluids in Mercury but what are heavy, and able to bear a great heat, and from which substances of great density may be nourished.

And why not, if God has placed different bodies at different distances from the sun, so as the denser bodies always possess the nearer places, and each body enjoys a degree of heat suitable to its condition, and proper for its nourishment? From this consideration it will best appear that the weights of all the

centre of force will cause a body to move in any given conic section, 114

“ a centripetal force that is as the cube of the ordinate tending to a vastly remote centre of force will cause a body to move in an hyperbola, 243

“ centrifugal force of bodies on the earth s equator, how great, 405

God, his nature, 506

Gravity mutual between the earth and its parts, 94

“ of a different nature from magnetical force, 397

“ the cause of it not assigned, 507

“ tends towards all the planets, 393

“ from the surfaces of the planets upwards decreases in the duplicate ratio of the distances from the

each other; which forces being unknown, philosophers have hitherto attempted the search of nature in vain; but I hope the principles here laid down will afford some light either to this or some truer method of philosophy.

In the publication of this work the most acute and universally learned Mr. Edmund Halley not only assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing; for when he had obtained of me my demonstrations of the figure of the celestial orbits, he continually pressed me to communicate the same

retained in its orbit by the force of gravity is enlarged on; and there are added new observations of Mr. Pound's of the proportion of the diameters of Jupiter to each other: there are, besides, added Mr. Kirk's observations of the comet in 1680; the orbit of that comet computed in an ellipsis by Dr. Halley; and the orbit of the comet in 1723, computed by Mr. Bradley.





BOOK I.





THE MATHEMATICAL PRINCIPLES

OF

NATURAL PHILOSOPHY.





* * *





DEFINITIONS.





DEFINITION I.

The quantity of matter is the measure of the same, arising from its density and bulk conjunctly.

Thus air of a

time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would descend through in the same given time.





SCHOLIUM.



The case of the 6th Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed); and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres.

Moreover, by means of the preceding Proposition and its Corollaries, we may discover the

progression, the densities AH, BI, CK, &c., will be in geometrical progression. And so in infinitum. Again; if the gravity of the particles of the fluid be the same at all distances, and the distances be in arithmetical progression, the densities will be in a geometrical progression as Dr. Halley has found. If the gravity be as the distance, and the squares of the distances be in arithmetical progression, the densities will be in geometrical progression. And so in infinitum. These things will be so, when the density of the fluid condensed by compression is as the force of compression; or,

of that simple pendulum; and this he did over and over every week for ten months together. And upon his return to France, comparing the length of that pendulum with the length of the pendulum at Paris (which was 3 Paris feet and 83⁄5 lines), he found it shorter by 1¼ 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 1⁄8 of an inch, or 1½ line; and to effect this, be cause the length of the screw at the

only to their least height; and their greatest height, if the moon declined towards the elevated pole, would happen at the 6th or 30th hour after the appulse of the moon to the meridian; and when the moon changed its declination, this flood would be changed into an ebb. An example of all which Dr. Halley has given us, from the observations of sea men in the port of Batsham, in the kingdom of Tunquin, in the latitude of 20° 50′ north. In that port, on the day which follows after the passage of the moon over the equator, the waters stagnate: when the moon declines to the north, they begin to flow

equation of the apogee; and this semi- annual equation in its greatest quantity comes to about 12° 18′, as nearly as I could collect from the phænomena. Our countryman, Horrox, was the first who advanced the theory of the moon's moving in an ellipsis about the earth placed in its lower focus. Dr. Halley improved the notion, by putting the centre of the ellipsis in an epicycle whose centre is uniformly revolved about the earth; and from the motion in this epicycle the mentioned inequalities in the progress and regress of the apogee, and in the quantity of eccentricity, do arise. Suppose the mean

draw the right lines Ff and Gg, and they will cut TA and τC in the points required, X and Z.

example.



Let the comet of the year 1680 be proposed. The following table shews the motion thereof, as observed by Flamsted, and calculated afterwards by him from his observations, and corrected by Dr. Halley from the same observations.

1680, Dec. 12

21

24

26

29

30

1681, Jan. 5

9

10

13

25

30

Feb. 2

5 Time Sun's

Longitude Comet's

Appar. True. Longitude. Lat. N.

h. ″

4.46

6.32½

6.12

5.14

7.55

8.02

5.51

6.49

5.54

6.56

7.44

8.07

6.20

6.50 h. ′ ″

4.46.0

6.36.59

6.17.52

ted. Longitude

observed. Latitude

observed Dif

Lo. Dif.

Lat.

Dec. 12 2792 ♑  6°.32′  8°.18½ ♑  6° 31½  8°.26 +1 − 7½

29 8403 ♓ 13.13⅔ 28.00 ♓ 13.11¾ 28.101⁄12 +2 −101⁄12

Feb.  5 16669 ♉ 17.00 15.29⅔ ♉ 16.59⅞ 15.27⅖ +0 + 2¼

Mar.  5 21737 29.19¾ 12.4 29.206⁄7 12. 3½ −1 +  ½

But afterwards Dr. Halley did determine the orbit to a greater accuracy by an arithmetical calculus than could be done by linear descriptions; and, retaining the place of the nodes in ♋ and ♑ 1° 53′, and the inclination of the plane of the orbit to the ecliptic 61° 20⅓′, as well as the time of the comet's being in

on the 6th and 11th O. S.; from its positions to the nearest fixed stars observed with sufficient accuracy, sometimes with a two feet, and sometimes with a ten feet telescope; from the difference of longitudes of Coburg and London, 11°; and from the places of the fixed stars observed by Mr. Pound, Dr. Halley has determined the places of the comet as follows:—



Nov. 3, 17h.2′, apparent time at London, the comet was in ♌ 29 deg. 51′, with 1 deg. 17′ 45″ latitude north.

November 5. 15h.58′ the comet was in ♍ 3° 23′, with 1° 6′ north lat.

November 10, 16h.31′, the comet was equally distant from two

there might be an error of 6 or 7 minutes, but hardly greater. The longitude of the comet, as found in the first and most accurate observation, being computed in the aforesaid parabolic orbit, comes out ♌ 29° 30′ 22″, its latitude north 1° 25′ 7″, and its distance from the sun 115546.

Moreover, Dr. Halley, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of September after Julius Cæsar was killed; An. Chr. 531, in the consulate of Lampadius and Orestes; An. Chr. 1106, in the month of February; and at the end of the year 1680; and that

by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayers ψ; and the other is the outmost of the wing, Bayer's θ. Whence the comet was then in ♎ 12° 46′ with latitude 50′ south. And I was informed by Dr. Halley, that on the same day at Boston in New England, in the latitude of 42½ deg. at 5h. in the morning (that is, at 9h.44′ in the morning at London), the comet was seen near ♎ 14°, with latitude 1° 30′ south.

Nov. 19, at 4½h. at Cambridge, the comet (by the observation of a young man) was distant from

I compared above with the observations, but likewise from that of the notable comet which appeared in the year 1664 and 1665, and was observed by Hevelius, who, from his own observations, calculated the longitudes and latitudes thereof, though with little accuracy. But from the same observations Dr. Halley did again compute its places; and from those new places determined its trajectory, finding its ascending node in ♊ 21° 13′ 55″; the inclination of the orbit to the plane of the ecliptic 21° 18′ 40″; the distance of its perihelion from the node, estimated in the comet's orbit, 49° 27′ 30″, its

at Dantzick, O. S.; and that the latus rectum of the parabola was 410286 such parts as the sun's mean distance from the earth is supposed to contain 100000. And how nearly the places of the comet computed in this orbit agree with the observations, will appear from the annexed table, calculated by Dr. Halley.

Appar. Time

at Dantzick. The observed Distances of the Comet from The observed Places. The Places

computed in

the Orb.

December °′″ °′″ °′″

d.h.′ The Lion's heart 46.24.20 Long. ♎  7.01.00 ♎  7. 1.29

 3.18.29½ The Virgin's spike 22.52.10 Lat. S. 21.39. 0 21.38.50

 4.18. 1½ The Lion's

the annual motion of the earth in the orbis magnus.

This theory is likewise confirmed by the motion of that comet, which in the year 1683 appeared retrograde, in an orbit whose plane contained almost a right angle with the plane of the ecliptic, and whose ascending node (by the computation of Dr. Halley) was in ♍ 23° 23′; the inclination of its orbit to the ecliptic 83° 11′; its perihelion in ♊ 25° 29′ 30″; its perihelion distance from the sun 56020 of such parts as the radius of the orbis magnus contains 100000; and the time of its perihelion July 2d.3h.50′. And the places thereof, computed by

was in ♍ 23° 23′; the inclination of its orbit to the ecliptic 83° 11′; its perihelion in ♊ 25° 29′ 30″; its perihelion distance from the sun 56020 of such parts as the radius of the orbis magnus contains 100000; and the time of its perihelion July 2d.3h.50′. And the places thereof, computed by Dr. Halley in this orbit, are compared with the places of the same observed by Mr. Flamsted, in the following table:—

1683

Eq. time. Sun's place Comet's

Long. com. Lat. Nor.

comput. Comet's

Long. obs'd Lat.Nor.

observ'd Diff.

Long. Diff.

Lat.

d. h. ′

July 13.12.55

15.11.15

17.10.20

23.13.40

2

- 1. 1

- 1.31 ′ ″

+ 0.07

+ 0.50

+ 0.30

- 1.14

-1. 7

- 0.27

- 2. 7

+ 1. 9

+ 1.30

+ 2. 0

- 0.27

- 5.32

- 0. 3

- 2. 4



-0. 3

- 0.28

+ 0.20

This theory is yet farther confirmed by the motion of that retrograde comet which appeared in the year 1682. The ascending node of this (by Dr. Halley's computation) was in ♉ 21° 16′ 30″; the inclination of its orbit to the plane of the ecliptic 17° 56′ 00″; its perihelion in ♒ 2° 52′ 50″; its perihelion distance from the sun 58328 parts, of which the radius of the orbis magnus contains 100000; the equal time of the comet's being in its

distance from the sun 998651 parts, of which the radius of the orbis magnus contains 1000000, and the equal time of its perihelion September 16d 16h.10′. The places of this comet computed in this orbit by Mr. Bradley, and compared with the places observed by himself, his uncle Mr. Pound, and Dr. Halley, may be seen in the following table.

1723

Eq. Time. Comet's

Long. obs. Lat. Nor.

obs. Comet's

Lon. com. Lat.Nor.

comp. Diff.

Lon. Diff.

Lat.

d. h. ′

Oct. 9.8. 5

10.6.21

12.7.22

14.8.57

15.6.35

21.6.22

22. 6.24

24.8. 2

29.8.56

30.6.20

Nov. 5.5.53

8.7. 6

14.6.20

20.7.45

Dec.

and so discover the periodic time of a comet's revolution in any orbit; whence, at last, we shall have the transverse diameters of their elliptic orbits and their aphelion distances.

That retrograde comet which appeared in the year 1607 described an orbit whose ascending node (according to Dr. Halley's computation) was in ♉ 20° 21′; and the inclination of the plane of the orbit to the plane of the ecliptic 17° 2′; whose perihelion was in ♒ 2° 16′; and its perihelion distance from the sun 58680 of such parts as the radius of the orbis magnus contains 100000; and the comet was in its perihelion

distance are rightly determined, it follows of necessity that Jupiter, by radii drawn to the sun, describes areas so conditioned as the hypothesis requires, that is, proportional to the times.

And the same thing may be concluded of Saturn from his satellite, by the observations of Mr. Huygens and Dr. Halley; though a longer series of observations is yet wanting to confirm the thing, and to bring it under a sufficiently exact computation.

For if Jupiter was viewed from the sun, it would never appear retrograde nor stationary, as it is seen sometimes from the earth, but always to go forward with a

earth must be first determined.

Mr. Flamsted (p. 387), by the micrometer, measured the diameter of Jupiter 40″ or 41″; the diameter of Saturn's ring 50″; and the diameter of the sun about 32′ 13″ (p. 387).

But the diameter of Saturn is to the diameter of the ring, according to Mr. Huygens and Dr. Halley, as 4 to 9; according to Galletius, as 4 to 10; and according to Hooke (by a telescope of 60 feet), as 5 to 12. And from the mean proportion, 5 to 12, the diameter of Saturn's body is inferred about 21″.

Such as we have said are the apparent magnitudes; but, because of the unequal refrangibility

the scattered light, which could not be seen before for the stronger light of the planet, when the planet is hid, appears every way farther spread. Lastly, from hence it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12″ or 15″; and Venus appeared to Mr. Crabtrie only 1′ 3″; to Horrox but 1′ 12″; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1′ 24″. Thus the apparent diameter of the moon, which in 1684, a few days

Flamsted observed), and therefore was much nearer to the sun: nay, it was even less than Mercury. For on the 17th of that month, when it was nearer to the earth, it appeared to Cassini through a telescope of 35 feet a little less than the globe of Saturn. On the 8th of this month, in the morning, Dr. Halley saw the tail, appearing broad and very short, and as if it rose from the body of the sun itself, at that time very near its rising. Its form was like that of an extraordinary bright cloud; nor did it disappear till the sun itself began to be seen above the horizon. Its splendor, therefore,

as it must have been if the true orbit in which it was carried was an ellipsis. The middle time between its ingress and egress was December 8d.2h. of the morning; and therefore at this time the comet ought to have been in its perihelion. And accordingly that very day, just before sunrising, Dr. Halley (as we said) saw the tail short and broad, but very bright, rising perpendicularly from the horizon. From the position of the tail it is certain that the comet had then crossed over the ecliptic, and got into north latitude, and therefore had passed by its perihelion, which lay on the other side

the united light of both globe and ring would be equal to the light of a globe whose diameter is 30″, it follows that the distance of the comet was to the distance of Saturn as 1 to inversely, and 12″ to 30 directly; that is, as 24 to 30, or 4 to 5. Again; the comet in the month of April 1665, as Hevelius informs us, excelled almost all the fixed stars in splendor, and even Saturn itself, as being of a much more vivid colour; for this comet was more lucid than that other which had appeared about the end of the preceding year, and had been compared to the stars of the first magnitude. The diameter

a great many such united into one.

Lastly; the same thing is inferred from the light of the heads, which increases in the recess of the comets from the earth towards the sun, and decreases in their return from the sun towards the earth; for so the comet of the year 1665 (by the observations of Hevelius), from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee; but yet the splendor of its head was daily in creasing, till, being hid under the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observations

time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee; but yet the splendor of its head was daily in creasing, till, being hid under the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observations of the same Hevelius), about the end of July, when it first appeared, moved at a very slow rate, advancing only about 40 or 45 minutes in its orb in a day's time; but from that time its diurnal motion was continually upon the increase, till September 4, when it arose to about 5 degrees; and therefore, in all this

time its diurnal motion was continually upon the increase, till September 4, when it arose to about 5 degrees; and therefore, in all this interval of time, the comet was approaching to the earth. Which is like wise proved from the diameter of its head, measured with a micrometer; for, August 6, Hevelius found it only 6′ 05″, including the coma, which, September 2 he observed to be 9′ 07″, and therefore its head appeared far less about the beginning than towards the end of the motion; though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the

it only 6′ 05″, including the coma, which, September 2 he observed to be 9′ 07″, and therefore its head appeared far less about the beginning than towards the end of the motion; though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hevelius declares. Wherefore in all this interval of time, on account of its recess from the sun, it decreased in splendor, notwithstanding its access towards the earth. The comet of the year 1618, about the middle of December, and that of the year 1680, about the end of the same month, did both move with

things with us.

The atmospheres of comets, in their descent towards the sun, by running out into the tails, are spent and diminished, and become narrower, at least on that side which regards the sun; and in receding from the sun, when they less run out into the tails, they are again enlarged, if Hevelius has justly marked their appearances. But they are seen least of all just after they have been most heated by the sun, and on that account then emit the longest and most resplendent tails; and, perhaps, at the same time, the nuclei are environed with a denser and blacker smoke in the lowermost

as the comet of 1680 was. And we read, in the Saxon Chronicle, of a like comet appearing in the year 1106, the star whereof was small and obscure (as that of 1680), but the splendour of its tail was very bright, and like a huge fiery beam stretched out in a direction between the east and north, as Hevelius has it also from Simeon, the monk of Durham. This comet appeared in the beginning of February, about the evening, and towards the south west part of heaven; from whence, and from the position of the tail, we infer that the head was near the sun. Matthew Paris says, It was distant from the sun by

such trajectories will always nearly agree with the phænomena, as appears not only from the parabolic trajectory of the comet of the year 1680, which I compared above with the observations, but likewise from that of the notable comet which appeared in the year 1664 and 1665, and was observed by Hevelius, who, from his own observations, calculated the longitudes and latitudes thereof, though with little accuracy. But from the same observations Dr. Halley did again compute its places; and from those new places determined its trajectory, finding its ascending node in ♊ 21° 13′ 55″; the inclination

of that star from the star A, that is, 52,′ 29″; and the difference of the longitude of the comet and the second star in Aries was 45′ or 46′, or, taking a mean quantity, 45′ 30″; and therefore the comet was in ♉ 0° 2′ 48″. From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54′. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion; and Hevelius, in the scheme of M. Auzout's observations which he constructed himself, corrected this irregular curvature, and so made the latitude of

quantity, 45′ 30″; and therefore the comet was in ♉ 0° 2′ 48″. From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54′. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion; and Hevelius, in the scheme of M. Auzout's observations which he constructed himself, corrected this irregular curvature, and so made the latitude of the comet 8° 55′ 30″. And, by farther correcting this irregularity, the latitude may become 8° 56, or 8° 57′.

This comet was also seen March 9, and at that time

fall in between them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to

of 8″ in breadth, when it should have appeared only of 3″ 14‴; and hence it is that the brighter fixed stars appear through the telescope as of 5″ or 6″ in diameter, and that with a good full light; but with a fainter light they appear to run out to a greater breadth. Hence, likewise, it was that Hevelius, by diminishing the aperture of the telescope, did cut off a great part of the light towards the circumference, and brought the disk of the star to be more distinctly defined, which, though hereby diminished, did yet appear as of 5″ or 6″ in diameter. But Mr. Huygens, only by clouding the

by the micrometer; for the scattered light, which could not be seen before for the stronger light of the planet, when the planet is hid, appears every way farther spread. Lastly, from hence it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12″ or 15″; and Venus appeared to Mr. Crabtrie only 1′ 3″; to Horrox but 1′ 12″; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1′ 24″. Thus the apparent diameter of the

from hence it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12″ or 15″; and Venus appeared to Mr. Crabtrie only 1′ 3″; to Horrox but 1′ 12″; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1′ 24″. Thus the apparent diameter of the moon, which in 1684, a few days both before and after the sun's eclipse, was measured at the observatory of Paris 31′ 30″, in the eclipse itself did not seem to exceed 30′ or 30′ 05″;

of the year 1607 and 1618, as their motions are defined by Kepler, passed between the sun and the earth; that of the year 1664 below the orbit of Mars; and that in 1680 below the orbit of Mercury, as its motion was defined by Sir Christopher Wren and others. By a like rectilinear hypothesis, Hevelius placed all the comets about which we have any observations below the orbit of Jupiter. It is a false notion, therefore, and contrary to astronomical calculation, which some have entertained, who, from the regular motion of the comets, either remove them into the regions of the fixed stars, or deny

in the middle, scarcely possessed the tenth part of this breadth, and was therefore only 11" or 12" broad; but the light and clearness of its head exceeded that of the year 1680, and was equal to that of the stars of the first or second magnitude. Moreover, the comet of the year 1665, in April, as Hevelius informs us, exceeded almost all the fixed stars in splendor, and even Saturn itself, as being of a much more vivid colour; for this comet was more lucid than that which appeared at the end of the foregoing year and was compared to the stars of the first magnitude. The diameter of the coma was

bright comets and the rising or setting sun is intimated (p. 494, 495). We may add to these the comet of the year 1101 or 1106, "the star of which was small and obscure (like that of 1680); but the splendour arising from it extremely bright, reaching like a fiery beam to the east and north," as Hevelius has it from Simeon, the monk of Durham. It appeared at the beginning of February about the evening in the south-west. From this and from the situation of the tail we may infer that the head was near the sun. Matthew Paris says, "it was about one cubit from the sun; from the third [or rather the

ever appeared there.

