In the preceding books I have presented principles of philosophya that are not, however, philosophical but strictly mathematical—that is, those on which the study of philosophy can be based. These principles are the laws and conditions of motions and of forces, which especially relate to philosophy. But in order to prevent these principles from seeming sterile, I have illustrated them with some philosophical scholiums [i.e., scholiums dealing with natural philosophy], treating topics that are general and that seem to be the most fundamental for philosophy, such as the density and resistance of bodies, spaces void of bodies, and the motion of light and sounds. It still remains for us to exhibit the system of the world from these same principles. On this subject I composed an earlier version of book 3 in popular form, so that it might be more widely read. But those who have not sufficiently grasped the principles set down here will certainly not perceive the force of the conclusions, nor will they lay aside the preconceptions to which they have become accustomed over many years; and therefore, to avoid lengthy disputations, I have translated the substance of the earlier version into propositions in a mathematical style, so that they may be read only by those who have first mastered the principles. But since in books 1 and 2 a great number of propositions occur which might be too time-consuming even for readers who are proficient in mathematics, I am unwilling to advise anyone to study every one of these propositions. It will be sufficient to read with care the Definitions, the Laws of Motion, and the first three sections of book 1, and then turn to this book 3 on the system of the world, consulting at will the other propositions of books 1 and 2 which are referred to here.
Proposition 1, Theorem 1
The forces by which the circumjovial planets [or satellites of Jupiter] are
continually drawn away from rectilinear motions and are maintained in their respective
orbits are directed to the center of Jupiter and are inversely as the squares of the
distances of their places from that center.
The first part of the proposition is evident from phen. 1 and from prop. 2 or prop. 3 of book 1, and the second part from phen. 1 and from corol. 6 to prop. 4 of book 1.
The same is to be understood for the planets that are Saturn’s companions [or satellites] by phen. 2.
Proposition 2, Theorem 2
The forces by which the primary planets are continually drawn away from
rectilinear motions and are maintained in their respective orbits are directed to the
sun and are inversely as the squares of their distances from its center.
The first part of the proposition is evident from phen. 5 and from prop. 2 of book 1, and the latter part from phen. 4 and from prop. 4 of the same book. But this second part of the proposition is proved with the greatest exactness from the fact that the aphelia are at rest. For the slightest departure from the ratio of the square would (by book 1, prop. 45, corol. 1) necessarily result in a noticeable motion of the apsides in a single revolution and an immense such motion in many revolutions.
Proposition 3, Theorem 3
The force by which the moon is maintained in its orbit is directed toward the
earth and is inversely as the square of the distance of its places from the center of
the earth.
The first part of this statement is evident from phen. 6 and from prop. 2 or prop. 3 of book 1, and the second part from the very slow motion of the moon’s apogee. For that motion, which in each revolution is only three degrees and three minutes forward [or in consequentia, i.e., in an easterly direction] can be ignored. For it is evident (by book 1, prop. 45, corol. 1) that if the distance of the moon from the center of the earth is to the semidiameter of the earth as D to 1, then the force from which such a motion may arise is inversely as D24/243, that is, inversely as that power of D of which the index is 24/243; that is, the proportion of the force to the distance is inversely as a little greater than the second power of the distance, but is 59¾ times closer to the square than to the cube. Now this motion of the apogee arises from the action of the sun (as will be pointed out below) and accordingly is to be ignored here. The action of the sun, insofar as it draws the moon away from the earth, is very nearly as the distance of the moon from the earth, and so (from what is said in book 1, prop. 45, corol. 2) is to the centripetal force of the moon as roughly 2 to 357.45, or 1 to 17829/40. And if so small a force of the sun is ignored, the remaining force by which the moon is maintained in its orbit will be inversely as D2. And this will be even more fully established by comparing this force with the force of gravity as is done in prop. 4 below.
COROLLARY. If the mean centripetal force by which the moon is maintained in its orbit is increased first in the ratio of 17729/40 to 17829/40, then also in the squared ratio of the semidiameter of the earth to the mean distance of the center of the moon from the center of the earth, the result will be the lunar centripetal force at the surface of the earth, supposing that that force, in descending to the surface of the earth, is continually increased in the ratio of the inverse square of the height.
Proposition 4, Theorem 4
The moon gravitates toward the earth and by the force of gravity is always drawn
back from rectilinear motion and kept in its orbit.
The mean distance of the moon from the earth in the syzygies is, according to Ptolemy and most astronomers, 59 terrestrial semidiameters, 60 according to Vendelin and Huygens, 60⅓ according to Copernicus, 60⅖ according to Street, and 56½ according to Tycho. But Tycho and all those who follow his tables of refractions, by making the refractions of the sun and moon (entirely contrary to the nature of light) be greater than those of the fixed stars—in fact greater by about four or five minutes—have increased the parallax of the moon by that many minutes, that is, by about a twelfth or fifteenth of the whole parallax. Let that error be corrected, and the distance will come to be roughly 60½ terrestrial semidiameters, close to the value that has been assigned by others. Let us assume a mean distance of 60 semidiameters in the syzygies; and also let us assume that a revolution of the moon with respect to the fixed stars is completed in 27 days, 7 hours, 43 minutes, as has been established by astronomers; and that the circumference of the earth is 123,249,600 Paris feet, according to the measurements made by the French. If now the moon is imagined to be deprived of all its motion and to be let fall so that it will descend to the earth with all that force urging it by which (by prop. 3, corol.) it is [normally] kept in its orbit, then in the space of one minute, it will by falling describe 151/12 Paris feet. This is determined by a calculation carried out either by using prop. 36 of book 1 or (which comes to the same thing) by using corol. 9 to prop. 4 of book 1. For the versed sine of the arc which the moon would describe in one minute of time by its mean motion at a distance of 60 semidiameters of the earth is roughly 151/12 Paris feet, or more exactly 15 feet, 1 inch, and 14/9 lines [or twelfths of an inch]. Accordingly, since in approaching the earth that force is increased as the inverse square of the distance, and so at the surface of the earth is 60 × 60 times greater than at the moon, it follows that a body falling with that force, in our regions, ought in the space of one minute to describe 60 × 60 × 151/12 Paris feet, and in the space of one second 151/12 feet, or more exactly 15 feet, 1 inch, and 14/9 lines. And heavy bodies do actually descend to the earth with this very force. For a pendulum beating seconds in the latitude of Paris is 3 Paris feet and 8½ lines in length, as Huygens observed. And the height that a heavy body describes by falling in the time of one second is to half the length of this pendulum as the square of the ratio of the circumference of a circle to its diameter (as Huygens also showed), and so is 15 Paris feet, 1 inch, 17/9 lines. And therefore that force by which the moon is kept in its orbit, in descending from the moon’s orbit to the surface of the earth, comes out equal to the force of gravity here on earth, and so (by rules 1 and 2) is that very force which we generally call gravity. For if gravity were different from this force, then bodies making for the earth by both forces acting together would descend twice as fast, and in the space of one second would by falling describe 30⅙ Paris feet, entirely contrary to experience.
This calculation is founded on the hypothesis that the earth is at rest. For if the earth and the moon move around the sun and in the meanwhile also revolve around their common center of gravity, then, the law of gravity remaining the same, the distance of the centers of the moon and earth from each other will be roughly 60½ terrestrial semidiameters, as will be evident to anyone who computes it. And the computation can be undertaken by book 1, prop. 60.
Scholium
The proof of the proposition can be treated more fully as follows. If several
moons were to revolve around the earth, as happens in the system of
Saturn or of Jupiter, their periodic times (by the argument of induction) would
observe the law which Kepler discovered for the planets, and therefore their
centripetal forces would be inversely as the squares of the distances from the
center of the earth, by prop. 1 of this book 3. And if the lowest of them were
small and nearly touched the tops of the highest mountains, its centripetal
force, by which it would be kept in its orbit, would (by the preceding
computation) be very nearly equal to the gravities of bodies on the tops of those
mountains. And this centripetal force would cause this little moon, if it were
deprived of all the motion with which it proceeds in its orbit, to descend to
the earth—as a result of the absence of the centrifugal force with which it
had remained in its orbit—and to do so with the same velocity with which
heavy bodies fall on the tops of those mountains, because the forces with
which they descend are equal. And if the force by which the lowest little
moon descends were different from gravity and that little moon also were
heavy toward the earth in the manner of bodies on the tops of mountains,
this little moon would descend twice as fast by both forces acting together.
Therefore, since both forces—namely, those of heavy bodies and those of the
moons—are directed toward the center of the earth and are similar to each
other and equal, they will (by rules 1 and 2) have the same cause. And
therefore that force by which the moon is kept in its orbit is the very one that we
generally call gravity. For if this were not so, the little moon at the top of a
mountain must either be lacking in gravity or else fall twice as fast as heavy
bodies generally do.
Proposition 5, Theorem 5
The circumjovial planets
[or satellites of Jupiter] gravitate toward Jupiter, the
circumsaturnian planets [or satellites of Saturn] gravitate toward Saturn, and the
circumsolar [or primary] planets gravitate toward the sun, and by the force of
their gravity they are always drawn back from rectilinear motions and kept in
curvilinear orbits.
For the revolutions of the circumjovial planets about Jupiter, of the circumsaturnian planets about Saturn, and of Mercury and Venus and the other circumsolar planets about the sun are phenomena of the same kind as the revolution of the moon about the earth, and therefore (by rule 2) depend on causes of the same kind, especially since it has been proved that the forces on which those revolutions depend are directed toward the centers of Jupiter, Saturn, and the sun, and decrease according to the same ratio and law (in receding from Jupiter, Saturn, and the sun) as the force of gravity (in receding from the earth).
COROLLARY 1. Therefore, there is gravity toward all planets universally. For no one doubts that Venus, Mercury, and the rest [of the planets, primary and secondary,] are bodies of the same kind as Jupiter and Saturn. And since, by the third law of motion, every attraction is mutual, Jupiter will gravitate toward all its satellites, Saturn toward its satellites, and the earth will gravitate toward the moon, and the sun toward all the primary planets.
COROLLARY 2. The gravity that is directed toward every planet is inversely as the square of the distance of places from the center of the planet.
COROLLARY 3. All the planets are heavy toward one another by corols. 1 and 2. And hence Jupiter and Saturn near conjunction, by attracting each other, sensibly perturb each other’s motions, the sun perturbs the lunar motions, and the sun and moon perturb our sea, as will be explained in what follows.
Scholium
Hitherto we have called “centripetal” that force by which celestial bodies
are kept in their orbits. It is now established that this force is gravity, and
therefore we shall call it gravity from now on. For the cause of the centripetal
force by which the moon is kept in its orbit ought to be extended to all the
planets, by rules 1, 2, and 4.
Proposition 6, Theorem 6
All bodies gravitate toward each of the planets, and at any given distance from
the center of any one planet the weight of any body whatever toward that planet
is proportional to the quantity of matter which the body contains.
Others have long since observed that the falling of all heavy bodies
toward the earth (at least on making an adjustment for the inequality of the
retardation that arises from the very slight resistance of the air) takes place
in equal times, and it is possible to discern that equality of the times, to a
very high degree of accuracy, by using pendulums. I have tested this with
gold, silver, lead, glass, sand, common salt, wood, water, and wheat. I got
two wooden boxes, round and equal. I filled one of them with wood, and
I suspended the same weight of gold (as exactly as I could) in the center
of oscillation of the other. The boxes, hanging by equal eleven-foot cords,
made pendulums exactly like each other with respect to their weight, shape,
and air resistance. Then, when placed close to each other [and set into
vibration], they kept swinging back and forth together with equal oscillations for
a very long time. Accordingly, the amount of matter in the gold (by book 2,
prop. 24, corols. 1 and 6) was to the amount of matter in the wood as the
action of the motive force upon all the gold to the action of the motive force
upon all the [added] wood—that is, as the weight of one to the weight of
the other. And it was so for the rest of the materials. In these experiments,
in bodies of the same weight, a difference of matter that would be even less
than a thousandth part of the whole could have been clearly noticed. Now,
there is no doubt that the nature of gravity toward the planets is the same as
toward the earth. For imagine our terrestrial bodies to be raised as far as the
orbit of the moon and, together with the moon, deprived of all motion, to
be released so as to fall to the earth simultaneously; and by what has already
been shown, it is certain that in equal times these falling terrestrial bodies
will describe the same spaces as the moon, and therefore that they are to the
quantity of matter in the moon as their own weights are to its weight.
Further, since the satellites of Jupiter revolve in times that are as the 3/2 power
of their distances from the center of Jupiter, their accelerative gravities
toward Jupiter will be inversely as the squares of the distances from the center
of Jupiter, and, therefore, at equal distances from Jupiter their accelerative
gravities would come out equal. Accordingly, in equal times in falling from
equal heights [toward Jupiter] they would describe equal spaces, just as
happens with heavy bodies on this earth of ours. And by the same argument
the circumsolar [or primary] planets, let fall from equal distances from the
sun, would describe equal spaces in equal times in their descent to the sun.
Moreover, the forces by which unequal bodies are equally accelerated are as
the bodies; that is, the weights [of the primary planets toward the sun] are as
the quantities of matter in the planets. Further, that the weights of Jupiter
and its satellites toward the sun are proportional to the quantities of their
matter is evident from the extremely regular motion of the satellites,
according to book 1, prop. 65, corol. 3. For if some of these were more strongly
attracted toward the sun in proportion to the quantity of their matter than
the rest, the motions of the satellites (by book 1, prop. 65, corol. 2) would be
perturbed by that inequality of attraction. If, at equal distances from the sun,
some satellite were heavier [or gravitated more] toward the sun in proportion
to the quantity of its matter than Jupiter in proportion to the quantity of its
own matter, in any given ratio, say d to e, then the distance between the
center of the sun and the center of the orbit of the satellite would always
be greater than the distance between the center of the sun and the center of
Jupiter and these distances would be to each other very nearly as the square
root of d to the square root of e, as I found by making a certain calculation.
And if the satellite were less heavy [or gravitated less] toward the sun in
that ratio of d to e, the distance of the center of the orbit of the satellite
from the sun would be less than the distance of the center of Jupiter from
the sun in that same ratio of the square root of d to the square root of e.
And so if, at equal distances from the sun, the accelerative gravity of any
satellite toward the sun were greater or smaller than the accelerative gravity
of Jupiter toward the sun, by only a thousandth of the whole gravity, the
distance of the center of the orbit of the satellite from the sun would be
greater or smaller than the distance of Jupiter from the sun by
of
the total distance, that is, by a fifth of the distance of the outermost satellite
from the center of Jupiter; and this eccentricity of the orbit would be very
sensible indeed. But the orbits of the satellites are concentric with Jupiter,
and therefore the accelerative gravities of Jupiter and of the satellites toward
the sun are equal to one another. And by the same argument the weights [or
gravities] of Saturn and its companions toward the sun, at equal distances
from the sun, are as the quantities of matter in them; and the weights of the
moon and earth toward the sun are either nil or exactly proportional to their
masses. But they do have some weight, according to prop. 5, corols. 1 and 3.
But further, the weights [or gravities] of the individual parts of each planet toward any other planet are to one another as the matter in the individual parts. For if some parts gravitated more, and others less, than in proportion to their quantity of matter, the whole planet, according to the kind of parts in which it most abounded, would gravitate more or gravitate less than in proportion to the quantity of matter of the whole. But it does not matter whether those parts are external or internal. For if, for example, it is imagined that bodies on our earth are raised to the orbit of the moon and compared with the body of the moon, then, if their weights were to the weights of the external parts of the moon as the quantities of matter in them, but were to the weights of the internal parts in a greater or lesser ratio, they would be to the weight of the whole moon in a greater or lesser ratio, contrary to what has been shown above.
COROLLARY 1. Hence, the weights of bodies do not depend on their forms and textures. For if the weights could be altered with the forms, they would be, in equal matter, greater or less according to the variety of forms, entirely contrary to experience.
COROLLARY 5. The force of gravity is of a different kind from the magnetic force. For magnetic attraction is not proportional to the [quantity of] matter attracted. Some bodies are attracted [by a magnet] more [than in proportion to their quantity of matter], and others less, while most bodies are not attracted [by a magnet at all]. And the magnetic force in one and the same body can be intended and remitted [i.e., increased and decreased] and is sometimes far greater in proportion to the quantity of matter than the force of gravity; and this force, in receding from the magnet, decreases not as the square but almost as the cube of the distance, as far as I have been able to tell from certain rough observations.
Proposition 7, Theorem 7
Gravity exists in all bodies universally and is proportional to the quantity of matter
in each.
We have already proved that all planets are heavy [or gravitate] toward one another and also that the gravity toward any one planet, taken by itself, is inversely as the square of the distance of places from the center of the planet. And it follows (by book 1, prop. 69 and its corollaries) that the gravity toward all the planets is proportional to the matter in them.
Further, since all the parts of any planet A are heavy [or gravitate] toward any planet B, and since the gravity of each part is to the gravity of the whole as the matter of that part to the matter of the whole, and since to every action (by the third law of motion) there is an equal reaction, it follows that planet B will gravitate in turn toward all the parts of planet A, and its gravity toward any one part will be to its gravity toward the whole of the planet as the matter of that part to the matter of the whole. Q.E.D.
COROLLARY 1. Therefore the gravity toward the whole planet arises from and is compounded of the gravity toward the individual parts. We have examples of this in magnetic and electric attractions. For every attraction toward a whole arises from the attractions toward the individual parts. This will be understood in the case of gravity by thinking of several smaller planets coming together into one globe and composing a larger planet. For the force of the whole will have to arise from the forces of the component parts. If anyone objects that by this law all bodies on our earth would have to gravitate toward one another, even though gravity of this kind is by no means detected by our senses, my answer is that gravity toward these bodies is far smaller than what our senses could detect, since such gravity is to the gravity toward the whole earth as [the quantity of matter in each of] these bodies to the [quantity of matter in the] whole earth.
COROLLARY 2. The gravitation toward each of the individual equal particles of a body is inversely as the square of the distance of places from those particles. This is evident by book 1, prop. 74, corol. 3.
Proposition 8, Theorem 8
If two globes gravitate toward each other, and their matter is homogeneous on all
sides in regions that are equally distant from their centers, then the weight of either
globe toward the other will be inversely as the square of the distance between the
centers.
After I had found that the gravity toward a whole planet arises from and is compounded of the gravities toward the parts and that toward each of the individual parts it is inversely proportional to the squares of the distances from the parts, I was still not certain whether that proportion of the inverse square obtained exactly in a total force compounded of a number of forces, or only nearly so. For it could happen that a proportion which holds exactly enough at very great distances might be markedly in error near the surface of the planet, because there the distances of the particles may be unequal and their situations dissimilar. But at length, by means of book 1, props. 75 and 76 and their corollaries, I discerned the truth of the proposition dealt with here.
COROLLARY 3. The densities of the planets can also be found. For the weights of equal and homogeneous bodies toward homogeneous spheres are, on the surfaces of the spheres, as the diameters of the spheres, by book 1, prop. 72; and therefore the densities of heterogeneous spheres are as those weights divided by the diameters of the spheres. Now, the true diameters of the sun, Jupiter, Saturn, and the earth were found to be to one another as 10,000, 997, 791, and 109, and the weights toward them are as 10,000, 943, 529, and 435 respectively, and therefore the densities are as 100, 94½, 67, and 400. The density of the earth that results from this computation does not depend on the parallax of the sun but is determined by the parallax of the moon and therefore is determined correctly here. Therefore the sun is a little denser than Jupiter, and Jupiter denser than Saturn, and the earth four times denser than the sun. For the sun is rarefied by its great heat. And the moon is denser than the earth, as will be evident from what follows [i.e., prop. 37, corol. 3].
cCOROLLARY 4. Therefore, other things being equal, the planets that are smaller are denser. For thus the force of gravity on their surfaces approaches closer to equality. But, other things being equal, the planets that are nearer to the sun are also denser; for example, Jupiter is denser than Saturn, and the earth is denser than Jupiter. The planets, of course, had to be set at different distances from the sun so that each one might, according to the degree of its density, enjoy a greater or smaller amount of heat from the sun.c If the earth were located in the orbit of Saturn, our water would freeze; in the orbit of Mercury, it would immediately go off in a vapor. For the light of the sun, to which its heat is proportional, is seven times denser in the orbit of Mercury than on earth, and I have found with a thermometer that water boils at seven times the heat of the summer sun. And there is no doubt that the matter of the planet Mercury is adapted to its heat and therefore is denser than this matter of our earth, since all denser matter requires a greater heat for the performance of the operations of nature.
Proposition 9, Theorem 9
In going inward from the surfaces of the planets, gravity decreases very nearly in
the ratio of the distances from the center.
If the matter of the planets were of uniform density, this proposition would hold true exactly, by book 1, prop. 73. Therefore the error is as great as can arise from the nonuniformity of the density.
Proposition 10, Theorem 10
The motions of the planets can continue in the heavens for a very long time.
In the scholium to prop. 40, book 2, it was shown that a globe of frozen
water moving freely in our air would, as a result of the resistance of the air,
lose 
of its motion in describing the length of its own semidiameter.
And the same proportion obtains very nearly in any globes, however large
they may be and however swift their motions. Now, I gather in the following
way that the globe of our earth is denser than if it consisted totally of water.
If this globe were wholly made of water, whatever things were rarer than
water would, because of their smaller specific gravity, emerge from the water
and float on the surface. And for this reason a globe made of earth that
was covered completely by water would emerge somewhere, if it were rarer
than water; and all the water flowing away from there would be gathered
on the opposite side. And this is the case for our earth, which is in great
part surrounded by seas. If the earth were not denser than the seas, it would
emerge from those seas and, according to the degree of its lightness, a part
of the earth would stand out from the water, while all those seas flowed to
the opposite side. By the same argument the spots on the sun are lighter than
the solar shining matter on top of which they float. And in whatever way the
planets were formed, at the time when the mass was fluid, all heavier matter
made for the center, away from the water. Accordingly, since the ordinary
matter of our earth at its surface is about twice as heavy as water, and a
little lower down, in mines, is found to be about three or four or even five
times heavier than water, it is likely that the total amount of matter in the
earth is about five or six times greater than it would be if the whole earth
consisted of water, especially since it has already been shown above that the
earth is about four times denser than Jupiter. Therefore, if Jupiter is a little
denser than water, then in the space of thirty days (during which this planet
describes a length of 459 of its semidiameters) it would, in a medium of
the same density as our air, lose almost a tenth of its motion. But since the
resistance of mediums decreases in the ratio of their weight and density (so
that water, which is 13⅗ times lighter than quicksilver, resists 13⅗ times
less; and air, which is 860 times lighter than water, resists 860 times less), it
follows that up in the heavens, where the weight of the medium in which the
planets move is diminished beyond measure, the resistance will nearly cease.
We showed in the scholium to prop. 22, book 2, that at a height of two
hundred miles above the earth, the air would be rarer than on the surface
of the earth in a ratio of 30 to 0.0000000000003998, or 75,000,000,000,000 to
1, roughly. And hence the planet Jupiter, revolving in a medium with the
same density as that upper air, would not, in the time of a million years,
lose a millionth of its motion as a result of the resistance of the medium.
In the spaces nearest to the earth, of course, nothing is found that creates
resistance except air, exhalations, and vapors. If these are exhausted with very
great care from a hollow cylindrical glass vessel, heavy bodies fall within the
glass vessel very freely and without any sensible resistance; gold itself and
the lightest feather, dropped simultaneously, fall with equal velocity and, in
falling through a distance of four or six or eight feet, reach the bottom at the
same time, as has been found by experiment. And therefore in the heavens,
which are void of air and exhalations, the
planets and comets, encountering
no sensible resistance, will move through those spaces for a very long time.
Hypothesis 1
The center of the system of the world is at rest.
No one doubts this, although some argue that the earth, others that the sun, is at rest in the center of the system. Let us see what follows from this hypothesis.
Proposition 11, Theorem 11
The common center of gravity of the earth, the sun, and all the planets is at rest.
For that center (by corol. 4 of the Laws) either will be at rest or will move uniformly straight forward. But if that center always moves forward, the center of the universe will also move, contrary to the hypothesis.
Proposition 12, Theorem 12
The sun is engaged in continual motion but never recedes far from the common
center of gravity of all the planets.
For since (by prop. 8, corol. 2) the matter in the sun is to the matter in Jupiter as 1,067 to 1, and the distance of Jupiter from the sun is to the semidiameter of the sun in a slightly greater ratio, the common center of gravity of Jupiter and the sun will fall upon a point a little outside the surface of the sun. By the same argument, since the matter in the sun is to the matter in Saturn as 3,021 to 1, and the distance of Saturn from the sun is to the semidiameter of the sun in a slightly smaller ratio, the common center of gravity of Saturn and the sun will fall upon a point a little within the surface of the sun. And continuing the same kind of calculation, if the earth and all the planets were to lie on one side of the sun, the distance of the common center of gravity of them all from the center of the sun would scarcely be a whole diameter of the sun. In other cases the distance between those two centers is always less. And therefore, since that center of gravity is continually at rest, the sun will move in one direction or another, according to the various configurations of the planets, but will never recede far from that center.
COROLLARY. Hence the common center of gravity of the earth, the sun, and all the planets is to be considered the center of the universe. For since the earth, sun, and all the planets gravitate toward one another and therefore, in proportion to the force of the gravity of each of them, are constantly put in motion according to the laws of motion, it is clear that their mobile centers cannot be considered the center of the universe, which is at rest. If that body toward which all bodies gravitate most had to be placed in the center (as is the commonly held opinion), that privilege would have to be conceded to the sun. But since the sun itself moves, an immobile point will have to be chosen for that center from which the center of the sun moves away as little as possible and from which it would move away still less, supposing that the sun were denser and larger, in which case it would move less.
Proposition 13, Theorem 13
The planets move in ellipses that have a focus in the center of the sun, and by
radii drawn to that center they describe areas proportional to the times.
We have already discussed these motions from the phenomena. Now that the principles of motions have been found, we deduce the celestial motions from these principles a priori. Since the weights of the planets toward the sun are inversely as the squares of the distances from the center of the sun, it follows (from book 1, props. 1 and 11, and prop. 13, corol. 1) that if the sun were at rest and the remaining planets did not act upon one another, their orbits would be elliptical, having the sun in their common focus, and they would describe areas proportional to the times. The actions of the planets upon one another, however, are so very small that they can be ignored, and they perturb the motions of the planets in ellipses about the mobile sun less (by book 1, prop. 66) than if those motions were being performed about the sun at rest.
