Gravitation and attraction direct all the planets in their courses.

Almost all of Descartes’ theory of gravity is based on the law of nature that any body moving in a curved line tends to move away from its center in a straight line that would touch the curve at one point. Such is the case of a sling that escapes from the hand, etc. All bodies, by turning with the earth, thus make an effort to move away from the center; but the subtle matter, making a much greater effort, was said to repel all other bodies.

It is easy to see that it was not for the subtle matter to make this greater effort, and to move away from the center of the so-called vortex rather than the other bodies; on the contrary, it was its nature (assuming it existed) to go to the center of its movement, and to let all the bodies that had more mass go to the circumference. This is in fact what happens on a table that turns in a circle, when, in a tube made in this table, several powders and several liquids of different specific gravities have been mixed: everything that has more mass moves away from the center; everything that has less mass approaches it. Such is the law of nature, and when Descartes made his so-called subtle matter circulate at the circumference, he began by violating this law of centrifugal forces, which he set as his first principle. It was in vain that he imagined that God had created dice turning on one another; that the scraping of these dice, which made up his subtle matter, escaping from all sides, thereby acquired more speed; that the center of a vortex became crusted, etc.; it was far from the case that these imaginations rectified this error.

Without losing more time fighting these beings of reason, let us follow the laws of mechanics that operate in nature. A body that moves circularly takes in this way, at each point of the curve it describes, a direction that would move it away from the circle, by making it follow a straight line.

This is true. But we must be careful that this body would only move away from the center in this way by this other great principle: that every body being of itself indifferent to rest and to movement, and having this inertia which is an attribute of matter, necessarily follows the line in which it is moved. Now, any body that turns around a center follows at every instant an infinitely small straight line, which would become an infinitely long straight line if it did not encounter any obstacles. The result of this principle, reduced to its just value, is therefore nothing other than that a body that follows a straight line will always follow a straight line: therefore another force is needed to make it describe a curve; therefore this other force, by which it describes the curve, would make it fall to the center at every instant, in case this straight-line projectile movement ceased. In truth, from moment to moment this body would go to A, to B, to C, if it escaped (figure 49).

But also from moment to moment it would fall back from A, from B, from C, to the center; because its movement is composed of two kinds of movements: the straight-line projectile movement, and the straight-line movement also imprinted by the centripetal force, a force by which it would go to the center. Thus, from the very fact that the body would describe these tangents A B C, it is demonstrated that there is a power that pulls it away from these tangents at the very instant it begins them. It is therefore absolutely necessary to consider any body moving in a curve as being moved by two powers, one of which is the one that would make it travel tangents, and which is called centrifugal force, or rather the force of inertia, of inactivity, by which a body always follows a straight line if it is not prevented; and the other force that pulls the body towards the center, which is called the centripetal force, and which is the true force[1].

From the establishment of this centripetal force, it follows first of all this demonstration that any mobile object that moves in a circle, or in an ellipse, or in any curve, moves around a center towards which it tends.

It also follows that this mobile object, whatever portions of the curve it travels, will describe, in its largest arcs and in its smallest arcs, equal areas in equal times. If, for example, a mobile object in one minute borders the space A C B (figure 51), which will contain one hundred miles of area, it must border in two minutes another space B C D of two hundred miles. This law, inviolably observed by all the planets, and unknown to all antiquity, was discovered, nearly one hundred and fifty years ago, by Kepler, who deserved the name of legislator in astronomy, despite his philosophical errors. He could not yet know the reason for this rule to which the celestial bodies are subject. Kepler’s extreme sagacity found the effect of which Newton’s genius found the cause.

Let the body A (figure 54) be moved in B in a very small space of time: at the end of a similar space, an equally continued movement (for there is no acceleration here) would make it come to G; but in B, it finds a force that pushes it in the line B H S: it therefore follows neither this path B H S, nor this path A B C: draw this parallelogram C D B H, then the mobile object being moved by the force B C, and by the force B H, it goes according to the diagonal B D; now this line B D and this line B A, conceived infinitely small, are the births of a curve, etc.; therefore this body must move in a curve.