Lastly, the same thing is inferred (p. 466, 467) from the light of the heads increasing in the recess of the comets from the earth towards the sun, and decreasing in their return from the sun towards the earth; for so the last comet of the year 1665 (by the observation of Hevelius), from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee: yet the splendor of its head was daily increasing, till, being hid by the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observation of the

from the time that it was first seen, was always losing of its apparent motion, and therefore had already passed its perigee: yet the splendor of its head was daily increasing, till, being hid by the sun's rays, the comet ceased to appear. The comet of the year 1683 (by the observation of the same Hevelius), about the end of July, when it first appeared, moved at a very slow rate, advancing only about 40 or 45 minutes in its orbit in a day's time. But from that time its diurnal motion was continually upon the increase till September 4, when it arose to about 5 degrees; and therefore in all this

time its diurnal motion was continually upon the increase till September 4, when it arose to about 5 degrees; and therefore in all this interval of time the comet was approaching to the earth. Which is likewise proved from the diameter of its head measured with a micrometer; for, August the 6th, Hevelius found it only 6' 5", including the coma: which, September 2, he observed 9' 7". And therefore its head appeared far less about the beginning than towards the end of its motion, though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same

found it only 6' 5", including the coma: which, September 2, he observed 9' 7". And therefore its head appeared far less about the beginning than towards the end of its motion, though about the beginning, because nearer to the sun, it appeared far more lucid than towards the end, as the same Hevelius declares. Wherefore in all this interval of time, on account of its recess from the sun, it decreased in splendor, notwithstanding its access towards the earth. The comet of the year 1618, about the middle of December, and that of the year 1680, about the end of the same month, did both move with

sun.





PHÆNOMENON IV.



That the fixed stars being at rest, the periodic times of the five primary planets, and (whether of the sun, about the earth, or) of the earth about the sun, are in the sesquiplicate proportion of their mean distances from the sun.



This proportion, first observed by Kepler, is now received by all astronomers; for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun. And as to the measures of the periodic times, all astronomers are agreed about them. But for the dimensions

by all astronomers; for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun. And as to the measures of the periodic times, all astronomers are agreed about them. But for the dimensions of the orbits, Kepler and Bullialdus, above all others, have determined them from observations with the greatest accuracy; and the mean distances corresponding to the periodic times differ but insensibly from those which they have assigned, and for the most part fall in between them; as we may see from the following

times with respect to the fixed stars, of the planets and earth revolving about the sun, in days and decimal parts of a day.



♄ ♃ ♂ ♁ ♀ ☿

10759,275. 4332,514. 686,9785. 365,2565. 224,6176. 87,9692.





The mean distances of the planets and of the earth from the sun.



♄ ♃ ♂

According to Kepler 951000. 519650. 152350.

” to Bullialdus 954198. 522520. 152350.

” to the periodic times 954006. 520096. 152369.



♁ ♀ ☿

According to Kepler 100000. 72400. 38806.

” to Bullialdus 100000. 72398. 38585.

” to the periodic times 100000. 72333. 38710.

As to Mercury and Venus, there can be no

10759,275. 4332,514. 686,9785. 365,2565. 224,6176. 87,9692.





The mean distances of the planets and of the earth from the sun.



♄ ♃ ♂

According to Kepler 951000. 519650. 152350.

” to Bullialdus 954198. 522520. 152350.

” to the periodic times 954006. 520096. 152369.



♁ ♀ ☿

According to Kepler 100000. 72400. 38806.

” to Bullialdus 100000. 72398. 38585.

” to the periodic times 100000. 72333. 38710.

As to Mercury and Venus, there can be no doubt about their distances from the sun; for they are determined by the elongations of those planets from the sun; and for the distances of the

SCHOLIUM.



The demonstration of this Proposition may be more diffusely explained after the following manner. Suppose several moons to revolve about the earth, as in the system of Jupiter or Saturn: the periodic times of these moons (by the argument of induction) would observe the same law which Kepler found to obtain among the planets; and therefore their centripetal forces would be reciprocally as the squares of the distances from the centre of the earth, by Prop. I, of this Book. Now if the lowest of these were very small, and were so near the earth as almost to touch the tops of the highest

The head of the former comet (according to the observations of Cysatus), December 1, appeared greater than the stars of the first magnitude; and, December 16 (then in the perigee), it was but little diminished in magnitude, but in the splendor and brightness of its light a great deal. January 7, Kepler, being uncertain about the head, left off observing. December 12, the head of the latter comet was seen and observed by Mr. Flamsted, when but 9 degrees distant from the sun; which is scarcely to be done in a star of the third magnitude. December 15 and 17, it appeared as a star of the third

again we have another argument proving the celestial spaces to be free, and without resistance, since in them not only the solid bodies of the planets and comets, but also the extremely rare vapours of comets tails, maintain their rapid motions with great freedom, and for an exceeding long time.

Kepler ascribes the ascent of the tails of the comets to the atmospheres of their heads; and their direction towards the parts opposite to the sun to the action of the rays of light carrying along with them the matter of the comets tails; and without any great incongruity we may suppose, that, in so free

little yellow; and in March 1573 it became ruddy, like Mars or Aldebaran: in May it turned to a kind of dusky whiteness, like that we observe in Saturn; and that colour it retained ever after, but growing always more and more obscure. Such also was the star in the right foot of Serpentarius, which Kepler's scholars first observed September 30, O.S. 1604, with a light exceeding that of Jupiter, though the night before it was not to be seen; and from that time it decreased by little and little, and in 15 or 16 months entirely disappeared. Such a new star appearing with an unusual splendor is said to

revolutions in curvilinear orbits, are questions which we do not know how the ancients explained; and probably it was to give some sort of satisfaction to this difficulty that solid orbs were introduced.

The later philosophers pretend to account for it either by the action of certain vortices, as Kepler and Des Cartes; or by some other principle of impulse or attraction, as Borelli, Hooke, and others of our nation; for, from the laws of motion, it is most certain that these effects must proceed from the action of some force or other.

But our purpose is only to trace out the quantity and

the centre of Jupiter, will be determined with a less error than 1′ 48″. But when the satellite is in the middle of the shadow, that longitude is the same with the heliocentric longitude of Jupiter; and, therefore, the hypothesis which Mr. Flamsted follows, viz., the Copernican, as improved by Kepler, and (as to the motion of Jupiter) lately corrected by himself, rightly represents that longitude within a less error than 1′ 48″; but by this longitude, together with the geocentric longitude, which is always easily found, the distance of Jupiter from the sun is determined; which must, therefore,

will be directed (by Prop. III, Cor. I) to the centre of the sun.

The distances of the planets from the sun come out the same, whether, with Tycho, we place the earth in the centre of the system, or the sun with Copernicus: and we have already proved that these distances are true in Jupiter.

Kepler and Bullialdus have, with great care (p. 388), determined the distances of the planets from the sun; and hence it is that their tables agree best with the heavens. And in all the planets, in Jupiter and Mars, in Saturn and the earth, as well as in Venus and Mercury, the cubes of their distances

ascribed to other causes which we shall afterwards explain. And thus we shall always find the said proportion to hold exactly; for the distances of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury, from the sun, drawn from the observations of astronomers, are, according to the computation of Kepler, as the numbers 951000, 519650, 152350, 100000, 72400, 38806; by the computation of Bullialdus, as the numbers 954198, 522520, 152350, 100000, 72398, 38585; and from the periodic times they come out 953806, 520116, 152399, 100000, 72333, 38710. Their distances, according to Kepler and Bullialdus,

computation of Kepler, as the numbers 951000, 519650, 152350, 100000, 72400, 38806; by the computation of Bullialdus, as the numbers 954198, 522520, 152350, 100000, 72398, 38585; and from the periodic times they come out 953806, 520116, 152399, 100000, 72333, 38710. Their distances, according to Kepler and Bullialdus, scarcely differ by any sensible quantity, and where they differ most the distances drawn from the periodic times, fall in between them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon

distances drawn from the periodic times, fall in between them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon

(by Cor. IV, Prop. III) that the circum-terrestrial force, compared with the circum-solar, is very small.

Ricciolus and Vendelin have severally tried to determine the sun's parallax from the moon's dichotomies observed by the telescope, and they agree that it does not exceed half a minute.

Kepler, from Tycho's observations and his own, found the parallax of Mars insensible, even in opposition to the sun, when that parallax is some thing greater than the sun's.

Flamsted attempted the same parallax with the micrometer in the perigeon position of Mars, but never found it above 25″; and

the comets is confirmed by the annual parallax of the orbit, in so far as the same is pretty nearly collected by the supposition that the comets move uniformly in right lines. The method of collecting the distance of a comet according to this hypothesis from four observations (first attempted by Kepler, and perfected by Dr. Wallis and Sir Christopher Wren) is well known; and the comets reduced to this regularity generally pass through the middle of the planetary region. So the comets of the year 1607 and 1618, as their motions are defined by Kepler, passed between the sun and the earth; that of

from four observations (first attempted by Kepler, and perfected by Dr. Wallis and Sir Christopher Wren) is well known; and the comets reduced to this regularity generally pass through the middle of the planetary region. So the comets of the year 1607 and 1618, as their motions are defined by Kepler, passed between the sun and the earth; that of the year 1664 below the orbit of Mars; and that in 1680 below the orbit of Mercury, as its motion was defined by Sir Christopher Wren and others. By a like rectilinear hypothesis, Hevelius placed all the comets about which we have any observations

more early, when yet nearer to the sun. The head of the former comet, according to the observations of Cysatus, Dec. 1, appeared greater than the stars of the first magnitude; and, Dec. 16 (being then in its perigee), of a small magnitude, and the splendor or clearness was much diminished. Jan. 7, Kepler, being uncertain about the head, left off observing. Dec. 12, the head of the last comet was seen and observed by Flamsted at the distance of 9 degrees from the sun, which a star of the third magnitude could hardly have been. December 15 and 17, the same appeared like a star of the third

a few miles in thickness, obscures and extinguishes the light not only of all the stars, but even of the moon itself; whereas the smallest stars are seen to shine through the immense thickness of the tails of comets, likewise illuminated by the sun, without the least diminution of their splendor.

Kepler ascribes the ascent of the tails of comets to the atmospheres of their heads, and their direction towards the parts opposite to the sun to the action of the rays of light carrying along with them the matter of the comets' tails; and without any great incongruity we may suppose that, in so free

move in ellipses having their focus in the centre of the primary, 413

“ “ by radii drawn to their primary describe areas proportional to the times, 386, 387, 390

“ “ revolve in periodic times that are in the sesquiplicate proportion of their distances from the primary, 386, 387

Problem Keplerian, solved by the trochoid and by approximations, 157 to 160

“ “ of the ancients, of four lines, related by Pappus, and attempted by Cartesius, by an algebraic calculus solved by a geometrical composition, 135

Projectiles move in parabolas when the resistance of the medium is taken away, 91, 115,

of light and the apparent diameter of a comet are given, its distance will be also given, by taking the distance of the comet to the distance of a planet in the direct proportion of their diameters and the reciprocal subduplicate proportion of their lights. Thus, in the comet of the year 1682, Mr. Flamsted observed with a telescope of 16 feet, and measured with a micrometer, the least diameter of its head, 2′ 00; but the nucleus or star in the middle of the head scarcely amounted to the tenth part of this measure; and therefore its diameter was only 11″ or 12″; but in the light and splendor of its

magnitude; and, December 16 (then in the perigee), it was but little diminished in magnitude, but in the splendor and brightness of its light a great deal. January 7, Kepler, being uncertain about the head, left off observing. December 12, the head of the latter comet was seen and observed by Mr. Flamsted, when but 9 degrees distant from the sun; which is scarcely to be done in a star of the third magnitude. December 15 and 17, it appeared as a star of the third magnitude, its lustre being diminished by the brightness of the clouds near the setting sun. December 26, when it moved with the greatest

17, the tail was seen by Ponthæus more than 15° long; Nov. 18, in New-England, the tail appeared 30° long, and directly opposite to the sun, extending itself to the planet Mars, which was then in ♍, 9° 54′: Nov. 19. in Maryland, the tail was found 15° or 20° long; Dec. 10 (by the observation of Mr. Flamsted), the tail passed through the middle of the distance intercepted between the tail of the Serpent of Ophiuchus and the star δ in the south wing of Aquila, and did terminate near the stars A, ω, b, in Bayer's tables. Therefore the end of the tail was in ♑ 19½°, with latitude about 34¼° north; Dec

perihelion in ♊ 25° 29′ 30″; its perihelion distance from the sun 56020 of such parts as the radius of the orbis magnus contains 100000; and the time of its perihelion July 2d.3h.50′. And the places thereof, computed by Dr. Halley in this orbit, are compared with the places of the same observed by Mr. Flamsted, in the following table:—

1683

Eq. time. Sun's place Comet's

Long. com. Lat. Nor.

comput. Comet's

Long. obs'd Lat.Nor.

observ'd Diff.

Long. Diff.

Lat.

d. h. ′

July 13.12.55

15.11.15

17.10.20

23.13.40

25.14.5

31.9.42

31.14.55

Aug. 2.14.56

4.10.49

6.10.9

9.10.26

15.14.1

of its orbit to the plane of the ecliptic 17° 56′ 00″; its perihelion in ♒ 2° 52′ 50″; its perihelion distance from the sun 58328 parts, of which the radius of the orbis magnus contains 100000; the equal time of the comet's being in its perihelion Sept. 4d.7h.39′. And its places, collected from Mr. Flamsted's observations, are compared with its places computed from our theory in the following table:—

1682

App. Time. Sun's place Comet's

Long. comp. Lat. Nor.

comp. Com. Long.

observed. Lat.Nor.

observ. Diff.

Long. Diff.

Lat.

d. h. ′

Aug. 19.16.38

20.15.38

21. 8.21

22. 8. 8

29.08.20

30.

centre of every planet, appears by Cor. VI, Prop. IV, Book 1; for the periodic times of the satellites of Jupiter are one to another (p. 386, 387) in the sesquiplicate proportion of their distances from the centre of this planet.

This proportion has been long ago observed in those satellites; and Mr. Flamsted, who had often measured their distances from Jupiter by the micrometer, and by the eclipses of the satellites, wrote to me, that it holds to all the accuracy that possibly can be discerned by our senses. And he sent me the dimensions of their orbits taken by the micrometer, and reduced to the mean

5

5⅔ 10

10

8

8⅔ 16

16

13

14 28

26

23

24⅔

After the invention of the micrometer:—

By Townley

" Flamsted

More accurately by the eclipses 5,51

5,31

5,578 8,78

8,85

8,876 13,47

13,98

14,159 24,72

24,23

24,903





And the periodic times of those satellites, by the observations of Mr. Flamsted, are 1d.18h.28′ 36″ | 3d.13h.17′ 54″ | 7d.3h.59′ 36″ | 16d.18h.5′ 13″, as above.

And the distances thence computed are 5,578 | 8,878 | 14,168 | 24,968, accurately agreeing with the distances by observation.

Cassini assures us (p. 388, 389) that the same proportion is observed in the

areas nearly uniform; but by radii drawn to the earth, it is sometimes swift, sometimes stationary, and sometimes retrograde.

That Jupiter, in a higher orb than Mars, is likewise revolved about the sun, with a motion nearly equable, as well in distance as in the areas described, I infer thus.

Mr. Flamsted assured me, by letters, that all the eclipses of the inner most satellite which hitherto have been well observed do agree with his theory so nearly, as never to differ therefrom by two minutes of time; that in the outmost the error is little greater; in the outmost but one, scarcely three times

and introduced by Mr. Romer. Supposing, then, that the theory differs by a less error than that of 2′ from the motion of the outmost satellite as hitherto described, and taking as the periodic time 16d. 18h.5′ 13″ to 2′ in time, so is the whole circle or 360° to the arc 1′ 48″, the error of Mr. Flamsted's computation, reduced to the satellite's orbit, will be less than 1′ 48″; that is, the longitude of the satellite, as seen from the centre of Jupiter, will be determined with a less error than 1′ 48″. But when the satellite is in the middle of the shadow, that longitude is the same with the

1′ 48″; that is, the longitude of the satellite, as seen from the centre of Jupiter, will be determined with a less error than 1′ 48″. But when the satellite is in the middle of the shadow, that longitude is the same with the heliocentric longitude of Jupiter; and, therefore, the hypothesis which Mr. Flamsted follows, viz., the Copernican, as improved by Kepler, and (as to the motion of Jupiter) lately corrected by himself, rightly represents that longitude within a less error than 1′ 48″; but by this longitude, together with the geocentric longitude, which is always easily found, the distance of

will accurately take place.

Now that this proportion has been established, we may compare the forces of the several planets among themselves (p. 391).

In the mean distance of Jupiter from the earth, the greatest elongation of the outmost satellite from Jupiter's centre (by the observations of Mr. Flamsted) is 8′ 13″; and therefore the distance of the satellite from the centre of Jupiter is to the mean distance of Jupiter from the centre of the sun as 124 to 52012, but to the mean distance of Venus from the centre of the sun as 124 to 7234; and their periodic times are 16¾d. and 224⅔d; and from

force 8500 times.

By the like computations I happened to discover an analogy, that is observed between the forces and the bodies of the planets; but, before I explain this analogy, the apparent diameters of the planets in their mean distances from the earth must be first determined.

Mr. Flamsted (p. 387), by the micrometer, measured the diameter of Jupiter 40″ or 41″; the diameter of Saturn's ring 50″; and the diameter of the sun about 32′ 13″ (p. 387).

But the diameter of Saturn is to the diameter of the ring, according to Mr. Huygens and Dr. Halley, as 4 to 9; according to Galletius,

diameters of the planets are to be diminished when without the sun, and to be augmented when within it, by some seconds. But the errors seem to be less than usual in the mensurations that are made by the micrometer. So from the diameter of the shadow, determined by the eclipses of the satellites, Mr. Flamsted found that the semi-diameter of Jupiter was to the greatest elongation of the outmost satellite as 1 to 24,903. Wherefore since that elongation is 8′ 13″, the diameter of Jupiter will be 39½″; and, rejecting the scattered light, the diameter found by the micrometer 40″ or 41″ will be reduced to

by astronomers: but all these follow from our principles in Cor. II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, Prop. LXVI, and are known really to exist in the heavens. And this may seen in that most ingenious, and if I mistake not, of all, the most accurate, hypothesis of Mr. Horrox, which Mr. Flamsted has fitted to the heavens; but the astronomical hypotheses are to be corrected in the motion of the nodes; for the nodes admit the greatest equation or prosthaphæresis in their octants, and this inequality is most conspicuous when the moon is in the nodes, and therefore also in the octants; and

but the diameter of the head was only 25", that is, almost the same with the diameter of a circle equal to Saturn and his ring. The coma or hair surrounding the head was about ten times as broad; namely, 41⁄6 min. Again; the least diameter of the hair of the comet of the year 1682, observed by Mr. Flamsted with a tube of 16 feet and measured with the micrometer, was 2' 0"; but the nucleus, or star in the middle, scarcely possessed the tenth part of this breadth, and was therefore only 11" or 12" broad; but the light and clearness of its head exceeded that of the year 1680, and was equal to that of

to it. So the comet that appeared Dec. 12 and 15, O.S. Anno 1679, at the time it emitted a very shining tail, whose splendor was equal to that of many stars like Jupiter, if their light were dilated and spread through so great a space, was, as to the magnitude of its nucleus, less than Jupiter (as Mr. Flamsted observed), and therefore was much nearer to the sun: nay, it was even less than Mercury. For on the 17th of that month, when it was nearer to the earth, it appeared to Cassini through a telescope of 35 feet a little less than the globe of Saturn. On the 8th of this month, in the morning, Dr.

square of the line AB. On the same Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks. By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early

of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a

of its gravity revolves in a circle concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given (by Cor. 9 of this Prop.). And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillatorio, has compared the force of gravity with the centrifugal forces of revolving bodies.

The preceding Proposition may be likewise demonstrated after this manner. In any circle suppose a polygon to be inscribed of any number of sides. And if a body,

  3,45.  8.  23,35.