Yet the action of Jupiter upon Saturn is not to be ignored entirely. For the
gravity toward Jupiter is to the gravity toward the sun (at equal distances)
as 1 to 1,067; and so in the conjunction of Jupiter and Saturn, since the
distance of Saturn from Jupiter is to the distance of Saturn from the sun
almost as 4 to 9, the gravity of Saturn toward Jupiter will be to the gravity
of Saturn toward the sun as 81 to 16 × 1,067, or roughly as 1 to 211. And
hence arises a perturbation of the orbit of Saturn in every conjunction of
this planet with Jupiter so sensible that astronomers have been at a loss
concerning it. According to the different situations of the planet Saturn in
these conjunctions, its eccentricity is sometimes increased and at other times
diminished, the aphelion sometimes is moved forward and at other times
perchance drawn back, and the mean motion is alternately accelerated and
retarded. Nevertheless, all the error in its motion around the sun, an error
arising from so great a force, can almost be avoided (except in the mean
motion) by putting the lower focus of its orbit in the common center of
gravity of Jupiter and the sun (by book 1, prop. 67); in which case, when that
error is greatest, it hardly exceeds two minutes. And the greatest error in the
mean motion hardly exceeds two minutes per year. But in the conjunction
of Jupiter and Saturn the accelerative gravities of the sun toward Saturn,
of Jupiter toward Saturn, and of Jupiter toward the sun are almost as 16,
81, and 
, or 156,609, and so the difference of the gravities
of the sun toward Saturn and of Jupiter toward Saturn is to the gravity of
Jupiter toward the sun as 65 to 156,609, or 1 to 2,409. But the greatest power
of Saturn to perturb the motion of Jupiter is proportional to this difference,
and therefore the perturbation of the orbit of Jupiter is far less than that of
Saturn’s. The perturbations of the remaining orbits are still less by far, except
that the orbit of the earth is sensibly perturbed by the moon. The common
center of gravity of the earth and the moon traverses an ellipse about the sun,
an ellipse in which the sun is located at a focus, and this center of gravity,
by a radius drawn to the sun, describes areas (in that ellipse) proportional to
the times; the earth, during this time, revolves around this common center
with a monthly motion.
Proposition 14, Theorem 14
The aphelia and nodes of the [planetary] orbits are at rest.
The aphelia are at rest, by book 1, prop. 11, as are also the planes of the orbits, by prop. 1 of the same book; and if these planes are at rest, the nodes are also at rest. But yet from the actions of the revolving planets and comets upon one another some inequalities will arise, which, however, are so small that they can be ignored here.
COROLLARY 1. The fixed stars also are at rest, because they maintain given positions with respect to the aphelia and nodes.
COROLLARY 2. And so, since the fixed stars have no sensible parallax arising from the annual motion of the earth, their forces will produce no sensible effects in the region of our system, because of the immense distance of these bodies from us. Indeed, the fixed stars, being equally dispersed in all parts of the heavens, by their contrary attractions annul their mutual forces, by book 1, prop. 70.
Scholium
Since the planets nearer to the sun (namely, Mercury, Venus, the earth, and
Mars) act but slightly upon one another because of the smallness of their
bodies [i.e., because their masses are small], their aphelia and nodes will be
at rest, except insofar as they are disturbed by the forces of Jupiter, Saturn,
and any bodies further away. And by the theory of gravity it follows that
their aphelia move slightly forward [or in consequentia] with respect to the
fixed stars, and do this as the 3/2 powers of the distances of these planets
from the sun. For example, if in a hundred years the aphelion of Mars is
carried forward [or in consequentia] 33′20″ with respect to the fixed stars,
then in a hundred years the aphelia of the earth, Venus, and Mercury will
be carried forward 17′40″, 10′53″, and 4′16″ respectively. And these motions
are ignored in this proposition because they are so small.
Proposition 15, Problem 1
To find the principal diameters of the [planetary] orbits.
These diameters are to be taken as the ⅔ powers of the periodic times by book 1, prop. 15; and then each one is to be increased in the ratio of the sum of the masses of the sun and each revolving planet to the first of two mean proportionals between that sum and the sun, by book 1, prop. 60.
Proposition 16, Problem 2
To find the eccentricities and aphelia of the [planetary] orbits.
The problem is solved by book 1, prop. 18.
Proposition 17, Theorem 15
The daily motions of the planets are uniform, and the libration of the moon arises
from its daily motion.
This is clear from the first law of motion and book 1, prop. 66, corol. 22. With respect to the fixed stars Jupiter revolves in 9h56m, Mars in 24h39m, Venus in about 23 hours, the earth in 23h56m, the sun in 25½ days, and the moon in 27d7h43m. That these things are so is clear from phenomena. With respect to the earth, the spots on the body of the sun return to the same place on the sun’s disc in about 27½ days; and therefore with respect to the fixed stars the sun revolves in about 25½ days. Now, since a lunar day (the moon revolving uniformly about its own axis) is a month long [i.e., is equal to a lunar month, the periodic time of the moon’s revolution in its orbit], the same face of the moon will always very nearly look in the direction of the further focus of its orbit, and therefore will deviate from the earth on one side or the other according to the situation of that focus. This is the moon’s libration in longitude; for the libration in latitude arises from the latitude of the moon and the inclination of its axis to the plane of the ecliptic. Mr. N. Mercator, in his book on astronomy, published in the beginning of the year 1676, set forth this theory of the moon’s libration more fully on the basis of a letter from me.
The outermost satellite of Saturn seems to revolve about its own axis with a motion similar to our moon’s, constantly presenting the same aspect toward Saturn. For in revolving about Saturn, whenever it approaches the eastern part of its own orbit, it is just barely seen and for the most part disappears from sight; and possibly this occurs because of certain spots in that part of its body which is then turned toward the earth, as Cassini noted. The outermost satellite of Jupiter also seems to revolve about its own axis with a similar motion, because in the part of its body turned away from Jupiter it has a spot which, whenever the satellite passes between Jupiter and our eyes, appears as if it were on the body of Jupiter.
Proposition 18, Theorem 16
The axes of the planets are smaller than the diameters that are drawn
perpendicularly to those axes.
If it were not for the daily circular motion of the planets, then, because the gravity of their parts is equal on all sides, they would have to assume a spherical figure. Because of that circular motion it comes about that those parts, by receding from the axis, endeavor to ascend in the region of the equator. And therefore if the matter is fluid, it will increase the diameters at the equator by ascending, and will decrease the axis at the poles by descending. Thus the diameter of Jupiter is found by astronomical observations to be shorter between the poles than from east to west. By the same argument, if our earth were not a little higher around the equator than at the poles, the seas would subside at the poles and, by ascending in the region of the equator, would flood everything there.
Proposition 19a, Problem 3
To find the proportion of a planet’s axis to the diameters perpendicular to that
axis.
b cOur fellow countryman Norwood, in about the year 1635, measured a distance of 905,751 London feet between London and York and observed the difference of latitudes between those places to be 2°28′ and thereby found the measure of one degree to be 367,196 London feet, that is, 57,300 Paris toises. Picard measured an arc of 1°22′55″ along the meridian between Amiens and Malvoisine and found an arc of one degree to be 57,060 Paris toises. The elder Cassini [Gian Domenico or Jean-Dominique] measured the distance along the meridian from the town of Collioure in Roussillon to the Paris observatory; and his son [Jacques] added the distance from the observatory to the tower of the city of Dunkerque. The total distance was 486,156½ toises, and the difference in latitudes between the town of Collioure and the city of Dunkerque was 8°31′11⅚″. Thus an arc of one degree comes out to be 57,061 Paris toises. And from these measures the circumference of the earth is found to be 123,249,600 Paris feet, and its semidiameter 19,615,800 feet, on the hypothesis that the earth is spherical.
A body revolving uniformly in a circle at a distance of 19,615,800 feet from the center, making a revolution in a single sidereal day of 23h56m4s, will describe an arc of 1,433.46 feet in the time of one second, an arc whose versed sine is 0.0523656 feet, or 7.54064 lines. And therefore the force by which heavy bodies descend at the latitude of Paris is to the dcentrifugald force of bodies on the equator (which arises from the daily motion of the earth) as 2,174 to 7.54064.
The centrifugal force of bodies on the earth’s equator is to the centrifugal force by which bodies recede rectilinearly from the earth at the latitude of Paris (48°50′10″) as the square of the radius to the square of the cosine of that latitude, that is, as 7.54064 to 3.267. Let this force be added to the force by which heavy bodies descend at the latitude of Paris; then a body falling at that latitude with the total force of gravity will, in the time of one second, describe 2,177.267 lines, or 15 Paris feet 1 inch and 5.267 lines. And the total force of gravity at that latitude will be to the ecentrifugale force of bodies on the earth’s equator as 2,177.267 to 7.54064 or 289 to 1.b
Therefore, if APBQ represents the figure of the earth, which is now no
longer spherical but generated by the rotation of an ellipse about its minor
axis PQ; and if ACQqca is a channel full of water,
going from the pole Qq to the center Cc and from
that center out to the equator Aa; then the weight
of the water in the leg ACca will have to be to the
weight of the water in the other leg QCcq as 289 to
288, because the centrifugal force arising from the
circular motion will sustain and take away one of
the 289 parts of weight of the water in the leg ACca, and consequently the
288 parts of weight of the water in the leg QCcq will sustain the 288 parts
remaining in the leg ACca. Further, on making the computation (according
to book 1, prop. 91, corol. 2), I find that if the earth were composed of
uniform matter and were deprived of all its motion, and its axis PQ were
to its diameter AB as 100 to 101, then the gravity in place Q toward the
earth would be to the gravity in the same place Q toward a sphere described
about the center C with a radius PC or QC as 126 to 125. And by the same
argument, the gravity in place A toward a spheroid generated by the rotation
of the ellipse APBQ about the axis AB is to the gravity in the same place A
toward a sphere described about a center C with a radius AC as 125 to 126.
Moreover, the gravity in place A toward the earth is a mean proportional
between the gravity toward the spheroid and the gravity toward the sphere,
because the sphere, when its diameter PQ is diminished in the ratio of 101
to 100, is transformed into the figure of the earth; and this figure, when
a third diameter (perpendicular to the two given diameters AB and PQ)
is diminished in the same ratio, is transformed into the said spheroid; and
the gravity in A, in either case, is diminished in very nearly the same ratio.
Therefore the gravity in A toward a sphere described about the center C with
a radius AC is to the gravity in A toward the earth as 126 to 125½; and
the gravity in place Q, toward a sphere described about the center C with a
radius QC, is to the gravity in place A, toward a sphere described about the
center C with a radius AC, in the ratio of the diameters (by book 1, prop. 72),
that is, as 100 to 101. Now let these three ratios (126 to 125, 126 to 125½,
and 100 to 101) be combined, and the gravity in place Q toward the earth
will become to the gravity in place A toward the earth as 126 × 126 × 100 to
125 × 125½ × 101, or as 501 to 500.
Now, since (by book 1, prop. 91, corol. 3) the gravity in either leg ACca
or QCcq of the channel is as the distance of places from the earth’s center, if
those legs are separated by transverse, equidistant surfaces into parts
proportional to the wholes, the weights of any number of these individual parts in
the leg ACca will be to the weights of the same number of individual parts
in the other leg as their magnitudes and accelerative gravities jointly, that is,
as 101 to 100 and 500 to 501, which is as 505 to 501. And accordingly, if the
centrifugal force of each part of the leg ACca (which force arises from the
daily motion) had been to the weight of the same part as 4 to 505, so that
it would take away four parts from the weight of each part (supposing it
to be divided into 505 parts), the weights would remain equal in each leg,
and therefore the fluid would stay at rest in equilibrium. But the centrifugal
force of each part is to the weight of the same part as 1 to 289; that is, the
fcentrifugalf force, which ought to have
been 
of the weight, is only 
of it. And therefore I say, according to the golden rule [or rule of three], that
if a centrifugal force of 
of the weight makes the height of the water in
the leg ACca exceed the height of the water in the leg QCcq by a hundredth
of its total height, the centrifugal force of 
of the weight will make the
excess of the height in the leg ACca be only 
of the height of the water
in the other leg QCcq. Therefore the diameter of the earth at the equator
is to its diameter through the poles as 230 to 229. And thus, since the mean
semidiameter of the earth, according to Picard’s measurement, is 19,615,800
Paris feet, or 3,923.16 miles (supposing a mile to be 5,000 feet), the earth will
be 85,472 feet or 171/10 miles higher at the equator than at the poles. And its
height at the equator will be about 19,658,600 feet, and at the poles will be
about 19,573,000 feet.
If a planet is larger or smaller than the earth, while its density and
periodic time of daily revolution remain the same, the ratio of centrifugal force
to gravity will remain the same, and therefore the ratio of the diameter
between the poles to the diameter at the equator will also remain the same.
But if the daily motion is accelerated or retarded in any ratio, the centrifugal
force will be increased or decreased in that same ratio squared, and
therefore the difference between the diameters will be increased or decreased very
nearly in the same squared ratio. And if the density of a planet is increased
or decreased in any ratio, the gravity tending toward the planet will also
be increased or decreased in the same ratio, and the difference between the
diameters in turn will be decreased in the ratio of the increase in the gravity
or will be increased in the ratio of the decrease in the gravity. Accordingly,
since the earth revolves [i.e., rotates] with respect to the fixed stars in 23h56m,
and Jupiter in 9h56m, and the squares of their periodic times are as 29 to
5, and the densities of these revolving bodies are as 400 to 94½, the
difference between the diameters of Jupiter will be to its smaller diameter as
to 1, or very nearly as 1 to 9⅓. Therefore Jupiter’s
diameter taken from east to west is to its diameter between the poles very nearly
as 10⅓ to 9⅓. gThus,
since its larger diameter is 37″, its smaller diameter
(which lies between the poles) will be 33″25‴. Because of the erratic light
let about 3″ be added, and the apparent diameters of this planet will come
out to be 40″ and 36″25‴, which are to each other nearly as 11⅙ to 10⅙.
This argument has been based on the hypothesis that the body of Jupiter is
uniformly dense. But if its body is denser toward the plane of the equator
than toward the poles, its diameters can be to each other as 12 to 11, or 13
to 12, or even 14 to 13. As a matter of fact, Cassini observed in the year
1691 that the diameter of Jupiter extending from east to west would exceed
its other diameter by about a fifteenth part of itself. Moreover, our fellow
countryman Pound, with a 123-foot-long telescope and the best
micrometer, measured the diameters of Jupiter in the year 1719 with the following
results.
Therefore the theory agrees with the phenomena. Further, the planets are more exposed to the heat of sunlight toward their equators and as a result hare somewhat more thoroughly heated thereh than toward the poles.
Proposition 20, Problem 4
To find and compare with one another the weights of bodies in different regions
of our earth.
Since the weights of the unequal legs of the water-channel ACQqca
are equal, and the weights of any parts that are proportional to the whole
legs and similarly situated in those legs are to one another as the weights
of the wholes, and thus are also equal to one another, the weights of parts
that are equal and similarly situated in the legs will be inversely as the
legs, that is, inversely as 230 to 229. This is likewise
the case for any homogeneous equal bodies that are
similarly situated in the legs of the channel. The
weights of these bodies are inversely as the legs,
that is, inversely as the distances of the bodies from
the earth’s center. Accordingly, if the bodies are in
the topmost parts of the channels, or on the surface
of the earth, their weights will be to one another inversely as their distances
from the center. And by the same argument, weights that are in any other
regions whatever, anywhere on the whole surface of the earth, are inversely as
the distances of those places from the center; and therefore, on the hypothesis
that the earth is a spheroid, the proportion of those weights is given.
From this the following theorem is deduced:a The increase of weight in
going from the equator to the poles is very nearly as the versed sine of twice
the latitude, or (which is the same) as the square of the sine of the latitude.
bAnd the arcs of degrees of latitude on a meridian are increased in about the
same ratio. Now, the latitude of Paris is 48°50′, the latitude of places on the
equator 00°00′, and that of places at the poles 90°; the versed sines of twice
those arcs of latitude are 11,334 and 00,000 and 20,000 (the radius being taken
to be 10,000); the gravity at the pole is to the gravity at the equator as 230 to
229; and the excess of the gravity at the pole to the gravity at the equator is
as 1 to 229. Hence the excess of the gravity at the latitude of Paris will be to
the gravity at the equator as 1 × 
to 229, or 5,667 to 2,290,000. And
therefore the total gravities in these places will be to each other as 2,295,667
to 2,290,000. And thus, since the lengths of pendulums oscillating with equal
periods are as the gravities, and at the latitude of Paris the length of a seconds
pendulum is 3 Paris feet and 8½ lines (or rather, because of the weight of
the air, 85/9 lines), the length of a pendulum at the equator will be shorter
than the length of a pendulum with the same period at Paris in the amount
of 1.087 lines. And a similar computation yields the following table.
Now some astronomers, sent to distant regions to make astronomical observations, have observed that their pendulum clocks went more slowly near the equator than in our regions. And indeed M. Richer first observed this in the year 1672 on the island of Cayenne. For while he was observing the transit of the fixed stars across the meridian in the month of August, he found that his clock was going more slowly than in its proper proportion to the mean motion of the sun, the difference being 2m28s every day. Then by constructing a simple pendulum that would oscillate in seconds as measured by the best clock, he noted the length of the simple pendulum, and he did this frequently, every week for ten months. Then, when he had returned to France, he compared the length of this pendulum with the length of a seconds pendulum at Paris (which was 3 Paris feet and 8⅗ lines long) and found that it was shorter than the Paris pendulum, the difference being 1¼ lines.d
M. Picard, at any rate, observed that an iron rod, which in wintertime when the weather was freezing was 1 foot long, came to be 1 foot and ¼ of a line long when heated by a fire. Later M. La Hire observed that an iron rod, which in an exactly similar winter was 6 feet long, came to be 6 feet and ⅔ of a line long when it was exposed to the summer sun. The heat [i.e., temperature] was greater in the first example than in the second, and in the second it was greater than that of the external parts of the human body. For metals grow extremely hot in the summer sun. But the pendulum rod in a pendulum clock is ordinarily never exposed to the heat of the summer sun, and never acquires a heat equal to that of the external surface of the human body. And, therefore, although a 3-foot-long pendulum rod in a clock will indeed be a little longer in summertime than in wintertime, this increase will scarcely surpass ¼ of 1 line. Accordingly, all of the difference in the length of pendulums with the same period in different regions cannot be attributed to differences in heat. Nor can this difference be attributed to errors made by the astronomers sent from France. For although their observations do not agree perfectly with one another, the errors are so small that they can be ignored. And in this they all agree: that at the equator, pendulums are shorter than pendulums with the same period at the Royal Observatory in Paris, ethe difference being neither less than 1¼ lines nor more than 2⅔ lines. By the observations of M. Richer made in Cayenne the difference was 1¼ lines. By those of M. Des Hayes that difference when corrected became 1½ or 1¾ lines. By the less accurate observations made by others, this difference came out as more or less 2 lines. And this discrepancy could have arisen partly from errors in observations, partly from the dissimilitude of the internal parts of the earth and from the height of mountains, and partly from the differences in heat [i.e., temperatures] of the air.
Proposition 21, Theorem 17
The equinoctial points regress, and the earth’s axis, by a nutation in every
annual revolution, inclines twice toward the ecliptic and twice returns to its former
position.
This is clear by book 1, prop. 66, corol. 20. This motion of nutation, however, must be very small—either scarcely or not at all perceptible.
Proposition 22, Theorem 18
All the motions of the moon and all the inequalities in its motions follow from
the principles that have been set forth.
That the major planets, while they are being carried about the sun, can carry other or minor planets [or satellites], revolving around them, and that those minor planets must revolve in ellipses having their foci in the centers of the major planets, is evident from book 1, prop. 65. Moreover, their motions will be perturbed in many ways by the sun’s action, and they will be influenced by those inequalities that are observed in our moon. Our moon, in any case (by book 1, prop. 66, corols. 2, 3, 4, and 5), moves more swiftly, and by a radius drawn to the earth describes an area greater for the time, and has a less curved orbit, and therefore approaches closer to the earth, in the syzygies than in the quadratures, except insofar as these effects are hindered by the motion of eccentricity. For the eccentricity is greatest (by book 1, prop. 66, corol. 9) when the moon’s apogee is in the syzygies, and least when it stands in the quadratures; and thus the moon in its perigee is swifter and closer to us, while in its apogee it is slower and more remote, in the syzygies than in the quadratures. Additionally, the apogee advances and the nodes regress, but with a nonuniform motion. And indeed the apogee (by prop. 66, corols. 7 and 8) advances more swiftly in its syzygies, regresses more slowly in the quadratures, and by the excess of the advance over the regression is annually carried forward [or in consequentia, i.e., from east to west in the direction of the signs]. But the nodes (by prop. 66, corol. 2) are at rest in their syzygies and regress most swiftly in the quadratures. The moon’s greatest latitude is also greater in its quadratures (by prop. 66, corol. 10) than in its syzygies, and (by prop. 66, corol. 6) the mean motion of the moon is slower in the earth’s perihelion than in its aphelion. And these are the more significant inequalities [of the moon’s motion] taken note of by astronomers.
There are also certain other inequalities not observed by previous astronomers, by which the lunar motions are so perturbed that until now these motions have not been reducible, by any law, to any definite rule. For the velocities or hourly motions of the moon’s apogee and nodes, and their equations, and also the difference between the greatest eccentricity in the syzygies and the least in the quadratures, and that inequality which is called the variation, are increased and decreased annually (by prop. 66, corol. 14) as the cube of the sun’s apparent diameter. And, additionally, the variation is increased or decreased very nearly as the square of the time between the quadratures (by book 1, lem. 10, corols. 1 and 2, and prop. 66, corol. 16), but in astronomical calculations this inequality is generally included under the moon’s prosthaphaeresis [or equation of the center] and confounded with it.
Proposition 23, Problem 5
To derive the unequal motions [i.e., the inequalities in the motions] of the satellites
of Jupiter and of Saturn from the motions of our moon.
From the motions of our moon the analogous motions of the moons or satellites of Jupiter are derived as follows. The mean motion of the nodes of Jupiter’s outermost satellite is (by book 1, prop. 66, corol. 16) to the mean motion of the nodes of our moon in a ratio compounded of the square of the ratio of the earth’s periodic time about the sun to Jupiter’s periodic time about the sun, and of the simple ratio of the satellite’s periodic time about Jupiter to the moon’s periodic time about the earth, and so in one hundred years that node completes 8°24′ backward [or in antecedentia, i.e., counter to the order of the signs]. The mean motions of the nodes of the inner satellites are (by the same corollary) to the motion of this outermost satellite as the periodic times of those inner satellites are to the periodic time of the outermost satellite and hence are given. Moreover (by the same corollary), the forward [or direct] motion of the upper apsis of each satellite [or its motion in consequentia] is to the backward [or retrograde] motion of its nodes [or the motion in antecedentia] as the motion of the apogee of our moon to the motion of its nodes, and hence is also given. However, the motion of the upper apsis found in this way must be decreased in the ratio of 5 to 9, or about 1 to 2, for a reason which would take too much time to explain here. The greatest equations of the nodes and upper apsis of each satellite are approximately to the greatest equations of the nodes and upper apsis of our moon respectively as the motions of the nodes and upper apsis of the satellites in the time of one revolution of the former equations are to the motions of the nodes and apogee of our moon in the time of one revolution of the latter equations. By the same corollary, the variation of a satellite as it would be observed from Jupiter is to the variation of our moon in the same proportion as the total motions of their nodes during the times in which respectively the satellite and our moon revolve as reckoned in relation to the sun; and therefore in the outermost satellite the variation does not exceed 5″12‴.
Proposition 24, Theorem 19
The ebb and flow of the sea arise from the actions of the sun and moon.
It is clear from book 1, prop. 66, corols. 19 and 20, that the sea should twice rise and twice fall in every day, lunar as well as solar, and also that the greatest height of the water, in deep and open seas, should occur less than six hours after the appulse of the luminaries to the meridian of a place, as happens in the whole eastern section of the Atlantic Ocean and the Ethiopic [or South Atlantic] Sea between France and the Cape of Good Hope, and also on the Chilean and Peruvian shore of the Pacific Ocean; on all these shores the tide comes in at about the second, third, or fourth hour, except in cases when the motion has been propagated from the deep ocean through shallow places and is delayed until the fifth, sixth, or seventh hour, or later. I number the hours from the appulse of either luminary to the meridian of a place, below the horizon as well as above, and by hours of a lunar day I mean twenty-fourths of that time in which the moon, by its apparent daily motion, returns to the meridian of the place. The force of the sun or moon to raise the sea is greatest in the very appulse of the luminary to the meridian of the place. But the force impressed upon the sea at that time remains for a while and is increased by a new force subsequently impressed, until the sea has ascended to its greatest height, which will happen in one or two hours, but more frequently at the shores in about three hours or even more if the sea is shallow.
Moreover, the two motions which the two luminaries excite will not be discerned separately but will cause what might be called a mixed motion. In the conjunction or the opposition of the luminaries their effects will be combined, and the result will be the greatest ebb and flow. In the quadratures the sun will raise the water while the moon depresses it and will depress the water while the moon raises it; and the lowest tide of all will arise from the difference between these two effects. And since, as experience shows, the effect of the moon is greater than that of the sun, the greatest height of the water will occur at about the third lunar hour. Outside of the syzygies and quadratures, the highest tide, which by the lunar force alone would always have to occur at the third lunar hour, and by the solar force alone at the third solar hour, will occur, as a result of the combining of the lunar and solar forces, at some intermediate time which is closer to the third lunar hour [than to the third solar hour]; and thus in the transit of the moon from the syzygies to the quadratures, when the third solar hour precedes the third lunar hour, the greatest height of the water will also precede the third lunar hour, and will do so by the greatest interval a little after the octants of the moon; and the highest tide will follow the third lunar hour with the same intervals in the transit of the moon from the quadratures to the syzygies. This is what happens in the open sea. For in the mouths of rivers the higher tides, other things being equal, will come to their peaks later.
Additionally, the effects of the luminaries depend on their distances from the earth. For at smaller distances their effects are greater, and at greater distances smaller, and this varies as the cubes of their apparent diameters. Therefore the sun in wintertime, when it is in its perigee, produces greater effects and makes the tides a little higher in the syzygies and a little lower (other things being equal) in the quadratures than in summertime; and the moon in its perigee every month produces higher tides than fifteen days before or after, when it is in its apogee. Accordingly, it happens that the two very highest tides do not follow each other in successive syzygies.
The effect of each luminary depends also on its declination, or distance from the equator. For if the luminary should be at one of the poles, it would constantly draw the individual parts of water, without intension and remission of action, and thus would produce no reciprocation of motion. Therefore the luminaries, in receding from the equator toward a pole, will lose their effects by degrees, and for this reason will produce lower tides in the solstitial syzygies than in the equinoctial syzygies. In the solstitial quadratures, however, they will produce higher tides than in the equinoctial quadratures, because the effect of the moon, which is now at the equator, most exceeds the effect of the sun. Therefore the highest tides occur at the syzygies of the luminaries, and the lowest at their quadratures, at about the times of either of the two equinoxes. And the highest tide in the syzygies is always acompanied by the lowest tide in the quadratures, as has been learned by experience. Moreover, as a result of the smaller distance of the sun from the earth in winter than in summer, it comes about that the highest and lowest tides more often precede the vernal equinox than follow it, and more often follow the autumnal equinox than precede it.