It must border equal spaces in equal times, for the space of the triangle S B A is equal to the space of the triangle S B D; these triangles are equal: therefore these areas are equal; therefore any body that travels equal areas in equal times in a curve makes its revolution around the center of the forces to which it tends; therefore the planets tend towards the sun, and not around the earth: for by taking the earth as the center, their areas are unequal with respect to the times; and by taking the sun as the center, these areas are always found to be proportional to the times, if you except the small disturbances caused by the very gravitation of the planets.

To also well understand what these areas proportional to the times are, and to see at a glance the advantage you get from this knowledge, look at the earth carried along in its ellipse around the sun S, its center (figure 55). When it goes from B to D, it sweeps as large a space as when it travels this large arc H K: the sector H K makes up in width what the sector B S D has in length. To make the area of these sectors equal in equal times, the body towards H K must go faster than towards B D. Thus the earth and any planet moves faster in its perihelion, which is the curve closest to the sun S, than in its aphelion, which is the curve furthest from this same focus S.

We therefore know what the center of a planet is, and what figure it describes in its orbit, by the areas it travels"; we know that any planet, when it is further from the center of its movement, gravitates less towards this center. Thus the earth being closer to the sun by one thirtieth and more, that is to say by twelve hundred thousand leagues, during our winter than during our summer, is also more attracted in winter; thus it goes faster then because of its curve; thus we have eight and a half more days of summer than of winter, and the sun appears in the northern signs eight and a half more days than in the southern ones. Since then every planet follows, with respect to the sun focus of its orbit, this law of gravitation that the moon experiences with respect to the earth, and to which all bodies are subject when falling on the earth, it is demonstrated that this gravitation, this attraction, acts on all the bodies that we know.

But another powerful demonstration of this truth is the law that all the planets follow respectively in their courses and in their distances; this is what we must examine closely.

↑ The 1738 editions also contained the following passage here: "This is how a body moved according to the horizontal line G E (figure 50), and according to the perpendicular line G F, obeys at every instant these two powers by traveling the diagonal G H."
This paragraph was deleted by Voltaire as of 1741. (B.)
↑ In the 1738 and 1741 editions, one also read here: "You will find the more extensive demonstration in the notes."
And one did in fact read in the notes the following two demonstrations:
Demonstration. That any mobile object attracted by a centripetal force describes in a curved line equal areas in equal times (figure 52).
"Any body moves with a uniform motion when there is no accelerating force: therefore the body A, moved in a straight line in the first time from A to B, will go in a similar time from B to C, from C to Z. These spaces being conceived as equal, the centripetal force, in the second time, gives this body in B some movement, and the body, instead of going to C, goes to H: what different direction did it have from B C? Draw the four lines C H, G B, C B, G H, the mobile object followed the diagonal B H of this parallelogram.
"Now, the two sides B C, B H of the parallelogram are in the same plane as the triangle A B S: therefore the forces are directed towards G S and towards the straight line A B C Z.
"The triangles S H B, S C B, are equal, since they are on the same base S B, and between the parallels H C, G B; but S B, A S, C B, are equal, having the same base and the same height: therefore S B, A S, H B, are also equal. "The same must be said of the triangles S T H, S D H: therefore all these triangles are equal. Diminish the height to infinity, the body, at each infinitely small moment, will describe the curve, of which all the lines tend to the point S: therefore in all cases the areas of these triangles are proportional to the times."
Demonstration. That any body, in a curve describing equal triangles around a point, is moved by the centripetal force around this point (figure 53).
"Let this curve be divided into equal parts A B, B H, H F, infinitely small, described in equal times; let the force be conceived to act at points B H F; let A B be extended to C, let B H be extended to T, the triangle S A B will be equal to the triangle S B H; for A B is equal to B C: therefore S B H is equal to S B C: therefore the force at B G is parallel to C H; but this line B G, parallel to C H, is the line B G S, tending to the center. The body at H is directed by the centripetal force along a line parallel to F T, just as at point B, it was directed by this same force in a line parallel to C H; now the line parallel to C H tends to S: therefore the line parallel to F T will also tend to S; therefore all the lines thus drawn will tend to the point S.
"Now conceive in S triangles similar to the ones above; the smaller these triangles above are, the more the triangles in S will approach a physical point, which point S will be the center of the forces."
These notes or demonstrations were not kept in the 1748 or 1756 editions. (B.)