The greatest elongation of the fourth satellite from Saturn's centre is commonly determined from the observations to be eight of those semi-diameters very nearly. But the greatest elongation of this satellite from Saturn's centre, when taken with an excellent micrometer in Mr. Huygens' telescope of 123 feet, appeared to be eight semi-diameters and 7⁄10 of a semi-diameter. And from this observation and the periodic times the distances of the satellites from Saturn's centre in semi-diameters of the ring are 2,1. 2,69. 3,75. 8,7. and 25,35. The diameter of Saturn observed in the

of time, to describe 151⁄12 of those feet; or more accurately 15 feet, 1 inch, and 1 line 4⁄9. And with this very force we actually find that bodies here upon earth do really descend; for a pendulum oscillating seconds in the latitude of Paris will be 3 Paris feet, and 8 lines ½ in length, as Mr. Huygens has observed. And the space which a heavy body describes by falling in one second of time is to half the length of this pendulum in the duplicate ratio of the circumference of a circle to its diameter (as Mr. Huygens has also shewn), and is therefore 15 Paris feet, 1 inch, 1 line 7⁄9. And

in the latitude of Paris will be 3 Paris feet, and 8 lines ½ in length, as Mr. Huygens has observed. And the space which a heavy body describes by falling in one second of time is to half the length of this pendulum in the duplicate ratio of the circumference of a circle to its diameter (as Mr. Huygens has also shewn), and is therefore 15 Paris feet, 1 inch, 1 line 7⁄9. And therefore the force by which the moon is retained in its orbit becomes, at the very surface of the earth, equal to the force of gravity which we observe in heavy bodies there. And therefore (by Rule I and II) the force by

any other distances from their centres, are (by this Prop.) greater or less in the reciprocal duplicate proportion of the distances. Thus from the periodic times of Venus, revolving about the sun, in 224d.16¾h, of the utmost circumjovial satellite revolving about Jupiter, in 16d.168⁄15h.; of the Huygenian satellite about Saturn in 15d.22⅔h.; and of the moon about the earth in 27d.7h.43′; compared with the mean distance of Venus from the sun, and with the greatest heliocentric elongations of the outmost circumjovial satellite from Jupiter's centre, 8′ 16″; of the Huygenian satellite from the centre

in 16d.168⁄15h.; of the Huygenian satellite about Saturn in 15d.22⅔h.; and of the moon about the earth in 27d.7h.43′; compared with the mean distance of Venus from the sun, and with the greatest heliocentric elongations of the outmost circumjovial satellite from Jupiter's centre, 8′ 16″; of the Huygenian satellite from the centre of Saturn, 3′ 4″; and of the moon from the earth, 10′ 33″: by computation I found that the weight of equal bodies, at equal distances from the centres of the sun, of Jupiter, of Saturn, and of the earth, towards the sun, Jupiter, Saturn, and the earth, were one to

longitude and distance are rightly determined, it follows of necessity that Jupiter, by radii drawn to the sun, describes areas so conditioned as the hypothesis requires, that is, proportional to the times.

And the same thing may be concluded of Saturn from his satellite, by the observations of Mr. Huygens and Dr. Halley; though a longer series of observations is yet wanting to confirm the thing, and to bring it under a sufficiently exact computation.

For if Jupiter was viewed from the sun, it would never appear retrograde nor stationary, as it is seen sometimes from the earth, but always to go

impulse of the same centripetal force as before, would, in one second of time, describe 151⁄12 Paris feet.

This we infer by a calculus formed upon Prop. XXXVI, and it agrees with what we observe in all bodies about the earth. For by the experiments of pendulums, and a computation raised thereon, Mr. Huygens has demonstrated that bodies falling by all that centripetal force with which (of whatever nature it is) they are impelled near the surface of the earth, do, in one second of time, describe 151⁄12 Paris feet.

But if the earth is supposed to move, the earth and moon together (by Cor. IV of the

distances from the earth must be first determined.

Mr. Flamsted (p. 387), by the micrometer, measured the diameter of Jupiter 40″ or 41″; the diameter of Saturn's ring 50″; and the diameter of the sun about 32′ 13″ (p. 387).

But the diameter of Saturn is to the diameter of the ring, according to Mr. Huygens and Dr. Halley, as 4 to 9; according to Galletius, as 4 to 10; and according to Hooke (by a telescope of 60 feet), as 5 to 12. And from the mean proportion, 5 to 12, the diameter of Saturn's body is inferred about 21″.

Such as we have said are the apparent magnitudes; but, because of the unequal

Hence, likewise, it was that Hevelius, by diminishing the aperture of the telescope, did cut off a great part of the light towards the circumference, and brought the disk of the star to be more distinctly defined, which, though hereby diminished, did yet appear as of 5″ or 6″ in diameter. But Mr. Huygens, only by clouding the eye-glass with a little smoke, did so effectually extinguish this scattered light, that the fixed stars appeared as mere points, void of all sensible breadth. Hence also it was that Mr. Huygens, from the breadth of bodies interposed to intercept the whole light of the

which, though hereby diminished, did yet appear as of 5″ or 6″ in diameter. But Mr. Huygens, only by clouding the eye-glass with a little smoke, did so effectually extinguish this scattered light, that the fixed stars appeared as mere points, void of all sensible breadth. Hence also it was that Mr. Huygens, from the breadth of bodies interposed to intercept the whole light of the planets, reckoned their diameters greater than others have measured them by the micrometer; for the scattered light, which could not be seen before for the stronger light of the planet, when the planet is hid, appears every

it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12″ or 15″; and Venus appeared to Mr. Crabtrie only 1′ 3″; to Horrox but 1′ 12″; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1′ 24″. Thus the apparent diameter of the moon, which in 1684, a few days both before and after the sun's eclipse, was measured at the observatory of Paris 31′ 30″, in the eclipse itself did not seem to exceed 30′ or 30′ 05″; and

the theory thereof confirmed by experiments of pendulums, 313 to 321

“ by experiments of falling bodies, 345 to 356

Rest, true and relative, 78

Rules of philosophy, 384

Satellites, the greatest heliocentric elongation of Jupiter's satellites, 387

“ the greatest heliocentric elongation of the Huygenian satellite from Saturn's centre, 398

“ the periodic times of Jupiter s satellites, and their distances from his centre, 386, 387

“ the periodic times of Saturn s satellites, and their distances from his centre, 387, 388

“ the inequalities of the motions of the satellites of Jupiter and Saturn

in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would descend through in the same given time.





SCHOLIUM.



The case of the 6th Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed); and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres.

Moreover, by means of the preceding Proposition and its Corollaries, we may

England, it was found in about ♏ 3°, and with almost the same latitude as before, that is, 1° 30′. The same day, at 5h. morning at Ballasore,ihe comet was observed in ♏ 1° 50′; and therefore at 5h. morning at London, the comet was ♏ 3° 5′ nearly. The same day, at 6½h. in the morning at London, Dr. Hook observed it in about ♏ 3° 30′, and that in the right line which passeth through Spica ♍ and Cor Leonis; not, indeed, exactly, but deviating a little from that line towards the north. Montenari likewise observed, that this day, and some days after, a right line drawn from the comet through Spica

comet through Spica passed by the south side of Cor Leonis at a very small distance therefrom. The right line through Cor Leonis and Spica ♍ did cut the ecliptic in ♍ 3° 46′ at an angle of 2° 51′; and if the comet had been in this line and in ♏ 3°, its latitude would have been 2° 26′; but since Hook and Montenari agree that the comet was at some small distance from this line towards the north, its latitude must have been something less. On the 20th, by the observation of Montenari, its latitude was almost the same with that of Spica ♍, that is, about 1° 30′. But by the agreement of Hook,

since Hook and Montenari agree that the comet was at some small distance from this line towards the north, its latitude must have been something less. On the 20th, by the observation of Montenari, its latitude was almost the same with that of Spica ♍, that is, about 1° 30′. But by the agreement of Hook, Montenari, and Ango, the latitude was continually increasing, and therefore must now, on the 22d, be sensibly greater than 1° 30′; and, taking a mean between the extreme limits but now stated, 2° 26′ and 1° 30′, the latitude will be about 1° 58′. Hook and Montenari agree that the tail of the

is, about 1° 30′. But by the agreement of Hook, Montenari, and Ango, the latitude was continually increasing, and therefore must now, on the 22d, be sensibly greater than 1° 30′; and, taking a mean between the extreme limits but now stated, 2° 26′ and 1° 30′, the latitude will be about 1° 58′. Hook and Montenari agree that the tail of the comet was directed towards Spica ♍, declining a little from that star towards the south according to Hook, but towards the north according to Montenari; and, therefore, that declination was scarcely sensible; and the tail, lying nearly parallel to the

be sensibly greater than 1° 30′; and, taking a mean between the extreme limits but now stated, 2° 26′ and 1° 30′, the latitude will be about 1° 58′. Hook and Montenari agree that the tail of the comet was directed towards Spica ♍, declining a little from that star towards the south according to Hook, but towards the north according to Montenari; and, therefore, that declination was scarcely sensible; and the tail, lying nearly parallel to the equator, deviated a little from the opposition of the sun towards the north.

Nov. 23, O. S. at 5h. morning, at Nuremberg (that is, at 4½h. at London),

stars.

Nov. 24, before sun-rising, the comet was seen by Montenari in ♏ 12° 52′ on the north side of the right line through Cor Leonis and Spica ♍, and therefore its latitude was something less than 2° 38′; and since the latitude, as we said, by the concurring observations of Montenari, Ango, and Hook, was continually increasing, therefore, it was now, on the 24th, something greater than 1° 58′; and, taking the mean quantity, may be reckoned 2° 18′, without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New

viz., of about 1°, or 1½°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and sometimes by Ponthæus and Cellius. The same day, at 5h. morning, at Ballasore, the comet was observed in ♏ 11° 45′; and, therefore, at 5h. morning at London, was in ♏ 13° nearly. And, by the theory, the comet was at that time in ♏ 13° 22′ 2″.

Nov. 25,

latitude of the star γ it was 1° 20′ 26″. Therefore if from the longitude of the star γ there be subducted the longitude 1° 20′ 26″, there will remain the longitude of the comet ♈ 27° 9′ 49″. M. Auzout, from this observation of his, placed the comet in ♈ 27° 0′, nearly; and, by the scheme in which Dr. Hooke delineated its motion, it was then in ♈ 26° 59′ 24″. I place it in ♈ 27° 4′ 46″, taking the middle between the two extremes.

From the same observations, M. Auzout made the latitude of the comet at that time 7° and 4′ or 5′ to the north; but he had done better to have made it 7° 3′ 29″, the

done better to have made it 7° 3′ 29″, the difference of the latitudes of the comet and the star γ being equal to the difference of the longitude of the stars γ and A.

February 22d.7h.30′ at London, that is, February 22d. 8h.46′ at Dantzick, the distance of the comet from the star A, according to Dr. Hooke's observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, delineated in like manner by M. Petit, was a fifth part of the distance between the star A and the first star of Aries, or 15′ 57″; and the distance of the comet from a right line joining the star A

in ♈ 28° 29′ 46″, with 8° 12′ 36″ north lat.

March 1, 7h at London, that is, March 1, 8h.16′ at Dantzick. the comet was observed near the second star in Aries, the distance between them being to the distance between the first and second stars in Aries, that is, to 1° 33′, as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8′ 16″ according to Dr. Hooke, or 8′ 5″ according to M. Gottignies; or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second

in Aries, the distance between them being to the distance between the first and second stars in Aries, that is, to 1° 33′, as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8′ 16″ according to Dr. Hooke, or 8′ 5″ according to M. Gottignies; or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1′ 35″ (in which he agrees very well with

or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1′ 35″ (in which he agrees very well with M. Auzout); or, according to Dr. Hooke, not quite so much, as perhaps only 1′. Wherefore if to the longitude of the first star in Aries we add 1′, and 8′ 10″ to its latitude, we shall have the longitude of the comet ♈ 29° 18′, with 8° 36′ 26″ north lat.

March 7, 7h.30′ at Paris (that is, March 7, 8h.37′ at Dantzick), from the

more obscure; and that, if its distance were 4 times greater, its light would be 256 times less; and therefore would be hardly perceivable to the naked eye. But now the comets often equal Saturn's light, without exceeding him in their apparent diameters. So the comet of the year 1668, according to Dr. Hooke's observations, equalled in brightness the light of a fixed star of the first magnitude; and its head, or the star in the middle of the coma, appeared, through a telescope of 15 feet, as lucid as Saturn near the horizon; but the diameter of the head was only 25", that is, almost the same with the

periodic times of the satellites of Jupiter.



1d.18h.27′.34″. 3d.13h.13′ 42″. 7d.3h.42′ 36″. 16d.16h.32′ 9″.



The distances of the satellites from Jupiter's centre.



From the observations of 1 2 3 4 semi-diameter of Jupiter.

Borelli 5⅔ 8⅔ 14 24⅔

Townly by the Microm. 5,52 8,78 13,47 24,72

Cassini by the Telescope 5 8 13 23

Cassini by the eclip. of the satel. 5⅔ 9 1423⁄60 253⁄10

From the periodic times 5,667 9,017 14,384 25,299

Mr. Pound has determined, by the help of excellent micrometers, the diameters of Jupiter and the elongation of its satellites after the following manner. The

Jupiter.



1d.18h.27′.34″. 3d.13h.13′ 42″. 7d.3h.42′ 36″. 16d.16h.32′ 9″.



The distances of the satellites from Jupiter's centre.



From the observations of 1 2 3 4 semi-diameter of Jupiter.

Borelli 5⅔ 8⅔ 14 24⅔

Townly by the Microm. 5,52 8,78 13,47 24,72

Cassini by the Telescope 5 8 13 23

Cassini by the eclip. of the satel. 5⅔ 9 1423⁄60 253⁄10

From the periodic times 5,667 9,017 14,384 25,299

Mr. Pound has determined, by the help of excellent micrometers, the diameters of Jupiter and the elongation of its satellites after the following manner. The greatest heliocentric elongation of the

PHÆNOMENON II.



That the circumsaturnal planets, by radii drawn to Saturn's centre, describe areas proportional to the times of description; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate proportion of their distances from its centre.



For, as Cassini from his own observations has determined, their distances from Saturn's centre and their periodic times are as follow.

The periodic times of the satellites of Saturn.



1d.21h.18′ 27″. 2d.17h.41′ 22″. 4d.12h.25′ 12″. 15d.22h.41′ 14″. 79d.7h.48′ 00″.



The distances of the satellites from

continually with the same face; for in its revolution round Saturn, as often as it comes to the eastern part of its orbit, it is scarcely visible, and generally quite disappears; which is like to be occasioned by some spots in that part of its body, which is then turned towards the earth, as M. Cassini has observed. So also the utmost satellite of Jupiter seems to revolve about its axis with a like motion, because in that part of its body which is turned from Jupiter it has a spot, which always appears as if it were in Jupiter's own body, whenever the satellite passes between Jupiter and our

difference of latitudes to be 2° 28′, determined the measure of one degree to be 367196 feet of London measure, that is 57300 Paris toises. M. Picart, measuring an arc of one degree, and 22′ 55″ of the meridian between Amiens and Malvoisine, found an arc of one degree to be 57060 Paris toises. M. Cassini, the father, measured the distance upon the meridian from the town of Collioure in Roussillon to the Observatory of Paris; and his son added the distance from the Observatory to the Citadel of Dunkirk. The whole distance was 486156½ toises and the difference of the latitudes of Collioure and

other as 11⅙ to 10⅙, very nearly. 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

(that is, 5h.10′ at London], by threads directed to the fixed stars, observed the comet in ♎ 8° 30′, with latitude 0° 40′ south. Their observations may be seen in a treatise which Ponthæus published concerning this comet. Cellius, who was present, and communicated his observations in a letter to Cassini saw the comet at the same hour in ♎ 8° 30′, with latitude 0° 30′ south. It was likewise seen by Galletius at the same hour at Avignon (that is, at 5h.42′ morning at London) in ♎ 8° without latitude. But by the theory the comet was at that time in ♎ 8° 16′ 45″, and its latitude was 0° 53′ 7″

time 1d.18h.28′ 36″ as 493½″ to 108 , neglecting those small fractions which, in observing, cannot be certainly determined.

Before the invention of the micrometer, the same distances were determined in semi-diameters of Jupiter thus:—

Distance of the 1st 2d 3d 4th

By Galileo

" Simon Marius

" Cassini

Borelli, more exactly 6

6

5

5⅔ 10

10

8

8⅔ 16

16

13

14 28

26

23

24⅔

After the invention of the micrometer:—

By Townley

" Flamsted

More accurately by the eclipses 5,51

5,31

5,578 8,78

8,85

8,876 13,47

13,98

14,159 24,72

24,23

24,903





And the periodic times of those

And the periodic times of those satellites, by the observations of Mr. Flamsted, are 1d.18h.28′ 36″ | 3d.13h.17′ 54″ | 7d.3h.59′ 36″ | 16d.18h.5′ 13″, as above.

And the distances thence computed are 5,578 | 8,878 | 14,168 | 24,968, accurately agreeing with the distances by observation.

Cassini assures us (p. 388, 389) that the same proportion is observed in the circum-saturnal planets. But a longer course of observations is required before we can have a certain and accurate theory of those planets.

In the circum-solar planets. Mercury and Venus, the same proportion holds with great

were dilated and spread through so great a space, was, as to the magnitude of its nucleus, less than Jupiter (as Mr. Flamsted observed), and therefore was much nearer to the sun: nay, it was even less than Mercury. For on the 17th of that month, when it was nearer to the earth, it appeared to Cassini through a telescope of 35 feet a little less than the globe of Saturn. On the 8th of this month, in the morning, Dr. Halley saw the tail, appearing broad and very short, and as if it rose from the body of the sun itself, at that time very near its rising. Its form was like that of an extraordinary

very remarkably from that time. The tail at the beginning extended itself from west to south, and in a situation almost parallel to the horizon, appearing like a shining beam 23 deg. in length. Afterwards, the light decreasing, its magnitude increased till the comet ceased to be visible; so that Cassini, at Bologna, saw it (Mar. 10, 11, 12) rising from the horizon 32 deg. in length. In Portugal it is said to have taken up a fourth part of the heavens (that is, 45 deg.), extending itself from west to east with a notable brightness; though the whole of it was not seen, because the head in this part

in resisting mediums, 252, 265, 281, 283, 345

Descent or Ascent rectilinear, the spaces described, the times of decryption, and the velocities acquired in such ascent or descent, compared, on the supposition of any kind of centripetal force, 160

Earth, its dimension by Norwood, by Picart, and by Cassini, 405

“ its figure discovered, with the proportion of its diameters, and the measure of the degrees upon the meridian, 405, 409

“ the excess of its height at the equator above its height at the poles, 407, 412

“ its greatest and least semi-diameter, 407

“ its mean semi-diameter, 407

“ the

will never produce any change in the positions or motions of the bodies among themselves.





SCHOLIUM.



Hitherto I have laid down such principles as have been received by mathematicians, and are confirmed by abundance of experiments. By the first two Laws and the first two Corollaries, Galileo discovered that the descent of bodies observed the duplicate ratio of the time, and that the motion of projectiles was in the curve of a parabola; experience agreeing with both, unless so far as these motions are a little retarded by the resistance of the air. When a body is falling, the uniform

the direct and inverse ratios) in the ratio of equality.

SCHOLIUM.



If the ellipsis, by having its centre removed to an infinite distance, de generates into a parabola, the body will move in this parabola; and the force, now tending to a centre infinitely remote, will become equable. Which is Galileo's theorem. And if the parabolic section of the cone (by changing the inclination of the cutting plane to the cone) degenerates into an hyperbola, the body will move in the perimeter of this hyperbola, having its centripetal force changed into a centrifugal force. And in like manner as in the

that is, to the term . Therefore the force sought is as , or, which is the same thing, as . As if the ordinate describe a parabola, m being = 2, and n = 1, the force will be as the given quantity 2B°, and therefore is given. Therefore with a given force the body will move in a parabola, as Galileo has demonstrated. If the ordinate describe an hyperbola, m being = 0 - 1, and n = 1, the force will be as 2A-3 or 2B3; and therefore a force which is as the cube of the ordinate will cause the body to move in an hyperbola. But leaving this kind of propositions, I shall go on to some others

the plane of incidence Aa in R; and let the line of emergence KI be produced and meet HM in L. About the centre L, with the interval LI, let a circle be described cutting both HM in P and Q, and MI produced in N; and, first, if the attraction or impulse be supposed uniform, the curve HI (by what Galileo has demonstrated) be a parabola, whose property is that of a rec- tangle under its given latus rectum and the line IM is equal to the square of HM; and moreover the line HM will be bisected in L. Whence if to MI there be let fall the perpendicular LO, MO, OR will be equal: and adding the equal

no farther towards the plane Ee. But neither can it proceed in the line of emergence Rd; because it is perpetually attracted or impelled towards the medium of incidence. It will return, therefore, between the planes Cc, Dd, describing an arc of a parabola QRq, whose principal vertex (by what Galileo has demonstrated) is in R, cutting the plane Cc in the same angle at q, that it did before at Q; then going on in the parabolic arcs qp, ph, &c., similar and equal to the former arcs QP, PH, &c., it will cut the rest of the planes in the same angles at p, h, &c., as it did before in P, H, &c., and

be expounded by . This decrement arises from the resistance which retards the body, and from the gravity which accelerates it. Gravity, in a falling body, which in its fall describes the space NI, produces a velocity with which it would be able to describe twice that space in the same time, as Galileo has demonstrated; that is, the velocity : but if the body describes the arc HI, it augments that arc only by the length HI - HN or ; and therefore generates only the velocity . Let this velocity be added to the beforementioned decrement, and we shall have the decrement of the velocity arising from

the third term for Roo. But since there are no more terms, the co-efficient S of the fourth term will vanish; and therefore the quantity , to which the density of the medium is proportional, will be nothing. Therefore, where the medium is of no density, the projectile will move in a parabola; as Galileo hath heretofore demonstrated. Q.E.I.