The effects of the luminaries depend also on the latitude of places. Let
ApEP represent the earth covered everywhere with deep waters, C its center,
P and p the poles, AE the equator, F any place not on the equator, Ff the
parallel of that place, Dd the parallel corresponding to it on the other side of
the equator, L the place that the
moon was occupying three hours
earlier, H the place on the earth
situated perpendicularly beneath L,
h the place opposite H, K and k
places 90 degrees distant from H
and h, CH and Ch the greatest
heights of the sea (measured from
the center of the earth), and CK and Ck the least heights. If an ellipse is
described with axes Hh, Kk, and then if by the revolution of this ellipse
about the major axis Hh a spheroid HPKhpk is described, this spheroid
will represent the figure of the sea very nearly, and CF, Cf, CD, Cd will
be the heights of the sea at places F, f, D, d. Further, if in the aforesaid
revolution of the ellipse any point N describes a circle NM which cuts
parallels Ff, Dd in any places R, T, and cuts the equator AE in S, CN
will be the height of the sea in all places R, S, T located on this circle.
Hence, in the daily revolution of any place F, the greatest flood tide will be
in F at the third hour after the appulse of the moon to the meridian above
the horizon; afterward, the greatest ebb tide will be in Q at the third hour
after the setting of the moon; then the greatest flood tide will be in f at the
third hour after the appulse of the moon to the meridian below the horizon;
finally, the greatest ebb tide will be in Q at the third hour after the rising
of the moon; and the latter flood tide in f will be smaller than the former
flood tide in F.
For the whole sea is divided into just two hemispherical flows [or flowing bodies of water], one in the hemisphere KHk verging to the north, the other in the opposite hemisphere Khk; and these may therefore be called the northern flow and the southern flow. These flowing bodies of waters, which are always opposite to each other, come by turns to the meridian of every single place, with an interval of twelve lunar hours between them. And since the northern regions partake more of the northern flow, and the southern regions more of the southern flow, higher and lower tides arise from them alternately, in every single place not on the equator in which the luminaries rise and set. Moreover, the higher tide, when the moon declines toward the vertex of the place, will occur at about the third hour after the appulse of the moon to the meridian above the horizon, and when the moon changes its declinationa, this higher tide will be turned into a lower one. And the greatest difference between these tides will occur at the times of the solstices, especially if the ascending node of the moon is in the first of Aries. Thus it has been found by experience that in winter, morning tides exceed evening tides and that in summer, evening tides exceed morning tides, at Plymouth by a height of about one foot, and at Bristol by a height of fifteen inches, according to the observations of Colepress and Sturmy.
Moreover, the motions hitherto described are changed somewhat by the force of reciprocation of the waters, by which a tide of the sea, even if the actions of the luminaries were to cease, would be able to persevere for a while. This conservation of impressed motion lessens the difference between alternate tides; and it makes the tides immediately after the syzygies higher and makes those immediately after the quadratures lower. Hence it happens that alternate tides at Plymouth and Bristol do not differ from each other by much more than a height of one foot or fifteen inches, and that the very highest tides in those same harbors are not the first tides after the syzygies but the third. All the motions are made slower also in their passing through shallows, to such an extent that the very highest tides, in certain straits and the mouths of rivers, are the fourth or even the fifth after the syzygies.
Further, it can happen that a tide is propagated from the ocean through different channels to the same harbor and passes more quickly through some channels than through others; in this case the same tide, divided into two or more tides arriving successively, can compose new motions of different kinds. Let us suppose that two equal tides come from different places to the same harbor and that the first precedes the second by a space of six hours and occurs at the third hour after the appulse of the moon to the meridian of the harbor. If the moon is on the equator at the time of this appulse to the meridian, then every six hours there will be equal flood tides coming upon corresponding equal ebb tides and causing those ebb tides to be balanced by the flood tides, and thus during the course of that day they will cause the water to stay quiet and still. If at that time the moon is declining from the equator, there will be alternately higher and lower tides in the ocean, as has been said; and from the ocean, two higher and two lower tides will each be alternately propagated toward this harbor. Moreover, the two greater flood tides will produce the highest water in the middle time between them; the greater and lesser flood tides will make the water ascend to its mean height in the middle time between them; and between the two lesser flood tides the water will ascend to its least height. Thus in the space of twenty-four hours, the water will only once reach its greatest height, not twice as usually happens, and will only once reach its least height; and the greatest height, if the moon is declining toward the pole above the horizon of the place, will occur at either the sixth or the thirtieth hour after the appulse of the moon to the meridian; and when the moon changes its declination, this flood tide will be changed into an ebb tide. An example of all these things has been given by Halley, on the basis of sailors’ observations, in Batsha harbor in the kingdom of Tonkin at a latitude of 20°50′ N. There the water stays still on the day following the transit of the moon over the equator; then, when the moon declines toward the north, the water begins to ebb and flow—not twice, as in other harbors, but only once every day; and the flood tide occurs at the setting of the moon, and the greatest ebb tide at its rising. This flood tide increases with the declination of the moon until the seventh or eighth day; then during the next seven days it decreases at the same rate at which it had previously increased. And when the moon changes its declination, the flood ceases and is then turned into an ebb. For thereafter the ebb tide occurs at the setting of the moon and the flood tide at its rising, until the moon again changes its declination. There are two different approaches from the ocean into this harbor and the neighboring channels, the one from the China Sea between the continent and the island of Leuconia, the other from the Indian Ocean between the continent and the island of Borneo. But whether there are tides coming through these channels in twelve hours from the Indian Ocean and in six hours from the China Sea, which thus occurring at the third and ninth lunar hours compound motions of this sort, or whether there is any other condition of those seas, I leave to be determined by observations of the neighboring shores.
Hitherto I have given the causes of the motions of the moon and seas. It is now proper to subjoin some things about the quantity of those motions.
Proposition 25, Problem 6
To find the forces of the sun that
perturb the motions of the moon.
Let S designate the sun, T the earth, P the moon, CADB the orbit of the moon. On SP take SK equal to ST; and let SL be to SK as SK2 to SP2, and draw LM parallel to PT; and if the accelerative gravity of the earth toward the sun is represented by the distance ST or SK, SL will be the accelerative gravity of the moon toward the sun. This is compounded of the parts SM and LM, of which LM and the part TM of SM perturb the motion of the moon, as has been set forth in book 1, prop. 66 and its corollaries. Insofar as the earth and moon revolve around their common center of gravity, the motion of the earth about that center will also be perturbed by entirely similar forces; but it is possible to refer the sums of the forces and the sums of the motions to the moon, and to represent the sums of the forces by the lines TM and ML that correspond to them. The force ML, in its mean quantity, is to the centripetal force by which the moon could revolve in its orbit, about an earth at rest at a distance PT, as the square of the ratio of the periodic time of the moon about the earth to that of the earth about the sun (by book 1, prop. 66, corol. 17), that is, as the square of the ratio of 27d7h43m to 365d6h9m, that is, as 1,000 to 178,725, or 1 to 17829/40. But we found in prop. 4 of this book 3 that if the earth and moon revolve about their common center of gravity, their mean distance from each other will be very nearly 60½ mean semidiameters of the earth. And the force by which the moon could revolve in orbit about the earth at rest at a distance PT of 60½ terrestrial semidiameters is to the force by which it could revolve in the same time at a distance of 60 semidiameters as 60½ to 60; and this force is to the force of gravity on the earth as 1 to 60 × 60 very nearly. And so the mean force ML is to the force of gravity on the surface of the earth as 1 × 60½ to 60 × 60 × 60 × 17829/40, or as 1 to 638,092.6. From this and from the proportion of the lines TM and ML, the force TM is also given; and these are the forces of the sun by which the motions of the moon are perturbed. Q.E.I.
Proposition 26, Problem 7
To find the hourly increase of the area that the moon, by a radius drawn to the
earth, describes in a circular orbit.
We have said that the area which the moon describes by a radius drawn
to the earth is proportional to the time, except insofar as the motion of the
moon is disturbed by the action of the sun. We propose to investigate here
the inequality of the moment, or of the hourly increase [under the foregoing
condition of disturbance]. To make the computation easier, let us imagine
that the orbit of the moon is circular, and let us ignore all inequalities with
the sole exception of the one under discussion here. Because of the enormous
distance of the sun, let us suppose also that the lines SP and ST are parallel
to each other. By this means the force LM will always be reduced to its mean
quantity TP, and so will the force TM be reduced to its mean quantity 3PK.
These forces (by corol. 2 of the laws of motion) compose the force TL; and
if a perpendicular LE is dropped to the radius TP, this force is resolved
into the forces TE and EL, of which TE, always acting along the radius
TP, neither accelerates nor retards the description of the area TPC made
by that radius TP; and EL, acting along the perpendicular to the radius,
accelerates or retards the description of the area, as much as it accelerates or
retards the moon. That acceleration of the moon, made in each individual
moment of time, in the transit of the moon from the quadrature C to the
conjunction A, is as the accelerating force itself EL, that is, as
.
Let the time be represented by the mean motion of the moon or (which
comes to about the same thing) by the angle CTP or by the arc CP. On CT
erect a normal CG (equal to CT). And when the quadrantal arc AC has
been divided into innumerable equal particles Pp, . . . , by which the same
innumerable quantity of equal particles of time can be represented, and when
a perpendicular pk has been drawn to CT, draw TG meeting KP and kp
(produced) in F and f; and FK will be equal to TK, and Kk will be to
PK as Pp to Tp, that is, in a given ratio; and therefore FK × Kk, or the
area FKkf, will be as , that is, as EL; and, by compounding,
the total area GCKF will be as the sum of all the forces EL impressed on
the moon in the total time CP, and so also as the velocity generated by this
sum, that is, as the acceleration of the description of the area CTP, or the
increase of its moment. The force by which the moon could revolve in its
periodic time CADB of 27d7h43m about the earth at rest, at the distance TP,
would make a body, by falling in the time CT, describe the space ½CT,
and at the same time acquire a velocity equal to the velocity with which
the moon moves in its orbit. This is evident from book 1, prop. 4, corol. 9.
However, since the perpendicular Kd dropped to TP is a third of EL, and
is equal to a half of TP or ML in the octants, the force EL in the octants
(where it is greatest) will exceed the force ML in the ratio of 3 to 2, and
so will be to that force by which the moon could revolve in its periodic
time about the earth at rest as 100 to ⅔ × 17,872½, or 11,915, and should
in the time CT generate a velocity which would be 
of the moon’s
velocity; but in the time CPA this force would generate a greater velocity in
the ratio of CA to CT or TP. Let the greatest force EL in the octants be
represented by the area FK × Kk equal to the rectangle ½TP × Pp. And
the velocity which that greatest force could generate in any time CP will be
to the velocity which any other lesser force EL generates, in the same time,
as the rectangle ½TP × CP to the area KCGF; but the velocities generated
in the whole time CPA will be to each other as the rectangle ½TP × CA
to the triangle TCG, or as the quadrantal arc CA to the radius TP. And so
(by book 5, prop. 9 of the Elements) the latter velocity generated in the whole
time will be of the velocity of the moon. Change this velocity of the
moon, which corresponds to the mean moment of the area, by adding and
subtracting half of the other velocity; and if the mean moment is represented
by the number 11,915, the sum 11,915 + 50 (or 11,965) will represent the
greatest moment of the area in the syzygy A, and the difference 11,915 − 50
(or 11,865) the least moment of the same area in the quadratures. Therefore
the areas which are described in equal times in the syzygies and quadratures
are to each other as 11,965 to 11,865. To the least moment 11,865 add the
moment that is to the difference (100) of the two above-mentioned moments
as the quadrilateral FKCG is to the triangle TCG or, which comes to the
same thing, as the square of the sine PK to the square of the radius TP (that
is, as Pd to TP); then the sum will represent the moment of the area when
the moon is in any intermediate place P.
All these things are so on the hypothesis that the sun and earth are at
rest, and that the moon has a synodic period of revolution of 27d7h43m. But
since the moon’s synodic period is actually 29d12h44m, the increments of the
moments should be increased in the ratio of the time, that is, in the ratio of
1,080,853 to 1,000,000. In this way the total increment, which was 
of
the mean moment, will now become 
of it. And so the moment of the
area in the quadrature of the moon will be to its moment in the syzygy as
11,023 − 50 to 11,023 + 50, or as 10,973 to 11,073; and to its moment, when
the moon is in any other intermediate place P, as 10,973 to 10,973 + Pd,
taking TP to be equal to 100.
Therefore the area that the moon, by a radius drawn to the earth, describes in every equal particle of time is very nearly as the sum of the number 219.46 and of the versed sine of twice the distance of the moon from the nearest quadrature, with respect to a circle whose radius is unity. These things are so when the variation in the octants is at its mean magnitude. But if the variation there is greater or less, that versed sine should be increased or decreased in the same ratio.
Proposition 27, Problem 8
From the hourly motion of the moon, to find its distance from the earth.
The area that the moon, by a radius drawn to the earth, describes in every moment of time is as the hourly motion of the moon and the square of the distance of the moon from the earth jointly. And therefore the distance of the moon from the earth is directly proportional to the square root of the area and inversely proportional to the square root of the hourly motion. Q.E.I.
COROLLARY 1. Hence the apparent diameter of the moon is given, since it is inversely as the distance of the moon from the earth. Let astronomers test how accurately this rule agrees with the phenomena.
COROLLARY 2. Hence also the lunar orbit can be defined more exactly from the phenomena than could have been done before now.
Proposition 28, Problem 9
To find the diameters of the orbit in which the moon would have to move, if there
were no eccentricity.
The curvature of the trajectory that a moving body describes, if it is
attracted in a direction which is everywhere perpendicular to that trajectory,
is as the attraction directly and the square of the velocity inversely. I reckon
the curvatures of lines as being among themselves in the ultimate ratio of
the sines or of the tangents of the angles of contact, with respect to equal
radii, when those radii are diminished indefinitely. Now, the attraction of
the moon toward the earth in the syzygies is the excess of its gravity toward
the earth over the solar force 2PK (as in the figure to prop. 25), by which
force the accelerative gravity of the moon toward the sun exceeds the
accelerative gravity of the earth toward the sun or is exceeded by it. In the
quadratures that attraction is the sum of the gravity of the moon toward
the earth and the solar force KT (which draws the moon toward the earth).
And these attractions, if 
is called N, are very nearly as
and 
, or as 178,725N × CT2 − 2000AT2 × CT
and 178,725N × AT2 + 1,000CT2 × AT. For if the accelerative gravity of
the moon toward the earth is represented by the number 178,725, then the
mean force ML, which in the quadratures is PT or TK and draws the moon
toward the earth, will be 1,000, and the mean force TM in the syzygies will
be 3,000; if the mean force ML is subtracted from that, there will remain the
force 2,000 by which the moon in the syzygies is drawn apart from the earth
and which I have called 2PK above. Now, the velocity of the moon in the
syzygies (A and B) is to its velocity in the quadratures (C and D) jointly as
CT is to AT and as the moment of the area that the moon (by a radius drawn
to the earth) describes in the syzygies is to the moment of that same area as
described in the quadratures, that is, as 11,073CT to 10,973AT. Take this
ratio squared inversely and the above ratio directly, and the curvature of the
moon’s orbit in the syzygies will become to its curvature in the quadratures
as 120,406,729 × 178,725AT2 × CT2 × N − 120,406,729 × 2,000AT4 × CT
to 122,611,329 × 178,725AT2 × CT2 × + 122,611,329 × 1,000CT4 × AT,
that is, as 2,151,969AT × CT × N − 24,081AT3 to 2,191,371AT × CT ×
N + 12,261CT3.
Since the figure of the lunar orbit is unknown, in its place let us assume
an ellipse DBCA, in whose center T the earth is placed, and let its major
axis DC lie between the quadratures and its
minor axis AB between the syzygies. And
since the plane of this ellipse revolves about
the earth with an angular motion, and since
the trajectory whose curvature we are
considering ought to be described in a plane
that is entirely devoid of any angular
motion, we must consider the figure that the
moon, while revolving in that ellipse,
describes in this place, that is, the figure Cpa,
whose individual points p are found by
taking any point P on the ellipse to represent
the place of the moon, and by drawing Tp
equal to TP in such a way that the angle PTp is equal to the apparent
motion of the sun since the time of quadrature C, or (which comes to almost
the same thing) in such a way that the angle CTp is to the angle CTP as
the time of a synodic revolution of the moon is to the time of a periodic
revolution, or as 29d12h44m to 27d7h43m. Therefore, take the angle CTa in
this same ratio to the right angle CTA, and let the length Ta be equal to
the length TA, then a will be the lower apsis and c the upper apsis of this
orbit Cpa. And by making calculations I find that the difference between
the curvature of the orbit Cpa at the vertex a and the curvature of the circle
described with center T and radius TA has a ratio to the difference between
the curvature of the ellipse at the vertex A and the curvature of that circle
which is equal to the ratio of the square of the angle CTP to the square of
the angle CTp and that the curvature of the ellipse at A is to the curvature
of that circle in the ratio of TA2 to TC2; and the curvature of that circle is
to the curvature of a circle described with center T and radius TC as TC
to TA; but this curvature is to the curvature of the ellipse at C in the ratio
of TA2 to TC2; and the difference between the curvature of the ellipse at
the vertex C and the curvature of this last circle is to the difference between
the curvature of the figure Tpa at the vertex C and the curvature of the
same circle in the ratio of the square of the angle CTp to the square of the
angle CTP. And these ratios are easily gathered from the sines of the
angles of contact and of the differences of the angles. Moreover, by comparing
these, the curvature of the figure Cpa at a comes out to its curvature at C
as AT3 + 
CT2 × AT to CT3 + AT2 × CT; where the factor
represents the difference of the squares of the angles CTP and CTp
divided by the square of the smaller angle CTP, or (which is the same) the
difference of the squares of the times 27d7h43m and 29d12h44m divided by the
square of the time 27d7h43m.
Therefore, since a designates the syzygy of the moon and C its quadrature, the proportion just found must be the same as the proportion of the curvature of the orbit of the moon in the syzygies to its curvature in the quadratures, which we found above. Accordingly, to find the proportion of CT to AT, I multiply the extremes by the means. And the resulting terms divided by AT × CT become 2,062.79CT4 − 2,151,969N × CT3 + 368,676N × AT × CT2 + 36,342AT2 × CT2 − 362,047N × AT2 × CT + 2,191,371N × AT3 + 4,051.4AT4 = 0. When I take the half-sum N of the terms AT and CT to be 1, and their half-difference to be x, there results CT = 1 + x and AT = 1 − x; and when these values are put into the equation and the resulting equation is resolved, x is found equal to 0.00719, and hence the semidiameter CT comes out 1.00719 and the semidiameter AT 0.99281. These numbers are very nearly as 701/24 and 691/24. Therefore the distance of the moon from the earth in the syzygies is to its distance in the quadratures (setting aside, that is, any consideration of eccentricity) as 691/24 to 701/24, or in round numbers as 69 to 70.
Proposition 29, Problem 10
To find the variation of the moon.
This inequality arises partly from the elliptical form of the orbit of the moon and partly from the inequality of the moments of the area that the moon describes by a radius drawn to the earth. If the moon P moved in the ellipse DBCA about the earth at rest in the center of the ellipse and, by a radius TP drawn to the earth, described the area CTP proportional to the time, and if furthermore the greatest semidiameter CT of the ellipse were to the least semidiameter TA as 70 to 69, then the tangent of the angle CTP would be to the tangent of the angle of the mean motion (reckoned from the quadrature C) as the semidiameter TA of the ellipse to its semidiameter TC, or as 69 to 70. Moreover, the description of the area CTP ought, in the progress of the moon from quadrature to syzygy, to be accelerated in such a way that the moment of this area in the syzygy of the moon will be to its moment in its quadrature as 11,073 to 10,973, and in such a way that the excess of the moment in any intermediate place P over the moment in the quadrature will be as the square of the sine of the angle CTP. And this will occur exactly enough if the tangent of angle CTP is diminished in the ratio of √10,973 to √11,073, or in the ratio of 68.6877 to 69. In this way the tangent of angle CTP will now be to the tangent of the mean motion as 68.6877 to 70; and the angle CTP in the octants, where the mean motion is 45°, will be found to be 44°27′28″, which, when subtracted from the angle of the mean motion of 45°, leaves the greatest variation 32′32″. These things would be so if the moon, in going from quadrature to syzygy, described an angle CTA of only 90°. But because of the motion of the earth, by which the sun is transferred forward [or in consequentia] in its apparent motion, the moon, before it reaches the sun, describes an angle CTa greater than a right angle, in the ratio of the time of a synodic revolution of the moon to the time of its periodic revolution, that is, in the ratio of 29d12h44m to 27d7h43m. And in this way all the angles about the center T are enlarged in the same ratio; and the greatest variation, which would otherwise be 32′32″, now increased in the same ratio, becomes 35′10″.
This is the magnitude of the greatest variation at the mean distance of the sun from the earth, ignoring the differences that can arise from the curvature of the earth’s orbit and the greater action of the sun upon the sickle-shaped and the new moon than upon the gibbous and the full moon. At other distances of the sun from the earth, the greatest variation is directly as the square of the time of synodic revolution and inversely as the cube of the distance of the sun from the earth. And therefore in the apogee of the sun the greatest variation is 33′14″, and in its perigee 37′11″ provided that the eccentricity of the sun is to the transverse semidiameter of the great orbit [i.e., the earth’s orbit] as 1615/16 to 1,000.
Hitherto we have investigated the variation in a noneccentric orbit, in which the moon in its octants is always at its mean distance from the earth. If the moon, because of its eccentricity, is more distant or less distant from the earth than if it were placed in this orbit, the variation can be a little greater or a little less than according to the rule asserted here; but I leave the excess or deficiency for astronomers to determine from phenomena.
Proposition 30, Problem 11
To find the hourly motion of the nodes of the moon in a circular
orbit.
Let S designate the sun, T the earth, P the moon, NPn the orbit of the moon, Npn the projection of the orbit in the plane of the ecliptic; N and n the nodes, nTNm the line of the nodes, indefinitely produced; PI and PK perpendiculars dropped to the lines ST and Qq; Pp a perpendicular dropped to the plane of the ecliptic; A and B the syzygies of the moon in the plane of the ecliptic; AZ a perpendicular to the line of the nodes Nn; Q and q the quadratures of the moon in the plane of the ecliptic; and pK a perpendicular to the line Qq, which lies between the quadratures. The force of the sun to perturb the motion of the moon has (by prop. 25) two components, one proportional to the line LM in the figure of that proposition, the other proportional to the line MT in that same figure. And the moon is attracted toward the earth by the first of these forces, and by the second it is attracted toward the sun along a line parallel to the straight line ST drawn from the earth to the sun. The first force LM acts in the plane of the moon’s orbit and therefore can make no change in the position of that plane. Therefore this force is to be ignored. The second force MT, by which the plane of the lunar orbit is perturbed, is the same as the force 3PK or 3IT. And this force (by prop. 25) is to the force by which the moon could revolve uniformly in a circle in its periodic time about the earth at rest as 3IT to the radius of the circle multiplied by the number 178.725, or as IT to the radius multiplied by 59.575. But in this calculation and in what follows, I consider all lines drawn from the moon to the sun to be parallel to the line drawn from the earth to the sun, because the inclination diminishes all effects in some cases nearly as much as it increases them in others; and we are here seeking the mean motions of the nodes, ignoring those niceties of detail which would make the calculation too cumbersome.
Now let PM represent the arc that the moon describes in a minimally
small given time, and ML the line-element one-half of which the moon could
describe in the same time by the impulse of the above-mentioned force 3IT.
Draw PL and MP, and produce them to m and l, and let them cut the plane
of the ecliptic there, and upon Tm drop the perpendicular PH. Since the
straight line ML is parallel to the plane of the ecliptic and so cannot meet
with the straight line ml (which lies in that plane) and yet these straight
lines lie in a common plane LMPml, these straight lines will be parallel, and
therefore the triangles LMP and lmP will be similar. Now, since MPm is
in the plane of the orbit in which the moon was moving while in place P,
the point m will fall upon the line Nn drawn through the nodes N and n of
that orbit. The force by which half of the line-element LM is generated—if
all of it were impressed all at once in place P—would generate that whole
line and would cause the moon to move in an arc whose chord would be
LP, and so would transfer the moon from the plane MPmT into the plane
LPlT; therefore the angular motion of the nodes that is generated by that
force will be equal to the angle mTl. Moreover, ml is to mP as ML is to MP,
and so, since MP is given (because the time is given), ml is as the rectangle
ML × mP, that is, as the rectangle IT × mP. And, provided that the angle Tml
is a right angle, the angle mTl is as 
, and therefore as 
,
that is (because Tm is to mP as TP is to PH), as 
; and so,
because TP is given, as IT × PH. But if the angle Tml or STN is oblique,
the angle mTl will be still smaller, in the ratio of the sine of the angle STN
to the radius, or of AZ to AT. Therefore the velocity of the nodes is as
IT × PH × AZ, or as the solid contained by [or the product of] the sines of
the three angles TPI, PTN, and STN.
If those angles are right angles, as happens when the nodes are in the quadratures and the moon is in the syzygy, the line-element ml will go off indefinitely and the angle mTl will become equal to the angle mPl. But in this case the angle mPl is to the angle PTM, which the moon describes about the earth in the same time by its apparent motion, as 1 to 59.575. For the angle mPl is equal to the angle LPM, that is, to the angle of the deflection of the moon from the straight-line path that the aforesaid solar force 3IT could generate by itself in that given time, if the gravity of the moon were then to cease; and the angle PTM is equal to the angle of the deflection of the moon from the straight-line path that the force by which the moon is kept in its orbit would generate in the same time, if the solar force 3IT were then to cease. And these forces, as we have said above, are to each other as 1 to 59.575. Therefore, since the mean hourly motion of the moon with respect to the fixed stars is 32′56″27‴12iv½, the hourly motion of the node in this case will be 33″10‴33iv12v. But in other cases this hourly motion will be to 33″10‴33iv12v as the solid contained by [or the product of] the sines of the three angles TPI, PTN, and STN (or the distance of the moon from the quadrature, of the moon from the node, and of the node from the sun) to the cube of the radius. And whenever the sign of any of the angles is changed from positive to negative and from negative to positive, retrograde motion will have to be changed into progressive motion, and progressive into retrograde. Hence it happens that the nodes advance whenever the moon is between either of the quadratures and the node nearest to that quadrature. In other cases, the nodes are retrograde, and they are carried backward [or in antecedentia] each month by the excess of the retrograde motion over the progressive.
