tion, then, is the weight of the sea about the equator lightened, CHAP. I. and thereby rendered susceptible of being supported at a higher level than at the poles, where no such counteracting force exists.

Our next inquiry is: what part of the whole is the part de- Ratio of the

stroyed? Or what part of (9+) is

Which, by common arithmetic, is,

?

D

diminution

computed.

c2 (3.1416)3 D ̄(3.1416)2 (7925)(5280)*

By the application of logarithms, we soon find the value of

this expression to be 288.4. Therefore, gD

1

1

+1 289.4°

We may now inquire, how rapidly the earth must revolve on its axis, so that the whole of gravity would be destroyed on the equator. That is, so that Fshall equal g. Equation c2

(1) then becomes, 9=D'

or c=√gD.

But as often as c is contained in the whole circumference, is the corresponding number of seconds in a revolution; that is, the time in seconds must correspond to the expression,

CHAP. I.

(58.) It is this centrifugal force itself that changed the shape of the earth, and made the equatorial diameter greater than the polar. Here, then, we have the same cause, exercising at once a direct and an indirect influence. The amount Rotation of the former (as we may see by the note) is easily calcuhas a direct lated; that of the latter is far more difficult, and requires a effect on gra- knowledge of the integral calculus, "But it has been clearly

and indirect

vity.

English and geogra phical miles.

treated by Newton, Maclaurin, Clairaut, and many other eminent geometers; and the result of their investigations is to show, that owing to the elliptic form of the earth alone, and independently of the centrifugal force, its attraction ought to increase the weight of a body, in going from the equator to the pole, by nearly its th part; which, together with the 2th part, due from centrifugal force, make the whole quantity th part; which corresponds with observations as deduced from the vibrations of pendulums."- See Natural

Philosophy.

C

E

Fig. 12.

F

590

(59.) The form of the earth is so nearly a sphere, that it is considered such, in geography, navigation, and in the general problems of astronomy.

The average length of a degree is 691 English miles; and, as this number is fractional, and inconvenient, navigators have taPcitly agreed to retain the ancient,

rough estimate of sixty miles to a degree; calling the mile a geographical mile. Therefore, the geographical mile is longer than the English mile.

=

D, in feet, (7925)(5280); g = 16.076. By the application of logarithms, we find this expression to be 5069 seconds, or 1 h. 24 m. 29 s.; which is about 17 times the rapidity of its present rotation.

In a subsequent portion of this work, we shall show how to arrive at this result by another principle, and through another operation.

As all meridians come together at the pole, it follows that CHAP. I. a degree, between the meridians, will become less and less as we approach the pole; and it is an interesting problem to trace the law of decrease.*

* This law of decrease will become apparent, by inspecting Fig. 12. Let EQ represent a degree, on the equator, and EQC a sector on the plane of the equator, and of course EC is at right angles to the axis CP. Let DFI be any plane parallel to EQC; then we shall have the following proportion: EC: DI :: EQ: DF.

In trigonometry, EC is known as the radius of the sphere; DI as the cosine of the latitude of the point D (the numerical values of sines and cosines, of all arcs, are given in trigonometrical tables): therefore we have the following rule, to compute the length of a degree between two meridians, on any parallel of latitude.

RULE.

·As radius is to the cosine of the latitude; so is the length of a degree on the equator, to the length of a parallel degree in that latitude.

Calling a degree, on the equator, 60 miles, what is the Example length of a degree of longitude, in latitude 42°?

At the latitude of 60°, the degree of longitude is 30 miles; the diminution is very slow near the equator, and very rapid

near the poles.

In navigation, the DF's are the known quantities ob- To reduce tained by the estimations from the log line, etc.; and the departure to longitude. navigator wishes to convert them into longitude, or, what

is the same thing, he wishes to find their values projected on

the equator, and he states the proportion thus:

DI : EC :: DF: EQ;

That is, as cosine of latitude is to radius, so is departure to difference of longitude.

CHAP. II.

Parallax in general.

CHAPTER II.

PARALLAX, GENERAL AND HORIZONTAL. RELATION BETWEEN
PARALLAX AND DISTANCE. REAL DIAMETER AND MAGNI-
TUDE OF THE MOON.

(60.) PARALLAX is a subject of very great importance in astronomy: it is the key to the measure of the planets-to their distances from the earth-and to the magnitude of the whole solar system.

Parallax is the difference in position, of any body, as seen from the center of the earth, and from its surface.

When a body is in the zenith of any observer, to him it has no parallax; for he sees it in the same place in the heavens, as though he viewed it from the center of the earth. The greatest possible parallax that a body can have, takes place when the body is in the horizon of the observer; and this parallax is called horizontal parallax. Hereafter, when we speak of the parallax of a body, horizontal parallax is to be understood, unless otherwise expressed.

A clear and summary illustration of parallax in general, is given by Fig. 13.

line CP, or Cp; from the observer at Z, it is seen in the

direction of ZP, or Zp; and the difference in direction, of CHAP. II. these two lines, is parallax. When P is in the zenith, there

is no parallax; when P is in the horizon, the angle ZP C is then greatest, and is the horizontal parallax.

rallax and

We now perceive that the horizontal parallax of any body Relation is equal to the apparent semidiameter of the earth, as seen from between pathe body. The greater the distance to the body, the less the distance. horizontal parallax; and when the distance is so great that the semidiameter of the earth would appear only as a point, then the body has no parallax. Conversely, if we can detect no sensible parallax, we know that the body must be at a vast distance from the earth, and the earth itself appear as a point from such a body, if, in fact, it were even visible.

Trigonometry gives the relation between the angles and sides of every conceivable triangle; therefore we know all about the horizontal triangle Z CP, when we know CZ and the angles. Calling the horizontal parallax of any body p, and the radius of the earth r, and the distance of the body from the center of the earth x (the radius of the table always R, or unity), then, by trigonometry, we have,

From this equation we have the following general rule, to find the distance to any celestial body:

Rule to

RULE. - Divide the radius of the tables by the sine of the horizontal parallax. Multiply that quotient by the semidiameter find the disof the earth, and the product will be the result.

This result will, of course, be in the same terms of linear measure as the semidiameter of the earth: that is, if r is in feet, the result will be in feet; ifr is in miles, the result will be in miles, etc.: but, for astronomy, our terrestrial measures are too diminutive, to be convenient (not to say inappropriate); and, for this reason, it is customary to call the semidiameter of the earth unity, and then the distance of any body from the earth is simply the quotient arising from dividing the radius by the sine of the horizontal parallax pertaining to

tances to the heavenly

bodies.