Example 3. Let the line AGK be an hyperbola, having its asymptote NX perpendicular to the horizontal plane AK, to find the density of the medium that will make a projectile move in that line.

Let MX be the other asymptote, meeting the ordinate DG produced in

and through the point I let there be drawn the right line KL parallel to the horizon and meeting the ice on both the sides thereof in K and L. Then the velocity of the water running out at the hole EF will be the same that it would acquire by falling from I through the space IG. Therefore, by Galileo's Theorems, IG will be to IH in the duplicate ratio of the velocity of the water that runs out at the hole to the velocity of the water in the circle AB, that is, in the duplicate ratio of the circle AB to the circle EF; those circles being reciprocally as the velocities of the water which in the

water will still run out with the same velocity as before, if the magnitude of the hole be the same. For though an heavy body takes a longer time in descending to the same depth, by an oblique line, than by a perpendicular line, yet in both cases it acquires in its descent the same velocity; as Galileo has demonstrated.

Case 3. The velocity of the water is the same when it runs out through a hole in the side of the vessel. For if the hole be small, so that the interval between the superficies AB and KL may vanish as to sense, and the stream of water horizontally issuing out may form a parabolic

an inch

0,75 of an inch

0,75 of an inch 4″

4-

4

4+

4

4+ 510 grains

642 grains

599 grains

515 grains

483 grains

641 grains 5,1 inches

5,2 inches

5,1 inches

5,0 inches

5,0 inches

5,2 inches 8"½

8

8

8¼

8½

8

But the times observed must be corrected; for the globes of mercury (by Galileo's theory), in 4 seconds of time, will describe 257 English feet, and 220 feet in only 3″42‴. So that the wooden table, when the pin was taken out, did not turn upon its hinges so quickly as it ought to have done; and the slowness of that revolution hindered the descent of the globes at the

16d 18h.05′ 13″ is to the time 1d.18h.28′ 36″ as 493½″ to 108 , neglecting those small fractions which, in observing, cannot be certainly determined.

Before the invention of the micrometer, the same distances were determined in semi-diameters of Jupiter thus:—

Distance of the 1st 2d 3d 4th

By Galileo

" Simon Marius

" Cassini

Borelli, more exactly 6

6

5

5⅔ 10

10

8

8⅔ 16

16

13

14 28

26

23

24⅔

After the invention of the micrometer:—

By Townley

" Flamsted

More accurately by the eclipses 5,51

5,31

5,578 8,78

8,85

8,876 13,47

13,98

14,159 24,72

24,23

24,903





And the

drawn off from a rectilinear motion, and retained in its orbit.



The mean distance of the moon from the earth in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60⅓; to Street, 602⁄5; and to Tycho, 56½. But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the fixed stars, and that by four or five minutes near the horizon, did thereby increase the moon's horizontal parallax by

a rectilinear motion, and retained in its orbit.



The mean distance of the moon from the earth in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60⅓; to Street, 602⁄5; and to Tycho, 56½. But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the fixed stars, and that by four or five minutes near the horizon, did thereby increase the moon's horizontal parallax by a like number

in the same place of the heavens; yet in the former case the tail of the comet (as well by my observations as by the observations of others) deviated from the opposition of the sun towards the north by an angle of 4½ degrees; whereas in the latter there was (according to the observations of Tycho) a deviation of 21 degrees towards the south. The refraction, therefore, of the heavens being thus disproved, it remains that the phænomena of the tails of comets must be derived from some reflecting matter.

And that the tails of comets do arise from their heads, and tend towards the parts

Gemma did not see upon the 8th of November, 1572, though he was observing that part of the heavens upon that very night, and the sky was perfectly serene; but the next night (November 9) he saw it shining much brighter than any of the fixed stars, and scarcely inferior to Venus in splendor. Tycho Brahe saw it upon the 11th of the same month, when it shone with the greatest lustre; and from that time he observed it to decay by little and little; and in 16 months' time it entirely disappeared. In the month of November, when it first appeared, its light was equal to that of Venus. In the month of

every such force as imaginary and precarious, and of no use in the phænomena of the heavens; and the whole remaining force by which Jupiter is impelled will be directed (by Prop. III, Cor. I) to the centre of the sun.

The distances of the planets from the sun come out the same, whether, with Tycho, we place the earth in the centre of the system, or the sun with Copernicus: and we have already proved that these distances are true in Jupiter.

Kepler and Bullialdus have, with great care (p. 388), determined the distances of the planets from the sun; and hence it is that their tables agree

decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes

the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes in the horizon, did thereby augment the horizontal parallax of the moon by about the like

31½') to be about 1⁄42 of the earth, as 43 to , or as about 128 to 127. And therefore the semi-diameter of the orbit, that is, the distance between the centres of the moon and earth, will in this case be 60½ semi-diameters of the earth, almost the same with that assigned by Copernicus, which the Tychonic observations by no means disprove; and, therefore, the duplicate proportion of the decrement of the force holds good in this distance. I have neglected the increment of the orbit which arises from the action of the sun as inconsiderable; but if that is subducted, the true distance will remain

IV, Prop. III) that the circum-terrestrial force, compared with the circum-solar, is very small.

Ricciolus and Vendelin have severally tried to determine the sun's parallax from the moon's dichotomies observed by the telescope, and they agree that it does not exceed half a minute.

Kepler, from Tycho's observations and his own, found the parallax of Mars insensible, even in opposition to the sun, when that parallax is some thing greater than the sun's.

Flamsted attempted the same parallax with the micrometer in the perigeon position of Mars, but never found it above 25″; and thence concluded

heavens; but the astronomical hypotheses are to be corrected in the motion of the nodes; for the nodes admit the greatest equation or prosthaphæresis in their octants, and this inequality is most conspicuous when the moon is in the nodes, and therefore also in the octants; and hence it was that Tycho, and others after him, referred this inequality to the octants of the moon, and made it menstrual; but the reasons by us adduced prove that it ought to be referred to the octants of the nodes, and to be made annual.

Beside those inequalities taken notice of by astronomers (p. 414, 445, 447,)

in the same place of the heavens; yet in the former case the tail of the comet (as well by my observations as by the observations of others) deviated from the opposition of the sun towards the north by an angle of 4½ degrees, whereas in the latter there was (according to the observation of Tycho) a deviation of 21 degrees towards the south. The refraction, therefore, of the heavens being thus disproved, it remains that the phænomena of the tails of comets must be derived from some reflecting matter. That vapours sufficient to fill such immense spaces may arise from the comet's

one and the same comet that appeared all that time, and also that the orbit of that comet is here rightly defined.

In the foregoing table we have omitted the observations of Nov. 16, 18, 20. and 23, as not sufficiently accurate, for at those times several persons had observed the comet. Nov. 17, O. S. Ponthæus and his companions, at 6h. in the morning at Rome (that is, 5h.10′ at London], by threads directed to the fixed stars, observed the comet in ♎ 8° 30′, with latitude 0° 40′ south. Their observations may be seen in a treatise which Ponthæus published concerning this comet. Cellius, who was present,

several persons had observed the comet. Nov. 17, O. S. Ponthæus and his companions, at 6h. in the morning at Rome (that is, 5h.10′ at London], by threads directed to the fixed stars, observed the comet in ♎ 8° 30′, with latitude 0° 40′ south. Their observations may be seen in a treatise which Ponthæus published concerning this comet. Cellius, who was present, and communicated his observations in a letter to Cassini saw the comet at the same hour in ♎ 8° 30′, with latitude 0° 30′ south. It was likewise seen by Galletius at the same hour at Avignon (that is, at 5h.42′ morning at London) in ♎ 8°

was likewise seen by Galletius at the same hour at Avignon (that is, at 5h.42′ morning at London) in ♎ 8° without latitude. But by the theory the comet was at that time in ♎ 8° 16′ 45″, and its latitude was 0° 53′ 7″ south.

Nov. 18, at 6h.30′ in the morning at Rome (that is, at 5h.40′ at London), Ponthæus observed the comet in ♎ 13° 30′, with latitude 1° 20′ south; and Cellius in ♎ 13° 30′, with latitude 1° 00 south. But at 5h.30′ in the morning at Avignon, Galletius saw it in ♎ 13° 00′, with latitude 1° 00 south. In the University of La Fleche, in France, at 5h. in the morning (that is, at 5h.9′

Nov. 20, Montenari, professor of astronomy at Padua, at 6h. in the morning at Venice (that is, 5h.10′ at London), saw the comet in ♎ 23°, with latitude 1° 30′ south. The same day, at Boston, it was distant from Spica ♍ by about 4° of longitude east, and therefore was in ♎ 23° 24′ nearly.

Nov. 21, Ponthæus and his companions, at 7¼h. in the morning, ob served the comet in ♎ 27° 50′, with latitude 1° 16′ south; Cellius, in ♎ 28°; P. Ango at 5h. in the morning, in ♎ 27° 45′; Montenari in ♎ 27° 51′. The same day, in the island of Jamaica, it was seen near the beginning of ♏, and of about the same

less than 2° 38′; and since the latitude, as we said, by the concurring observations of Montenari, Ango, and Hook, was continually increasing, therefore, it was now, on the 24th, something greater than 1° 58′; and, taking the mean quantity, may be reckoned 2° 18′, without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or 1½°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as

than 1° 58′; and, taking the mean quantity, may be reckoned 2° 18′, without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or 1½°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and

are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and sometimes by Ponthæus and Cellius. The same day, at 5h. morning, at Ballasore, the comet was observed in ♏ 11° 45′; and, therefore, at 5h. morning at London, was in ♏ 13° nearly. And, by the theory, the comet was at that time in ♏ 13° 22′ 2″.

Nov. 25, before sunrise, Montenari observed the comet in ♏ 17¾′ nearly; and

its place March 5; and V its place March 9. In determining the length of the tail, I made the following observations.

Nov. 4 and 6, the tail did not appear; Nov. 11, the tail just begun to shew itself, but did not appear above ½ deg. long through a 10 feet telescope; Nov. 17, the tail was seen by Ponthæus more than 15° long; Nov. 18, in New-England, the tail appeared 30° long, and directly opposite to the sun, extending itself to the planet Mars, which was then in ♍, 9° 54′: Nov. 19. in Maryland, the tail was found 15° or 20° long; Dec. 10 (by the observation of Mr. Flamsted), the tail passed

Sagitta, nor did it reach much farther; terminating in ♒ 4°, with latitude 42½° north nearly. But these things are to be understood of the length of the brighter part of the tail; for with a more faint light, observed, too, perhaps, in a serener sky, at Rome, Dec. 12, 5h.40′, by the observation of Ponthæus, the tail arose to 10° above the rump of the Swan, and the side thereof towards the west and towards the north was 45′ distant from this star. But about that time the tail was 3° broad towards the upper end; and therefore the middle thereof was 2° 15′ distant from that star towards the south, and

directed to the bright star in the eastern shoulder of Auriga, and therefore deviated from the opposition of the sun towards the north by an angle of 10°. Lastly, Feb. 10, with a telescope I observed the tail 2° long; for that fainter light which I spoke of did not appear through the glasses. But Ponthæus writes, that, on Feb. 7, he saw the tail 12° long. Feb. 25, the comet was without a tail, and so continued till it disappeared.



Now if one reflects upon the orbit described, and duly considers the other appearances of this comet, he will be easily satisfied that the bodies of comets are solid,

and whose ordinate BD will be as the square of the line AB. On the same Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks. By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis,

did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with

and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the

centripetal force, describes in any time, is a mean proportional between the diameter of the circle, and the space which the same body falling by the same given force would descend through in the same given time.





SCHOLIUM.



The case of the 6th Corollary obtains in the celestial bodies (as Sir Christopher Wren, Dr. Hooke, and Dr. Halley have severally observed); and therefore in what follows, I intend to treat more at large of those things which relate to centripetal force decreasing in a duplicate ratio of the distances from the centres.

Moreover, by means of the preceding Proposition and its

cutting CE in i, we may produce iE to V, so as EV may be to Ei as FH to HI, and then draw Vf parallel to BD. It will come to the same, if about the centre i with an interval IH, we describe a circle cutting BD in X, and produce iX to Y so as iY may be equal to IF, and then draw Yf parallel to BD.

Sir Christopher Wren and Dr. Wallis have long ago given other solutions of this Problem.





PROPOSITION XXIX. PROBLEM XXI.



To describe a trajectory given in kind, that may be cut by four right lines given by position, into parts given in order, kind, and proportion.



Suppose a trajectory is to be described that

from any place to the centre, and the time equal to it in which the body revolving uniformly about the centre of the globe at any distance describes an arc of a quadrant. For this time (by Case 2) is to the time of half the oscillation in any cycloid QRS as 1 to .

Cor. 2. Hence also follow what Sir Christopher Wren and M. Huygens have discovered concerning the vulgar cycloid. For if the diameter of the globe be infinitely increased, its sphaerical superficies will be changed into a plane, and the centripetal force will act uniformly in the direction of lines perpendicular to that plane, and this cycloid of

our's will become the same with the common cycloid. But in that case the length of the arc of the cycloid between that plane and the describing point will become equal to four times the versed sine of half the arc of the wheel between the same plane and the describing point, as was discovered by Sir Christopher Wren. And a pendulum between two such cycloids will oscillate in a similar and equal cycloid in equal times, as M. Huygens demonstrated. The descent of heavy bodies also in the time of one oscillation will be the same as M. Huygens exhibited.

The propositions here demonstrated are adapted to the true

parallax of the orbit, in so far as the same is pretty nearly collected by the supposition that the comets move uniformly in right lines. The method of collecting the distance of a comet according to this hypothesis from four observations (first attempted by Kepler, and perfected by Dr. Wallis and Sir Christopher Wren) is well known; and the comets reduced to this regularity generally pass through the middle of the planetary region. So the comets of the year 1607 and 1618, as their motions are defined by Kepler, passed between the sun and the earth; that of the year 1664 below the orbit of Mars; and that in

generally pass through the middle of the planetary region. So the comets of the year 1607 and 1618, as their motions are defined by Kepler, passed between the sun and the earth; that of the year 1664 below the orbit of Mars; and that in 1680 below the orbit of Mercury, as its motion was defined by Sir Christopher Wren and others. By a like rectilinear hypothesis, Hevelius placed all the comets about which we have any observations below the orbit of Jupiter. It is a false notion, therefore, and contrary to astronomical calculation, which some have entertained, who, from the regular motion of the comets, either

incidence, will be equal to the same also at last. Q.E.D.





SCHOLIUM.



These attractions bear a great resemblance to the reflexions and refractions of light made in a given ratio of the secants, as was discovered by Snellius; and consequently in a given ratio of the sines, as was exhibited by Des Cartes. For it is now certain from the phenomena of Jupiter's Satellites, confirmed by the observations of different astronomers, that light is propagated in succession, and requires about seven or eight minutes to travel from the sun to the earth. Moreover, the rays of light that are in our air (as

in D; that point D will touch the curve sought CDE, and, by touching it any where at pleasure, will determine that curve. Q.E.I.

Cor. 1. By causing the point A or B to go off sometimes in infinitum, and sometimes to move towards other parts of the point C, will be obtained all those figures which Cartesius has exhibited in his Optics and Geometry relating to refractions. The invention of which Cartesius having thought fit to conceal, is here laid open in this Proposition.

Cor. 2. If a body lighting on any superficies CD in the direction of a right line AD, drawn according to any law, should emerge

determine that curve. Q.E.I.

Cor. 1. By causing the point A or B to go off sometimes in infinitum, and sometimes to move towards other parts of the point C, will be obtained all those figures which Cartesius has exhibited in his Optics and Geometry relating to refractions. The invention of which Cartesius having thought fit to conceal, is here laid open in this Proposition.

Cor. 2. If a body lighting on any superficies CD in the direction of a right line AD, drawn according to any law, should emerge in the direction of another right line DK; and from the point C there be drawn curve lines CP, CQ,

of all bodies within the reach of our experiments; and therefore (by Rule III) to be affirmed of all bodies whatsoever. If the æther, or any other body, were either altogether void of gravity, or were to gravitate less in proportion to its quantity of matter, then, because (according to Aristotle, Des Cartes, and others) there is no diiference betwixt that and other bodies but in mere form of matter, by a successive change from form to form, it might be changed at last into a body of the same condition with those which gravitate most in proportion to their quantity of matter; and, on the other hand,

in curvilinear orbits, are questions which we do not know how the ancients explained; and probably it was to give some sort of satisfaction to this difficulty that solid orbs were introduced.

The later philosophers pretend to account for it either by the action of certain vortices, as Kepler and Des Cartes; or by some other principle of impulse or attraction, as Borelli, Hooke, and others of our nation; for, from the laws of motion, it is most certain that these effects must proceed from the action of some force or other.

But our purpose is only to trace out the quantity and properties of this

388

“ its distance from the sun, 389

“ the motion of its aphelion, 405

Matter, its quantity of matter defined, 73

“ its vis insita defined, 74

“ its impressed force defined, 74

“ its extension, hardness, impenetrability, mobility, vis inertiæ, gravity, how discovered, 385

“ subtle matter of Descartes inquired into, 320

Mechanical Powers explained and demonstrated, 94

Mercury, its periodic time, 388

“ its distance from the sun, 389

“ the motion of its aphelion, 405

Method of first and last ratios, 95

“ of transforming figures into others of the same analytical order, 141

“ of fluxions,

mediums, 287

“ of funependulous bodies in resisting mediums, 304

“ and resistance of fluids, 323

“ propagated through fluids, 356

“ of fluids after the manner of a vortex, or circular, 370

Motions, composition and resolution of them, 84

Ovals for optic uses, the method of finding them which Cartesius concealed, 246

“ a general solution of Cartesius's problem, 247, 248

Orbits, the invention of those which are described by bodies going off from a given place with a given velocity according to a given right line, when the centripetal force is reciprocally as the square of the distance, and the

mediums, 304

“ and resistance of fluids, 323

“ propagated through fluids, 356

“ of fluids after the manner of a vortex, or circular, 370

Motions, composition and resolution of them, 84

Ovals for optic uses, the method of finding them which Cartesius concealed, 246

“ a general solution of Cartesius's problem, 247, 248

Orbits, the invention of those which are described by bodies going off from a given place with a given velocity according to a given right line, when the centripetal force is reciprocally as the square of the distance, and the absolute quantity of that force is known, 123

“

to the times, 386, 387, 390

“ “ revolve in periodic times that are in the sesquiplicate proportion of their distances from the primary, 386, 387

Problem Keplerian, solved by the trochoid and by approximations, 157 to 160

“ “ of the ancients, of four lines, related by Pappus, and attempted by Cartesius, by an algebraic calculus solved by a geometrical composition, 135

Projectiles move in parabolas when the resistance of the medium is taken away, 91, 115, 243, 273

“ their motions in resisting mediums, 255, 268

Pulses of the air, by which sounds are propagated, their intervals or breadths

of mediums is somewhat more largely handled than before; and new experiments of the resistance of heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on; and there are added new observations of Mr. Pound's of the proportion of the diameters of Jupiter to each other: there are, besides, added Mr. Kirk's observations of the comet in 1680; the orbit of that comet computed in an ellipsis by Dr. Halley; and the orbit of the comet in 1723, computed by Mr. Bradley.





BOOK I.





THE MATHEMATICAL

from Jupiter's centre.



From the observations of 1 2 3 4 semi-diameter of Jupiter.