COROLLARY 1. Hence, if from the ends P and M of the minimally small given arc PM, the perpendiculars PK and Mk are dropped to the line Qq that joins the quadratures, and these perpendiculars are produced until they cut the line of the nodes Nn in D and d, then the hourly motion of the nodes will be as the area MPDd and the square of the line AZ jointly. For let PK, PH, and AZ be the above-mentioned three sines—namely, PK the sine of the distance of the moon from the quadrature, PH the sine of the distance of the moon from the node, and AZ the sine of the distance of the node from the sun—then the velocity of the node will be as the solid [or product] PK × PH × AZ. But PT is to PK as PM to Kk, and so, because PT and PM are given, Kk is proportional to PK. Also, AT is to PD as AZ to PH, and therefore PH is proportional to the rectangle PD × AZ; and, combining these ratios, PK × PH is as the solid Kk × PD × AZ, and PK × PH × AZ is as Kk × PD × AZ2, that is, as the area PDdM and AZ2 jointly. Q.E.D.
COROLLARY 2. In any given position of the nodes, the mean hourly motion is half of their hourly motion in the moon’s syzygies, and thus is to 16″55‴16iv36v as the square of the sine of the distance of the nodes from the syzygies is to the square of the radius, or as AZ2 to AT2. For if the moon traverses the semicircle QAq with uniform motion, the sum of all the areas PDdM during the time in which the moon goes from Q to M will be the area QMdE, which is terminated at the tangent QE of the circle; and in the time in which the moon reaches point n, that sum will be the total area EQAn, which the line PD describes; then as the moon goes from n to q, the line PD will fall outside the circle and will describe the area nqe (which is terminated at the tangent qe of the circle)—which, since the nodes were previously retrograde but now are progressive, must be subtracted from the former area, and (since it is equal to the area QEN) will leave the semicircle NQAn. Therefore, during the time in which the moon describes a semicircle, the sum of all the areas PDdM is the area of that semicircle; and in the time in which the moon describes a circle, the sum of all those areas is the area of the whole circle. But the area PDdM, when the moon is in the syzygies, is the rectangle of the arc PM and the radius PT; and in the time in which the moon describes a circle, the sum of all the areas that are equal to this one is the rectangle of the whole circumference and the radius of the circle; and this rectangle, since it is equal to two circles, is twice as large as the former rectangle. Accordingly, if the nodes moved with the same velocity uniformly continued that they have in the lunar syzygies, they would describe a space twice as large as the space which they really describe; and therefore the mean motion—with which, if it were continued uniformly, they would describe the space that they really cover with their nonuniform motion—is one-half of the motion which they have in the moon’s syzygies. Hence, since the greatest hourly motion of the nodes, if the nodes are in the quadratures, is 33″10‴33iv12v, their mean hourly motion in this case will be 16″35‴16iv36v. And since the hourly motion of the nodes is always as AZ2 and the area PDdM jointly, and therefore the hourly motion of the nodes in the moon’s syzygies is as AZ2 and the area PDdM jointly, that is (because the area PDdM described in the syzygies is given), as AZ2, the mean motion will also be as AZ2; and hence this motion, when the nodes are outside the quadratures, will be to 16″35‴16iv36v as AZ2 to AT2. Q.E.D.
Proposition 31, Problem 12
To find the hourly motion of the nodes of the moon in an elliptical orbit.
Let Qpmaq represent an ellipse described with a major axis Qq and a minor axis ab, QAqB a circle circumscribed about this ellipse, T the earth in the common center of both, S the sun, p the moon moving in the ellipse, and pm the arc that the moon describes in a minimally small given particle of time, N and n the nodes joined by the line Nm, pK and mk perpendiculars dropped to the axis Qq and produced on both sides until they meet the circle at P and M and the line of the nodes at D and d. And if the moon, by a radius drawn to the earth, describes an area proportional to the time, the hourly motion of the node in the ellipse will be as the area pDdm and AZ2 jointly.
To demonstrate this, let PF touch the circle at P and, produced, meet
TN at F; let pf touch the ellipse at p and, produced, meet the same TN at
f; and let these tangents come together on the axis TQ at Y. And let ML
designate the space that the moon, revolving in a circle, would describe by a
transverse motion under the action and impulse of the aforesaid force 3IT or
3PK, while it describes the arc PM; and let ml designate the space that the
moon, revolving in an ellipse, could describe in the same time, also under
the action of the force 3IT or 3PK. Further, let Lp and lp be produced
until they meet the plane of the ecliptic at G and g; and let FG and fg
be drawn, of which let FG produced cut pf, pg, and TQ at c, e, and R
respectively; and let fg produced cut TQ at r. Then, since the force 3IT
or 3PK in the circle is to the force 3IT or 3pK in the ellipse as PK is to
pK, or as AT to aT, the space ML generated by the first force will be to
the space ml generated by the second force as PK to pK, that is (because
the figures PYKp and FYRc are similar), as FR to cR. Moreover, ML is
to FG (because the triangles PLM and PGF are similar) as PL to PG, that
is (because Lk, PK, and GR are parallel), as pl to pe, that is (because the
triangles plm and cpe are similar), as lm to ce; and thus LM is to lm, or
FR is to cR, as FG is to ce. And therefore if fg were to ce as fY to cY,
that is, as fr to cR (that is, as fr to FR and FR to cR jointly, that is, as fT
to FT and FG to ce jointly), then, since the ratio FG to ce taken away from
both sides leaves the ratios fg to FG and fT to FT, the ratio fg to FG
would be as fT to FT, and so the angles that FG and fg would subtend at
the earth T would be equal to each other. But these angles (by what we have
set forth in the preceding prop. 30) are the motions of the nodes in the time
in which the moon traverses the arc PM in the circle, and the arc pm in the
ellipse; and therefore the motions of the nodes in the circle and in the ellipse
would be equal to each other. These things would be so, if only fg were
to ce as fY to cY, that is, if fg were equal to
. But because the
triangles fgp and cep are similar, fg is to ce as
fp to cp, and so fg is equal
to 
; and therefore the angle that fg really subtends is to the former
angle that FC subtends (that is, the motion of the nodes in the ellipse is to
the motion of the nodes in the circle) as this fg or
to the former
fg or ,
that is, as fp × cY to fY × cp, or as fp to fY and cY to
cp; that is (if ph, parallel to TN, meets FP at h), as Fh to FY and FY to
FP; that is, as Fh to FP or Dp to DP, and so as the area Dpmd to the area
DPMd. And therefore, since (by prop. 30, corol. 1) the latter area and AZ2
jointly are proportional to the hourly motion of the nodes in the circle, the
former area and AZ2 jointly will be proportional to the hourly motion of the
nodes in the ellipse. Q.E.D.
COROLLARY. Therefore, since in any given position of the nodes, the sum of all the areas pDdm, in the time in which the moon goes from the quadrature to any place m, is the area mpQEd, which is terminated at the tangent QE of the ellipse, and the sum of all those areas in a complete revolution is the area of the whole ellipse, the mean motion of the nodes in the ellipse will be to the mean motion of the nodes in the circle as the ellipse to the circle, that is, as Ta to TA, or as 69 to 70. And therefore, since (by prop. 30, corol. 2) the mean hourly motion of the nodes in the circle is to 16″35‴16iv36v as AZ2 to AT2 if the angle 16″21‴3iv30v is taken to the angle 16″35‴16iv36v as 69 to 70, the mean hourly motion of the nodes in the ellipse will be to 16″21‴3iv30v as AZ2 to AT2, that is, as the square of the sine of the distance of the node from the sun to the square of the radius.
But the moon, by a radius drawn to the earth, describes an area more swiftly in the syzygies than in the quadratures, and on that account the time is shortened in the syzygies and lengthened in the quadratures, and along with the time the motion of the nodes is increased and decreased. Now, the moment of an area in the quadratures of the moon was to its moment in the syzygies as 10,973 to 11,073; and therefore the mean motion in the octants is to the excess in the syzygies and to the deficiency in the quadratures as the half-sum 11,023 of the numbers is to their half-difference 50. Accordingly, since the time of the moon in each equal particle of its orbit is inversely as its velocity, the mean time in the octants will be to the excess of time in the quadratures and its deficiency in the syzygies, arising from this cause, as 11,023 to 50 very nearly. With regard to positions of the moon between the quadratures and the syzygies, I find that the excess of the moments of the area in any one of these positions over the least moment in the quadratures is very nearly as the square of the sine of the distance of the moon from the quadratures; and therefore the difference between the moment in any position and the mean moment in the octants is as the difference between the square of the sine of the distance of the moon from the quadratures and the square of the sine of 45°, or half of the square of the radius; and the increase of the time in any one of the positions between the octants and the quadratures, and its decrease between the octants and the syzygies, is in the same ratio. But the motion of the nodes, in the time in which the moon traverses each equal particle of its orbit, is accelerated or retarded as the square of the time.
For that motion, while the moon traverses PM, is (other things being equal) as ML, and ML is in the squared ratio of the time. Therefore, the motion of the nodes in the syzygies, a motion completed in the time in which the moon traverses given particles of its orbit, is diminished as the square of the ratio of the number 11,073 to the number 11,023; and the decrement is to the remaining motion as 100 to 10,973 and to the total motion as 100 to 11,073 very nearly. But the decrement in positions between the octants and syzygies and the increment in positions between the octants and quadratures are to this decrement very nearly as [i] the total motion in those positions to the total motion in the syzygies and as [ii] the difference between the square of the sine of the distance of the moon from the quadrature and half of the square of the radius to half of the square of the radius, jointly. Hence, if the nodes are in the quadratures and two positions are taken equally distant from the octant, one on one side and one on the other, and another two are taken at the same distance from the syzygy and from the quadrature, and if from the decrements of the motions in the two positions between the syzygy and octant are subtracted the increments of the motions in the remaining two positions that are between the octant and quadrature, the remaining decrement will be equal to the decrement in the syzygy, as will be easily apparent upon examination. And accordingly the mean decrement, which must be subtracted from the mean motion of the nodes, is a fourth of the decrement in the syzygy. The total hourly motion of the nodes in the syzygies (when it was supposed the moon described, by a radius drawn to the earth, an area proportional to the time) was 32″42‴7iv. And according to what we have just said, the decrement of the motion of the nodes, in the time when the moon—now moving more swiftly—describes the same space, is to this motion as 100 to 11,073; and so the decrement is 17‴43iv11v, of which a fourth (4‴25iv48v) subtracted from the mean hourly motion found above (16″21‴3iv30v) leaves 16″16‴37iv42v, the corrected mean hourly motion.
If the nodes are beyond the quadratures and two places equally distant from the syzygies are considered, one on one side and one on the other, the sum of the motions of the nodes when the moon is in these positions will be to the sum of their motions when the moon is in the same positions and the nodes are in the quadratures as AZ2 to AT2. And the decrements of the motions, arising from the causes just now set forth, will be to each other as the motions themselves, and therefore the remaining motions will be to each other as AZ2 to AT2, and the mean motions will be as the remaining motions. Therefore the corrected mean hourly motion, in any given situation of the nodes, is to 16″16‴37iv42v as AZ2 to AT2, that is, as the square of the sine of the distance of the nodes from the syzygies to the square of the radius.
Proposition 32, Problem 13
To find the mean motion of the nodes of the moon.
The mean annual motion is the sum of all the mean hourly motions in a year. Suppose that the node is in N and that as each hour is completed, it is drawn back into its former place so that, notwithstanding its own proper motion, it always maintains some given position with respect to the fixed stars. And suppose that during this same time the sun S, as a result of the motion of the earth, advances from the node and completes its apparent annual course with a uniform apparent motion. Moreover, let Aa be the minimally small given arc that the straight line TS, always drawn to the sun, describes in a minimally small given time by its intersection with the circle NAn; then (by what has already been shown) the mean hourly motion will be as AZ2, that is (because AZ and ZY are proportional), as the rectangle of AZ and ZY, that is, as the area AZYa. And the sum of all the mean hourly motions from the beginning will be as the sum of all the areas aYZA, that is, as the area NAZ. Moreover, the greatest area AZYa is equal to the rectangle of the arc Aa and the radius of the circle; and therefore the sum of all such rectangles in the whole circle will be to the sum of the same number of greatest rectangles as the area of the whole circle to the rectangle of the whole circumference and the radius, that is, as 1 to 2. Now, the hourly motion corresponding to the greatest rectangle was 16″16‴37iv42v, and this motion, in a whole sidereal year of 365d6h9m, adds up to 39°38′7″50‴. And so half of this, 19°49′3″55‴, is the mean motion of the nodes that corresponds to the whole circle. And the motion of the nodes in the time during which the sun goes from N to A is to 19°49′3″55‴ as the area NAZ is to the whole circle.
These things are so on the hypothesis that each hour the node is drawn
back to its former place, in such a way that when a whole year is completed,
the sun returns to the same node from which it had initially departed. But
as a result of the motion of that node, it comes about that the sun returns
to the node more quickly; and now this shortening of the time must be
computed. Since in a total year the sun travels through 360°, and in the same
time the node with its greatest motion would travel through 39°38′7″50‴,
or 39.6355°, and the mean motion of the node in any place N is to its
mean motion in its quadratures as AZ2 to AT2, the motion of the sun will
be to the motion of the node in N as 360AT2 to 39.6355AZ2, that is, as
9.0827646AT2 to AZ2. Hence, if the circumference NAn of the whole circle
is divided into equal particles Aa, then the time in which the sun traverses
the particle Aa (the circle being at rest) will be to the time in which it
traverses the same particle (if the circle revolves along with the nodes about
the center T) inversely as 9.0827646AT2 to 9.0827646AT2 + AZ2. For the
time is inversely as the velocity with which the particle [of arc] is traversed,
and this velocity is the sum of the velocities of the sun and of the node. Let
the sector NTA represent the time in which the sun, without the motion of
the node, would traverse the arc NA, and let the particle ATa of the sector
represent the particle of time in which it would traverse the minimally small
arc Aa; furthermore, drop a perpendicular aY to Nn and on AZ take dZ of
a length such that the rectangle of dZ and ZY is to the particle ATa of the
sector as AZ2 is to 9.0827646AT2 + AZ2 (that is, such that dZ is to ½AZ
as AT2 is to 9.0827646AT2 + AZ2); then the rectangle of dZ and ZY will
designate the decrement of time arising from the motion of the node during
the total time in which the arc Aa is traversed. And if the point d touches
the curve NdGn,a
the curvilinear area NdZ will be the total decrement
in the time in which the whole arc NA is traversed; and therefore the excess
of the sector NAT over the area NAZ will be that total time. And since the
motion of the node in a smaller time is smaller in proportion to the time,
the area AaYZ also will have to be diminished in the same proportion. And
this will happen if on AZ the length eZ is taken, which is to the length AZ
as AZ2 is to 9.0827646AT2 + AZ2. For thus the rectangle of eZ and ZY
will be to the area AZYa as the decrement of the time in which the arc Aa
is traversed is to the total time in which it would be traversed if the node
were at rest; and therefore that rectangle will correspond to the decrement
of the motion of the node. And if the point e touches the curve
NeFn,b
the total area NeZ, which is the sum of all the decrements of that motion,
will correspond to the total decrement in the time during which the arc AN
is traversed, and the remaining area NAe will correspond to the remaining
motion, which is the true motion of the node in the time in which the total
arc NA is traversed by the joint motions of the sun and the node. Now, the
area of the semicircle is to the area of the figure NeFn, found by the method
of infinite series, nearly as 793 to 60. And the motion that corresponds to
the whole circle was 19°49′3″55‴, and therefore the motion that corresponds
to double the figure NeFn is 1°29′58″2‴. Subtracting this from the former
motion leaves 18°19′5″53‴, the total motion of the node with respect to the
fixed stars between its successive conjunctions with the sun; and this motion,
subtracted from the annual motion of the sun of 360°, leaves 341°40′54″7‴,
the motion of the sun between the same conjunctions. And this motion is to
the annual motion of 360° as the motion of the node just found (18°19′5″53‴)
to its annual motion, which will therefore be 19°18′1″23‴. This is the mean
motion of the nodes in a sidereal year. From the astronomical tables this is
19°21′21″50‴. The difference is less than
of the total motion and seems
to arise from the eccentricity of the moon’s orbit and its inclination to the
plane of the ecliptic. By the eccentricity of the orbit, the motion of the nodes
is too much accelerated; and on the other hand, by its inclination it is retarded
somewhat, and reduced to its correct velocity.
Proposition 33, Problem 14
To find the true motion of the nodes of the moon.
In the time which is as the area NTA ‒ NdZ (in the preceding figure), that motion is as the area NAe, and hence is given. But because the calculation is too difficult, it is preferable to use the following construction of the problem. With center C and any interval CD as radius, describe a circle BEFD. Produce DC to A so that AB is to AC as the mean motion is to half of the true mean motion when the nodes are in the quadratures (that is, as 19°18′1″23‴ to 19°49′3″55‴); and thus BC will be to AC as the difference of the motions (0°31′2″32‴) to the latter motion (19°49′3″55‴), that is, as 1 to 383/10. Next, through point D draw the indefinite line Gg, touching the circle in D; and let the angle BCE or BCF be taken equal to twice the distance of the sun from the place of the node, as found from the mean motion, and let AE or AF be drawn cutting the perpendicular DG in G. The true motion of the nodes will be found if now an angle is taken that is to the total motion of the node between its syzygies (that is, to 9°11′3″) as the tangent DG is to the total circumference of the circle BED, and if that angle (for which the angle DAG can be used) is added to the mean motion of the nodes when the nodes are passing from quadratures to syzygies and is subtracted from the same mean motion when they are passing from syzygies to quadratures. For the true motion thus found will agree very nearly with the true motion which results from representing the time by the area NTA — NdZ and the motion of the node by the area NAe, as will be evident to anyone considering the matter and performing the computations. This is the semimonthly equation of the motion of the nodes. There is also a monthly equation, but it is not at all needed in order to find the latitude of the moon. For, since the variation of the inclination of the moon’s orbit to the plane of the ecliptic is subject to a double inequality, one semimonthly and the other monthly, the monthly inequality of the variation and the monthly equation of the nodes so moderate and correct each other that both can be ignored in determining the latitude of the moon.
COROLLARY. From this and the preceding proposition it is clear that the nodes are stationary in their syzygies; in the quadratures, however, they regress by an hourly motion of 16″19‴26iv. It is also clear that the equation of the motion of nodes in the octants is 1°30′. This all squares exactly with celestial phenomena.
Scholium
J. Machin, Gresham Professor of Astronomy, and Henry Pemberton, M.D.,
have independently found the motion of the nodes by yet another method.
Some mention of the latter’s method has been made elsewhere. And the
papers (which I have seen) of both men contained two propositions, which
agreed with each other. Here I shall present Mr. Machin’s paper, since it was
the first to come into my hands.
ON THE MOTION OF THE NODES OF THE MOON
Proposition 1
The mean motion of the sun from the node is defined by a mean geometrical proportional between the mean motion of the sun and that mean motion with which the sun recedes most swiftly from the node in the quadratures.
Let T be the place where the earth is, Nn the line of the nodes of the moon at any given time, KTM a line drawn at right angles to this line, and TA a straight line revolving around the center with the angular velocity with which the sun and the node recede from each other, in such a way that the angle between the straight line Nm (which is at rest) and TA (which is revolving) is always equal to the distance between the places of the sun and of the node. Now, if any straight line TK is divided into parts TS and SK, which are to each other as the hourly mean motion of the sun is to the hourly mean motion of the nodes in the quadratures, and if the straight line TH is taken so as to be a mean proportional between the part TS and the whole TK, this straight line among the rest will be proportional to the mean motion of the sun from the node.
For describe a circle NKnM with center T and radius TK, and with the same center and the semiaxes TH and TN describe an ellipse NHnL, and in the time in which the sun recedes from the node through the arc Na, if the straight line Tba is drawn, the area of the sector NTa will represent the sum of the motions of the node and sun in that same time. Therefore let Aa be the minimally small arc that the straight line Tba—revolving according to the aforesaid law—uniformly describes in a given particle of time, and the minimally small sector TAa will be as the sum of the velocities whereby the sun and the node are carried separately in that time. The velocity of the sun, however, is nearly uniform, since its small inequality introduces scarcely any variation in the mean motion of the nodes. The other part of this sum, namely the velocity of the node in its mean quantity, increases in receding from the syzygies as the square of the sine of its distance from the sun (by Principia, book 3, prop. 31, corol.) and, when it is greatest in the quadratures to the sun at K, has the same ratio to the velocity of the sun that SK has to TS; that is, it is as (the difference of the squares of TK and TH or) the rectangle KH × HM is to TH2. But the ellipse NBH divides the sector ATa, which represents the sum of these two velocities, into two parts ABba and BTb, which are proportional to the velocities. For, produce BT to the circle in β, and, from point B to the major axis, drop a perpendicular BG which, produced in both directions, meets the circle in points F and f. Then, since the space ABba is to the sector TBb as the rectangle AB × Bβ to BT2 (for that rectangle is equal to the difference of the squares of TA and TB because the straight line Aβ is cut equally at T and unequally at B), this ratio—when the space ABba is greatest at K—will be the same as the ratio of the rectangle KH × HM to HT2; but the greatest mean velocity of the node was [previously shown to be] in this ratio to the velocity of the sun. Therefore, in the quadratures, the sector ATa is divided into parts proportional to the velocities. And since the rectangle KH × HM is to HT2 as FB × Bf to BG2 and since the rectangle AB × Bβ is equal to the rectangle FB × Bf, it follows that the area-element ABba when it is greatest will be to the remaining sector TBb as the rectangle AB × Bβ to BG2. But the ratio of the area-elements was always as the rectangle AB × Bβ to BT2; and therefore the area-element ABba in the place A is smaller than the corresponding area-element in the quadratures, in the ratio of BG2 to BT2, that is, as the square of the sine of the distance of the sun from the node. And accordingly the sum of all the area-elements ABba (namely, the space ABN) will be as the motion of the node in the time in which the sun departs from the node and passes through the arc NA. And the remaining space (namely, the elliptical sector NTB) will be as the mean motion of the sun in the same time. And, therefore, since the mean annual motion of the node is the motion that it makes in the time during which the sun has completed its period, the mean motion of the node from the sun will be to the mean motion of the sun itself as the area of the circle to the area of the ellipse, that is, as the straight line TK to the straight line TH (which is the mean proportional between TK and TS); or, which comes to the same thing, as the mean proportional TH to the straight line TS.
Proposition 2
Given the mean motion of the nodes of the moon, to find the true motion.
Let the angle A be the distance of the sun from the mean place of the node, or the mean motion of the sun from the node. Then if angle B is taken so that its tangent is to the tangent of angle A as TH to TK—that is, as the square root of the ratio of the mean hourly motion of the sun to the mean hourly motion of the sun from the node when the node is in the quadratures—that same angle B will be the distance of the sun from the true place of the node. For draw FT, and (by the proof of the previous proposition) the angle FTN will be the distance of the sun from the mean place of the node, while the angle ATN will be the distance from the true place, and the tangents of these angles are to each other as TK to TH.
COROLLARY. Hence the angle FTA is the equation of the moon’s nodes, and the sine of this angle, when it is greatest in the octants, is to the radius as KH to TK + TH. And the sine of this equation in any other place A is to the greatest sine as the sine of the sum of the angles FTN + ATN is to the radius—that is, nearly as the sine of 2FTN (that is, twice the distance of the sun from the mean place of the node) is to the radius.
Scholium
If the mean hourly motion of the nodes in the quadratures is 16″16‴37iv42v (that is, 39°38′7′50‴ in a whole sidereal year), then TH will be to TK as the square root of the ratio of the number 9.0827646 to the number 10.0827646, that is, as 18.6524761 to 19.6524761. And therefore TH is to HK as 18.6524761 to 1, that is, as the motion of the sun in a sidereal year to the mean motion of the node, which is 19°18′1″23⅔‴.
But if the mean motion of the nodes of the moon in twenty Julian years is 386°50′15″, as is deduced from observations used in the theory of the moon, the mean motion of the nodes in a sidereal year will be 19°20′31″58‴. And TH will be to HK as 360° to 19°20′31′58‴, that is, as 18.61214 to 1. Hence the mean hourly motion of the nodes in the quadratures will come out 16″18‴48iv. And the greatest equation of the nodes in the octants will be l°29′57″.
Proposition 34, Problem 15
To find the hourly variation of the inclination of the lunar orbit to the plane of
the ecliptic.
Let A and a represent the syzygies, Q and q the quadratures, N and
n the nodes, P the place of the moon in its orbit, p the projection of that
place on the plane of the ecliptic, and mTl the momentaneous motion of the
nodes as above. Drop the perpendicular PG to the line Tm, join pG and
produce it until it meets Tl in g, and also join Pg; then the angle PGp will
be the inclination of the moon’s orbit to the plane of the ecliptic when the
moon is in P, and the angle Pgp will be the inclination of the same orbit
after a moment of time has been completed; and thus the angle GPg will be
the momentaneous variation of the inclination. But this angle GPg is to the
angle GTg as TG to PG and Pp to PG jointly. And therefore, if an hour is
substituted for the moment of time, then—since the angle GTg (by prop. 30)
is to the angle 33″10‴33iv as IT × PG × AZ to AT3—the angle GPg (or
the hourly variation of the inclination) will be to the angle 33″10‴33iv as
IT × AZ × TG × 
to AT3. Q.E.I.
These things are so on the hypothesis that the moon revolves uniformly in a circular orbit. But if that orbit is elliptical, the mean motion of the nodes will be diminished in the ratio of the minor axis to the major axis, as has been set forth above. And the variation of the inclination will also be diminished in the same ratio.
COROLLARY 1. If the perpendicular TF is erected on Nn, and pM is the
hourly motion of the moon in the plane of the ecliptic, and if the
perpendiculars pK and Mk are dropped to QT and produced in both directions
to meet TF at H and h, then IT will be to AT as Kk to Mp, and TG to
Hp as TZ to AT, and so IT × TG will be equal to 
, that
is, equal to the area HpMh multiplied by the ratio 
; and therefore the
hourly variation of the inclination will be to 33″10‴33iv as HpMh multiplied
by AZ × × is to AT3.
COROLLARY 2. And so, if the earth and the nodes, as each hour is
completed, were drawn back from their new places and were always restored
instantly to their former places, so that their given position remained
unchanged throughout an entire periodic month, the total variation of the
inclination during the time of that month would be to 33″10‴33iv as the sum
of all the areas HpMh which are generated during a revolution of the point
p (these areas being summed according to their proper signs + and −)
multiplied by AZ × TZ × is to Mp × AT3, that is, as the whole circle QAqa
multiplied by AZ × TZ × is to Mp × AT3, that is, as the circumference
QAqa multiplied by AZ × TZ × is to 2Mp × AT2.
