Angular Momentum

Proposition 16: Every body that moves in a circle (e.g., a stone in a sling) is continuously determined to continue in motion at a tangent to that circle.

Proof: A body that moves in a circle is continuously prevented by an external force from continuing to move in a straight line (Cor. previous Prop.).

If this force ceases, the body will of itself proceed to move in a straight line (Prop. 1 5).

A body that moves in a circle is determined by an external cause to move at a tangent to the circle.

If you deny this, suppose that a stone at B is determined (e.g., by a sling) to move not along the tangent BD but along another line conceived as drawn without or within the circle from the same point.

When the sling is supposed to be coming from L toward B, let this line be BF.

If on the other hand the sling is supposed to be coming from C toward B, let this line be BG.

If BH is the line drawn from the center through the circumference, which it cuts at B, I understand the angle GBH to be equal to the angle FBH.

But if the stone at B is determined to move toward F by the sling moving in a circle from L toward B, then it necessarily follows (Ax. 1 8) that when the sling 0 moves with a contrary determination from C toward B, the stone will be determined to proceed to move in line with BF with a contrary determination and will therefore tend not toward G but toward K.

This is contrary to our hypothesis.

No line except a tangent can be drawn through point B making equal adjacent angles, DBH, ABH, with the line BH,4 there can be no line but a tangent that can preserve the same hypothesis, whether the sling moves from L to B or from C to B. And so the stone can tend to move along no line but the tangent. Q.E.D.

Proof 2: Instead of a circle, conceive a hexagon ABH inscribed in a circle, and a body C at rest on one side, AB.

Then conceive thata ruler DBE, whose one end I suppose to be fixed at the center D while the other end is free, moves about the center D, continuously cutting the line AB.

If the ruler DBE, conceived to move in this way, meets the body C just when the ruler cuts the line AB at right angles, by its impact the ruler will determine the body C to proceed to move along the line FBAG toward G, that is, along the side AB produced indefinitely.

But because we have chosen a hexagon at random, the same must be affirmed of any other figure that we conceive can be inscribed in a circle, namely, that when a body C, at rest on one side of the figure, is struck by the ruler DBE just when the ruler cuts that side at right angles, it will be determined by that ruler to proceed to move along that side produced indefinitely.

Archimedes defines a circle as a rectilinear shape having an infinite number of sides.

Let us conceive, then, a circle instead of a hexagon.

Whenever the ruler DBE meets the body C, it always meets it just when it cuts some side of such a figure at right angles, and thus will never meet the body C without at the same time determining it to proceed to move along that side produced indefinitely.

And because any side produced in either direction must always fall outside the figure, that side produced indefinitely will be the tangent to a figure of an infinite number of sides, that is, a circle. If, then, instead of a ruler

⁴ It is evident from Propositions 18 and 19 of Book 3 of the Elements.

⁵ 5 [Spinoza’s diagram is mislabeled. A must be at the corner of the hexagon between B and G. In this alternative proof Spinoza substitutes the Archlmedean concept of a cucle for the Euclidean and proVides a duect proof rather than a reductio argument.]

We conceive a sling moving in a circle, this will continuously determine the stone to proceed to move at a tangent. Q.E.D.

Both of these proofS can be adapted to any curvilinear figure.

Proposition 17: Every body that moves in a circle endeavors to move away from the center of the circle that it describes.

Proof: As long as a body moves in a circle, it is being compelled by some external cause.

If this ceases, it at once proceeds to move at a tangent to the circle (previous Prop.).

All the points of this tangent, except that which touches the circle, fall outside the circle (Prop. 16 Book 3 Elements) and are therefore further distant from the center.

Therefore when a stone moving in a circle in a sling EA is at point A, it endeavors to continue in a line, all of whose points are farther distant from the center E than any points on the circumference LAB.

This is nothing other than to endeavor to move away from the center of the circle that it describes. Q.E.D.

Proposition 18: If a body A moves toward a body B, which is at rest, and B loses nothing of its state of rest in spite of the impetus of body A, then neither will A lose anything of its motion, but will retain entirely the same quantity of motion that it had before.

Proof: Assume that body A loses some of its motion without transferring the lost motion to B. This will reduce the amount of motion in Nature than before, which is absurd (Prop. 13 Part 2).

The proof proceeds in the same way with respect to the state of rest of body B. Therefore, if Body A does not transfer anything to Body B, then B will retain all its rest, and A, all its motion. Q.E.D.