Borelli 5⅔ 8⅔ 14 24⅔

Townly by the Microm. 5,52 8,78 13,47 24,72

Cassini by the Telescope 5 8 13 23

Cassini by the eclip. of the satel. 5⅔ 9 1423⁄60 253⁄10

From the periodic times 5,667 9,017 14,384 25,299

Mr. Pound has determined, by the help of excellent micrometers, the diameters of Jupiter and the elongation of its satellites after the following manner. The greatest heliocentric elongation of the fourth satellite from Jupiter's centre was taken with a micrometer in a 15 feet telescope, and at the mean

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 13¾ to 12¾

315⁄7. HO was to HI as 7 to 6, and, produced, did pass between the stars D and E, so as the distance of the star D from this right line was 1⁄6CD. LM was to LN as 2 to 9, and, produced, did pass through the star H. Thus were the positions of the fixed stars determined in respect of one another.



Mr. Pound has since observed a second time the positions of those fixed stars amongst themselves, and collected their longitudes and latitudes according to the following table.

The

fixed

stars. Their

Longitudes Latitude

North. The

fixed

stars. Their

Longitudes Latitude

North.



A

B

C

E

F

G

being then extremely near the horizon, was scarcely discernible, and therefore its place could not be determined with that certainty as in the foregoing observations.

Prom these observations, by constructions of figures and calculations, I deduced the longitudes and latitudes of the comet; and Mr. Pound, by correcting the places of the fixed stars, hath determined more correctly the places of the comet, which correct places are set down above. Though my micrometer was none of the best, yet the errors in longitude and latitude (as derived from my observations) scarcely exceed one minute. The comet

month, on the 6th and 11th O. S.; from its positions to the nearest fixed stars observed with sufficient accuracy, sometimes with a two feet, and sometimes with a ten feet telescope; from the difference of longitudes of Coburg and London, 11°; and from the places of the fixed stars observed by Mr. Pound, Dr. Halley has determined the places of the comet as follows:—



Nov. 3, 17h.2′, apparent time at London, the comet was in ♌ 29 deg. 51′, with 1 deg. 17′ 45″ latitude north.

November 5. 15h.58′ the comet was in ♍ 3° 23′, with 1° 6′ north lat.

November 10, 16h.31′, the comet was equally distant

20″. Its perihelion distance from the sun 998651 parts, of which the radius of the orbis magnus contains 1000000, and the equal time of its perihelion September 16d 16h.10′. The places of this comet computed in this orbit by Mr. Bradley, and compared with the places observed by himself, his uncle Mr. Pound, and Dr. Halley, may be seen in the following table.

1723

Eq. Time. Comet's

Long. obs. Lat. Nor.

obs. Comet's

Lon. com. Lat.Nor.

comp. Diff.

Lon. Diff.

Lat.

d. h. ′

Oct. 9.8. 5

10.6.21

12.7.22

14.8.57

15.6.35

21.6.22

22. 6.24

24.8. 2

29.8.56

30.6.20

Nov. 5.5.53

8.7. 6

for the periodic times are the same, and the dimensions of the orbits are the same, whether the sun revolves about the earth, or the earth about the sun. And as to the measures of the periodic times, all astronomers are agreed about them. But for the dimensions of the orbits, Kepler and Bullialdus, above all others, have determined them from observations with the greatest accuracy; and the mean distances corresponding to the periodic times differ but insensibly from those which they have assigned, and for the most part fall in between them; as we may see from the following table.

The

stars, of the planets and earth revolving about the sun, in days and decimal parts of a day.



♄ ♃ ♂ ♁ ♀ ☿

10759,275. 4332,514. 686,9785. 365,2565. 224,6176. 87,9692.





The mean distances of the planets and of the earth from the sun.



♄ ♃ ♂

According to Kepler 951000. 519650. 152350.

” to Bullialdus 954198. 522520. 152350.

” to the periodic times 954006. 520096. 152369.



♁ ♀ ☿

According to Kepler 100000. 72400. 38806.

” to Bullialdus 100000. 72398. 38585.

” to the periodic times 100000. 72333. 38710.

As to Mercury and Venus, there can be no doubt about their distances from the sun; for

224,6176. 87,9692.





The mean distances of the planets and of the earth from the sun.



♄ ♃ ♂

According to Kepler 951000. 519650. 152350.

” to Bullialdus 954198. 522520. 152350.

” to the periodic times 954006. 520096. 152369.



♁ ♀ ☿

According to Kepler 100000. 72400. 38806.

” to Bullialdus 100000. 72398. 38585.

” to the periodic times 100000. 72333. 38710.

As to Mercury and Venus, there can be no doubt about their distances from the sun; for they are determined by the elongations of those planets from the sun; and for the distances of the superior planets, all dispute is cut off

directed (by Prop. III, Cor. I) to the centre of the sun.

The distances of the planets from the sun come out the same, whether, with Tycho, we place the earth in the centre of the system, or the sun with Copernicus: and we have already proved that these distances are true in Jupiter.

Kepler and Bullialdus have, with great care (p. 388), determined the distances of the planets from the sun; and hence it is that their tables agree best with the heavens. And in all the planets, in Jupiter and Mars, in Saturn and the earth, as well as in Venus and Mercury, the cubes of their distances are as the

the said proportion to hold exactly; for the distances of Saturn, Jupiter, Mars, the Earth, Venus, and Mercury, from the sun, drawn from the observations of astronomers, are, according to the computation of Kepler, as the numbers 951000, 519650, 152350, 100000, 72400, 38806; by the computation of Bullialdus, as the numbers 954198, 522520, 152350, 100000, 72398, 38585; and from the periodic times they come out 953806, 520116, 152399, 100000, 72333, 38710. Their distances, according to Kepler and Bullialdus, scarcely differ by any sensible quantity, and where they differ most the distances drawn from

of Kepler, as the numbers 951000, 519650, 152350, 100000, 72400, 38806; by the computation of Bullialdus, as the numbers 954198, 522520, 152350, 100000, 72398, 38585; and from the periodic times they come out 953806, 520116, 152399, 100000, 72333, 38710. Their distances, according to Kepler and Bullialdus, scarcely differ by any sensible quantity, and where they differ most the distances drawn from the periodic times, fall in between them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre

periodic times, fall in between them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of

first of Aries. So it is found by experience that the morning tides in winter 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 at Bristol by the height of 15 inches, according to the observations of Colepress and Sturmy.

But the motions which we have been describing suffer some alteration from that force of reciprocation, which the waters, being once moved, retain a little while by their vis insita. Whence it comes to pass that the tides may continue for some time, though the actions of the luminaries should

proportion is to be collected from the proportion of the motions of the sea, which are the effects of those forces. Before the mouth of the river Avon, three miles below Bristol, the height of the ascent of the water in the vernal and autumnal syzygies of the luminaries (by the observations of Samuel Sturmy) amounts to about 45 feet, but in the quadratures to 25 only. The former of those heights arises from the sum of the aforesaid forces, the latter from their difference. If, therefore, S and L are supposed to represent respectively the forces of the sun and moon while they are in the equator, as

by more than 7 or 8 feet. Suppose the greatest difference of those heights to be 9 feet, and L + S will be to L - S as 20½ to 11½, or as 41 to 23; a proportion that agrees well enough with the former. But because of the great tide at Bristol, we are rather to depend upon the observations of Sturmy; and, therefore, till we procure something that is more certain, we shall use the proportion of 9 to 5.

But because of the reciprocal motions of the waters, the greatest tides do not happen at the times of the syzygies of the luminaries, but, as we have said before, are the third in order after

the greatest tides do not happen at the times of the syzygies of the luminaries, but, as we have said before, are the third in order after the syzygies; or (reckoning from the syzygies) follow next after the third appulse of the moon to the meridian of the place after the syzygies; or rather (as Sturmy observes) are the third after the day of the new or full moon, or rather nearly after the twelfth hour from the new or full moon, and therefore fall nearly upon the forty-third hour after the new or full of the moon. But in this port they fall out about the seventh hour after the appulse of the

node of the moon is about the first of Aries. So the morning tides in winter exceed those of the evening, and the evening tides exceed those of the morning in summer; at Plymouth by the height of one foot, but at Bristol by the height of 15 inches, according to the observations of Colepress and Sturmy.

But the motions which we have been describing suffer some alteration from that force of reciprocation which the waters [having once received] retain a little while by their vis insita; whence it comes to pass that the tides may continue for some time, though the actions of the luminaries should

obstructs the ingress of the waters from the sea, and promotes their egress to the sea, making the ingress later and slower, and the egress sooner and faster; and hence it is that the reflux is of longer duration that the influx, especially far up the rivers, where the force of the sea is less. So Sturmy tells us, that in the river Avon, three miles below Bristol, the water flows only five hours, but ebbs seven; and without doubt the difference is yet greater above Bristol, as at Caresham or the Bath. This difference does likewise depend upon the quantity of the flux and reflux; for the more

then by means of this rule predict the quantities of the tides to every syzygy and quadrature.

At the mouth of the river Avon, three miles below Bristol (p. 450 to 453), in spring and autumn, the whole ascent of the water in the conjunction or opposition of the luminaries (by the observation of Sturmy) is about 45 feet, but in the quadratures only 25. Because the apparent diameters of the luminaries are not here determined, let us assume them in their mean quantities, as well as the moon's declination in the equinoctial quadratures in its mean quantity, that is, 23½°; and the versed sine of

of 7 feet, with a micrometer and threads placed in the focus of the telescope; by which instruments we determined the positions both of the fixed stars among themselves, and of the comet in respect of the fixed stars. Let A represent the star of the fourth magnitude in the left heel of Perseus (Bayer's ο), B the following star of the third magnitude in the left foot (Bayer's ζ), C a star of the sixth magnitude (Bayer's n) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, α, β, γ, δ, other smaller stars in the same foot; and let p, P, Q, R, S, T, V, X, represent the places

by which instruments we determined the positions both of the fixed stars among themselves, and of the comet in respect of the fixed stars. Let A represent the star of the fourth magnitude in the left heel of Perseus (Bayer's ο), B the following star of the third magnitude in the left foot (Bayer's ζ), C a star of the sixth magnitude (Bayer's n) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, α, β, γ, δ, other smaller stars in the same foot; and let p, P, Q, R, S, T, V, X, represent the places of the comet in the observations above set down; and, reckoning the

positions both of the fixed stars among themselves, and of the comet in respect of the fixed stars. Let A represent the star of the fourth magnitude in the left heel of Perseus (Bayer's ο), B the following star of the third magnitude in the left foot (Bayer's ζ), C a star of the sixth magnitude (Bayer's n) in the heel of the same foot, and D, E, F, G, H, I, K, L, M, N, O, Z, α, β, γ, δ, other smaller stars in the same foot; and let p, P, Q, R, S, T, V, X, represent the places of the comet in the observations above set down; and, reckoning the distance AB of 807⁄12 parts, AC was 52¼ of those

the comet as follows:—



Nov. 3, 17h.2′, apparent time at London, the comet was in ♌ 29 deg. 51′, with 1 deg. 17′ 45″ latitude north.

November 5. 15h.58′ the comet was in ♍ 3° 23′, with 1° 6′ north lat.

November 10, 16h.31′, the comet was equally distant from two stars in ♌ which are σ and τ in Bayer; but it had not quite touched the right line that joins them, but was very little distant from it. In Flamsted's catalogue this star σ was then in ♍ 14° 15′, with 1 deg. 41′ lat. north nearly, and τ in ♍ 17° 3½′, with 0 deg. 34′ lat. south; and the middle point between those stars was ♍ 15° 39¼′,

of La Fleche, in France, at 5h. in the morning (that is, at 5h.9′ at London), it was seen by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayers ψ; and the other is the outmost of the wing, Bayer's θ. Whence the comet was then in ♎ 12° 46′ with latitude 50′ south. And I was informed by Dr. Halley, that on the same day at Boston in New England, in the latitude of 42½ deg. at 5h. in the morning (that is, at 9h.44′ in the morning at London), the comet was seen near ♎ 14°, with latitude 1° 30′

19. in Maryland, the tail was found 15° or 20° long; Dec. 10 (by the observation of Mr. Flamsted), the tail passed through the middle of the distance intercepted between the tail of the Serpent of Ophiuchus and the star δ in the south wing of Aquila, and did terminate near the stars A, ω, b, in Bayer's tables. Therefore the end of the tail was in ♑ 19½°, with latitude about 34¼° north; Dec 11, it ascended to the head of Sagitta (Bayer's α, β), terminating in ♑ 26° 43′, with latitude 38° 34′ north; Dec. 12, it passed through the middle of Sagitta, nor did it reach much farther; terminating in ♒

of the distance intercepted between the tail of the Serpent of Ophiuchus and the star δ in the south wing of Aquila, and did terminate near the stars A, ω, b, in Bayer's tables. Therefore the end of the tail was in ♑ 19½°, with latitude about 34¼° north; Dec 11, it ascended to the head of Sagitta (Bayer's α, β), terminating in ♑ 26° 43′, with latitude 38° 34′ north; Dec. 12, it passed through the middle of Sagitta, nor did it reach much farther; terminating in ♒ 4°, with latitude 42½° north nearly. But these things are to be understood of the length of the brighter part of the tail; for with a

Nov. 17, O. S. Ponthæus and his companions, at 6h. in the morning at Rome (that is, 5h.10′ at London], by threads directed to the fixed stars, observed the comet in ♎ 8° 30′, with latitude 0° 40′ south. Their observations may be seen in a treatise which Ponthæus published concerning this comet. Cellius, who was present, and communicated his observations in a letter to Cassini saw the comet at the same hour in ♎ 8° 30′, with latitude 0° 30′ south. It was likewise seen by Galletius at the same hour at Avignon (that is, at 5h.42′ morning at London) in ♎ 8° without latitude. But by the theory the

morning at London) in ♎ 8° without latitude. But by the theory the comet was at that time in ♎ 8° 16′ 45″, and its latitude was 0° 53′ 7″ south.

Nov. 18, at 6h.30′ in the morning at Rome (that is, at 5h.40′ at London), Ponthæus observed the comet in ♎ 13° 30′, with latitude 1° 20′ south; and Cellius in ♎ 13° 30′, with latitude 1° 00 south. But at 5h.30′ in the morning at Avignon, Galletius saw it in ♎ 13° 00′, with latitude 1° 00 south. In the University of La Fleche, in France, at 5h. in the morning (that is, at 5h.9′ at London), it was seen by P. Ango, in the middle between two small stars,

the comet in ♎ 23°, with latitude 1° 30′ south. The same day, at Boston, it was distant from Spica ♍ by about 4° of longitude east, and therefore was in ♎ 23° 24′ nearly.

Nov. 21, Ponthæus and his companions, at 7¼h. in the morning, ob served the comet in ♎ 27° 50′, with latitude 1° 16′ south; Cellius, in ♎ 28°; P. Ango at 5h. in the morning, in ♎ 27° 45′; Montenari in ♎ 27° 51′. The same day, in the island of Jamaica, it was seen near the beginning of ♏, and of about the same latitude with Spica ♍, that is, 2° 2′. The same day, at 5h. morning, at Ballasore, in the East Indies (that is, at

of Montenari, Ango, and Hook, was continually increasing, therefore, it was now, on the 24th, something greater than 1° 58′; and, taking the mean quantity, may be reckoned 2° 18′, without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or 1½°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by

and, taking the mean quantity, may be reckoned 2° 18′, without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or 1½°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and sometimes by

rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and sometimes by Ponthæus and Cellius. The same day, at 5h. morning, at Ballasore, the comet was observed in ♏ 11° 45′; and, therefore, at 5h. morning at London, was in ♏ 13° nearly. And, by the theory, the comet was at that time in ♏ 13° 22′ 2″.

Nov. 25, before sunrise, Montenari observed the comet in ♏ 17¾′ nearly; and Cellius

and Cellius. The same day, at 5h. morning, at Ballasore, the comet was observed in ♏ 11° 45′; and, therefore, at 5h. morning at London, was in ♏ 13° nearly. And, by the theory, the comet was at that time in ♏ 13° 22′ 2″.

Nov. 25, before sunrise, Montenari observed the comet in ♏ 17¾′ nearly; and Cellius observed at the same time that the comet was in a right line between the bright star in the right thigh of Virgo and the southern scale of Libra; and this right line cuts the comet's way in ♏ 18° 36′. And, by the theory, the comet was in ♏ 18⅓° nearly.

From all this it is plain that these

distance of the stars γ and A, that is, 1° 19′ 46″ of a great circle; and therefore in the parallel of the latitude of the star γ it was 1° 20′ 26″. Therefore if from the longitude of the star γ there be subducted the longitude 1° 20′ 26″, there will remain the longitude of the comet ♈ 27° 9′ 49″. M. Auzout, from this observation of his, placed the comet in ♈ 27° 0′, nearly; and, by the scheme in which Dr. Hooke delineated its motion, it was then in ♈ 26° 59′ 24″. I place it in ♈ 27° 4′ 46″, taking the middle between the two extremes.

From the same observations, M. Auzout made the latitude of the

of the comet ♈ 27° 9′ 49″. M. Auzout, from this observation of his, placed the comet in ♈ 27° 0′, nearly; and, by the scheme in which Dr. Hooke delineated its motion, it was then in ♈ 26° 59′ 24″. I place it in ♈ 27° 4′ 46″, taking the middle between the two extremes.

From the same observations, M. Auzout made the latitude of the comet at that time 7° and 4′ or 5′ to the north; but he had done better to have made it 7° 3′ 29″, the difference of the latitudes of the comet and the star γ being equal to the difference of the longitude of the stars γ and A.

February 22d.7h.30′ at London, that is,

being equal to the difference of the longitude of the stars γ and A.

February 22d.7h.30′ at London, that is, February 22d. 8h.46′ at Dantzick, the distance of the comet from the star A, according to Dr. Hooke's observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, delineated in like manner by M. Petit, was a fifth part of the distance between the star A and the first star of Aries, or 15′ 57″; and the distance of the comet from a right line joining the star A and the first of Aries was a fourth part of the same fifth part, that is, 4′; and therefore the

or 8′ 5″ according to M. Gottignies; or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1′ 35″ (in which he agrees very well with M. Auzout); or, according to Dr. Hooke, not quite so much, as perhaps only 1′. Wherefore if to the longitude of the first star in Aries we add 1′, and 8′ 10″ to its latitude, we shall have the longitude of the comet ♈ 29° 18′, with 8° 36′ 26″ north lat.

March 7, 7h.30′ at Paris (that is, March 7, 8h.37′ at

so much, as perhaps only 1′. Wherefore if to the longitude of the first star in Aries we add 1′, and 8′ 10″ to its latitude, we shall have the longitude of the comet ♈ 29° 18′, with 8° 36′ 26″ north lat.

March 7, 7h.30′ at Paris (that is, March 7, 8h.37′ at Dantzick), from the observations of M. Auzout, the distance of the comet from the second star in Aries was equal to the distance of that star from the star A, that is, 52,′ 29″; and the difference of the longitude of the comet and the second star in Aries was 45′ or 46′, or, taking a mean quantity, 45′ 30″; and therefore the comet was in ♉ 0°

in Aries was equal to the distance of that star from the star A, that is, 52,′ 29″; and the difference of the longitude of the comet and the second star in Aries was 45′ or 46′, or, taking a mean quantity, 45′ 30″; and therefore the comet was in ♉ 0° 2′ 48″. From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54′. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion; and Hevelius, in the scheme of M. Auzout's observations which he constructed himself, corrected this irregular

the comet was in ♉ 0° 2′ 48″. From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54′. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion; and Hevelius, in the scheme of M. Auzout's observations which he constructed himself, corrected this irregular curvature, and so made the latitude of the comet 8° 55′ 30″. And, by farther correcting this irregularity, the latitude may become 8° 56, or 8° 57′.

This comet was also seen March 9, and at that time its place must have been in

quality of all bodies within the reach of our experiments; and therefore (by Rule III) to be affirmed of all bodies whatsoever. If the æther, or any other body, were either altogether void of gravity, or were to gravitate less in proportion to its quantity of matter, then, because (according to Aristotle, Des Cartes, and others) there is no diiference betwixt that and other bodies but in mere form of matter, by a successive change from form to form, it might be changed at last into a body of the same condition with those which gravitate most in proportion to their quantity of matter; and, on the

of heaven; from whence, and from the position of the tail, we infer that the head was near the sun. Matthew Paris says, It was distant from the sun by about a cubit, from, three of the clock (rather six) till nine, putting forth a long tail. Such also was that most resplendent comet described by Aristotle, lib. 1, Meteor. 6. The head whereof could not be seen, because it had set before the sun, or at least was hid under the sun's rays; but next day it was seen as well as might be; for, having left the sun but a very little way, it set immediately after it. And the scattered light of the head,

sun, or at least was hid under the sun's rays; but next day it was seen as well as might be; for, having left the sun but a very little way, it set immediately after it. And the scattered light of the head, obscured by the too great splendour (of the tail) did not yet appear. But afterwards (as Aristotle says) when the splendour (of the tail) was now diminished (the head of), the comet recovered its native brightness; and the splendour (of its tail) reached now to a third part of the heavens (that is, to 60°). This appearance was in the winter season (an. 4, Olymp. 101), and, rising to Orion's

quiescent in the centre, some at a swifter, others at a slower rate.