COROLLARY 3. Accordingly, in a given position of the nodes, the mean
hourly variation, from which, continued uniformly for a month, that monthly
variation could be generated, is to 33″10‴33iv as
AZ × TZ × to 2AT2,
or as Pp × 
to PG × 4AT, that is (since Pp is to PG as the sine
of the above-mentioned inclination to the radius, and is to 4AT
as the sine of twice the angle ATn to four times the radius), as the sine of
that same inclination multiplied by the sine of twice the distance of the nodes
from the sun to four times the square of the radius.
COROLLARY 4. Since the hourly variation of the inclination, when the
nodes are in the quadratures, is (by this proposition) to the angle 33310433iv
as IT × AZ × TG × to
AT3, that is, as 
to 2AT,
that is, as the sine of twice the distance of the moon from the quadratures
multiplied by is to twice the radius, it follows that the sum of all the
hourly variations, in the time in which the moon in this situation of the nodes
passes from quadrature to syzygy (that is, in the space of 177⅙ hours), will
be to the sum of the same number of angles 33″10‴33iv, or 5,878″, as the
sum of all the sines of twice the distance of the moon from the quadratures
multiplied by is to the sum of the same number of diameters; that is,
as the diameter multiplied by is to the circumference; that is, if the
inclination is 5°1′, as 7 × 
to 22, or 278 to 10,000. And accordingly
the total variation, composed of the sum of all the hourly variations in the
aforesaid time, is 163″, or 2′43″.
Proposition 35, Problem 16
To find the inclination of the moon’s orbit to the plane of the ecliptic at a given
time.
Let AD be the sine of the greatest inclination, and AB the sine of the least inclination. Bisect BD in C, and with center C and radius BC describe a circle BGD. On AC take CE in the ratio to EB that EB has to 2BA. Now if, for the given time, the angle AEG is set equal to twice the distance of the nodes from the quadratures, and the perpendicular GH is dropped to AD, then AH will be the sine of the required inclination.
For GE2 is equal to GH2 + HE2 = BH × HD + HE2 = HB × BD +
HE2 − BH2 = HB × BD + BE2 − 2BH × BE = BE2 + 2EC × BH =
2EC × AB + 2EC × BH = 2EC × AH. And thus, since 2EC is given,
GE2 is as AH. Now let AEg represent twice the distance of the nodes from
the quadratures after some given moment of time has been completed, and
the arc Gg (because the angle GEg is given) will be as the distance GE.
Moreover, Hh is to Gg as GH to GC, and therefore Hh is as the solid [or
product] GH × Gg, or GH × GE; that is, as
× GE2 or
× AH,
that is, as AH and the sine of the angle AEG jointly. Therefore, if AH,
in any given case, is the sine of the inclination, it will be increased by the
same increments as the sine of the inclination, by corol. 3 of the preceding
prop. 34, and therefore will always remain equal to that sine. But when the
point G falls upon either point B or D, AH is equal to this sine and therefore
remains always equal to it. Q.E.D.
In this demonstration, I have supposed that the angle BEG, which is twice the distance of the nodes from the quadratures, increases uniformly. For there is no time to consider all the minute details of inequalities. Now suppose that the angle BEG is a right angle and that in this case Gg is the hourly increment of twice the distance of the nodes and sun from each other; then (by corol. 3 of prop. 34) the hourly variation of the inclination in the same case will be to 33″10‴33iv as the solid [or product] of the sine AH of the inclination and the sine of the right angle BEG (which is twice the distance of the nodes from the sun) is to four times the square of the radius; that is, as the sine AH of the mean inclination to four times the radius; that is (since that mean inclination is about 5°8½′), as its sine (896) to four times the radius (40,000), or as 224 to 10,000. And the total variation, corresponding to BD, the difference of the sines, is to that hourly variation as the diameter BD to the arc Gg; that is, as the diameter BD to the semicircumference BGD and the time of 2,0797/10 hours (during which the node goes from the quadratures to the syzygies) to 1 hour jointly; that is, as 7 to 11 and 2,0797/10 to 1. Therefore, if all the ratios are combined, the total variation BD will become to 33″10‴33iv as 224 × 7 × 2,0797/10 to 110,000, that is, as 29,645 to 1,000, and hence that variation BD will come out 16′23½″.
This is the greatest variation of the inclination insofar as the place of the moon in its orbit is not considered. For if the nodes are in the syzygies, the inclination is not at all changed by the various positions of the moon. But if the nodes are in the quadratures, the inclination is less when the moon is in the syzygies than when it is in the quadratures, by a difference of 2′43″, as we have indicated in corol. 4 of prop. 34. And the total mean variation BD, diminished when the moon is in its quadratures by 1′21½″ (half of this excess), becomes 15′2″; while in the syzygies it is increased by the same amount and becomes 17′45″. Therefore, if the moon is in the syzygies, the total variation in the passage of the nodes from quadratures to syzygies will be 17′45″; and so if the inclination, when the nodes are in the syzygies, is 5°17′20″, it will be 4°59′35″ when the nodes are in the quadratures and the moon in the syzygies. And that these things are so is confirmed by observations.
If now it is desired to find the inclination of the orbit when the moon is in the syzygies and the nodes are in any position whatever, let AB become to AD as the sine of 4°59′35″ is to the sine of 5°17′20″, and take the angle ABG equal to twice the distance of the nodes from the quadratures; then AH will be the sine of the required inclination. The inclination of the orbit is equal to this inclination when the moon is 90° distant from the nodes. In other positions of the moon, the monthly inequality that occurs in the variation of the inclination is compensated for in the calculation of the latitude of the moon (and, in a manner, canceled) by the monthly inequality in the motion of the nodes (as we have said above) and thus can be neglected in calculating that latitude.
Proposition 36, Problem 17
To find the force of the sun to move the sea.
The sun’s force ML or PT to perturb the motions of the moon, in the moon’s quadratures, was (by prop. 25 of this book 3) to the force of gravity here on earth as 1 to 638,092.6. And the force TM—LN or 2PK in the moon’s syzygies is twice as great. Now these forces, in the descent to the surface of the earth, are diminished in the ratio of the distances from the center of the earth, that is, in the ratio 60½ to 1; and so the first force on the surface of the earth is to the force of gravity as 1 to 38,604,600. By this force the sea is depressed in places that are 90 degrees distant from the sun. By the other force, which is twice as great, the sea is elevated both in the region directly under the sun and in the region opposite to the sun. The sum of these forces is to the force of gravity as 1 to 12,868,200. And since the same force arouses the same motion, whether it depresses the water in the regions that are 90 degrees distant from the sun or elevates the water in regions under the sun and opposite to the sun, this sum will be the total force of the sun to agitate the sea, and it will have the same effect as if all of it elevated the sea in regions under the sun and opposite it and had no action at all in regions that are 90 degrees distant from the sun.
This is the force of the sun to put the sea in motion in any given place when the sun is in the Zenith of the place as well as at its mean distance from the earth. In other positions of the sun, the force for raising the sea is directly as the versed sine of twice the altitude of the sun above the horizon of the place and inversely as the cube of the distance of the sun from the earth.
COROLLARY. The centrifugal force of the parts of the earth, arising from the daily motion of the earth (a force that is to the force of gravity as 1 to 289), causes the height of the water under the equator to exceed its height under the poles by a measure of 85,472 Paris feet (as was seen above in prop. 19); therefore, the solar force with which we have been dealing (since it is to the force of gravity as 1 to 12,868,200 and so to that centrifugal force as 289 to 12,868,200 or 1 to 44,527) will cause the height of the water in regions directly under the sun and directly opposite to the sun to exceed its height in places that are 90 degrees distant from the sun by a measure of only 1 Paris foot and 111/30 inches. For this measure is to the measure of 85,472 feet as 1 to 44,527.
Proposition 37, Problem 18
To find the force of the moon to move the sea.
COROLLARY 1. Since the water acted on by the force of the sun ascends to a height of 1 foot and 111/30 inches, by the force of the moon it will ascend to a height of 8 feet and 75/22 inches, and by both forces to a height of 10½ feet, and when the moon is in its perigee the water will ascend to a height of 12½ feet and beyond, especially when the tide is made greater by winds. And so great a force is more than sufficient to give rise to all the motions of the sea and corresponds exactly to the quantity of the motions. For in seas that extend widely from east to west, as in the Pacific Ocean and the parts of the Atlantic Ocean and the Ethiopic [or South Atlantic] Sea, which are outside the tropics, the water is generally raised to a height of 6, 9, 12, or 15 feet. And in the Pacific Ocean, which is deeper and wider, the tides are said to be greater than in the Atlantic Ocean and the Ethiopic Sea. For, to have the tide be full, the width of the sea from east to west should be no less than 90 degrees. In the Ethiopic Sea the ascent of the water within the tropics is less than in the temperate zones, because of the narrowness of the sea between Africa and the southern part of America. In the middle of the sea the water cannot rise unless it simultaneously falls on both shores, both the eastern and the western; nevertheless, in our narrow seas, the water ought to rise alternately on the two shores, that is, rise on one shore while it falls on the other. For this reason the ebb and flow are generally very small in islands that are farthest from the shores. In certain harbors, where the water is compelled to flow in and flow out with great impetus through shallow places, so as to fill and empty bays alternately, the ebb and flow must be greater than usual, as at Plymouth and Chepstow Bridge in England, at Mont-Saint-Michel and the city of Avranches in Normandy, at Cambay and Pegu bin the East Indies.b In these places the sea, coming in and going back out with great velocity, at times inundates the shores and at other times leaves them dry for many miles. And the impetus of flowing in or going back out cannot be broken before the water is raised or depressed to 30, 40, or 50 feet and more. And the same is true of oblong and shallow straits, such as the Straits of Magellan and that channel by which England is surrounded [presumably, the channel and seas, but not the ocean, bordering England]. The tide in harbors and straits of this sort is increased beyond measure by the impetus of running in and back. But on shores that face the deep and open sea with a steep descent, where the water can be raised and can fall without the impetus of flowing out and coming back, the magnitude of the tide corresponds to the forces of the sun and moon.
COROLLARY 3.c Since the force of the moon to move the sea is to the similar force of the sun as 4.4815 to 1, and since those forces (by book 1, prop. 66, corol. 14) are as the densities of the bodies of the moon and sun and the cubes of their apparent diameters jointly, the density of the moon will be to the density of the sun as 4.4815 to 1 directly and as the cube of the diameter of the moon to the cube of the diameter of the sun inversely, that is (since the apparent mean diameters of the moon and the sun are 31′16½″ and 32′12″), as 4,891 to 1,000. Now, the density of the sun was to the density of the earth as 1,000 to 4,000, and therefore the density of the moon is to the density of the earth as 4,891 to 4,000, or 11 to 9. Therefore the body of the moon is denser and more earthy than our earth.
COROLLARY 7. And the mean distance of the center of the moon from the center of the earth (in the octants of the moon) will be nearly 60⅖ greatest semidiameters of the earth. For the greatest semidiameter of the earth was 19,658,600 Paris feet, and the mean distance between the centers of the earth and the moon, which consists of 60⅖ such semidiameters, is equal to 1,187,379,440 feet. And this distance (by the preceding corollary) is to the distance of the center of the moon from the common center of gravity of the earth and the moon as 40.788 to 39.788; and hence the latter distance is 1,158,268,534 feet. And since the moon revolves with respect to the fixed stars in 27d7h434/9m, the versed sine of the angle that the moon describes in the time of one minute is 12,752,341, the radius being 1,000,000,000,000,000. And the radius is to this versed sine as 1,158,268,534 feet to 14.7706353 feet. The moon, therefore, falling toward the earth under the action of that force with which it is kept in its orbit, will in the time of one minute describe 14.7706353 feet. And by increasing this force in the ratio of 17829/40 to 17729/40, the total force of gravity in the orbit of the moon will be found by prop. 3, corol. [of this book 3]. And falling toward the earth under the action of this force, the moon will describe 14.8538067 feet in the time of one minute. And at 1/60 of the distance of the moon from the center of the earth, that is, at a distance of 197,896,573 feet from the center of the earth, a heavy body—falling in the time of one second—will likewise describe 14.8538067 feet. eAnd so, at a distance of 19,615,800 feet (which is the mean semidiameter of the earth), a heavy body in falling will describe—in the time of one second—15.11175 feet, or 15 feet 1 inch and 41/11 lines. This will be the descent of bodies at a latitude of 45 degrees. And by the foregoing table, presented in prop. 20, the descent will be a little greater at the latitude of Paris by about ⅔ of a line. Therefore, by this computation, heavy bodies falling in a vacuum at the latitude of Paris will—in the time of one second—describe approximately 15 Paris feet 1 inch and 425/33 lines. And if gravity is diminished by taking away the centrifugal force that arises from the daily motion of the earth at that latitude, heavy bodies falling there will—in the time of one second—describe 15 feet 1 inch and 1½ lines. And that heavy bodies do fall with this velocity at the latitude of Paris has been shown above in props. 4 and 19 [of this book 3].e
COROLLARY 8. fThe mean distance between the centers of the earth and the moon in the syzygies of the moon is 60 greatest semidiameters of the earth, taking away roughly 1/30 of a semidiameter. And in the moon’s quadratures, the mean distance between these centers is 605/6 semidiameters of the earth. For these two distances are to the mean distance of the moon in the octants as 69 and 70 to 69½, by prop. 28.f
Proposition 38, Problem 19
To find the figure of the body of the moon.
If the body of the moon were fluid like our sea, the force of the earth to elevate that fluid in both the nearest and farthest parts would be to the force of the moon by which our sea is raised in the regions both under the moon and opposite to the moon as the accelerative gravity of the moon toward the earth is to the accelerative gravity of the earth toward the moon and as the diameter of the moon is to the diameter of the earth, jointly—that is, as 39.788 to 1 and 100 to 365 jointly, or as 1,081 to 100. Hence, since our sea is raised by the force of the moon to 8⅗ feet, the lunar fluid would have to be raised by the force of the earth to 93 feet. And for this reason the figure of the moon would be a spheroid, the greatest diameter of which, produced, would pass through the center of the earth and would exceed by 186 feet the diameters perpendicular to that one. Therefore, it is just such a figure that the moon has and must have had from the beginning. Q.E.I.
COROLLARY. And hence it happens that the same face of the moon is always turned toward the earth. For in any other position, the moon cannot remain at rest, but by a motion of oscillation will always return to this position. But those oscillations would nevertheless be extremely slow because the forces producing them are small in magnitude; so that the face of the moon that should always look toward the earth can (for the reason given in prop. 17) be turned toward the other focus of the moon’s orbit and not at once be drawn back from there and turned toward the earth.
aLemma 1
Let APEp represent the earth, uniformly dense, with a center C and poles P and
p and equator AE, and suppose a sphere Papeb to be described with center C and
radius CP. Let QR be the plane on which a straight line drawn from the center
of the sun to the center of the earth stands perpendicularly. Then, if the individual
particles of the whole exterior earth PapAPepE, which is higher than the sphere
just described, endeavor to recede in both directions from the plane QR, and the
endeavor of each particle is as its distance from the plane, I say, first of all, that the
total force and efficacy of all the particles that lie in the circle of the equator AE
(disposed uniformly outside the globe, in the manner of a ring completely
encircling that globe) to rotate the earth around its center will be to the total force and
efficacy of the same number of particles standing at point A of the equator (which
is most distant from the plane QR) to move the earth with a similar circular
motion around its center as 1 is to 2. And that circular motion will be performed
around an axis lying in the common section of the equator and the plane QR.
For let IK be any smaller circle parallel to the equator AE, and let
L and l be any two equal particles situated in this circle outside the globe
Pape.c And if perpendiculars LM and
lm are dropped to the plane QR,
which is perpendicular to a radius
drawn to the sun, the total forces with
which the particles recede from the
plane QR will be proportional to the
perpendiculars LM and lm. Now,
let the straight line Ll be parallel
to the plane Pape; bisect Ll at X;
through the point X draw Nn parallel
to the plane QR and meeting the
perpendiculars LM and lm at N and n; and drop a perpendicular XY to
the plane QR. Then the contrary forces of the particles L and l to rotate
the earth in opposite directions are as LM × MC and lm × mC, that is, as
LN × MC + NM × MC and in × mC − nm × mC, or LN × MC + NM × MC and
LN × mC − NM × mC; and their difference LN × Mm − NM × (MC + mC)
is the force of both particles taken together to rotate the earth. The positive
part of this difference, LN × Mm or 2LN × NX, is to the force 2AH × HC
of two particles of the same magnitude located at A as LX2 to AC2. And the
negative part, NM × (MC + mC) or 2XY × CY is to the force 2AH × HC
of the same particles located at A as CX2 to AC2. And accordingly the
difference of the parts, that is, the force of the two particles L and l (taken
together) to rotate the earth, is to the force of two particles equal to those and
standing in the place A, likewise to rotate the earth, as LX2 − CX2 to AC2.
But if the circumference IK of the circle IK is divided into innumerable
equal particles L, all the LX2 will be to as many IX2 as 1 to 2 (by lem. 1),
and to this same number of AC2 as IX2 to 2 AC2; and just as many CX2 will
be to the same number of AC2 as 2CX2 to 2AC2. Therefore the combined
forces of all the particles in the circumference of the circle IK are to the
combined forces of as many particles in the place A as IX2 − 2CX2 to 2AC2,
and therefore (by lem. 1) to the combined forces of as many particles in the
circumference of the circle AE as IX2 − 2CX2 to AC2.
Now, if the diameter Pp of the sphered is divided into innumerable equal parts, on which the same number of circles IK stand, the matter in the perimeter of each circle IK will be as IX2; and so the force of that matter to rotate the earth will be as IX2 multiplied by IX2 − 2CX2. And the force of the same matter, if it stood in the perimeter of the circle AE, would be as IX2 multiplied by AC2. And therefore the force of all the particles of the total matter standing outside the globe in the perimeters of all the circles is to the force of as many particles standing in the perimeter of the greatest circle AE as all the IX2 multiplied by IX2 − 2CX2 to as many IX2 multiplied by AC2, that is, as all the AC2 − CX2 multiplied by AC2 − 3CX2 to as many AC2 − CX2 multiplied by AC2, that is, as all the AC4 − 4AC2 × CX2 + 3CX4 to as many AC4 − AC2 × CX2, that is, as the total fluent quantity whose fluxione is AC4 − 4AC2 × CX2 + 3CX4 to the total fluent quantity whose fluxion is AC4 − AC2 × CX2; and accordingly, by the method of fluxions, as AC4 × CX − 4/3AC2 × CX3 + ⅗CX5 to AC4 × CX − ⅓AC2 × CX3, that is, if the whole of Cp or AC is written in place of CX, as 4/15 AC5 to ⅔AC5, or as 2 to 5. Q.E.D.
Proposition 39, Problem 20
To find the precession of the equinoxes.
The mean hourly motion of the nodes of the moon in a circular orbit was, for the nodes in the quadratures, 16″35‴16iv36v, and half of this, 8″17‴38iv18v, is (for the reasons explained above [at the end of corol. 2 to prop. 30]) the mean hourly motion of the nodes in such an orbit; and in a whole sidereal year the mean motion adds up to 20°11′46″ [see beginning of prop. 32]. Therefore, since in a year the nodes of the moon would, in such an orbit, move backward [or in antecedentia] through 20°11′46″; and since, if there were more moons, the motion of the nodes of each (by book 1, prop. 66, corol. 16) would be as the periodic times; it follows that if the moon revolved near the surface of the earth in the space of a sidereal day, the annual motion of the nodes would be to 20°11′46″ as a sidereal day of 23h56m is to the periodic time of the moon, 27d7h43m—that is, as 1,436 to 39,343. And the same is true of the nodes of a ring of moons surrounding the earth, whether those moons do not touch one another, or whether they become liquid and take the form of a continuous ring, or finally whether that ring becomes rigid and inflexible.
Let us imagine therefore that this ring, as to its quantity of matter, is equal to all of the earth PapAPepE that lies outside of the globe Papea (as in the figure to lem. 2). This globe is to the earth that lies outside of it as aC2 to AC2 − aC2, that is (since the earth’s smaller semidiameter PC or aC is to its greater semidiameter AC as 229 to 230), as 52,441 to 459. Hence, if this ring girded the earth along the equator and both together revolved about the diameter of the ring, the motion of the ring would be to the motion of the interior globe (by lem. 3 of this third book) as 459 to 52,441 and 1,000,000 to 925,275 jointly, that is, as 4,590 to 485,223; and so the motion of the ring would be to the sum of the motions of the ring and globe as 4,590 to 489,813. Hence, if the ring adheres to the globe and communicates to the globe its own motion with which its nodes or equinoctial points regress, the motion that will remain in the ring will be to its former motion as 4,590 to 489,813, and therefore the motion of the equinoctial points will be diminished in the same ratio. Therefore the annual motion of the equinoctial points of a body composed of the ring and the globe will be to the motion 20°11′46″ as 1,436 to 39,343 and 4,590 to 489,813 jointly, that is, as 100 to 292,369. But the forces by which the nodes of the moons [i.e., a ring of moons] regress (as I have explained above), and so by which the equinoctial points of the ring regress (that is, the forces 3IT in the figure to prop. 30), are—in the individual particles—as the distances of those particles from the plane QR, and it is with these forces that the particles recede from the plane; and therefore (by lem. 2), if the matter of the ring were scattered over the whole surface of the globe, as in the configuration PapAPepE, so as to constitute that exterior part of the earth, the total force and efficacy of all the particles to rotate the earth about any diameter of the equator, and thus to move the equinoctial points, would come out less than before in the ratio of 2 to 5. And hence the annual regression of the equinoxes would now be to 20°11′46″ as 10 to 73,092, and accordingly would become 9″56‴50iv.
We have now described the system of the sun, the earth, the moon, and the planets; something must still be said about comets.
Lemma 4
The comets are higher than the moon and move in the planetary regions.
Just as the lack of diurnal parallax requires that comets be located beyond the sublunar regions, so the fact that comets have an annual parallax is convincing evidence that they descend into the regions of the planets. For comets which move forward according to the order of the signs are all, toward the end of their visibility, either slower than normal or retrograde if the earth is between them and the sun, but swifter than they should be if the earth is approaching opposition. And conversely those comets that move contrary to the order of the signs are swifter than they should be, at the end of their visibility when the earth is between them and the sun, and slower than they should be or retrograde if the earth is on the opposite side of the sun. This happens principally as a result of the motion of the earth in its different positions [with respect to the comets], just as is the case for the planets, which, according as the motion of the earth is either in the same direction or in an opposite one, are sometimes retrograde, and sometimes seem to advance more slowly and at other times more swiftly. If the earth goes in the same direction as the comet and by its angular motion is carried about the sun so much more swiftly that a straight line continually drawn through the earth and the comet converges toward the regions beyond the comet, then the comet as seen from the earth will appear to be retrograde because of its slower motion; but if the earth is going more slowly, the motion of the comet (taking away the motion of the earth) becomes at least slower. But if the earth goes in a direction opposite to the comet’s motion, the motion of the comet will as a result appear speeded up. And from the acceleration or retardation or retrograde motion, the distance of the comet may be ascertained in the following way.
Let 
QA, QB, and QC be three observed longitudes of a comet
at the beginning of its [visible] motion, and let QF be its last observed
longitude, just as the comet ceases to be seen. Draw the straight line ABC,
whose parts AB and BC placed between the straight lines QA and QB,
and between the straight lines QB and QC, are to each other as the times
between the first three observations. Let AC be produced to G, so that AG
is to AB as the time between the first and the last observation is to the time
between the first and the second observation, and let QG be joined. Then,
if the comet moved uniformly in a straight line and the earth were either
at rest or also moved forward in a straight line with uniform motion, the
angle QG would be the longitude of the comet at the time of the last
observation. Therefore, the angle FQG, which is the difference between the
longitudes, arises from the inequality of the motions of the comet and of the
earth. And this angle, if the earth and comet move in opposite directions, is
added to the angle QG, and thus makes the apparent motion of the comet
swifter; but if the comet is going in the same direction as the earth, this angle
is subtracted from that same angle QG and makes the motion of the comet
either slower or possibly retrograde, as I have just explained. Therefore this
angle arises chiefly from the motion of the earth and on that account is rightly
regarded as the parallax of the comet, ignoring, of course, any increase or
decrease in it which could arise from the nonuniform motion of the comet
in its own orbit. And the distance of the comet may be ascertained from this
parallax in the following manner.
Let S represent the sun, acT the earth’s orbit, a the place of the earth
in the first observation, c the place of the earth in the third observation,
T the place of the earth in the
last observation, and let T be a
straight line drawn toward the
beginning of Aries. Let angle TV
be taken equal to angle QF,
that is, equal to the longitude of
the comet when the earth is in T.
Let ac be drawn and produced
to g, so that ag is to ac as AG
to AC; then g will be the place
which the earth would reach at
the time of the last observation,
with its motion uniformly continued in the straight line ac. And so if g
is drawn parallel to T and the angle gV is taken equal to the angle
QG, this angle gV will be equal to the longitude of the comet as seen
from place g, and the angle TVg will be the parallax that arises from the
transfer of the earth from place g to place T; and accordingly V will be the
place of the comet in the plane of the ecliptic. And this place V is ordinarily
lower than the orbit of Jupiter.
The same may be ascertained from the curvature of the path of comets. These bodies go almost in great circles as long as they move more swiftly, but at the end of their course, when that part of their apparent motion which arises from parallax has a greater proportion to the total apparent motion, they tend to deviate from such circles, and whenever the earth moves in one direction, they tend to go off in the opposite direction. Because this deviation corresponds to the motion of the earth, it arises chiefly from parallax, and its extraordinary quantity, according to my computation, has placed disappearing comets quite far below Jupiter. Hence it follows that when comets are closer to us, in their perigees and perihelions, they very often descend below the orbits of Mars and of the inferior planets.
