he year 1531, when one was seen by Appian; again, he reckoned
1456, 1380, 1305. All these appearances were the same comet, in all
probability, returning every seventy-five or seventy-six years. The
period was easily allowed to be not exact, because of perturbing
planets. He then predicted its return for 1758, or perhaps 1759, a date
he could not himself hope to see. He lived to a great age, but he died
sixteen years before this date.

As the time drew nigh, three-quarters of a century afterwards,
astronomers were greatly interested in this first cometary prediction,
and kept an eager look-out for "Halley's comet." Clairaut, a most
eminent mathematician and student of Newton, proceeded to calculate out
more exactly the perturbing influence of Jupiter, near which it had
passed. After immense labour (for the difficulty of the calculation was
extreme, and the mass of mere figures something portentous), he
predicted its return on the 13th of April, 1759, but he considered that
he might have made a possible error of a month. It returned on the 13th
of March, 1759, and established beyond all doubt the rule of the
Newtonian theory over comets.

[Illustration: FIG. 71.--Well-known model exhibiting the oblate
spheroidal form as a consequence of spinning about a central axis. The
brass strip _a_ looks like a transparent globe when whirled, and bulges
out equatorially.]

No. 10. Applying the idea of centrifugal force to the earth considered
as a rotating body, he perceived that it could not be a true sphere, and
calculated its oblateness, obtaining 28 miles greater equatorial than
polar diameter.

Here we return to one of the more simple deductions. A spinning body of
any kind tends to swell at its circumference (or equator), and shrink
along its axis (or poles). If the body is of yielding material, its
shape must alter under the influence of centrifugal force; and if a
globe of yielding substance subject to known forces rotates at a
definite pace, its shape can be calculated. Thus a plastic sphere the
size of the earth, held together by its own gravity, and rotating once a
day, can be shown to have its equatorial diameter twenty-eight miles
greater than its polar diameter: the two diameters being 8,000 and 8,028
respectively. Now we have no guarantee that the earth is of yielding
material: for all Newton could tell it might be extremely rigid. As a
matter of fact it is now very nearly rigid. But he argued thus. The
water on i