However, it was agreed on both sides that the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was of a later date, introduced by Eudoxus, Calippus, and Aristotle; when the ancient philosophy began to decline, and to give place to the new prevailing fictions of the Greeks.

But, above all things, the phænomena of comets can by no means consist with the notion of solid orbs. The Chaldeans, the most learned astronomers of their time, looked upon the comets

but afterwards, when the head begins to appear, and is got farther from the sun, that splendor always decreases, and turns by degrees into a paleness like to that of the milky way, but much more sensible at first; after that vanishing gradually. Such was that most resplendent comet described by Aristotle, Lib. 1, Meteor. 6. "The head thereof could not be seen, because it set before the sun, or at least was hid under the sun's rays; but the next day it was seen as well as might be; for, having left the sun but a very little way, it set immediately after it; and the scattered light of the head

or at least was hid under the sun's rays; but the next day it was seen as well as might be; for, having left the sun but a very little way, it set immediately after it; and the scattered light of the head obscured by the too great splendour (of the tail) did not yet appear. But afterwards (says Aristotle), when the splendour of the tail was now diminished (the head of), the comet recovered its native brightness. And the splendour of its tail reached now to a third part of the heavens (that is, to 60°). It appeared in the winter season, and, rising to Orion's girdle, there vanished away." Two

latitude 0° 40′ south. Their observations may be seen in a treatise which Ponthæus published concerning this comet. Cellius, who was present, and communicated his observations in a letter to Cassini saw the comet at the same hour in ♎ 8° 30′, with latitude 0° 30′ south. It was likewise seen by Galletius at the same hour at Avignon (that is, at 5h.42′ morning at London) in ♎ 8° without latitude. But by the theory the comet was at that time in ♎ 8° 16′ 45″, and its latitude was 0° 53′ 7″ south.

Nov. 18, at 6h.30′ in the morning at Rome (that is, at 5h.40′ at London), Ponthæus observed the comet in

time in ♎ 8° 16′ 45″, and its latitude was 0° 53′ 7″ south.

Nov. 18, at 6h.30′ in the morning at Rome (that is, at 5h.40′ at London), Ponthæus observed the comet in ♎ 13° 30′, with latitude 1° 20′ south; and Cellius in ♎ 13° 30′, with latitude 1° 00 south. But at 5h.30′ in the morning at Avignon, Galletius saw it in ♎ 13° 00′, with latitude 1° 00 south. In the University of La Fleche, in France, at 5h. in the morning (that is, at 5h.9′ at London), it was seen by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of

38′; and since the latitude, as we said, by the concurring observations of Montenari, Ango, and Hook, was continually increasing, therefore, it was now, on the 24th, something greater than 1° 58′; and, taking the mean quantity, may be reckoned 2° 18′, without any considerable error. Ponthæus and Galletius will have it that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or 1½°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the

that the latitude was now decreasing; and Cellius, and the observer in New England, that it continued the same, viz., of about 1°, or 1½°. The observations of Ponthæus and Cellius are more rude, especially those which were made by taking the azimuths and altitudes; as are also the observations of Galletius. Those are better which were made by taking the position of the comet to the fixed stars by Montenari, Hook, Ango, and the observer in New England, and sometimes by Ponthæus and Cellius. The same day, at 5h. morning, at Ballasore, the comet was observed in ♏ 11° 45′; and, therefore, at 5h. morning

Mr. Flamsted (p. 387), by the micrometer, measured the diameter of Jupiter 40″ or 41″; the diameter of Saturn's ring 50″; and the diameter of the sun about 32′ 13″ (p. 387).

But the diameter of Saturn is to the diameter of the ring, according to Mr. Huygens and Dr. Halley, as 4 to 9; according to Galletius, as 4 to 10; and according to Hooke (by a telescope of 60 feet), as 5 to 12. And from the mean proportion, 5 to 12, the diameter of Saturn's body is inferred about 21″.

Such as we have said are the apparent magnitudes; but, because of the unequal refrangibility of light, all lucid points are

for the scattered light, which could not be seen before for the stronger light of the planet, when the planet is hid, appears every way farther spread. Lastly, from hence it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12″ or 15″; and Venus appeared to Mr. Crabtrie only 1′ 3″; to Horrox but 1′ 12″; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1′ 24″. Thus the apparent diameter of the moon, which in

quiescent in the fluid that carries it.

Cor. 2. And if the vortex be of an uniform density, the same body may revolve at any distance from the centre of the vortex.





SCHOLIUM.



Hence it is manifest that the planets are not carried round in corporeal vortices; for, according to the Copernican hypothesis, the planets going round the sun revolve in ellipses, having the sun in their common focus; and by radii drawn to the sun describe areas proportional to the times. But now the parts of a vortex can never revolve with such a motion. Let AD, BE, CF, represent three orbits described about the sun S,

and by the force of gravity is continually drawn off from a rectilinear motion, and retained in its orbit.



The mean distance of the moon from the earth in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60⅓; to Street, 602⁄5; and to Tycho, 56½. But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the fixed stars, and that by four or five minutes near the horizon, did thereby

phænomena of the heavens; and the whole remaining force by which Jupiter is impelled will be directed (by Prop. III, Cor. I) to the centre of the sun.

The distances of the planets from the sun come out the same, whether, with Tycho, we place the earth in the centre of the system, or the sun with Copernicus: and we have already proved that these distances are true in Jupiter.

Kepler and Bullialdus have, with great care (p. 388), determined the distances of the planets from the sun; and hence it is that their tables agree best with the heavens. And in all the planets, in Jupiter and Mars, in Saturn

of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes in the horizon, did thereby augment

mean apparent diameter 31½') to be about 1⁄42 of the earth, as 43 to , or as about 128 to 127. And therefore the semi-diameter of the orbit, that is, the distance between the centres of the moon and earth, will in this case be 60½ semi-diameters of the earth, almost the same with that assigned by Copernicus, which the Tychonic observations by no means disprove; and, therefore, the duplicate proportion of the decrement of the force holds good in this distance. I have neglected the increment of the orbit which arises from the action of the sun as inconsiderable; but if that is subducted, the true

BD will be as the square of the line AB. On the same Laws and Corollaries depend those things which have been demonstrated concerning the times of the vibration of pendulums, and are confirmed by the daily experiments of pendulum clocks. By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was

Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise

iE to V, so as EV may be to Ei as FH to HI, and then draw Vf parallel to BD. It will come to the same, if about the centre i with an interval IH, we describe a circle cutting BD in X, and produce iX to Y so as iY may be equal to IF, and then draw Yf parallel to BD.

Sir Christopher Wren and Dr. Wallis have long ago given other solutions of this Problem.





PROPOSITION XXIX. PROBLEM XXI.



To describe a trajectory given in kind, that may be cut by four right lines given by position, into parts given in order, kind, and proportion.



Suppose a trajectory is to be described that may be similar

by the annual parallax of the orbit, in so far as the same is pretty nearly collected by the supposition that the comets move uniformly in right lines. The method of collecting the distance of a comet according to this hypothesis from four observations (first attempted by Kepler, and perfected by Dr. Wallis and Sir Christopher Wren) is well known; and the comets reduced to this regularity generally pass through the middle of the planetary region. So the comets of the year 1607 and 1618, as their motions are defined by Kepler, passed between the sun and the earth; that of the year 1664 below the orbit

gravitates towards the earth, and by the force of gravity is continually drawn off from a rectilinear motion, and retained in its orbit.



The mean distance of the moon from the earth in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60⅓; to Street, 602⁄5; and to Tycho, 56½. But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the fixed stars, and that by four or five minutes near

duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes in the horizon,

than the absolute circum-saturnal.

From the regularity of the heliocentric and irregularity of the geocentric motions of Venus, of Jupiter, and the other planets, it is evident (by Cor. IV, Prop. III) that the circum-terrestrial force, compared with the circum-solar, is very small.

Ricciolus and Vendelin have severally tried to determine the sun's parallax from the moon's dichotomies observed by the telescope, and they agree that it does not exceed half a minute.

Kepler, from Tycho's observations and his own, found the parallax of Mars insensible, even in opposition to the sun, when that parallax

From which distances, together with the periodic times, by the method above explained, it is easy to infer that the absolute circum-solar force is greater than the absolute circum-terrestrial force at least 229400 times.

And though we were only certain, from the observations of Ricciolus and Vendelin, that the sun's parallax was less than half a minute, yet from this it will follow that the absolute circum-solar force exceeds the absolute circum-terrestrial force 8500 times.

By the like computations I happened to discover an analogy, that is observed between the forces and the bodies of the

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

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 difference not less than 1¼ line, nor greater than 2⅔ lines. By the observations of M. Richer, in the island of Cayenne, the difference was 1¼ line. That difference being corrected by those of M. des Hayes, becomes 1½ line or 1¾ 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

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 1⁄12, which agrees very well with 187⁄1000 line 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

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 1⁄12, which agrees very well with 187⁄1000 line 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

an inch, or 1½ 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 1682, M. Varin and M. des Hayes found the length of a simple pendulum vibrating in seconds at the Royal Observatory of Paris to be 3 feet and 85⁄9 lines. And by the same method in the island of Goree, they found the length of an isochronal pendulum to be 3 feet and 65⁄9 lines, differing from the former by two lines. And in the same year, going to the islands of Guadaloupe and Martinico, they found that the length of an

3 feet and 65⁄9 lines, differing from the former by two lines. And in the same year, going to the islands of Guadaloupe and Martinico, they found that the length of an isochronal pendulum in those islands was 3 feet and 6½ 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 together the clock agreed 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

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 difference not less than 1¼ line, nor greater than 2⅔ lines. By the observations of M. Richer, in the island of Cayenne, the difference was 1¼ line. That difference being corrected by those of M. des Hayes, becomes 1½ line or 1¾ line. By the less accurate observations of others, the same was

at noon, anno 1700, O.S. viz. The mean motion of the sun ♑ 20° 43′ 40″, and of its apogee ♋ 7° 44′ 30″; the mean motion of the moon ♒ 15° 21′ 00″; of its apogee, ♊ 8° 20′ 00″; and of its ascending node ♌ 27° 24′ 20″; and the difference of meridians betwixt the Observatory at Greenwich and the Royal Observatory at Paris, 0h.9′ 20″: but the mean motion of the moon and of its apogee are not yet obtained with sufficient accuracy.





PROPOSITION XXXVI. PROBLEM XVII.



To find the force of the sun to move the sea.



The sun's force ML or PT to disturb the motions of the moon, was (by Prop. XXV.) in the moon's

commonly are inseparable from refraction; and the distinct transmission of the light of the fixed stars and planets to us is a demonstration that the æther or celestial medium is not endowed with any refractive power: for as to what is alleged, that the fixed stars have been sometimes seen by the Egyptians environed with a Coma or Capitlitium, because that has but rarely happened, it is rather to be ascribed to a casual refraction of clouds; and so the radiation and scintillation of the fixed stars to tin refractions both of the eyes and air; for upon laying a telescope to the eye, those radiations

of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early observers of the heavens; and from them, probably, this philosophy was spread abroad among other nations; for from them it was, and the nations about them, that the Greeks, a people of themselves more addicted to the study of philology than of nature, derived their first, as well as

other nations; for from them it was, and the nations about them, that the Greeks, a people of themselves more addicted to the study of philology than of nature, derived their first, as well as soundest, notions of philosophy; and in the vestal ceremonies we may yet trace the ancient spirit of the Egyptians; for it was their way to deliver their mysteries, that is, their philosophy of things above the vulgar way of thinking, under the veil of religious rites and hieroglyphic symbols.

It is not to be denied but that Anaxagoras, Democritus, and others, did now and then start up, who would have it that

use to be inseparable from refraction; and the distinct transmission of the light of the fixed stars and planets to us is a demonstration that the æther or celestial medium is not endowed with any refractive power. For as to what is alledged that the fixed stars have been sometimes seen by the Egyptians environed with a coma or capillitium because that has but rarely happened, it is rather to be ascribed to a casual refraction of clouds, as well as the radiation and scintillation of the fixed stars to the refractions both of the eyes and air; for upon applying a telescope to the eye, those

assisted me with his pains in correcting the press and taking care of the schemes, but it was to his solicitations that its becoming public is owing; for when he had obtained of me my demonstrations of the figure of the celestial orbits, he continually pressed me to communicate the same to the Royal Society, who afterwards, by their kind encouragement and entreaties, engaged me to think of publishing them. But after I had begun to consider the inequalities of the lunar motions, and had entered upon some other things relating to the laws and measures of gravity, and other forces: and the figures that

of pendulum clocks. By the same, together with the third Law, Sir Christ. Wren, Dr. Wallis, and Mr. Huygens, the greatest geometers of our times, did severally determine the rules of the congress and reflexion of hard bodies, and much about the same time communicated their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums,

their discoveries to the Royal Society, exactly agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the resistance of the air as to the elastic force of the

are equal, draw the right line TZ so as to bisect the right line AB; then find the triangle ATZ as above. Q.E.I.



Case 3. If all the three are equal, the point Z will be placed in the centre of a circle that passes through the points A, B, C. Q.E.I.

This problematic Lemma is likewise solved in Apollonius's Book of Tactions restored by Vieta.





PROPOSITION XXI. PROBLEM XIII.



About a given focus to describe a trajectory that shall pass through given points and touch right lines given by position.



Let the focus S, the point P, and the tangent TR be given, and suppose that the other focus H

Produce PO to K, so that OK may be equal to PO, and OK will be an ordinate on the other side of that diameter. Since, therefore, the points A, B, P and K are placed in the conic section, and PK cuts AB in a given angle, the rectangle PQK (by Prop. XVII., XIX., XXI. and XXIII., Book III., of Apollonius's Conics) will be to the rectangle AQB in a given ratio. But QK and PR are equal, as being the differences of the equal lines OK, OP, and OQ, OR; whence the rectangles PQK and PQ PR are equal; and therefore the rectangle PQ PR is to the rectangle AQB, that is, to the rectangle PS PT in a given

and H; unless, perhaps, the angle AGB is a right angle, and at the same time BG² equal to the rectangle AGH, in which case the locus will be a circle.

And so we have given in this Corollary a solution of that famous Problem of the ancients concerning four lines, begun by Euclid, and carried on by Apollonius; and this not an analytical calculus, but a geometrical composition, such as the ancients required.





LEMMA XX.



If the two opposite angular points A and P of any parallelogram ASPQ touch any conic section in the points A and P; and the sides AQ, AS of one of those angles, indefinitely

and so it manifestly appears from the following table.

The periodic times of the satellites of Jupiter.



1d.18h.27′.34″. 3d.13h.13′ 42″. 7d.3h.42′ 36″. 16d.16h.32′ 9″.



The distances of the satellites from Jupiter's centre.



From the observations of 1 2 3 4 semi-diameter of Jupiter.

Borelli 5⅔ 8⅔ 14 24⅔

Townly by the Microm. 5,52 8,78 13,47 24,72

Cassini by the Telescope 5 8 13 23

Cassini by the eclip. of the satel. 5⅔ 9 1423⁄60 253⁄10

From the periodic times 5,667 9,017 14,384 25,299

Mr. Pound has determined, by the help of excellent micrometers, the diameters of Jupiter and

explained; and probably it was to give some sort of satisfaction to this difficulty that solid orbs were introduced.

The later philosophers pretend to account for it either by the action of certain vortices, as Kepler and Des Cartes; or by some other principle of impulse or attraction, as Borelli, Hooke, and others of our nation; for, from the laws of motion, it is most certain that these effects must proceed from the action of some force or other.

But our purpose is only to trace out the quantity and properties of this force from the phænomena (p. 218), and to apply what we discover in

36″ as 493½″ to 108 , neglecting those small fractions which, in observing, cannot be certainly determined.

Before the invention of the micrometer, the same distances were determined in semi-diameters of Jupiter thus:—

Distance of the 1st 2d 3d 4th

By Galileo

" Simon Marius

" Cassini

Borelli, more exactly 6

6

5

5⅔ 10

10

8

8⅔ 16

16

13

14 28

26

23

24⅔

After the invention of the micrometer:—

By Townley

" Flamsted

More accurately by the eclipses 5,51

5,31

5,578 8,78

8,85

8,876 13,47

13,98

14,159 24,72

24,23

24,903





And the periodic times of those satellites, by

the difference. But he shortened the rod of his clock by more than the 1⁄8 of an inch, or 1½ 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 1682, M. Varin and M. des Hayes found the length of a simple pendulum vibrating in seconds at the Royal Observatory of Paris to be 3 feet and 85⁄9 lines. And by the same method in the island of Goree, they found the length of an isochronal pendulum to be 3 feet and 65⁄9 lines, differing from the former by two lines. And in the

3⅔ lines, than at Paris. He had done better to have reckoned those differences 1⅓ and 25⁄9: for these differences correspond to the differences 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 6½ lines; that in the island of St. Christophers it was 3 feet and 6¾ lines; and in the island of St. Domingo 3 feet and 7

that isochronal pendulums are shorter under the equator than at the Royal Observatory of Paris, by a difference not less than 1¼ line, nor greater than 2⅔ lines. By the observations of M. Richer, in the island of Cayenne, the difference was 1¼ line. That difference being corrected by those of M. des Hayes, becomes 1½ line or 1¾ 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

is about the first of Aries. So it is found by experience that the morning tides in winter 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 at Bristol by the height of 15 inches, according to the observations of Colepress and Sturmy.

But the motions which we have been describing suffer some alteration from that force of reciprocation, which the waters, being once moved, retain a little while by their vis insita. Whence it comes to pass that the tides may continue for some time, though the actions of the luminaries

latter from their difference. If, therefore, S and L are supposed to represent respectively the forces of the sun and moon while they are in the equator, as well as in their mean distances from the earth, we shall have L + S to L - S as 45 to 25, or as 9 to 5.

At Plymouth (by the observations of Samuel Colepress) the tide in its mean height rises to about 16 feet, and in the spring and autumn the height thereof in the syzygies may exceed that in the quadratures by more than 7 or 8 feet. Suppose the greatest difference of those heights to be 9 feet, and L + S will be to L - S as 20½ to 11½, or as 41 to 23;

if the ascending node of the moon is about the first of Aries. So the morning tides in winter exceed those of the evening, and the evening tides exceed those of the morning in summer; at Plymouth by the height of one foot, but at Bristol by the height of 15 inches, according to the observations of Colepress and Sturmy.

But the motions which we have been describing suffer some alteration from that force of reciprocation which the waters [having once received] retain a little while by their vis insita; whence it comes to pass that the tides may continue for some time, though the actions of the

And those inequalities, by the Corollaries we have named, are very great, and generate the principal which I call the semi-annual equation of the apogee; and this semi- annual equation in its greatest quantity comes to about 12° 18′, as nearly as I could collect from the phænomena. Our countryman, Horrox, was the first who advanced the theory of the moon's moving in an ellipsis about the earth placed in its lower focus. Dr. Halley improved the notion, by putting the centre of the ellipsis in an epicycle whose centre is uniformly revolved about the earth; and from the motion in this epicycle the

is hid, appears every way farther spread. Lastly, from hence it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12″ or 15″; and Venus appeared to Mr. Crabtrie only 1′ 3″; to Horrox but 1′ 12″; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1′ 24″. Thus the apparent diameter of the moon, which in 1684, a few days both before and after the sun's eclipse, was measured at the observatory of Paris 31′ 30″, in the

taken notice of by astronomers: but all these follow from our principles in Cor. II, III, IV, V, VI, VII, VIII, IX, X, XI, XII, XIII, Prop. LXVI, and are known really to exist in the heavens. And this may seen in that most ingenious, and if I mistake not, of all, the most accurate, hypothesis of Mr. Horrox, which Mr. Flamsted has fitted to the heavens; but the astronomical hypotheses are to be corrected in the motion of the nodes; for the nodes admit the greatest equation or prosthaphæresis in their octants, and this inequality is most conspicuous when the moon is in the nodes, and therefore also in

a direction between the east and north, as Hevelius has it also from Simeon, the monk of Durham. This comet appeared in the beginning of February, about the evening, and towards the south west part of heaven; from whence, and from the position of the tail, we infer that the head was near the sun. Matthew Paris says, It was distant from the sun by about a cubit, from, three of the clock (rather six) till nine, putting forth a long tail. Such also was that most resplendent comet described by Aristotle, lib. 1, Meteor. 6. The head whereof could not be seen, because it had set before the sun, or at least

from it extremely bright, reaching like a fiery beam to the east and north," as Hevelius has it from Simeon, the monk of Durham. It appeared at the beginning of February about the evening in the south-west. From this and from the situation of the tail we may infer that the head was near the sun. Matthew Paris says, "it was about one cubit from the sun; from the third [or rather the sixth] to the ninth hour sending out a long stream of light." The comet of 1264, in July, or about the solstice, preceded the rising sun, sending out its beams with a great light towards the west as far as the middle of the

a little above the horizon: but as the sun went forwards it retired every day farther from the horizon, till it passed by the very middle of the heavens. It is said to have been at the beginning large and bright, having a large coma, which decayed from day to day. It is described in Append. Matth. Paris, Hist. Ang. after this manner: "An. Christi 1265, there appeared a comet so wonderful, that none then living had ever seen the like; for, rising from the east with a great brightness, it extended itself with a great light as far as the middle of the hemisphere towards the west." The Latin original

lat.