The nearness of comets is confirmed also from the light of their heads. For the brightness of a heavenly body illuminated by the sun and going off into distant regions is diminished as the fourth power of the distance; that is, it is diminished as the square because of the increased distance of the body from the sun and diminished as the square again because of the diminished apparent diameter. Thus, if both the quantity of light [i.e., brightness] and the apparent diameter of the comet are given, its distance will be found by taking its distance to the distance of some planet directly in the ratio of diameter to diameter and inversely as the square root of the ratio of light to light. Thus, as observed by Flamsteed through a sixteen-foot telescope and measured with a micrometer, the least diameter of the comaa of the comet of the year 1682 equaled 2′0″, while the nucleus or star in the middle of the head occupied scarcely a tenth of this width and therefore was only 11″ or 12″ wide. But in the light and brilliance of its head it surpassed the head of the comet of the year 1680 and rivaled stars of the first or second magnitude. Let us suppose that Saturn with its ring was about four times brighter; then, because the light of the ring almost equals the light of the globe within it, and the apparent diameter of the globe is about 21″, so that the light of the globe and the ring together would equal the light of a globe whose diameter was 30″, it follows that the distance of the comet will be to the distance of Saturn as 1 to √4 inversely and 12″ to 30″ directly, that is, as 24 to 30 or as 4 to 5. Again, on the authority of Hevelius, the comet of April 1665 surpassed in its brilliance almost all the fixed stars, and even Saturn itself (that is, by reason of its far more vivid color). Indeed, this comet was brighter than the one which had appeared at the end of the preceding year and was comparable to stars of the first magnitude. The width of the comet’s coma was about 6′, but the nucleus, when compared with the planets by the aid of a telescope, was clearly smaller than Jupiter and was judged to be sometimes smaller than the central body of Saturn and sometimes equal to it. Further, since the diameter of the coma of comets rarely exceeds 8′ or 12′, and the diameter of the nucleus or central star is about a tenth or perhaps a fifteenth of the diameter of the coma, it is evident that such stars generally have the same apparent magnitude as the planets. Hence, since their light can often be compared to the light of Saturn and sometimes surpasses it, it is manifest that all the comets in their perihelions should be placed either below Saturn or not far above. Those who banish the comets almost to the region of the fixed stars are, therefore, entirely wrong; certainly in such a situation, they would not be illuminated by our sun any more than the planets in our solar system are illuminated by the fixed stars.
In treating these matters, we have not been considering the obscuring of comets by that very copious and thick smoke by which the head is surrounded, always gleaming dully as if through a cloud. For the darker the body is rendered by this smoke, the closer it must approach to the sun for the amount of light reflected from it to rival that of the planets. This makes it likely that the comets descend far below the sphere of Saturn, as we have proved from their parallax.
But this same result is, to the highest degree, confirmed from their tails. These arise either from reflection by the smoke scattered through the aether or from the light of the head. In the first case the distance of the comets must be diminished, since otherwise the smoke always arising from the head would be propagated through spaces far too great, with such a velocity and expansion as to be unbelievable. In the second case, all the light of both the tail and the coma must be ascribed to the nucleus of the head. Therefore, if we suppose that all this light is united and condensed within the disc of the nucleus, then certainly that nucleus, whenever it emits a very large and very bright tail, will far surpass in its brilliance even Jupiter itself. Therefore, if it has a smaller apparent diameter and is sending forth more light, it will be much more illuminated by the sun and thus will be much closer to the sun. By the same argument, furthermore, the heads ought to be located below the orbit of Venus, when they are hidden under the sun and emit tails both very great and very bright like fiery beams, as they do sometimes. For if all of that light were understood to be gathered together into a single star, it would sometimes surpass Venus itself, not to say several Venuses combined.
Finally, the same thing may be ascertained from the light of the heads, which increases as comets recede from the earth toward the sun and decreases as they recede from the sun toward the earth. Thus the latter comet of 1665 (according to the observations of Hevelius), from the time when it began to be seen, was always decreasing in its apparent motion and therefore had already passed its perigee; but the splendor of its head nevertheless increased from day to day until the comet, concealed by the sun’s rays, ceased to be visible. The comet of 1683 (also according to the observations of Hevelius) at the end of July, when it was first sighted, was moving very slowly, advancing about 40′ or 45′ in its orbit each day. From that time its daily motion kept increasing continually until 4 September, when it came to about 5°. Therefore, in all this time the comet was approaching the earth. This is gathered also from the diameter of the head, as measured with a micrometer, since Hevelius found it to be on 6 August only 6′5″ including the coma, but on 2 September 9′7″. Therefore the head appeared far smaller at the beginning than at the end of the motion; yet at the beginning the head showed itself far brighter in the vicinity of the sun than toward the end of its motion, as Hevelius also reports. Accordingly, in all this time, because of its receding from the sun, it decreased with respect to its light, notwithstanding its approach to the earth.
The comet of 1618, about the middle of December, and that of 1680, about the end of the same month, were moving very swiftly and therefore were then in their perigees. Yet the greatest splendor of their heads occurred about two weeks earlier, when they had just emerged from the sun’s rays, and the greatest splendor of their tails occurred a little before that, when they were even nearer to the sun. The head of the first of these comets, according to the observations of [Johann Baptist] Cysat, seemed on 1 December to be greater than stars of the first magnitude, and on 16 December (being now in its perigee) it had failed little in magnitude, but very much in the splendor or clarity of its light. On 7 January Kepler, being uncertain about its head, brought his observing to an end. On 12 December the head of the second of these comets was sighted, and was observed by Flamsteed at a distance of 9° from the sun, a thing which would scarcely have been possible in a star of the third magnitude. On 15 and 17 December it appeared as a star of the third magnitude, since it was diminished by the brightness of clouds near the setting sun. On 26 December, moving with the greatest speed and being almost in its perigee, it was less than the mouth of Pegasus, a star of the third magnitude. On 3 January it appeared as a star of the fourth magnitude, on 9 January as a star of the fifth magnitude, and on 13 January it disappeared from view, as a result of the splendor of the crescent moon. On 25 January it scarcely equaled stars of the seventh magnitude. If equal times are taken on both sides of the perigee (before and after), then the head, being placed at those times in distant regions, ought to have shone with equal brilliance because of its equal distances from the earth, but it appeared brightest in the region [on the side of the perigee] toward the sun and disappeared on the other side of the perigee. Therefore from the great difference of light in these two situations, it is concluded that there is a great nearness of the sun and the comet in the first of these situations. For the light of comets tends to be regular and be greatest when the heads move most swiftly, and accordingly are in their perigees, except insofar as this light becomes greater in the vicinity of the sun.
COROLLARY 1. Therefore comets shine by the sun’s light reflected from them.
COROLLARY 2. From what has been said it will also be understood why comets appear so frequently in the region of the sun. If they were visible in the regions far beyond Saturn, they would have to appear more often in the parts of the sky opposite to the sun. For those that were in these parts would be nearer to the earth; and the sun, being in between, would obscure the others. Yet in running through the histories of comets, I have found that four or five times more have been detected in the hemisphere toward the sun than in the opposite hemisphere, besides without doubt not a few others which the sun’s light hid from view. Certainly, in their descent to our regions comets neither emit tails nor are so brightly illuminated by the sun that they show themselves to the naked eye so as to be discovered before they are closer to us than Jupiter itself. But by far the greater part of the space described about the sun with so small a radius is situated on the side of the earth that faces the sun, and comets are generally more brightly illuminated in that greater part, since they are much closer to the sun.
COROLLARY 3. Hence also it is manifest that the heavens are lacking in resistance. For the comets, following paths that are oblique and sometimes contrary to the course of the planets, move in all directions very freely and preserve their motions for a very long time even when these are contrary to the course of the planets. Unless I am mistaken, comets are a kind of planet and revolve in their orbits with a continual motion. For there seems to be no foundation for the allegation of some writers, basing their argument on the continual changes of the heads, that comets are meteors. The heads of comets are encompassed with huge atmospheres, and the atmospheres must be denser as one goes lower. Therefore, it is in these clouds, and not in the very bodies of the comets, that those changes are seen. Thus, if the earth were viewed from the planets, it would doubtless shine with the light of its own clouds, and its solid body would be almost hidden beneath the clouds. Thus, the belts of Jupiter are formed in the clouds of that planet, since they change their situation relative to one another, and the solid body of Jupiter is seen with greater difficulty through those clouds. And the bodies of comets must be much more hidden beneath their atmospheres, which are both deeper and thicker.
Proposition 40, Theorem 20
Comets move in conics having their foci in the center of the sun, and by radii
drawn to the sun, they describe areas proportional to the times.
This is evident by corol. 1 to prop. 13 of the first book compared with props. 8, 12, and 13 of the third book.
COROLLARY 1. Hence, if comets revolve in orbits, these orbits will be ellipses, and the periodic times will be to the periodic times of the planets as the 3/2 powers of their principal axes. And therefore comets, for the most part being beyond the planets and on that account describing orbits with greater axes, will revolve more slowly. For example, if the axis of the orbit of a comet were four times greater than the axis of the orbit of Saturn, the time of a revolution of the comet would be to the time of a revolution of Saturn (that is, to 30 years) as 4√4 (or 8) to 1, and accordingly would be 240 years.
COROLLARY 2. But these orbits will be so close to parabolas that parabolas can be substituted for them without sensible errors.
COROLLARY 3. And therefore (by book 1, prop. 16, corol. 7) the velocity of every comet will always be to the velocity of any planet, [considered to be] revolving in a circle about the sun, very nearly as the square root of twice the distance of the planet from the center of the sun to the distance of the comet from the center of the sun. Let us take the radius of the earth’s orbit (or the greatest semidiameter of the ellipse in which the earth revolves) to be of 100,000,000 parts; then the earth will describe by its mean daily motion 1,720,212 of these parts, and by its hourly motion 71,675½ parts. And therefore the comet, at the same mean distance of the earth from the sun, and having a velocity that is to the velocity of the earth as √2 to 1, will describe by its daily motion 2,432,747 of these parts, and by its hourly motion 101,364½ parts. But at greater or smaller distances, both the daily and the hourly motion will be to this daily and hourly motion as the square root of the ratio of the distances inversely, and therefore is given.
COROLLARY 4. Hence, if the latus rectum of a parabola is four times greater than the radius of the earth’s orbit, and if the square of that radius is taken to be 100,000,000 parts, the area that the comet describes each day by a radius drawn to the sun will be l,216,373½ parts, and in each hour that area will be 50,682¼ parts. But if the latus rectum is greater or smaller in any ratio, then the daily and hourly area will be greater or smaller, as the square root of that ratio.
Lemma 5
To find a parabolic curve that will pass through any number of given points.
Let the points be A, B, C, D, E, F, . . . , and from them to any straight line HN, given in position, drop the perpendiculars AH, BI, CK, DL, EM, FN, . . . .
CASE 1. If the intervals HI, IK, KL, . . . between the points H, I, K, L, M, N are equal, take the first differences b, b2, b3, b4, b5, . . . of the perpendiculars AH, BI, CK, . . . ; the second differences c, c2, c3, c4, . . . ; the third differences d, d2, d3, . . . ; that is, in such a way that AH − BI = b, BI − CK = b2, CK − DL = b3, DL + EM = b4 −EM + FN = b5, . . . , then b − b2 = c, . . . , and go on in this way to the last difference, which here is f. Then, if any perpendicular RS is erected, which is to be an ordinate to the required curve, in order to find its length, suppose each of the intervals HI, IK, KL, LM, . . . to be unity, and let AH be equal to a, −HS = p, ½p × (−IS) = q, ⅓q × (+SK) = r, ¼r × (+SL) = s, ⅕s × (+SM) = t, proceeding, that is, up to the penultimate perpendicular ME, and prefixing negative signs to the terms HS, IS, . . . , which lie on the same side of the point S as A, and positive signs to the terms SK, SL, . . . , which lie on the other side of the point S. Then if the signs are observed exactly, RS will be = a + bp + cq + dr + es + ft + . . . .
CASE 2. But if the intervals HI, IK, . . . between the points H, I, K,
L, . . . are unequal, take b, b2, b3, b4, b5, . . . , the first differences of the
perpendiculars AH, BI, CK, . . . divided by the intervals between the
perpendiculars; take c, c2, c3, c4, . . . , the second differences divided by each two
intervals; d, d2, d3, . . . , the third differences divided by each three intervals; e,
e2, . . . , the fourth differences divided by each four intervals, and so on—that
is, in such a way that 
, . . . , and
then 
, . . . , and afterward
. . . . When these differences have been found, let AH
be equal to a, −HS = p, p × (−IS) = q, q × (+SK) = r, r × (+SL) = s,
s × (+SM) = t, proceeding, that is, up to the penultimate perpendicular
ME; then the ordinate RS will be = a + bp + cq + dr + es + ft + . . . .
COROLLARY. Hence the areas of all curves can be found very nearly. For if several points are found of any curve which is to be squared [i.e., any curve whose area is desired] and a parabola is understood to be drawn through them, the area of this parabola will be very nearly the same as the area of that curve which is to be squared. Moreover, a parabola can always be squared geometrically by methods which are very well known.
Lemma 6
From several observed places of a comet, to find its place at any given intermediate
time.
Let HI, IK, KL, LM represent the times between the observations (in the figure to lem. 5), HA, IB, KC, LD, ME five observed longitudes of the comet, and HS the given time between the first observation and the required longitude. Then, if a regular curve ABCDE is understood to be drawn through the points A, B, C, D, E, and if the ordinate RS is found by the above lemma, RS will be the required longitude.
By the same method the latitude at a given time is found from five observed latitudes.
If the differences of the observed longitudes are small, say only 4 or 5 degrees, three or four observations would suffice for finding the new longitude and latitude. But if the differences are greater, say 10 or 20 degrees, five observations must be used.
Lemma 7
To draw a straight line BC through a given point P, so that the parts PB and
PC of that line, cut off by two straight lines AB and AC, given in position, have
a given ratio to each other.
From that point P draw any straight line PD to either of the straight lines, say AB, and produce PD toward the other straight line AC as far as E, so that PE is to PD in the given ratio. Let EC be parallel to AD; and if CPB is drawn, PC will be to PB as PE to PD. Q.E.F.
Lemma 8
Let ABC be a parabola with focus S. Let the segment ABCI be cut off by the
chord AC (which is bisected at I), let its diameter be Iμ, and let its vertex be μ.
On Iμ produced, take μO equal to half of Iμ. Join OS and produce it to ξ, so
that Sξ is equal to 2SO. Then, if a comet B moves in the arc CBA, and if ξB
is drawn cutting AC in E, I say that the point E will cut off from the chord AC
the segment AE very nearly proportional to the time.
For join EO, cutting the parabolic arc ABC in Y, and draw μX so as to touch the same arc in the vertex μ and meet EO in X; then the curvilinear area AEXμA will be to the curvilinear area ACYμA as AE to AC. And thus, since triangle ASE is in the same ratio to triangle ASC as the ratio of those curvilinear areas, the total area ASEXμA will be to the total area ASCYμA as AE to AC. Moreover, since ξO is to SO as 3 to 1, and EO is in the same ratio to XO, SX will be parallel to EB; and therefore, if BX is joined, the triangle SEB will be equal to the triangle XEB. Thus, if the triangle EXB is added to the area ASEXμA and from that sum the triangle SEB is taken away, there will remain the area ASBXμA equal to the area ASEXμA, and thus it will be to the area ASCYμA as AE to AC. But the area ASBYμA is very nearly equal to the area ASBXμA, and the area ASBYμA is to the area ASCYμA as the time in which the arc AB is described to the time of describing the total arc AC. And thus AE is to AC very nearly in the ratio of the times. Q.E.D.
COROLLARY. When point B falls upon the vertex μ of the parabola, AE is to AC exactly in the ratio of the times.
Scholium
If μξ is joined, cutting AC at δ, and if ξn, which is to μB as 27MI to
16Mμ, is taken in this line, then when Bn is drawn it will cut the chord
AC more nearly in the ratio of the lines than before. But the point n is to
be taken so as to lie beyond point ξ if point B is more distant than point μ
from the principal vertex of the parabola; and contrariwise if B is less distant
from that vertex.
Lemma 9
The straight lines Iμ and μM and the length
are equal to one another.
For 4Sμ is the latus rectum of a parabola, extending to the vertex μ.
Lemma 10
Let Sμ be produced to N and P, so that μN is one-third of μI, and so that
SP is to SN as SN to Sμ. Then, in the time in which a comet describes the arc
AμC, it would—if it moved forward always with the velocity that it has at a
height equal to SP—describe a length equal to the chord AC.
For if the comet were to move forward in the same time uniformly
in the straight line that touches the parabola at μ, and with the
velocity that it has in μ, then the area that
it would describe by a radius drawn to
point S would be equal to the parabolic
area ASCμ. And hence the space
determined by the length described along the
tangent and the length Sμ would be to
the space determined by the lengths AC
and SM as the area ASCμ to the
triangle ASC, that is, as SN to SM.
Therefore, AC is to the length described along the tangent as Sμ to SN. But the
velocity of the comet at the height SP is (by book 1, prop. 16, corol. 6) to its
velocity at the height Sμ as the square root of the ratio of SP to Sμ inversely,
that is, in the ratio of Sμ to SN; hence the length described in the same time
with this velocity will be to the length described along the tangent as Sμ to
SN. Therefore, since AC and the length described with this new velocity are
in the same ratio to the length described along the tangent, they are equal to
each other. Q.E.D.
COROLLARY. Therefore, in that same time, the comet, with the velocity that it has at the height Sμ + ⅔Iμ, would describe the chord AC very nearly.
Lemma 11
Suppose a comet, deprived of all motion, to be let fall from the height SN or
Sμ + ⅓Iμ, so as to fall toward the sun, and suppose this comet to be urged
toward the sun always by that force, uniformly continued, by which it is urged at
the beginning. Then in half of the time in which the comet describes the arc AC
in its orbit, it would—in this descent toward the sun—describe a space equal to
the length Iμ.
For by lem. 10, in the same time in which the comet describes the
parabolic arc AC, it will—with the velocity that it has at the height
SP—describe the chord AC; and hence (by book 1, prop. 16, corol. 7), revolving
by the force of its own gravity, it would—in that same time, in a circle whose
semidiameter was SP—describe an arc whose length would be to the chord
AC of the parabolic arc in the ratio of 1 to √2. And therefore, falling from
the height SP toward the sun with the weight that it has toward the sun at
that height, it would in half that time (by book 1, prop. 4, corol. 9) describe
a space equal to the square of half of that chord, divided by four times the
height SP, that is, the space 
. Thus, since the weight of the comet toward
the sun at the height SN is to its weight toward the sun at the height SP
as SP to Sμ, the comet—falling toward the sun with the weight that it has
at the height SN—will in the same time describe the space
, that is, a
space equal to the length Iμ or Mμ. Q.E.D.
Proposition 41, Problem 21
To determine the trajectory of a comet moving in a parabola, from three given
observations.
Having tried many approaches to this exceedingly difficult problem, I devised certain problems [i.e., propositions] in book 1 which are intended for its solution. But later on, I conceived the following slightly simpler solution.
Let three observations be chosen, distant from one another by nearly equal intervals of time. But let that interval of time when the comet moves more slowly be a little greater than the other, that is, so that the difference of the times is to the sum of the times as the sum of the times to more or less six hundred days, or so that the point E (in the figure to lem. 8) falls very nearly on the point M and deviates from there toward I rather than toward A. If such observations are not at hand, a new place of the comet must be found by the method of lem. 6.
Let S represent the sun; T, t, and τ three places of the earth in its orbit; TA, tB, and τC three observed longitudes of the comet; V the time between the first observation and the second; W the time between the second and the third; X the length that the comet could describe in that total time [V + W] with the velocity that it has in the mean distance of the earth from the sun (which length is to be found by the method of book 3, prop. 40, corol. 3); and let tV be a perpendicular to the chord Tτ. In the mean observed longitude tB, let the point B be taken anywhere at all for the place of the comet in the plane of the ecliptic, and from there toward the sun S draw line BE so as to be to the sagitta tV as the contenta of SB and St2 to the cube of the hypotenuse of the right-angled triangle whose sides are SB and the tangent of the latitude of the comet in the second observation to the radius tB. And through point E (by lem. 7 of this third book) draw the straight line AEC so that its parts AE and EC, terminated in the straight lines TA and τC, are to each other as the times V and W. Then A and C will be the places of the comet in the plane of the ecliptic in the first and third observations very nearly, provided that B is its correctly assumed place in the second observation.
Upon AC, bisected in I, erect a perpendicular Ii. Through point B let a line Bi be imagined,b drawn parallel to AC. Let Si be a line imagined as cutting AC at λ, and complete the parallelogram iIλμ. Take Iσ equal to 3Iλ, and through the sun S draw the dotted line σξ equal to 3Sσ + 3iλ. And after deleting the letters A, B, C, and I, let a new imagined line BE be drawn from the point B toward the point ξ so that it is to the former line BE as the square of the distance BS to the quantity Sμ + ⅓iλ. And through point E again draw the straight line AEC according to the same rule as before, that is, so that its parts AE and EC are to each other as the times V and W between observations. Then A and C will be the places of the comet more exactly.
Upon AC, bisected in I, erect the perpendiculars AM, CN, and IO, so that, of these perpendiculars, AM and CN are the tangents of the latitudesc in the first and third observations (to the radii TA and τC). Join MN, cutting IO in O. Construct the rectangle iIλμ as before. On IA produced, take ID equal to Sμ + ⅔iλ. Then on MN, toward N, take MP so that it is to the length X found above as the square root of the ratio of the mean distance of the earth from the sun (or of the semidiameter of the earth’s orbit) to the distance OD. If point P falls upon point N, then A, B, and C will be three places of the comet, through which its orbit is to be described in the plane of the ecliptic. But if point P does not fall upon point N, then on the straight line AC take CG equal to NP, in such a way that points G and P lie on the same side of the straight line NC.
Using the same method by which points E, A, C, and G were found from the assumed point B, find from other points b and β (assumed in any way whatever) the new points e, a, c, g, and ε, α, κ, γ. Then if the circumference of circle Ggγ is drawn through G, g, and γ, cutting the straight line τC in Z, Z will be a place of the comet in the plane of the ecliptic. And if on AC, ac, and ακ, there are taken AF, af, and αφ, equal respectively to CG, cg, and κγ, and if the circumference of a circle Ffφ is drawn through points F, f, and φ, cutting the straight line AT in X, then point X will be another place of the comet in the plane of the ecliptic. At the points X and Z, erect the tangents of the latitudes of the comet (to the radii TX and τZ), and two places of the comet in its orbit will be found. Finally (by book 1, prop. 19), let a parabola with focus S be described through those two places; this parabola will be the trajectory of the comet. Q.E.I.
The demonstration of this construction follows from the lemmas, since the straight line AC is cut in E in the ratio of the times, by lem. 7, as required by lem. 8; and since BE, by lem. 11, is that part of the straight line BS or Bξ which lies in the plane of the ecliptic between the arc ABC and the chord AEC; and since MP (by lem. 10, corol.) is the length of the chord of the arc that the comet must describe in its orbit between the first observation and the third, and therefore would be equal to MN, provided that B is a true place of the comet in the plane of the ecliptic.
But it is best not to choose the points B, b, and β any place whatever, but to take them as close to true as possible. If the angle AQt, at which the projection of the orbit described in the plane of the ecliptic cuts the straight line tB, is known approximately, imagine the straight line AC drawn at that angle so that it is to 4/3Tτ as the square root of the ratio of SQ to St. And by drawing the straight line SEB, so that its part EB is equal to the length Vt, point B will be determined, which may be used the first time around. Then, after deleting the straight line AC and drawing AC anew according to the preceding construction, and after additionally finding the length MP, take point b on tB according to the rule that if TA and τC cut each other in Y, the distance Yb is to the distance YB in a ratio compounded of the ratio of MP to MN and the square root of the ratio of SB to Sb. And the third point β will have to be found by the same method, if it is desired to repeat the operation for the third time. But by this method two operations would, for the most part, be sufficient. For if the distance Bb happens to be very small, then after the points F, f and G, g have been found, the straight lines Ff and Gg (when drawn) will cut TA and τC in the required points X and Z.
EXAMPLE. Let the comet of 1680 be proposed as the example. The following table shows its motion as observed by Flamsteed and as calculated by him from these observations, and corrected by Halley on the basis of the same observations.
To these add certain observations of my own.
These observations were made with a seven-foot telescope, and a micrometer the threads of which were placed in the focus of the telescope; and with these instruments we determined both the positions of the fixed stars in relation to one another and the positions of the comet in relation to 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 the star of the sixth magnitude in the heel of the same foot (Bayer’s n) and D, E, F, G, H, I, K, L, M, N, O, Z, α, β, γ, and δ other smaller stars in the same foot. And let p, P, Q, R, S, T, V, and X be the places of the comet in the observations described above; and, the distance AB being reckoned at 807/12 parts, AC was 52¼ parts, BC 58⅚, AD 575/12, BD 826/11, CD 23⅔, AE 294/7, CE 57½, DE 4911/12, AI 277/12, BI 52⅙, CI 367/12, DI 535/11, AK 38⅔, BK 43, CK 315/9, FK 29, FB 23, FC 36¼, AH 186/7, DH 50⅞, BN 465/12, CN 31⅓, BL 455/12, NL 315/7. HO was to HI as 7 to 6 and, when produced, passed between stars D and E in such a way that the distance of star D from this straight line was ⅙CD. LM was to LN as 2 to 9 and, when produced, passed through star H. Thus the positions of the fixed stars in relation to one another were determined.
Finally our fellow countryman Pound again observed the positions of these fixed stars in relation to one another and recorded their longitudes and latitudes, as in the following table.
I observed the positions of the comet in relation to these fixed stars as follows.
On Friday, 25 February (O.S.), at 8h30m P.M., the distance of the comet, which was at p, from star E was less than 3/13 AE, and greater than ⅕AE, and thus was approximately 3/14AE; and the angle ApE was somewhat obtuse, but almost a right angle. For if a perpendicular were dropped from A to pE, the distance of the comet from that perpendicular was ⅕pE.
On the same night at 9h30m, the distance of the comet (which was at P)
from star E was greater than 
AE and less than 
AE, and thus was
very nearly 
AE, or 8/39AE. And the distance of the comet from a
perpendicular dropped from star A to the straight line PE was ⅘PE.
On Sunday, 27 February, at 8h15m P.M., the distance of the comet (which was at Q) from star O equaled the distance between stars O and H; and the straight line QO, produced, passed between stars K and B. Because of intervening clouds, I could not determine the position of this straight line more exactly.
On Tuesday, 1 March, at 11h P.M., the comet (which was at R) lay exactly between stars K and C; and the part CR of the straight line CRK was a little greater than ⅓CK and a little smaller than ⅓CK + ⅛CR, and thus was equal to ⅓CK + 1/16CR, or 16/45CK.
On Wednesday, 2 March, at 8h P.M., the distance of the comet (which was at S) from star C was very close to 4/9FC. The distance of star F from the straight line CS, produced, was 1/24FC, and the distance of star B from that same straight line was five times greater than the distance of star F. Also, the straight line NS, produced, passed between stars H and I and was five or six times nearer to star H than to star I.
On Saturday, 5 March, at 11h30m P.M. (when the comet was at T), the straight line MT was equal to ½ML, and the straight line LT, produced, passed between B and F four or five times closer to F than to B, cutting off from BF a fifth or sixth part of it toward F. And MT, produced, passed outside the space BF on the side of star B and was four times closer to star B than to star F. M was a very small star that could scarcely be seen through the telescope, and L was a greater star, of about the eighth magnitude.
On Monday, 7 March, at 9h30m P.M. (when the comet was at V), the straight line Vα, produced, passed between B and F, cutting off 1/10BF from BF on the side of F, and was to the straight line Vβ as 5 to 4. And the distance of the comet from the straight line αβ was ½Vβ.
On Wednesday, 9 March, at 8h30m P.M. (when the comet was at X), the straight line γX was equal to ¼γδ, and a perpendicular dropped from star δ to the straight line γX was ⅖γδ.
On the same night at 12h (when the comet was at Y), the straight line γY was equal to ⅓γδ or a little smaller, say 5/16γδ, and a perpendicular dropped from star δ to the straight line γY was equal to about ⅙ or 1/7γδ. But the comet could scarcely be discerned because of its nearness to the horizon, nor could its place be determined so surely as in the preceding observations.
From observations of this sort, by constructions of diagrams, and by calculations, I found the longitudes and latitudes of the comet, and from the corrected places of the fixed stars our fellow countryman Pound corrected the places of the comet, and these corrected places are given above. I used a crudely made micrometer, but nevertheless the errors of longitudes and latitudes (insofar as they come from my observations) scarcely exceed one minute. Moreover, the comet (according to my observations) at the end of its motion began to decline noticeably toward the north from the parallel which it had occupied at the end of February.
Now, in order to determine the orbit of the comet, I selected—from the
observations hitherto described—three that Flamsteed made, on 21
December, 5 January, and 25 January. From these observations I found St to be of
9,842.1 parts, and Vt to be of 455 parts (10,000 such parts being the
semidiameter of the earth’s orbit). Then for the first operation, assuming tB to be
of 5,657 parts, I found SB to be of 9,747, BE the first time 412, Sμ 9,503, iλ
413; BE the second time 421, OD 10,186, X 8,528.4, MP 8,450, MN 8,475,
NP 25. Hence for the second operation I reckoned the distance tb to be
5,640. And by this operation I found at last the distance TX to be 4,775 and
the distance τZ to be 11,322. In determining the orbit from these distances,
I found the descending node in 
1°53′ and the ascending node in 
1°53′,
and the inclination of its plane to the plane of the ecliptic to be 61°20⅓′. I
found that its vertex (or the perihelion of the comet) was 8°38′ distant from
the node and was in 
27°43′ with a latitude 7°34′ S; and that its latus
rectum was 236.8, and that the area described each day by a radius drawn to the
sun was 93,585, supposing the square of the semidiameter of the earth’s orbit
to be 100,000,000; and I found that the comet had advanced in this orbit in
the order of the signs, and was on 8 December 0h4m A.M. in the vertex of the
orbit or the perihelion. I made all these determinations graphically by a scale
of equal parts and by chords of angles, taken from the table of natural sines,
constructing a fairly large diagram, that is, one in which the semidiameter
of the earth’s orbit (of 10,000 parts) was equal to 16⅓ inches of an English
foot.
Finally, in order to establish whether the comet moved truly in the orbit thus found, I calculated—partly by arithmetical and partly by graphical operations—the places of the comet in this orbit at the times of certain observations, as can be seen in the following table.
On 3 November 17h2m, apparent time at London, the comet was in
29°51′ with latitude 1°17′45″ N.
On 5 November 15h58m, the comet was
in 
3°23′ with latitude 1°6′ N.
eIn this table we have omitted the observations made on 16, 18, 20, and
23 November as being less exact. Yet the comet was observed at these times
also.e In fact, [Giuseppe Dionigi] Ponteo and his associates, on 17 November
(O.S.) at 6h A.M. in Rome, that is, at 5h10m London time, using threads
applied to the fixed stars, observed the comet in 
8°30′ with latitude 0°40′ S.
Their observations may be found in the treatise which Ponteo published
about this comet. [Marco Antonio] Cellio, who was present and sent his own
observations in a letter to Mr. Cassini, saw the comet at the same hour in
8°30′ with latitude 0°30′ S. At the same hour
Gallet in Avignon (that is, at
5h42m A.M. London time) saw the comet in 8° with null latitude; at which
time, according to the theory, the comet was in 8°16′45″ with latitude
0°53′7″ S.
On 18 November at 6h30m A.M. in Rome (that is, at 5h40m London time)
Ponteo saw the comet in 3°30′ with latitude 1°20′ S; Cellio saw it in
13°30′ with latitude 1°00′ S. Moreover, Gallet at 5h30m A.M. in Avignon
saw the comet in 13°00′ with latitude l°00′ S. And the Reverend Father
Ango at the College of La Flèche in France, at 5h A.M. (that is, at 5h9m
London time), saw the comet midway between two small stars, of which
one is the middle star of three in a straight line in the southern hand of
Virgo, Bayer’s ψ, and the other is the outermost star of the wing, Bayer’s ϑ.
Thus the comet was then in 12°46′ with latitude 50′ S. On the same day
at Boston in New England, at a latitude of 42½°, at 5h A.M. (that is, 9h44m
London time), the comet was seen near 14° with latitude 1°30′ S, as I was
informed by the distinguished Halley.
On 19 November at 4h30m A.M. in Cambridge, the comet (according to
the observation of a certain young man) was about 2 degrees distant from
Spica Virginis toward the northwest. And Spica was in 19°23′47″ with
latitude 2°1′59″ S. On the same day at 5h A.M. at Boston in New England,
the comet was 1 degree distant from Spica Virginis, the difference of latitudes
being 40 minutes. On the same day on the island of Jamaica, the comet was
about 1 degree distant from Spica. On the same day Mr. Arthur Storer,
at the Patuxent River, near Hunting Creek in Maryland, which borders on
Virginia, at latitude 38½°, at 5h A.M. (that is, 10h London time), saw the comet
above Spica Virginis and almost conjoined with Spica, the distance between
them being about ¾ of a degree. And comparing these observations with
one another, I gather that at 9h44m in London the comet was in 18°50′
with latitude roughly l°25′ S. And by the theory the comet was then in
18°52′15″ with latitude 1°26′54″
S.f
On 20 November, Mr. Geminiano Montanari, professor of astronomy in
Padua, at 6h A.M. in Venice (that is, 5h10m London time), saw the comet in
23° with latitude l°30′ S. On the same day at Boston the comet was distant
from Spica by 4 degrees of longitude eastward and so was approximately in
23°24′.
On 21 November, Ponteo and his associates at 7h15m A.M. observed the
comet in 27°50′ with latitude 1°16′ S, Cellio in
28°, Ango at 5h A.M. in
27°45′, Montanari in 27°51′. On the same day on the island of Jamaica
the comet was seen near the beginning of Scorpio and had roughly the same
latitude as Spica Virginis, that is, 2°2′. On the same day at 5h A.M. at Balasore
in the East Indies (that is, at 11h20m the preceding night, London time) the
comet was distant 7°35′ eastward from Spica Virginis. It was in a straight line
between Spica and the scale [or pan of the Balance] and so was in 26°58′
with latitude roughly 1°11′ S, and after 5 hours and 40 minutes (that is, at 5h
A.M. London time) was in 28°12′ with latitude 1°16′ S. And by the theory
the comet was then in 28°10′36″ with latitude 1°53′35″ S.
On 22 November, the comet was seen by Montanari in 
2°33′, while at
Boston in New England it appeared in approximately 3°, with about the
same latitude as before, that is, 1°30′. On the same day at 5h A.M. at Balasore
the comet was observed in 1°50′, and so at 5h A.M. in London the comet
was approximately in 3°5′. On the same day at London at 6h30m A.M.
our fellow countryman Hooke saw the comet in approximately 3°30′, on
a straight line that passes between Spica Virginis and the heart of Leo, not
exactly indeed, but deviating a little from that line toward the north.
Montanari likewise noted that a line drawn from the comet through Spica passed,
on this day and the following days, through the southern side of the heart
of Leo, there being a very small interval between the heart of Leo and this
line. The straight line passing through the heart of Leo and Spica Virginis
cut the ecliptic in 3°46′, at an angle of 2°51′. And if the comet had been
located in this line in 3°, its latitude would have been 2°26′. But since
the comet, by the agreement of Hooke and Montanari, was at some distance
from this line toward the north, its latitude was a little less. On the 20th,
according to the observation of Montanari, its latitude almost equaled the
latitude of Spica Virginis and was roughly 1°30′; and by the agreement of
Hooke, Montanari, and Ango, the latitude was continually increasing and
so now (on the 22d) was sensibly greater than 1°30′. And the mean latitude
between the limits now established, 2°26′ and 1°30′, will be roughly 1°58′.
The tail of the comet, by the agreement of Hooke and Montanari, was
directed toward Spica Virginis, declining somewhat from that star—southward
according to Hooke, northward according to Montanari; and so that
declination was hardly perceptible, and the tail, being almost parallel to the equator,
was deflected somewhat northward from the opposition of the sun.
On 23 November (O.S.) at 5h A.M. at Nuremberg (that is, at 4h30m
London time) Mr. [Johann Jacob] Zimmermann saw the comet in 8°8′ with
latitude 2°31′ S, determining its distances from the fixed stars.
On 24 November before sunrise the comet was seen by Montanari in
12°52′ on the northern side of a straight line drawn through the heart of
Leo and Spica Virginis, and so had a latitude a little less than 2°38′. This
latitude (as we have said), according to the observations of Montanari, Ango,
and Hooke, was continually increasing, and so it was now (on the 24th) a little
greater than 1°58′, and at its mean magnitude can be taken as 2°18′ without
perceptible error. Ponteo and Gallet would have the latitude decreased now,
and Cellio and the observer in New England would have it retained at
about the same magnitude, namely 1 or 1½ degrees. The observations of
Ponteo and Cellio are rather crude, especially those that were made by taking
azimuths and altitudes, and so are those of Gallet; better are the ones that
were made by means of the positions of the comet in relation to fixed stars by
Montanari, Hooke, Ango, and the observer in New England, and sometimes
by Ponteo and Cellio. On the same day at 5h A.M. at Balasore, the comet was
observed in 11°45′, and so at 5h A.M. at London it was nearly in 13°.
And by the theory the comet was at that time in 13°22′42″.
On 25 November before sunrise Montanari observed the comet
approximately in 17¾°. And Cellio observed at the same time that the comet
was in a straight line between the bright star in the right thigh of Virgo and
the southern scale of Libra, and this straight line cuts the path of the comet
in 18°36′. And by the theory the comet was at that time approximately in
18⅓°.
Therefore these observations agree with the theory insofar as they agree with one another, and by such agreement they prove that it was one and the same comet that appeared in the whole time from the 4th of November to the 9th of March. The trajectory of this comet cut the plane of the ecliptic twice and therefore was not rectilinear. It cut the ecliptic not in opposite parts of the heavens, but at the end of Virgo and at the beginning of Capricorn, at points separated by an interval of about 98 degrees; and thus the course of the comet greatly deviated from a great circle. For in November its course declined by at least 3 degrees from the ecliptic toward the south, and afterward in December verged from the ecliptic 29 degrees toward the north: the two parts of its orbit, in which the comet tended toward the sun and returned from the sun, declining from each other by an apparent angle of more than 30 degrees, as Montanari observed. This comet moved through nine signs, namely from the last degree of Leo to the beginning of Gemini, besides [that part of] the sign of Leo through which it moved before it began to be seen; and there is no other theory according to which a comet may travel over so great a part of the heaven with a motion according to some rule. Its motion was extremely nonuniform. For about the 20th of November it described approximately 5 degrees per day; then, with a retarded motion between 26 November and 12 December, that is, during 15½ days, it described only 40 degrees; and afterward, with its motion accelerated again, it described about 5 degrees per day until its motion began to be retarded again. And the theory that corresponds exactly to so nonuniform a motion through the greatest part of the heavens, and that observes the same laws as the theory of the planets, and that agrees exactly with exact astronomical observations cannot fail to be true.
Furthermore, it seemed appropriate to show the trajectory that the comet described and the actual tail that it projected in different positions, as in the accompanying figure, in the plane of the trajectory; in this figure, ABC denotes the trajectory of the comet, D the sun, DE the axis of the trajectory, DF the line of nodes, GH the intersection of the sphere of the earth’s orbit with the plane of the trajectory, I the place of the comet on 4 November 1680, K its place on 11 November, L its place on 19 November, M its place on 12 December, N its place on 21 December, O its place on 29 December, P its place on 5 January of the following year, Q its place on 25 January, R its place on 5 February, S its place on 25 February, T its place on 5 March, and V its place on 9 March. I used the following observations in determining the tail.
On 4 and 6 November the tail was not yet visible. On 11 November
the tail, which had now begun to be seen, was observed through a ten-foot
telescope to be no more than half a degree long. On 17 November the tail
was observed by Ponteo to be more than 15 degrees long. On 18 November
the tail was seen in New England to be 30 degrees long and directly opposite
to the sun, and it was extended out to the star ♂ [i.e., the planet Mars],
which was then in 9°54′. On 19 November, in Maryland, the tail was
seen to be 15 or 20 degrees long. On 10 December the tail (according to the
observations of Flamsteed) was passing through the middle of the distance
between the tail of Serpens (the Serpent of Ophiuchus) and the star δ in the
southern wing of Aquila and terminated near the stars A, ω, b in Bayer’s
tables. Therefore the end of the comet’s tail was in 19½° with a latitude
of about 34¼° N. On 11 December the tail was rising as far as the head of
Sagitta (Bayer’s α, β), terminating in 26°43′, with a latitude of 38°34′ N.
On 12 December the tail was passing through the middle of Sagitta and did
not extend very much further, terminating in 
4°, with a latitude of about
42½° N.
These things are to be understood of the length of the brighter part of
the tail. For when the light was fainter and the sky perhaps clearer, on 12
December at 5h40m in Rome, the tail was observed by Ponteo to extend to 10
degrees beyond the uropygium of Cygnus [i.e., the rump of the Swan], and its
side toward the northwest terminated 45 minutes from this star. Moreover, in
those days the tail was 3 degrees wide near its upper end, and so the middle
of it was 2°15′ distant from that star toward the south, and its upper end was
in 
22° with a latitude of 61° N. And hence the tail was about 70 degrees
long.
On 21 December the tail rose almost to Cassiopeia’s Chair, being equally
distant from β and Schedar [= α Cassiopeiae] and having a distance from
each of them equal to their distance from each other, and so terminating in
24° with a latitude of 47½°. On 29 December the tail was touching Scheat,
which was situated to the left of it, and exactly filled the space between
the two stars in the northern foot of Andromeda; it was 54 degrees long;
accordingly it terminated in 
19° with a latitude of 35°. On 5 January the
tail touched the star π in the breast of Andromeda on the right side and
the star μ in the girdle on the left side, and (according to our observations)
was 40 degrees long; but it was curved, and its convex side faced to the
south. Near the head of the comet, the tail made an angle of 4 degrees with
the circle passing through the sun and the head of the comet; but near the
other end, it was inclined to that circle at an angle of 10 or 11 degrees, and
the chord of the tail contained an angle of 8 degrees with that circle. On
13 January the tail was visible enough between Alamech and Algol [= β
Persei], but it ended in a very faint light toward the star κ in Perseus’s side.
The distance of the end of the tail from the circle joining the sun and the
comet was 3°50′, and the inclination of the chord of the tail to that circle
was 8½ degrees. On 25 and 26 January the tail shone with a faint light to a
length of 6 or 7 degrees; and, a night or so later, when the sky was extremely
clear, it attained a length of 12 degrees and a little more, with a light that
was very faint and scarcely to be perceived. But its axis was directed exactly
toward the bright star in the eastern shoulder of Auriga, and accordingly
declined from the opposition of the sun toward the north at an angle of 10
degrees. Finally on 10 February, my eyes armed [with a telescope], I saw
the tail to be 2 degrees long. For the fainter light mentioned above was not
visible through the glasses. But Ponteo writes that on 7 February he saw the
tail with a length of 12 degrees. On 25 February and thereafter, the comet
appeared without a tail.
Whoever considers the orbit just described and turns over in his mind the other phenomena of this comet will without difficulty agree that the bodies of comets are solid, compact, fixed, and durable, like the bodies of planets. For if comets were nothing other than vapors or exhalations of the earth, the sun, and the planets, this one ought to have been dissipated at once during its passage through the vicinity of the sun. For the heat of the sun is as the density of its rays, that is, inversely as the square of the distance of places from the sun. And thus, since the distance of the comet from the center of the sun on 8 December, when it was in its perihelion, was to the distance of the earth from the center of the sun as approximately 6 to 1,000, the heat of the sun on the comet at that time was to the heat of the summer sun here on earth as 1,000,000 to 36, or as 28,000 to 1. But the heat of boiling water is about three times greater than the heat that dry earth acquires in the summer sun, as I have found [by experiment]; and the heat of incandescent iron (if I conjecture correctly) is about three or four times greater than the heat of boiling water; and hence the heat that dry earth on the comet would have received from the sun’s rays, when it was in its perihelion, would be about two thousand times greater than the heat of incandescent iron. But with so great a heat, vapors and exhalations, and all volatile matter, would have to have been consumed and dissipated at once.
Therefore the comet, in its perihelion, received an immense heat at [i.e., when near] the sun, and it can retain that heat for a very long time. For a globe of incandescent iron, one inch wide, standing in the air would scarcely lose all its heat in the space of one hour. But a larger globe would preserve its heat for a longer time in the ratio of its diameter, because its surface (which is the measure according to which it is cooled by contact with the surrounding air) is smaller in that ratio with respect to the quantity of hot matter it contains. And so a globe of incandescent iron equal to this earth of ours—that is, more or less 40,000,000 feet wide—would scarcely cool off in as many days, or about 50,000 years. Nevertheless, I suspect that the duration of heat is increased in a smaller ratio than that of the diameter because of some latent causes, and I wish that the true ratio might be investigated by experiments.
Further, it should be noted that in December, when the comet had just become hot at the sun, it was emitting a far larger and more splendid tail than it had done earlier in November, when it had not yet reached its perihelion. And, universally, the greatest and brightest tails all arise from comets immediately after their passage through the region of the sun. Therefore the heating up of the comet is conducive to a great size of its tail, and from this I believe it can be concluded that the tail is nothing other than extremely thin vapor that the head or nucleus of the comet emits by its heat.
There are indeed three opinions about the tails of comets: that the tails are the brightness of the sun’s light propagated through the translucent heads of comets; that the tails arise from the refraction of light in its progress from the head of the comet to the earth; and finally that these tails are a cloud or vapor continually rising from the head of the comet and going off in a direction away from the sun. The first opinion is held by those who are not yet instructed in the science of optics. For beams of sunlight are not seen in a dark room except insofar as the light is reflected from particles of dust and smoke always flying about through the air, and for this reason in air darkened with thicker smoke the beams of sunlight appear brighter and strike the eye more strongly, while in clearer air these beams are fainter and are perceived with greater difficulty, but in the heavens, where there is no matter to reflect these beams of sunlight, they cannot be seen at all. Light is not seen insofar as it is in the beam, but only to the degree that it is reflected to our eyes; for vision results only from rays that impinge upon the eyes. Therefore some reflecting matter must exist in the region of the tail, since otherwise the whole sky, illuminated by the light of the sun, would shine uniformly.
The second opinion is beset with many difficulties. The tails are never
variegated in color, and yet colors are generally the inseparable concomitants
of refractions. The light of the fixed stars and the planets which is
transmitted to us is distinct [i.e., clearly defined]; this demonstrates that the celestial
medium is empowered with no refractive force. It is said that the Egyptians
sometimes saw the fixed stars surrounded by a head of hair, but this happens
very rarely, and so it must be ascribed to some chance refraction by clouds.
The radiation and scintillation of the fixed stars also should be referred to
refractions both by the eyes and by the tremulous air, since they disappear
when these stars are viewed through telescopes. By the tremor of the air and
of the ascending vapors it happens that rays are easily turned aside
alternately from the narrow space of the pupil of the eye but not at all from the
wider aperture of the objective lens of a telescope. Thus it is that scintillation
is generated in the former case while it ceases in the latter; and the
cessation of scintillation in the latter case demonstrates the regular transmission
of light through the heavens without any sensible refraction. And to counter
the argument that tails are not generally seen in comets when their light is
not strong enough, for the reason that the secondary rays do not then have
enough force to affect the eyes, and that this is why the tails of the fixed stars
are not seen, it should be pointed out that the light of the fixed stars can be
increased more than a hundred times by means of telescopes, and yet no tails
are seen. The planets also shine with more light, but they have no tails; and
often comets have the greatest tails when the light of their heads is faint and
exceedingly dull. For such was the case for the comet of 1680; in December,
at a time when the light from its head scarcely equaled stars of the second
magnitude, it was emitting a tail of notable splendor as great as 40, 50, 60, or
70 degrees in length and more. Afterward, on 27 and 28 January, the head
appeared as a star of only the seventh magnitude, but the tail extended to
6 or 7 degrees in length with a very faint light that was sensible enough;
and with a very dim light, which could scarcely be seen, it stretched out as
far as 12 degrees or a little further, as was said above. But even on 9 and
10 February, when the head had ceased to be seen by the naked eye, the
tail—when I viewed it through a telescope—was 2 degrees long. Further, if
the tail arose from refraction by celestial matter, and if it deviated from the
opposition of the sun in accordance with the form of the heavens, then, in
the same regions of the heavens, that deviation ought always to take place
in the same direction. But the comet of 1680, on 28 December at 8h30m P.M.
London time, was in 8°41′ with a latitude of 28°6′ N, the sun being in
18°26′. And the comet of 1577, on 29 December, was in 8°41′ with a
latitude of 28°40′ N, the sun again being in approximately 18°26′. In both
cases the earth was in the same place and the comet appeared in the same
part of the sky; yet in the former case the tail of the comet (according to
my observations and those made by others) was declining by an angle of 4½
degrees from the opposition of the sun toward the north, but in the latter
case (according to the observations of Tycho) the declination was 21 degrees
toward the south. Therefore, since refraction by the heavens has been
rejected, the remaining possibility is to derive the phenomena of comets’ tails
from some matter that reflects light.
Moreover, the laws which the tails of comets observe prove that these tails arise from the heads and ascend into regions turned away from the sun. For example, if the tails lie in planes of the comets’ orbits which pass through the sun, they always deviate from being directly opposite the sun and point toward the region which the heads, advancing in those orbits, have left behind. Again, to a spectator placed in those planes, the tails appear in regions directly turned away from the sun; while for observers not in those planes, the deviation gradually begins to be perceived and appears greater from day to day. Furthermore, other things being equal, the deviation is less when the tail is more oblique to the orbit of the comet, and also when the head of the comet approaches closer to the sun, especially if the angle of deviation is taken near the head of the comet. And besides, the tails that do not deviate appear straight, while those that do deviate are curved. Again, this curvature is greater when the deviation is greater, and more sensible when the tail, other things being equal, is longer; for in shorter tails the curvature is scarcely noticed. Then, too, the angle of deviation is smaller near the head of the comet and larger near the other extremity of the tail; and thus the convex side of the tail faces the direction from which the deviation is made and which is along a straight line drawn from the sun through the head of the comet indefinitely. Finally, the tails that are more extended and wider and that shine with a more vigorous light are a little more resplendent on their convex sides and are terminated by a less indistinct limit than on their concave sides. For all these reasons, then, the phenomena of the tail depend on the motion of the head and not on the region of the sky in which the head is seen; and therefore these phenomena do not come about through refraction by the heavens, but arise from the head supplying the matter. For as in our air the smoke of any ignited body seeks to ascend and does so either perpendicularly (if the body is at rest) or obliquely (if the body is moving sideways), so in the heavens, where bodies gravitate toward the sun, smoke and vapors must ascend with respect to the sun (as has already been said) and move upward either directly, if the smoking body is at rest, or obliquely, if the body by advancing always leaves the places from which the higher parts of the vapor have previously ascended. And the swifter the ascent of the vapor, the less the obliquity, namely in the vicinity of the sun and near the smoking body. Moreover, as a result of this difference in obliquity, the column of vapor will be curved; and since the vapor on that side of the column in the direction of the comet’s motion is a little more recent [i.e., more recently exhaled], so also the column will be somewhat more dense on that same side, and therefore will reflect light more abundantly and will be terminated by a less indistinct limit. I add nothing here concerning sudden and uncertain agitations of the tails, nor concerning their irregular shapes (which are sometimes described), because either these effects may arise from changes in our air and the motions of the clouds that may obscure those tails in one part or the other; or, perhaps, these effects may arise because some parts of the Milky Way may be confused with the tails as they pass by and may be considered as if they were parts of the tails.
Moreover, the rarity of our own air makes it understandable that vapors sufficient to fill such immense spaces can arise from the atmospheres of comets. For the air near the surface of the earth occupies a space about 850 times greater than water of the same weight, and thus a cylindrical column of air 850 feet high has the same weight as a foot-high column of water of the same width. Further, a column of air rising to the top of our atmosphere is equal in weight to a column of water about 33 feet high; and therefore if the lower part, 850 feet high, of the whole air column is taken away, the remaining upper part will be equal in weight to a column of water 32 feet high. And hence (by a rule confirmed by many experiments, that the compression of air is as the weight of the incumbent atmosphere and that gravity is inversely as the square of the distance of places from the center of the earth), by making a computation using the corollary of prop. 22, book 2, I found that air, at a height above the surface of the earth of one terrestrial semidiameter, is rarer than here on earth in a far greater ratio than that of all space below the orbit of Saturn to a globe described with a diameter of one inch. And thus a globe of our air one inch wide, with the rarity that it would have at the height of one terrestrial semidiameter, would fill all the regions of the planets as far out as the sphere of Saturn and far beyond. Accordingly, since still higher air becomes immensely rare and since the comag or atmosphere of a comet is (as reckoned from the center) about ten times higher than the surface of the nucleus is, and the tail then ascends even higher, the tail will have to be exceedingly rare. And even if, because of the much thicker atmosphere of comets and the great gravitation of bodies toward the sun and the gravitation of the particles of air and vapors toward one another, it can happen that the air in the celestial spaces and in the tails of comets is not so greatly rarefied, it is nevertheless clear from this computation that a very slight quantity of air and vapors is abundantly sufficient to produce all those phenomena of the tails. For the extraordinary rarity of the tails is also evident from the fact that stars shine through them. The terrestrial atmosphere, shining with the light of the sun, by its thickness of only a few miles obscures and utterly extinguishes the light not only of all the stars but also of the moon itself; yet the smallest stars are known to shine, without any loss in their brightness, through the immense thickness of the tails, which are likewise illuminated by the light of the sun. Nor is the brightness of most cometary tails generally greater than that of our air reflecting the light of the sun in a beam, one or two inches wide, let into a dark room.