March 1, 7h at London, that is, March 1, 8h.16′ at Dantzick. the comet was observed near the second star in Aries, the distance between them being to the distance between the first and second stars in Aries, that is, to 1° 33′, as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8′ 16″ according to Dr. Hooke, or 8′ 5″ according to M. Gottignies; or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth

them being to the distance between the first and second stars in Aries, that is, to 1° 33′, as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8′ 16″ according to Dr. Hooke, or 8′ 5″ according to M. Gottignies; or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1′ 35″ (in which he agrees very well with M. Auzout); or, according to Dr.

that is, to 1° 33′, as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8′ 16″ according to Dr. Hooke, or 8′ 5″ according to M. Gottignies; or, taking a mean between both, 8′ 10″. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1′ 35″ (in which he agrees very well with M. Auzout); or, according to Dr. Hooke, not quite so much, as perhaps only 1′. Wherefore if to the longitude

them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the

times greater than the absolute circum-saturnal.

From the regularity of the heliocentric and irregularity of the geocentric motions of Venus, of Jupiter, and the other planets, it is evident (by Cor. IV, Prop. III) that the circum-terrestrial force, compared with the circum-solar, is very small.

Ricciolus and Vendelin have severally tried to determine the sun's parallax from the moon's dichotomies observed by the telescope, and they agree that it does not exceed half a minute.

Kepler, from Tycho's observations and his own, found the parallax of Mars insensible, even in opposition to the sun, when

than 29 to 7233.

From which distances, together with the periodic times, by the method above explained, it is easy to infer that the absolute circum-solar force is greater than the absolute circum-terrestrial force at least 229400 times.

And though we were only certain, from the observations of Ricciolus and Vendelin, that the sun's parallax was less than half a minute, yet from this it will follow that the absolute circum-solar force exceeds the absolute circum-terrestrial force 8500 times.

By the like computations I happened to discover an analogy, that is observed between the forces and the

Newton





1846







NEWTON'S PRINCIPIA.





* * *



THE



MATHEMATICAL PRINCIPLES



OF



NATURAL PHILOSOPHY,





BY SIR ISAAC NEWTON;





TRANSLATED INTO ENGLISH BY ANDREW MOTTE.





TO WHICH IS ADDED



NEWTON'S SYSTEM OF THE WORLD;





With a Portrait taken from the Bust in the Royal Observatory at Greenwich.





FIRST AMERICAN EDITION, CAREFULLY REVISED AND CORRECTED,

WITH A LIFE OF THE AUTHOR, BY N. W. CHITTENDEN, M. A., &c.





* * *





NEW-YORK ·

PUBLISHED BY DANIEL ADEE, 45 LIBERTY STREET.



* * *

(not individually listed)

Dedication 3

Introduction to the American Edition 5

Life of Sir

eclipses.

But the theory of the moon ought to be examined and proved from the phenomena, first in the syzygies, then in the quadratures, and last of all in the octants; and whoever pleases to undertake the work will find it not amiss to assume the following mean motions of the sun and moon at the Royal Observatory of Greenwich, to the last day of December at noon, anno 1700, O.S. viz. The mean motion of the sun ♑ 20° 43′ 40″, and of its apogee ♋ 7° 44′ 30″; the mean motion of the moon ♒ 15° 21′ 00″; of its apogee, ♊ 8° 20′ 00″; and of its ascending node ♌ 27° 24′ 20″; and the difference of meridians betwixt the

of equinoxes) 455

Lemmas IV-XI, Propositions XL-XLII (Comets) 460

General Scholium 503



The System of the World. 511



Index to the Principia. 575





(introduction of the American edition was removed)


THE PRINCIPIA.





THE AUTHOR'S PREFACE





Since the ancients (as we are told by Pappus), made great account of the science of mechanics in the investigation of natural things; and the moderns, laying aside substantial forms and occult qualities, have endeavoured to subject the phænomena of nature to the laws of mathematics, I have in this treatise cultivated mathematics so far as it

describe areas proportional to the times, 386, 387, 390

“ “ revolve in periodic times that are in the sesquiplicate proportion of their distances from the primary, 386, 387

Problem Keplerian, solved by the trochoid and by approximations, 157 to 160

“ “ of the ancients, of four lines, related by Pappus, and attempted by Cartesius, by an algebraic calculus solved by a geometrical composition, 135

Projectiles move in parabolas when the resistance of the medium is taken away, 91, 115, 243, 273

“ their motions in resisting mediums, 255, 268

Pulses of the air, by which sounds are propagated, their

use in determining and demonstrating any other thing that is likewise geometrical.

It may also be objected, that if the ultimate ratios of evanescent quantities are given, their ultimate magnitudes will be also given: and so all quantities will consist of indivisibles, which is contrary to what Euclid has demonstrated concerning incommensurables, in the 10th Book of his Elements. But this objection is founded on a false supposition. For those ultimate ratios with which quantities vanish are not truly the ratios of ultimate quantities, but limits towards which the ratios of quantities decreasing

between the points A and H; unless, perhaps, the angle AGB is a right angle, and at the same time BG² equal to the rectangle AGH, in which case the locus will be a circle.

And so we have given in this Corollary a solution of that famous Problem of the ancients concerning four lines, begun by Euclid, and carried on by Apollonius; and this not an analytical calculus, but a geometrical composition, such as the ancients required.





LEMMA XX.



If the two opposite angular points A and P of any parallelogram ASPQ touch any conic section in the points A and P; and the sides AQ, AS of one of

the force of the sphærical superficies FE which is generated by the revolution of the arc FE, and is cut any where, as in r, by the line de, the annular part of the superficies generated by the revolution of the arc rE will be as the lineola Dd, the radius of the sphere PE remaining the same; as Archimedes has demonstrated in his Book of the Sphere and Cylinder. And the force of this superficies exerted in the direction of the lines PE or Pr situate all round in the conical superficies, will be as this annular superficies itself; that is as the lineola Dd, or, which is the same, as the rectangle

superficies EFG described from the centre P, and let the segment be divided into the parts BREFGS, FEDG. Let us suppose that segment to be not a purely mathematical but a physical superficies, having some, but a perfectly inconsiderable thickness. Let that thickness be called O, and (by what Archimedes has demonstrated) that superficies will be as PF DF O. Let us suppose besides the attractive forces of the particles of the sphere to be reciprocally as that power of the distances, of which n is index; and the force with which the superficies EFG attracts the body P will be (by Prop. LXXIX) as ,

defect of this pressure it comes to pass that their resistance is a little greater than in a duplicate ratio of their velocity.

So that the theory agrees with the phænomena of bodies falling in water. It remains that we examine the phænomena of bodies falling in air.

Exper. 13. From the top of St. Paul's Church in London, in June 1710, there were let fall together two glass globes, one full of quicksilver, the other of air; and in their fall they described a height of 220 English feet. A wooden table was suspended upon iron hinges on one side, and the other side of the same was supported by a

of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men,

appears from the following table.

The periodic times of the satellites of Jupiter.



1d.18h.27′.34″. 3d.13h.13′ 42″. 7d.3h.42′ 36″. 16d.16h.32′ 9″.



The distances of the satellites from Jupiter's centre.



From the observations of 1 2 3 4 semi-diameter of Jupiter.

Borelli 5⅔ 8⅔ 14 24⅔

Townly by the Microm. 5,52 8,78 13,47 24,72

Cassini by the Telescope 5 8 13 23

Cassini by the eclip. of the satel. 5⅔ 9 1423⁄60 253⁄10

From the periodic times 5,667 9,017 14,384 25,299

Mr. Pound has determined, by the help of excellent micrometers, the diameters of Jupiter and the elongation of its

the invention of the micrometer, the same distances were determined in semi-diameters of Jupiter thus:—

Distance of the 1st 2d 3d 4th

By Galileo

" Simon Marius

" Cassini

Borelli, more exactly 6

6

5

5⅔ 10

10

8

8⅔ 16

16

13

14 28

26

23

24⅔

After the invention of the micrometer:—

By Townley

" Flamsted

More accurately by the eclipses 5,51

5,31

5,578 8,78

8,85

8,876 13,47

13,98

14,159 24,72

24,23

24,903





And the periodic times of those satellites, by the observations of Mr. Flamsted, are 1d.18h.28′ 36″ | 3d.13h.17′ 54″ | 7d.3h.59′ 36″ | 16d.18h.5′ 13″, as above.

And the

PROPOSITION IV. THEOREM IV.



That the moon gravitates towards the earth, and by the force of gravity is continually drawn off from a rectilinear motion, and retained in its orbit.



The mean distance of the moon from the earth in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60⅓; to Street, 602⁄5; and to Tycho, 56½. But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the

most the distances drawn from the periodic times, fall in between them.

That the circum-terrestrial force likewise decreases in the duplicate proportion of the distances, I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon

thereto.



Our countryman, Mr. Norwood, measuring a distance of 905751 feet of London measure between London and York, in 1635, and observing the difference of latitudes to be 2° 28′, determined the measure of one degree to be 367196 feet of London measure, that is 57300 Paris toises. M. Picart, measuring an arc of one degree, and 22′ 55″ of the meridian between Amiens and Malvoisine, found an arc of one degree to be 57060 Paris toises. M. Cassini, the father, measured the distance upon the meridian from the town of Collioure in Roussillon to the Observatory of Paris; and his son added

of the pendulum before computed. And therefore 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 su 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 lire, was lengthened into one foot and ¼ 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

then in their perigees; but the greatest splendor of their heads was seen two weeks before, when they had just got clear of the sun's rays; and the greatest splendor of their tails a little more early, when yet nearer to the sun. The head of the former comet (according to the observations of Cysatus), December 1, appeared greater than the stars of the first magnitude; and, December 16 (then in the perigee), it was but little diminished in magnitude, but in the splendor and brightness of its light a great deal. January 7, Kepler, being uncertain about the head, left off observing. December 12,

then in their perigees; but the greatest splendor of their heads was seen two weeks before, when they had just got clear of the sun's rays; and the greatest splendor of their tails a little more early, when yet nearer to the sun. The head of the former comet, according to the observations of Cysatus, Dec. 1, appeared greater than the stars of the first magnitude; and, Dec. 16 (being then in its perigee), of a small magnitude, and the splendor or clearness was much diminished. Jan. 7, Kepler, being uncertain about the head, left off observing. Dec. 12, the head of the last comet was seen and

30′, with latitude 1° 20′ south; and Cellius in ♎ 13° 30′, with latitude 1° 00 south. But at 5h.30′ in the morning at Avignon, Galletius saw it in ♎ 13° 00′, with latitude 1° 00 south. In the University of La Fleche, in France, at 5h. in the morning (that is, at 5h.9′ at London), it was seen by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayers ψ; and the other is the outmost of the wing, Bayer's θ. Whence the comet was then in ♎ 12° 46′ with latitude 50′ south. And I was informed by Dr. Halley,

with latitude 1° 30′ south. The same day, at Boston, it was distant from Spica ♍ by about 4° of longitude east, and therefore was in ♎ 23° 24′ nearly.

Nov. 21, Ponthæus and his companions, at 7¼h. in the morning, ob served the comet in ♎ 27° 50′, with latitude 1° 16′ south; Cellius, in ♎ 28°; P. Ango at 5h. in the morning, in ♎ 27° 45′; Montenari in ♎ 27° 51′. The same day, in the island of Jamaica, it was seen near the beginning of ♏, and of about the same latitude with Spica ♍, that is, 2° 2′. The same day, at 5h. morning, at Ballasore, in the East Indies (that is, at 11h.20′ of the night

matter in the head at first, which was afterwards gradually spent.

And, which farther makes for the same purpose, I find, that the heads of other comets, which did put forth tails of the greatest bulk and splendor, have appeared but obscure and small. For in Brazil, March 5, 1668, 7h. P. M., St. N. P. Valentinus Estancius saw a comet near the horizon, and towards the south west, with a head so small as scarcely to be discerned, but with a tail above measure splendid, so that the reflection thereof from the sea was easily seen by those who stood upon the shore; and it looked like a fiery beam extended 23° in length

a bright light, its head and tail being extremely resplendent. The length of the tail, which was at first 20 or 30 deg., increased till December 9, when it arose to 75 deg,, but with a light much more faint and dilute than at the beginning. In the year 1668, March 5, N. S., about 7 in the evening, P. Valent. Estancius, being in Brazil, saw a comet near the horizon in the south-west. Its head was small, and scarcely discernible, but its tail extremely bright and refulgent, so that the reflection of it from the sea was easily seen by those who stood upon the shore. This great splendor lasted but three days,

And we read, in the Saxon Chronicle, of a like comet appearing in the year 1106, the star whereof was small and obscure (as that of 1680), but the splendour of its tail was very bright, and like a huge fiery beam stretched out in a direction between the east and north, as Hevelius has it also from Simeon, the monk of Durham. This comet appeared in the beginning of February, about the evening, and towards the south west part of heaven; from whence, and from the position of the tail, we infer that the head was near the sun. Matthew Paris says, It was distant from the sun by about a cubit, from, three of the clock

the rising or setting sun is intimated (p. 494, 495). We may add to these the comet of the year 1101 or 1106, "the star of which was small and obscure (like that of 1680); but the splendour arising from it extremely bright, reaching like a fiery beam to the east and north," as Hevelius has it from Simeon, the monk of Durham. It appeared at the beginning of February about the evening in the south-west. From this and from the situation of the tail we may infer that the head was near the sun. Matthew Paris says, "it was about one cubit from the sun; from the third [or rather the sixth] to the ninth hour sending out a

of the stars γ and A.

February 22d.7h.30′ at London, that is, February 22d. 8h.46′ at Dantzick, the distance of the comet from the star A, according to Dr. Hooke's observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, delineated in like manner by M. Petit, was a fifth part of the distance between the star A and the first star of Aries, or 15′ 57″; and the distance of the comet from a right line joining the star A and the first of Aries was a fourth part of the same fifth part, that is, 4′; and therefore the comet was in ♈ 28° 29′ 46″, with 8° 12′

distance of that star from the star A, that is, 52,′ 29″; and the difference of the longitude of the comet and the second star in Aries was 45′ or 46′, or, taking a mean quantity, 45′ 30″; and therefore the comet was in ♉ 0° 2′ 48″. From the scheme of the observations of M. Auzout, constructed by M. Petit, Hevelius collected the latitude of the comet 8° 54′. But the engraver did not rightly trace the curvature of the comet's way towards the end of the motion; and Hevelius, in the scheme of M. Auzout's observations which he constructed himself, corrected this irregular curvature, and so made the

of the laws by which this electric and elastic Spirit operates.





END OF THE MATHEMATICAL PRINCIPLES.





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↑ Dr. Pocock derives the Latin word Deus from the Arabic du (in the oblique case di), which signifies Lord. And in this sense princes are called gods, Psal. lxxxii. ver. 6; and John x. ver. 35. And Moses is called a god to his brother Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the

the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men, and other parts of the world, to be

this electric and elastic Spirit operates.





END OF THE MATHEMATICAL PRINCIPLES.





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↑ Dr. Pocock derives the Latin word Deus from the Arabic du (in the oblique case di), which signifies Lord. And in this sense princes are called gods, Psal. lxxxii. ver. 6; and John x. ver. 35. And Moses is called a god to his brother Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in

Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and

and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14.

and in the vestal ceremonies we may yet trace the ancient spirit of the Egyptians; for it was their way to deliver their mysteries, that is, their philosophy of things above the vulgar way of thinking, under the veil of religious rites and hieroglyphic symbols.

It is not to be denied but that Anaxagoras, Democritus, and others, did now and then start up, who would have it that the earth possessed the centre of the world, and that the stars of all sorts were revolved towards the west about the earth quiescent in the centre, some at a swifter, others at a slower rate.

However, it was agreed on

a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early observers of the heavens; and from them, probably, this philosophy was spread abroad among other nations; for from them it was, and the nations about them, that the Greeks, a people of themselves more addicted to the study of philology than of nature, derived their first, as well as soundest, notions of philosophy; and in the vestal ceremonies we may yet trace the ancient spirit of the Egyptians; for it was their way to deliver their mysteries, that is, their

the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was of a later date, introduced by Eudoxus, Calippus, and Aristotle; when the ancient philosophy began to decline, and to give place to the new prevailing fictions of the Greeks.

But, above all things, the phænomena of comets can by no means consist with the notion of solid orbs. The Chaldeans, the most learned astronomers of their time, looked upon the comets (which of ancient times before had been numbered among the celestial bodies) as a particular sort of planets,

change the number of the propositions and the citations. I heartily beg that what I have here done may be read with candour; and that the defects in a subject so difficult be not so much reprehended as kindly supplied, and investigated by new endeavours of my readers.

ISAAC NEWTON.



Cambridge. Trinity College May 8, 1686

In the second edition the second section of the first book was enlarged. In the seventh section of the second book the theory of the resistances of fluids was more accurately investigated, and confirmed by new experiments. In the third book the moon's theory and the praecession of the

heavy bodies falling in air are added. In the third book, the argument to prove that the moon is retained in its orbit by the force of gravity is enlarged on; and there are added new observations of Mr. Pound's of the proportion of the diameters of Jupiter to each other: there are, besides, added Mr. Kirk's observations of the comet in 1680; the orbit of that comet computed in an ellipsis by Dr. Halley; and the orbit of the comet in 1723, computed by Mr. Bradley.





BOOK I.





THE MATHEMATICAL PRINCIPLES

OF

NATURAL PHILOSOPHY.





* * *





DEFINITIONS.





DEFINITION I.

The quantity of

agreeing among themselves as to those rules. Dr. Wallis, indeed, was something more early in the publication; then followed Sir Christopher Wren, and, lastly, Mr. Huygens. But Sir Christopher Wren confirmed the truth of the thing before the Royal Society by the experiment of pendulums, which Mr. Mariotte soon after thought fit to explain in a treatise entirely upon that subject. But to bring this experiment to an accurate agreement with the theory, we are to have a due regard as well to the resistance of the air as to the elastic force of the concurring bodies. Let the spherical bodies A, B be

and, the angle of emergence remaining all along equal to the angle of incidence, will be equal to the same also at last. Q.E.D.





SCHOLIUM.



These attractions bear a great resemblance to the reflexions and refractions of light made in a given ratio of the secants, as was discovered by Snellius; and consequently in a given ratio of the sines, as was exhibited by Des Cartes. For it is now certain from the phenomena of Jupiter's Satellites, confirmed by the observations of different astronomers, that light is propagated in succession, and requires about seven or eight minutes to travel

from the phenomena of Jupiter's Satellites, confirmed by the observations of different astronomers, that light is propagated in succession, and requires about seven or eight minutes to travel from the sun to the earth. Moreover, the rays of light that are in our air (as lately was discovered by Grimaldus, by the admission of light into a dark room through a small hole, which I have also tried) in their passage near the angles of bodies, whether transparent or opaque (such as the circular and rectangular edges of gold, silver and brass coins, or of knives, or broken pieces of stone or glass), are

are given, the moments of the extremes will be as those extremes. The same is to be understood of the sides of any given rectangle.

Cor. 3. And if the sum or difference of two squares is given, the moments of the sides will be reciprocally as the sides.





SCHOLIUM.



In a letter of mine to Mr. J. Collins, dated December 10, 1672, having described a method of tangents, which I suspected to be the same with Slusius's method, which at that time was not made public, I subjoined these words: This is one particular, or rather a Corollary, of a general method, which extends itself, without any

given rectangle.