The space of time in which the vapor ascends from the head to the end of the tail can more or less be found by drawing a straight line from the end of the tail to the sun and noting the place where this straight line cuts the trajectory. For if the vapor has been ascending in a straight line away from the sun, then the vapor that is now in the end of the tail must have begun to ascend from the head at the time when the head was in that place of intersection. But the vapor does not ascend in a straight line away from the sun, but rather ascends obliquely, since the vapor retains the motion of the comet which it had before its ascent and this motion is compounded with the motion of its own ascent. And therefore the solution of the problem will be nearer the true one if the straight line that cuts the orbit is drawn parallel to the length of the tail, or rather (because of the curvilinear motion of the comet) if it diverges from the line of the tail. In this way I found that the vapor that was in the end of the tail on 25 January had begun to ascend from the head before 11 December and thus had spent more than forty-five days in its total ascent. But all of the tail that appeared on 10 December had ascended in the space of those two days that had elapsed after the time of the perihelion of the comet. The vapor, therefore, rose most swiftly at the beginning of its ascent, in the vicinity of the sun, and afterward proceeded to ascend with a motion always retarded by the vapor’s own gravity; and as the vapor ascended, it increased the length of the tail. The tail, however, as long as it was visible, consisted of almost all the vapor which had ascended from the comet’s head since the time of the comet’s perihelion; and that vapor which was the first to ascend, and which composed the end of the tail, did not disappear from view until its distance both from the sun which illuminated it and from our eyes became too great for it to be seen any longer. Hence it happens, also, that in other comets which have short tails, those tails do not rise up with a swift and continual motion from the heads of the comets and soon disappear, but are permanent columns of vapors and exhalations (propagated from the heads by a very slow motion that lasts many days) which, by sharing in the motion that the heads had at the beginning of the exhalations of the vapors, continue to move along through the heavens together with the heads. And hence again it may be concluded that the celestial spaces are lacking in any force of resisting, since in them not only the solid bodies of the planets and comets but also the rarest vapors of the tails move very freely and preserve their extremely swift motions for a very long time.
The ascent of the tails of comets from the atmospheres of the heads and the movement of the tails in directions away from the sun are ascribed by Kepler to the action of rays of light that carry the matter of the tail along with them. And it is not altogether unreasonable to suppose that in very free [or empty] spaces, the extremely thin upper air should yield to the action of the rays, despite the fact that gross substances in the very obstructed regions here on earth cannot be sensibly propelled by the rays of the sun. Someone else believes that there can be particles with the property of levity as well as gravity and that the matter of the tails levitates and through its levitation ascends away from the sun. But since the gravity of terrestrial bodies is as the quantity of matter in the bodies and thus, if the quantity of matter remains constant, cannot be intended and remitted [or increased and decreased], I suspect that this ascent arises rather from the rarefaction of the matter of the tails. Smoke ascends in a chimney by the impulse of the air in which it floats. This air, rarefied by heat, ascends because of its diminished specific gravity and carries along with it the entangled smoke. Why should the tail of a comet not ascend away from the sun in the same manner? For the sun’s rays do not act on the mediums through which they pass except in reflection and refraction. The reflecting particles, warmed by this action, will warm the aethereal upper air in which they are entangled. This will become rarefied on account of the heat communicated to it; and because its specific gravity, with which it was formerly tending toward the sun, is diminished by this rarefaction, it will ascend and will carry with it the reflecting particles of which the tail is composed. This ascent of the vapors is also increased by the fact that they revolve about the sun and endeavor by this action to recede from the sun, while the atmosphere of the sun and the matter of the heavens are either completely at rest or revolve more slowly only by the motion that they have received from the rotation of the sun.
These are the causes of the ascent of tails of comets in the vicinity of the sun, where the orbits are more curved, and the comets are within the denser (and, on that account, heavier) atmosphere of the sun and soon emit extremely long tails. For the tails which arise at that point, by conserving their motion and meanwhile gravitating toward the sun, will move about the sun in ellipses as the heads of the comets do; and by that motion they will always accompany the heads and will very freely adhere to them. For the gravity of the vapors toward the sun will no more cause the tails to fall afterward from the heads toward the sun than the gravity of the heads can cause them to fall from the tails. By their common gravity they will either fall simultaneously and together toward the sun or will be simultaneously retarded in their ascent; and therefore this gravity does not hinder the tails and heads of comets from very easily acquiring (whether from the causes already described or any others whatsoever), and afterward very freely preserving, any position in relation to one another.
The tails that are formed when comets are in their perihelia will therefore go off into distant regions together with their heads, and either will return to us from there together with the heads after a long series of years or rather, having been rarefied there, will disappear by degrees. For afterward, in the descent of the heads toward the sun, new little tails should be propagated from the heads with a slow motion, and thereupon should be immeasurably increased in the perihelia of those comets which descend as far as the atmosphere of the sun. For vapor in those very free spaces becomes continually rarefied and dilated. For this reason it happens that every tail at its upper extremity is broader than near the head of the comet. Moreover, it seems reasonable that by this rarefaction the vapor—continually dilated—is finally diffused and scattered throughout the whole heavens, and then is by degrees attracted toward the planets by its gravity and mixed with their atmospheres. For just as the seas are absolutely necessary for the constitution of this earth, so that vapors may be abundantly enough aroused from them by the heat of the sun, which vapors either—being gathered into clouds—fall in rains and irrigate and nourish the whole earth for the propagation of vegetables, or—being condensed in the cold peaks of mountains (as some philosophize with good reason)—run down into springs and rivers; so for the conservation of the seas and fluids on the planets, comets seem to be required, so that from the condensation of their exhalations and vapors, there can be a continual supply and renewal of whatever liquid is consumed by vegetation and putrefaction and converted into dry earth. For all vegetables grow entirely from fluids and afterward, in great part, change into dry earth by putrefaction, and slime is continually deposited from putrefied liquids. Hence the bulk of dry earth is increased from day to day, and fluids—if they did not have an outside source of increase—would have to decrease continually and finally to fail. Further, I suspect that that spirit which is the smallest but most subtle and most excellent part of our air, and which is required for the life of all things, comes chiefly from comets.
In the descent of comets to the sun, their atmospheres are diminished by running out into tails and (certainly in that part which faces toward the sun) are made narrower; and, in turn, when comets are receding from the sun, and when they are now running out less into tails, they become enlarged, if Hevelius has correctly noted their phenomena. Moreover, these atmospheres appear smallest when the heads, after having been heated by the sun, have gone off into the largest and brightest tails, and the nuclei are surrounded in the lowest parts of their atmospheres by smoke possibly coarser and blacker. For all smoke produced by great heat is generally coarser and blacker. Thus, at equal distances from the sun and the earth, the head of the comet which we have been discussing appeared darker after its perihelion than before. For in December it was generally compared to stars of the third magnitude, but in November to stars of the first magnitude and the second magnitude. And those who saw both describe the earlier appearance as a greater comet. For a certain young man of Cambridge, who saw this comet on 19 November, found its light, however leaden and pale, to be equal to Spica Virginis and to shine more brightly than afterward. And on 20 November (O.S.) the comet appeared to Montanari greater than stars of the first magnitude, its tail being 2 degrees long. And Mr. Storer, in a letter that came into our hands, wrote that in December, at a time when the largest and brightest tail was being emitted, the head of the comet was small and in visible magnitude was far inferior to the comet which had appeared in November before sunrise. And he conjectured that the reason for this was that in the beginning the matter of the head was more copious and had been gradually consumed.
It seems to pertain to the same point that the heads of other comets that emitted very large and very bright tails appeared rather dull and very small. For on 5 March 1668 (N.S.) at 7h P.M., the Reverend Father Valentin Stansel, in Brazil, saw a comet very close to the horizon toward the southwest with a very small head that was scarcely visible, but with a tail so shining beyond measure that those who were standing on the shore easily saw its appearance reflected from the sea. In fact it had the appearance of a brilliantly shining torch with a length of 23 degrees, verging from west to south and almost parallel to the horizon. But so great a splendor lasted only three days, decreasing noticeably immediately afterward; and meanwhile, as its splendor was decreasing, the tail was increasing in size. Thus in Portugal the tail is said to have occupied almost a quarter of the sky—that is, 45 degrees—stretched out from west to east with remarkable splendor, and yet not all of the tail was visible, since in those regions the head was always hidden below the horizon. From the increase of the size of the tail and the decrease of the splendor, it is manifest that the head was receding from the sun and had been nearest to the sun at the beginning of its visibility, as was the case for the comet of 1680. And in the Anglo-Saxon Chronicle, one reads about a similar comet of 1106, “of which the star was small and dim h(as was that of 1680),h but the splendor that came out of it stretched out extremely bright and like a huge torch toward the northeast” as Hevelius also has it from Simeon the Monk of Durham. This comet appeared at the beginning of February, and thereafter was seen at about evening toward the southwest. And from this and from the position of the tail it is concluded that the head was near the sun. “Its distance from the sun,” says Matthew of Paris, “was about one cubit, as from the third hour (more correctly, the sixth) until the ninth hour it emitted a long ray from itself.” Such also was that fiery comet described by Aristotle (Meteor. 1.6), “whose head, on the first day, was not seen because it had set before the sun, or at least was hidden under the sun’s rays; but on the following day, it was seen as much as it could be. For it was distant from the sun by the least possible distance, and soon set. Because of the excessive burning (of the tail, that is), the scattered fire of the head did not yet appear, but as time went on,” says Aristotle, “since (the tail) was now flaming less, the comet’s own face came back to (the head). And it extended its splendor as far as a third of the sky (that is, to 60 degrees). Moreover, it appeared in the winter (in the 4th year of the 101st Olympiad) and, ascending up to Orion’s belt, vanished there.”
The comet of 1618 which emerged out of the sun’s rays with a very large tail seemed to equal stars of the first magnitude, or even to surpass them a little, but a number of greater comets have appeared which had shorter tails. Some of these are said to have equaled Jupiter, others Venus or even the moon.
We said that comets are a kind of planet revolving about the sun in very eccentric orbits. And just as among the primary planets (which have no tails) those which revolve in smaller orbits closer to the sun are generally smaller, so it seems reasonable also that the comets which approach closer to the sun in their perihelia are for the most part smaller, since otherwise they would act on the sun too much by their attraction. I leave the transverse diameters of the orbits and the periodic times of revolution of the comets to be determined by comparing comets that return in the same orbits after long intervals of time. Meanwhile the following proposition may shed some light on this matter.
Proposition 42, Problem 22
To correct a comet’s trajectory that has been found [by the method of prop. 41].
OPERATION 1. Assume the position of the plane of the trajectory, as found by prop. 41, and select three places of the comet which have been determined by very accurate observations and which are as greatly distant from one another as possible; let A be the time between the first and second observations, and B the time between the second and third. The comet should be in its perigee in one of these places, or at least not far from perigee. From these apparent places find, by trigonometric operations, three true places of the comet in that assumed plane of the trajectory. Then through those places thus found, describe a conic about the center of the sun as focus, by arithmetical operations made along the lines of prop. 21, book 1; and let D and E be areas of the conic which are bounded by radii drawn from the sun to those places—namely, D the area between the first and second observations, and E the area between the second and third. And let T be the total time in which the total area D + E should be described by the comet, with the velocity as found by prop. 16, book 1.
OPERATION 2. Let the longitude of the nodes of the plane of the trajectory be increased by adding 20 or 30 minutes (which can be called P) to that longitude; but keep constant the inclination of that plane to the plane of the ecliptic. Then from the three aforesaid observed places of the comet, let three true places of the comet be found in this new plane (as in oper. 1); and also the orbit passing through those places, two of its areas (which can be called d and e) described between observations, and the total time t in which the total area d + e should be described.
OPERATION 3. Keep constant the longitude of the nodes in the first operation, and let the inclination of the plane of the trajectory to the plane of the ecliptic be increased by adding 20 or 30 minutes (which can be called Q) to that inclination. Then from the aforesaid three observed apparent places of the comet, let three true places be found in this new plane; and also the orbit passing through those places, two of its areas (which can be called δ and ϵ) described between observations, and the total time τ in which the total area δ + ϵ should be described.
Now take C so as to be to 1 as A to B, and take G to 1 as D to E, and
g to 1 as d to e, and γ to 1 as δ to ϵ, and let S be the true time between the
first and third observations; and carefully observing the signs + and −, seek
the numbers m and n, by the rule that 2G − 2C = mG − mg + nG − nγ; and
2T − 2S = mT − mt + nT − nτ. And if, in the first operation, I designates
the inclination of the plane of the trajectory to the plane of the ecliptic,
and K the longitude of either node, I + nQ will be the true inclination of
the plane of the trajectory to the plane of the ecliptic, and K + mP will
be the true longitude of the node. And finally if in the first, second, and
third operations, the quantities R, r, and ρ designate the latera recta of the
trajectory, and the quantities 
the transverse diameters [or latera
transversa] respectively, R + mr − mR + nρ − nR will be the true latus
rectum, and 
will be the true transverse diameter
of the trajectory that the comet describes. And given the transverse diameter,
the periodic time of the comet is also given. Q.E.I.
But the periodic times of revolving comets, and the transverse diameters [latera transversa] of their orbits, will by no means be determined exactly enough except by the comparison with one another of comets that appear at diverse times. If several comets are found, after equal intervals of times, to have described the same orbit, it will have to be concluded that all these are one and the same comet revolving in the same orbit. And then finally from the times of their revolutions the transverse diameters of the orbits will be given, and from these diameters the elliptical orbits will be determined.
To this end, therefore, the trajectories of several comets should be
calculated on the hypothesis that they are parabolic. For such trajectories will
always agree very nearly with the phenomena. This is clear not only from
the parabolic trajectory of the comet of 1680, which I compared above with
the observations, but also from the trajectory of that remarkable comet which
appeared in 1664 and 1665 and was observed by Hevelius. He calculated the
longitudes and latitudes of this comet from his own observations, but not
very accurately. From the same observations our own Halley calculated the
places of this comet anew, and then finally he determined the trajectory of the
comet from the places thus calculated. And he found its ascending node in
21°13′55″, the inclination of its orbit to the plane of the ecliptic 21°18′40″,
the distance of its perihelion from the node in the orbit 49°27′30″. The
perihelion in 8°40′30″ with heliocentric latitude 16°1′45″ S. The comet in its
perihelion on 24 November, 11h52m P.M. mean time [lit. equated time] at
London, or 13h8m (O.S.) at Gdansk, and the latus rectum of the parabola 410,286,
the mean distance of the earth from the sun being 100,000. How exactly the
calculated places of the comet in this orbit agree with the observations will
be evident from the following table calculated by Halley [p. 932].
In February, in the beginning of 1665, the first star of Aries, which I
shall from here on call γ, was in 28°30′15″ with latitude 7°8′58″ N. The
second star of Aries was in 29°17′18″ with latitude 8°28′16″ N. And a
certain other star of the seventh magnitude, which I shall call A, was in
28°24′45″ with latitude 8°28′33″ N. And on 7 February at 7h30m Paris
time (that is, 7 February at 8h30m Gdansk time) (O.S.), the comet made a
right triangle with those stars γ and A, with the right angle at γ. And the
distance of the comet from the star γ was equal to the distance between
the stars γ and A, that is, 1°19′46″ along a great circle, and therefore it
was 1°20′26″ in the parallel of the latitude of the star γ. Therefore, if the
longitude 1°20′26″ is taken away from the longitude of the star γ, there will
remain the longitude of the comet 27°9′49″. Auzout, who had made this
observation, put the comet in roughly 27°0′. And from the diagram with
which Hooke delineated its motion, it was then in 26°59′24″. Taking the
mean, I have put it in 27°4′46″. From the same observation, Auzout took
the latitude of the comet at that time to be 7° and 4′ or 5′ toward the north.
He would have put it more correctly at 7°3′29″, since the difference of the
latitudes of the comet and of the star γ was equal to the difference of the
longitudes of the stars γ and A.
On 22 February at 7h30m in London (that is, 22 February at 8h46m
Gdansk time), the distance of the comet from the star A, according to Hooke’s
observation (which he himself delineated in a diagram) and also according to
Auzout’s observations (delineated in a diagram by Petit), was a fifth of the
distance between the star A and the first star of Aries, or 15′57″. And the
distance of the comet from the line joining the star A and the first star of
Aries was a fourth of that same fifth part, that is, 4′. And hence the comet
was in 28°29′46″ with latitude 8°12′36″ N.
On 1 March at 7h0m at London (that is, 1 March at 8h16m Gdansk time),
the comet was observed near the second star of Aries, the distance between
them being to the distance between the first and second stars of Aries, that
is, to 1°33′, as 4 to 45 according to Hooke, or as 2 to 23 according to [Gilles
François] Gottigniez. Accordingly, the distance of the comet from the second
star of Aries was 8°16″ according to Hooke, or 8′5″ according to Gottigniez;
or, taking the mean, was 8′10″. And according to Gottigniez the comet had
now just gone beyond the second star of Aries by about a space of a fourth
or a fifth of the course completed in one day, that is, roughly 1′35″ (and
Auzout agrees well enough with this), or a little less according to Hooke,
say 1′. Therefore, if 1′ is added to the longitude of the first star of Aries, and
8′10″ to its latitude, the longitude of the comet will be found to be 29°18′,
and its latitude 8°36′26″ N.
On 7 March at 7h30m in Paris (that is, 7 March at 8h37m Gdansk time),
the distance of the comet from the second star of Aries, according to Auzout’s
observations, was equal to the distance of the second star of Aries from the
star A, that is, 52′29″. And the difference between the longitudes of the comet
and of the second star of Aries was 45′ or 46′ or, taking the mean, 45′30″.
And therefore the comet was in 0°2′48″. From the diagram of Auzout’s
observations that Petit constructed, Hevelius determined the latitude of the
comet to be 8°54′. But the engraver curved the path of the comet irregularly
toward the end of its motion, and Hevelius corrected the irregular curving
in a diagram of Auzout’s observations drawn by Hevelius himself, and thus
made the latitude of the comet 8°55′30″. And by correcting the irregularity
a little more, the latitude can come out to be 8°56′, or 8°57′.
This comet was also seen on 9 March and then must have been located
in 0°18’ with latitude roughly 9°3½′ N.
This comet was visible for three months in all, during which time it passed through about six signs, completing about 20 degrees in each day. Its path deviated considerably from a great circle, being curved northward; and toward the end, its motion changed from retrograde to direct. And notwithstanding so unusual a path, the theory agrees with the observations from beginning to end no less exactly than theories of the planets tend to agree with observations of them, as will be clear upon examination of the table. Nevertheless, roughly 2 minutes must be subtracted when the comet was swiftest, and this will result by taking away 12 seconds from the angle between the ascending node and the perihelion, or by making that angle 49°27′18″. The annual parallax of each of the two comets (both this one and the previous one) was quite pronounced, and as a result it gave proof of the annual motion of the earth in its orbit.
The theory is confirmed also by the motion of the comet that appeared
in 1683. It had a retrograde motion in an orbit whose plane contained almost
a right angle with the plane of the ecliptic. Its ascending node (by Halley’s
calculation) was in 23°23′; the inclination of its orbit to the ecliptic 83°11′;
its perihelion in 25°29′30″; its perihelial distance from the sun 56,020, the
radius of the earth’s orbit being taken at 100,000, and the time of its perihelion
2 July 3h50m. And the places of the comet in this orbit, as calculated by Halley
and compared with the places observed by Flamsteed, are displayed in the
following table.
The theory is confirmed also by the motion of the retrograde comet
that appeared in 1682. Its ascending node (by Halley’s calculation) was in
21°16′30″. The inclination of the orbit to the plane of the ecliptic 17°56′0″.
Its perihelion in 2°52′50″. Its perihelial distance from the sun 58,328, the
radius of the earth’s orbit being 100,000. And the perihelion 4 September
7h39m mean [lit. equated] time. And the places calculated from Flamsteed’s
observations and compared with the places calculated by the theory are shown
in the following table.
By these examples it is more than sufficiently evident that the motions of comets are no less exactly represented by the theory that we have set forth than the motions of planets are generally represented by planetary theories. And therefore the orbits of comets can be calculated by this theory, and the periodic time of a comet revolving in any orbit whatever can then be determined, and finally the transverse diameters [lit. latera transversa] of their elliptical orbits and their aphelian distances will become known.
The retrograde comet that appeared in 1607 described an orbit whose
ascending node (according to Halley’s calculation) was in 20°21′; the
inclination of the plane of its orbit to the plane of the ecliptic was 17°2′; its
perihelion was in 2°16′; and its perihelial distance from the sun was 58,680,
the radius of the earth’s orbit being 100,000. And the comet was in its
perihelion on 16 October at 3h50m. This orbit agrees very closely with the orbit of
the comet that was seen in 1682. If these two comets should be one and the
same, this comet will revolve in a space of seventy-five years and the major
axis of its orbit will be to the major axis of the earth’s orbit as ∛(75 × 75)
to 1, or roughly 1,778 to 100. And the aphelial distance of this comet from
the sun will be to the mean distance of the earth from the sun as roughly
35 to 1. And once these quantities are known, it will not be at all difficult
to determine the elliptical orbit of this comet. What has just been said will
be found to be true if the comet returns hereafter in this orbit in a space of
seventy-five years. The other comets seem to revolve in a greater time and
to ascend higher.
But because of the great number of comets, and the great distance of their aphelia from the sun, and the long time that they spend in their aphelia, they should be disturbed somewhat by their gravities toward one another, and hence their eccentricities and times of revolutions ought sometimes to be increased a little and sometimes decreased a little. Accordingly, it is not to be expected that the same comet will return exactly in the same orbit, and with the same periodic times. It is sufficient if no greater changes are found to occur than those that arise from the above-mentioned causes.
And hence a reason appears why comets are not restricted to the zodiac as planets are, but depart from there and are carried with various motions into all regions of the heavens—namely, for this purpose, that in their aphelia, when they move most slowly, they may be as far distant from one another as possible and may attract one another as little as possible. And this is the reason why comets that descend the lowest, and so move most slowly in their aphelia, should also ascend to the greatest heights.
The comet that appeared in 1680 was distant from the sun in its perihelion by less than a sixth of the sun’s diameter; and because its velocity was greatest in that region and also because the atmosphere of the sun has some density, the comet must have encountered some resistance and must have been somewhat slowed down and must have approached closer to the sun; and by approaching closer to the sun in every revolution, it will at length fall into the body of the sun. But also, in its aphelion, when it moves most slowly, the comet can sometimes be slowed down by the attraction of other comets and as a result fall into the sun. So also fixed stars, which are exhausted bit by bit in the exhalation of light and vapors, can be renewed by comets falling into them and then, kindled by their new nourishment, can be taken for new stars. Of this sort are those fixed stars that appear all of a sudden, and that at first shine with maximum brilliance and subsequently disappear little by little. Of such sort was the star that Cornelius Gemma saw in Cassiopeia’s Chair on 9 November 1572; it was shining brighter than all the fixed stars, scarcely inferior to Venus in its brilliance. But he did not see it at all on 8 November, when he was surveying that part of the sky on a clear night. Tycho Brahe saw this same star on the 11th of that month, when it shone with the greatest splendor; and he observed it decreasing little by little after that time, and he saw it disappearing after the space of sixteen months. In November, when it first appeared, it equaled Venus in brightness. In December, somewhat diminished, it equaled Jupiter. In January 1573 it was less than Jupiter and greater than Sirius, and it became equal to Sirius at the end of February and the beginning of March. In April and May it was equal to stars of the second magnitude; in June, July, and August, to stars of the third magnitude; in September, October, and November, to stars of the fourth magnitude; in December and in January 1574, to stars of the fifth magnitude; and in February, to stars of the sixth magnitude; and in March, it vanished from sight. Its color at the start was clear, whitish, and bright; afterward it became yellowish, and in March of 1573 reddish like Mars or the star Aldebaran, while in May it took on a livid whiteness such as we see in Saturn, and it maintained this color up to the end, yet all the while becoming fainter. Such also was the star in the right foot of Serpentarius, the beginning of whose visibility was observed by the pupils of Kepler in 1604, on 30 September (O.S.); they saw it exceeding Jupiter in its light, although it had not been visible at all on the preceding night. And from that time it decreased little by little and in the space of fifteen or sixteen months vanished from sight. It was when such a new star appeared shining beyond measure that Hipparchus is said to have been stimulated to observe the fixed stars and to put them into a catalog. But fixed stars that alternately appear and disappear, and increase little by little, and are hardly ever brighter than fixed stars of the third magnitude, seem to be of another kind and, in revolving, seem to show alternately a bright side and a dark side. And the vapors that arise from the sun and the fixed stars and the tails of comets can fall by their gravity into the atmospheres of the planets and there be condensed and converted into water and humid spirits, and then—by a slow heat—be transformed gradually into salts, sulphurs, tinctures, slime, mud, clay, sand, stones, corals, and other earthy substances.
, and 
. And when the distances are increased
or decreased, the weights are decreased or increased as the squares of the
distances. The weights of equal bodies toward the sun, Jupiter, Saturn, and the
earth at distances of 10,000, 997, 791, and 109 respectively from their centers
(and hence their weights on the surfaces) will be as 10,000, 943, 529, and 435.
What the weights of bodies are on the surface of the moon will be shown
below.a
, and
respectively. If the parallax
of the sun is taken as greater or less than 10″30‴, the quantity of matter in
the earth will have to be increased or decreased in the cubed ratio.b
lines already found from the theory. Moreover, Richer
repeated his observations in Cayenne every week during a ten-month period,
and compared the lengths he found there for a pendulum consisting of an
iron rod with its lengths similarly found in France [i.e., with its lengths
adjusted in Paris so as to have the same period]. This diligence and caution
seem to have been lacking in other observers. If his observations are to be
trusted, the earth will be higher at the equator than at the poles by an excess
of about seventeen miles, as came out above by the
theory.b e
than in its aphelion, and does so inversely
as the cube of the distance of the earth
from the sun. Because the equation of the
center of the sun is comprehended in the
annual argument, the center of the moon’s
orbit moves more swiftly in the epicycle
BDA inversely as the square of the distance of the earth from the sun.
In order for the center of the moon’s orbit to move still more swiftly,
inversely in the simple ratio of the distance, draw a straight line DE from
the center D of the orbit toward the apogee of the moon, or parallel to
the straight line TC, and take the angle EDF equal to the excess of the
above-mentioned annual argument over the distance of the apogee of the
moon from the perigee of the sun in a forward direction [or in consequentia];
or, which is the same, take the angle CDF equal to the complement of the
true anomaly of the sun to 360°. And let DF be to DC jointly as twice
the eccentricity of the earth’s orbit is to the mean distance of the sun from
the earth and as the daily mean motion of the sun from the apogee of the
moon is to the daily mean motion of the sun from its own apogee, that is,
as 33⅞ to 1,000 and 52′27″16‴ to 59′8″10‴ jointly, or as 3 to 100. And
suppose that the center of the moon’s orbit is located in point F and revolves
in an epicycle whose center is D and whose radius is DF, while the point
D advances in the circumference of the circle DABD. For in this manner
the velocity with which the center of the moon’s orbit will move in a certain
curved line described about the center C will be very nearly inversely as the
cube of the distance of the sun from the earth, as it ought to be.