Cor. 3. And if the sum or difference of two squares is given, the moments of the sides will be reciprocally as the sides.





SCHOLIUM.



In a letter of mine to Mr. J. Collins, dated December 10, 1672, having described a method of tangents, which I suspected to be the same with Slusius's method, which at that time was not made public, I subjoined these words: This is one particular, or rather a Corollary, of a general method, which extends itself, without any troublesome calculation, not only to the drawing of tangents to any curve lines, whether geometrical or mechanical, or

8″ 12‴

7″ 42‴

7″ 42‴

7″ 57‴

8″ 12‴

7″ 42‴ 226 feet 11 inch.

230 feet 9 inch.

227 feet 10 inch.

224 feet 5 inch.

225 feet 5 inch

230 feet 7 inch. 6 feet 11 inch

10 feet 9 inch

7 feet 0 inch

4 feet 5 inch

5 feet 5 inch

10 feet 7 inch



Exper. 14. Anno 1719, in the month of July, Dr. Desaguliers made some experiments of this kind again, by forming hogs' bladders into spherical orbs; which was done by means of a concave wooden sphere, which the bladders, being wetted well first, were put into. After that being blown full of air, they were obliged to fill up the spherical cavity that

and, on the other hand, swifter in the summer.

Now by experiments it actually appears that sounds do really advance in one second of time about 1142 feet of English measure, or 1070 feet of French measure.

The velocity of sounds being known, the intervals of the pulses are known also. For M. Sauveur, by some experiments that he made, found that an open pipe about five Paris feet in length gives a sound of the same tone with a viol-string that vibrates a hundred times in one second. Therefore there are near 100 pulses in a space of 1070 Paris feet, which a sound runs over in a second of time;

of gravity is continually drawn off from a rectilinear motion, and retained in its orbit.



The mean distance of the moon from the earth in the syzygies in semi-diameters of the earth, is, according to Ptolemy and most astronomers, 59; according to Vendelin and Huygens, 60; to Copernicus, 60⅓; to Street, 602⁄5; and to Tycho, 56½. But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed the refractions of the fixed stars, and that by four or five minutes near the horizon, did thereby increase the moon's

will deviate a little to one side and to the other from the earth in the lower focus; and this is the libration in longitude; for the libration in latitude arises from the moon's latitude, and the inclination of its axis to the plane of the ecliptic. This theory of the libration of the moon, Mr. N. Mercator in his Astronomy, published at the beginning of the year 1676, explained more fully out of the letters I sent him. The utmost satellite of Saturn seems to revolve about its axis with a motion like this of the moon, respecting Saturn continually with the same face; for in its revolution round

about the equator than at the poles, the seas would subside about the poles, and, rising towards the equator, would lay all things there under water.





PROPOSITION XIX. PROBLEM III.



To find the proportion of the axis of a planet to the diameter, perpendicular thereto.



Our countryman, Mr. Norwood, measuring a distance of 905751 feet of London measure between London and York, in 1635, and observing the difference of latitudes to be 2° 28′, determined the measure of one degree to be 367196 feet of London measure, that is 57300 Paris toises. M. Picart, measuring an arc of one degree, and 22′

without marking the difference. But he shortened the rod of his clock by more than the 1⁄8 of an inch, or 1½ 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 1682, M. Varin and M. des Hayes found the length of a simple pendulum vibrating in seconds at the Royal Observatory of Paris to be 3 feet and 85⁄9 lines. And by the same method in the island of Goree, they found the length of an isochronal pendulum to be 3 feet and 65⁄9 lines, differing from the former by two

they found the length of an isochronal pendulum to be 3 feet and 65⁄9 lines, differing from the former by two lines. And in the same year, going to the islands of Guadaloupe and Martinico, they found that the length of an isochronal pendulum in those islands was 3 feet and 6½ 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 together the clock agreed with the motion of the sun. In November following, upon his arrival at Lisbon, he found his clock to go slower than

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 6½ lines; that in the island of St. Christophers it was 3 feet and 6¾ lines; and in the island of St. Domingo 3 feet and 7 lines.

And in the year 1704, P. Feuillé, at Puerto Bello in America, found that the length of the pendulum vibrating in seconds was 3 Paris feet, and only 57⁄12 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

the su 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 lire, was lengthened into one foot and ¼ 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 ⅔ 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

are quiescent in their syzygies, but regressive in their quadratures, by an hourly motion of 16″ 19‴ 26iv.; and that the equation of the motion of the nodes in the octants is 1° 30′; all which exactly agree with the phænomena of the heavens.





SCHOLIUM.



Mr. Machin, Astron., Prof. Gresh., and Dr. Henry Pemberton, separately found out the motion of the nodes by a different method. Mention has been made of this method in another place. Their several papers, both of which I have seen, contained two Propositions, and exactly agreed with each other in both of them. Mr. Machin's paper coming first to my hands,

-1.3

-1.28

+1.59

+1.45

+0.56

+0.32

+0.10

+0.33

-1.20

-2.10

-2.42

-0.41

-2.49

+0.35 ′ ″

-2.0

+1.7

-0.25

+0.44

+0.12

-0.33

+0.8

+0.58

+0.18

+0.2

+1.25

-0.11

+0.14

+2.0

+1.10

+2.14

This comet also appeared in the November before, and at Coburg, in Saxony, was observed by Mr. Gottfried Kirch, on the 4th of that month, on the 6th and 11th O. S.; from its positions to the nearest fixed stars observed with sufficient accuracy, sometimes with a two feet, and sometimes with a ten feet telescope; from the difference of longitudes of Coburg and London, 11°; and from the places of the fixed

being computed in the aforesaid parabolic orbit, comes out ♌ 29° 30′ 22″, its latitude north 1° 25′ 7″, and its distance from the sun 115546.

Moreover, Dr. Halley, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of September after Julius Cæsar was killed; An. Chr. 531, in the consulate of Lampadius and Orestes; An. Chr. 1106, in the month of February; and at the end of the year 1680; and that with a long and remarkable tail, except when it was seen after Cæsar's death, at which time, by reason of the inconvenient situation of the earth,

♌ 29° 30′ 22″, its latitude north 1° 25′ 7″, and its distance from the sun 115546.

Moreover, Dr. Halley, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of September after Julius Cæsar was killed; An. Chr. 531, in the consulate of Lampadius and Orestes; An. Chr. 1106, in the month of February; and at the end of the year 1680; and that with a long and remarkable tail, except when it was seen after Cæsar's death, at which time, by reason of the inconvenient situation of the earth, the tail was not so conspicuous), set himself to find

its latitude north 1° 25′ 7″, and its distance from the sun 115546.

Moreover, Dr. Halley, observing that a remarkable comet had appeared four times at equal intervals of 575 years (that is, in the month of September after Julius Cæsar was killed; An. Chr. 531, in the consulate of Lampadius and Orestes; An. Chr. 1106, in the month of February; and at the end of the year 1680; and that with a long and remarkable tail, except when it was seen after Cæsar's death, at which time, by reason of the inconvenient situation of the earth, the tail was not so conspicuous), set himself to find out an

18, at 6h.30′ in the morning at Rome (that is, at 5h.40′ at London), Ponthæus observed the comet in ♎ 13° 30′, with latitude 1° 20′ south; and Cellius in ♎ 13° 30′, with latitude 1° 00 south. But at 5h.30′ in the morning at Avignon, Galletius saw it in ♎ 13° 00′, with latitude 1° 00 south. In the University of La Fleche, in France, at 5h. in the morning (that is, at 5h.9′ at London), it was seen by P. Ango, in the middle between two small stars, one of which is the middle of the three which lie in a right line in the southern hand of Virgo, Bayers ψ; and the other is the outmost of the wing, Bayer's θ. Whence the

at that time in ♎ 19° 23′ 47″, with latitude 2° 1′ 59″ south. The same day, at 5h. in the morning, at Boston in New England, the comet was distant from Spica ♍ 1°, with the difference of 40′ in latitude. The same day, in the island of Jamaica, it was about 1° distant from Spica ♍. The same day, Mr. Arthur Storer, at the river Patuxent, near Hunting Creek, in Maryland, in the confines of Virginia, in lat. 38½°, at 5 in the morning (that is, at 10h. at London), saw the comet above Spica ♍, and very nearly joined with it, the distance between them being about ¾ of one deg. And from these observations

with it, the distance between them being about ¾ of one deg. And from these observations compared, I conclude, that at 9h.44′ at London the comet was in ♎ 18° 50′, with about 1° 25′ latitude south. Now by the theory the comet was at that time in ♎ 18° 52′ 15″, with 1° 26′ 54″ lat. south.

Nov. 20, Montenari, professor of astronomy at Padua, at 6h. in the morning at Venice (that is, 5h.10′ at London), saw the comet in ♎ 23°, with latitude 1° 30′ south. The same day, at Boston, it was distant from Spica ♍ by about 4° of longitude east, and therefore was in ♎ 23° 24′ nearly.

Nov. 21, Ponthæus and his companions, at 7¼h. in the

but towards the north according to Montenari; and, therefore, that declination was scarcely sensible; and the tail, lying nearly parallel to the equator, deviated a little from the opposition of the sun towards the north.

Nov. 23, O. S. at 5h. morning, at Nuremberg (that is, at 4½h. at London), Mr. Zimmerman saw the comet in ♏ 8° 8′, with 2° 31′ south lat. its place being collected by taking its distances from fixed stars.

Nov. 24, before sun-rising, the comet was seen by Montenari in ♏ 12° 52′ on the north side of the right line through Cor Leonis and Spica ♍, and therefore its latitude was

+ 3.49

+ 1. 2

+ 1.22

- 0.1

+ 0. 6 ′ ″

+ 0.12

+ 1.50

+ 2.26

+ 1.30

+ 3.42

- 0.34

+ 3.11

+ 1.48

- 0.43

+ 0. 3

+ 0.45

This theory is also confirmed by the retrograde motion of the comet that appeared in the year 1723. The ascending node of this comet (according to the computation of Mr. Bradley, Savilian Professor of Astronomy at Oxford) was in ♈ 14° 16′. The inclination of the orbit to the plane of the ecliptic 49° 59′. Its perihelion was in ♉ 12° 15′ 20″. Its perihelion distance from the sun 998651 parts, of which the radius of the orbis magnus contains 1000000, and the equal time of its perihelion September 16d 16h.10′. The

supply of new fuel those old stars, acquiring new splendor, may pass for new stars. Of this kind are such fixed stars as appear on a sudden, and shine with a wonderful brightness at first, and afterwards vanish by little and little. Such was that star which appeared in Cassiopeia's chair; which Cornelius Gemma did not see upon the 8th of November, 1572, though he was observing that part of the heavens upon that very night, and the sky was perfectly serene; but the next night (November 9) he saw it shining much brighter than any of the fixed stars, and scarcely inferior to Venus in splendor. Tycho Brahe

first observed September 30, O.S. 1604, with a light exceeding that of Jupiter, though the night before it was not to be seen; and from that time it decreased by little and little, and in 15 or 16 months entirely disappeared. Such a new star appearing with an unusual splendor is said to have moved Hipparchus to observe, and make a catalogue of, the fixed stars. As to those fixed stars that appear and disappear by turns, and increase slowly and by degrees, and scarcely ever exceed the stars of the third magnitude, they seem to be of another kind, which revolve about their axes, and, having a light and

for by the hypothesis of vortices; for comets are carried with very eccentric motions through all parts of the heavens indifferently, with a freedom that is incompatible with the notion of a vortex.

Bodies projected in our air suffer no resistance but from the air. Withdraw the air, as is done in Mr. Boyle's vacuum, and the resistance ceases; for in this void a bit of line down and a piece of solid gold descend with equal velocity. And the parity of reason must take place in the celestial spaces above the earth's atmosphere; in which spaces, where there is no air to resist their motions, all bodies

are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic Spirit operates.





END OF THE MATHEMATICAL PRINCIPLES.





* * *



↑ Dr. Pocock derives the Latin word Deus from the Arabic du (in the oblique case di), which signifies Lord. And in this sense princes are called gods, Psal. lxxxii. ver. 6; and John x. ver. 35. And Moses is called a god to his brother Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the

END OF THE MATHEMATICAL PRINCIPLES.





* * *



↑ Dr. Pocock derives the Latin word Deus from the Arabic du (in the oblique case di), which signifies Lord. And in this sense princes are called gods, Psal. lxxxii. ver. 6; and John x. ver. 35. And Moses is called a god to his brother Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales,

THE MATHEMATICAL PRINCIPLES.





* * *



↑ Dr. Pocock derives the Latin word Deus from the Arabic du (in the oblique case di), which signifies Lord. And in this sense princes are called gods, Psal. lxxxii. ver. 6; and John x. ver. 35. And Moses is called a god to his brother Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil,

35. And Moses is called a god to his brother Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver.

Aaron, and a god to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x

to Pharaoh (Exod. iv. ver. 16; and vii. ver. 1). And in the same sense the souls of dead princes were formerly, by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal.

by the Heathens, called gods, but falsely, because of their want of dominion.

↑ This was the opinion of the Ancients. So Pythagoras, in Cicer. de Nat. Deor. lib. i Thales, Anaxagoros, Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver.

Virgil, Georg. lib. iv. ver. 220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped; but erroneously.

220; and Æneid, lib. vi. ver. 721. Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped; but erroneously.





THE SYSTEM OF THE WORLD.





* *

Philo Allegor, at the beginning of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped; but erroneously.





THE SYSTEM OF THE WORLD.





* * *





It was the ancient

of lib. i. Aratus, in his Phænom. at the beginning. So also the sacred writers; as St. Paul, Acts, xvii. ver 27, 28. St. John's Gosp. chap. xiv. ver. 2. Moses. in Deut. iv. ver. 39; and x ver. 14. David, Psal. cxxxix. ver. 7, 8, 9. Solomon, 1 Kings, viii. ver. 27. Job, xxii. ver. 12, 13, 14. Jeremiah, xxiii. ver. 23, 24. The Idolaters opposed the sun, moon, and stars, the souls of men, and other parts of the world, to be parts of the Supreme God, and therefore to be worshipped; but erroneously.





THE SYSTEM OF THE WORLD.





* * *





It was the ancient opinion of not a few, in the

as one of the planets, described an annual course about the sun, while by a diurnal motion it was in the mean time revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a

the planets, described an annual course about the sun, while by a diurnal motion it was in the mean time revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of

an annual course about the sun, while by a diurnal motion it was in the mean time revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of

sun, while by a diurnal motion it was in the mean time revolved about its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the

its own axis; and that the sun, as the common fire which served to warm the whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early

the whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early observers of the heavens; and from them, probably, this philosophy

whole, was fixed in the centre of the universe.

This was the philosophy taught of old by Philolaus, Aristarchus of Samos, Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early observers of the heavens; and from them, probably, this philosophy was spread abroad

Plato in his riper years, and the whole sect of the Pythagoreans; and this was the judgment of Anaximander, more ancient than any of them; and of that wise king of the Romans, Numa Pompilius, who, as a symbol of the figure of the world with the sun in the centre, erected a temple in honour of Vesta, of a round form, and ordained perpetual fire to be kept in the middle of it.

The Egyptians were early observers of the heavens; and from them, probably, this philosophy was spread abroad among other nations; for from them it was, and the nations about them, that the Greeks, a people of

vestal ceremonies we may yet trace the ancient spirit of the Egyptians; for it was their way to deliver their mysteries, that is, their philosophy of things above the vulgar way of thinking, under the veil of religious rites and hieroglyphic symbols.

It is not to be denied but that Anaxagoras, Democritus, and others, did now and then start up, who would have it that the earth possessed the centre of the world, and that the stars of all sorts were revolved towards the west about the earth quiescent in the centre, some at a swifter, others at a slower rate.

However, it was agreed on both sides that

the west about the earth quiescent in the centre, some at a swifter, others at a slower rate.

However, it was agreed on both sides that the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was of a later date, introduced by Eudoxus, Calippus, and Aristotle; when the ancient philosophy began to decline, and to give place to the new prevailing fictions of the Greeks.

But, above all things, the phænomena of comets can by no means consist with the notion of solid orbs. The Chaldeans, the most learned astronomers of their time,

about the earth quiescent in the centre, some at a swifter, others at a slower rate.

However, it was agreed on both sides that the motions of the celestial bodies were performed in spaces altogether free and void of resistance. The whim of solid orbs was of a later date, introduced by Eudoxus, Calippus, and Aristotle; when the ancient philosophy began to decline, and to give place to the new prevailing fictions of the Greeks.

But, above all things, the phænomena of comets can by no means consist with the notion of solid orbs. The Chaldeans, the most learned astronomers of their time, looked

solid orbs was of a later date, introduced by Eudoxus, Calippus, and Aristotle; when the ancient philosophy began to decline, and to give place to the new prevailing fictions of the Greeks.

But, above all things, the phænomena of comets can by no means consist with the notion of solid orbs. The Chaldeans, the most learned astronomers of their time, looked upon the comets (which of ancient times before had been numbered among the celestial bodies) as a particular sort of planets, which, describing very eccentric orbits, presented themselves to our view only by turns, viz., once in a revolution,

13″ is to the time 1d.18h.28′ 36″ as 493½″ to 108 , neglecting those small fractions which, in observing, cannot be certainly determined.

Before the invention of the micrometer, the same distances were determined in semi-diameters of Jupiter thus:—

Distance of the 1st 2d 3d 4th

By Galileo

" Simon Marius

" Cassini

Borelli, more exactly 6

6

5

5⅔ 10

10

8

8⅔ 16

16

13

14 28

26

23

24⅔

After the invention of the micrometer:—

By Townley

" Flamsted

More accurately by the eclipses 5,51

5,31

5,578 8,78

8,85

8,876 13,47

13,98

14,159 24,72

24,23

24,903





And the periodic times of

greater; that in the innermost but one the difference is indeed much greater, yet so as to agree as nearly with his computations as the moon does with the common tables; and that he computes those eclipses only from the mean motions corrected by the equation of light discovered and introduced by Mr. Romer. Supposing, then, that the theory differs by a less error than that of 2′ from the motion of the outmost satellite as hitherto described, and taking as the periodic time 16d. 18h.5′ 13″ to 2′ in time, so is the whole circle or 360° to the arc 1′ 48″, the error of Mr. Flamsted's computation,

I infer thus.

The mean distance of the moon from the centre of the earth, is, in semi-diameters of the earth, according to Ptolemy, Kepler in his Ephemerides, Bullialdus, Hevelius, and Ricciolus, 59; according to Flamsted, 59⅓; according to Tycho, 56½; to Vendelin, 60; to Copernicus, 60⅓; to Kircher, 62½ ( p . 391, 392, 393).

But Tycho, and all that follow his tables of refraction, making the refractions of the sun and moon (altogether against the nature of light) to exceed those of the fixed stars, and that by about four or five minutes in the horizon, did thereby augment the horizontal

the planet, when the planet is hid, appears every way farther spread. Lastly, from hence it is that the planets appear so small in the disk of the sun, being lessened by the dilated light. For to Hevelius, Galletius, and Dr. Halley, Mercury did not seem to exceed 12″ or 15″; and Venus appeared to Mr. Crabtrie only 1′ 3″; to Horrox but 1′ 12″; though by the mensurations of Hevelius and Hugenius without the sun's disk, it ought to have been seen at least 1′ 24″. Thus the apparent diameter of the moon, which in 1684, a few days both before and after the sun's eclipse, was measured at the observatory of

now diminished (the head of), the comet recovered its native brightness. And the splendour of its tail reached now to a third part of the heavens (that is, to 60°). It appeared in the winter season, and, rising to Orion's girdle, there vanished away." Two comets of the same kind are described by Justin, Lib. 37, which, according to his account, "shined so bright, that the whole heaven seemed to be on fire; and by their greatness filled up a fourth part of the heavens, and by their splendour exceeded that of the sun." By which last words a near position of these bright comets and the rising or

to this number of parts, in Tab. 1, we find 30 days. Again; the comet of 1682 entered the sphere of the orbis magnus about Aug. 11, and arrived at its perihelion about Sep. 16, being then distant from the sun by about 350 parts, to which, in Tab. I, belong 33½ days. Lastly; that memorable comet of Regiomontanus, which in 1472 was carried through the circum-polar parts of our northern hemisphere with such rapidity as to describe 40 degrees in one day, entered the sphere of the orbis magnus Jan 21, about the time that it was passing by the pole, and, hastening from thence towards the sun, was hid under the