154 Earth

We have:

This represents the value of an investment of $1 after 1

year accruing continuously compounded interest at an

interest rate of rpercent per annum.

One can use the fourth definition to prove that eis

an

IRRATIONAL NUMBER

. One begins by assuming, to

the contrary, that eis a fraction of the form and

multiplies the formula presented in definition 4 by q!.

The fractional parts that remain cannot add to a

whole number.

In 1873 French mathematician Charles Hermite

(1822–1901) proved that eis a

TRANSCENDENTAL

NUMBER

.

Earth Our planet, the third from the Sun, is often

assumed to have the shape of a perfect

SPHERE

, but

detailed measurements show it to be the shape of an

oblate spheroid with equatorial radius 3,963 miles

(6,378 km) and polar radius 3,950 miles (6,357 km).

Its mean orbital distance from the Sun is 9.296 ×107

miles (1.496 ×108km), one “astronomical unit,” and

its mass is 1.317 ×1025 lb (5.976 ×1024 kg).

The Earth takes 23 hours, 56 minutes, and 4.1 sec-

onds to complete a sidereal day, that is, one full rota-

tion about its axis as measured relative to the fixed

stars. The solar day, the time it takes for a point Pon

the surface of the Earth initially facing the Sun to

return to that position is, by definition, precisely 24

hours. The difference in time measurements is

explained by the fact that the Earth advances in its

orbit as it rotates: the Earth must turn slightly more

than 360°to bring Pback to face the Sun as the planet

moves forward.

The first known attempt to calculate the circum-

ference of the Earth was made by the Greek scholar

E

RATOSTHENES OF

C

YRENE

(ca. 275–195

B

.

C

.

E

.).

Eratosthenes observed that at the summer solstice,

noon on June 21 of every year, the Sun shone directly

to the bottom of a well in the city of Syene (present

day Aswan). This meant that the Sun was directly

overhead at this time. He also noted that at the same

time in Alexandria, a city approximately 500 miles

due north of Syene, objects cast shadows, meaning

that the Sun was not directly overhead at this loca-

tion. These observations provided Eratosthenes the

means to compute the radius of the Earth. He

assumed that the Sun was sufficiently far away from

our planet that the rays of light that reach us from it

can be regarded as essentially parallel. Eratosthenes

measured the angle cast by shadows at Alexandria

(by using an object of known height and measuring

the length of the shadows cast) and found the angle

to be 1/50 of a full turn, about 7.2°. The circumfer-

ence of the Earth follows: if 1/50th of a full turn cor-

responds to a distance of 500 miles, then a full turn

must correspond to a distance of 50 ×500 = 25,000

miles.

Geographical Coordinates

The lines of latitude and longitude form a system of

COORDINATES

on the Earth. The “parallels of lati-

tude” are circles parallel to the equator, labeled

according to a measurement of angle. Stated pre-

cisely, if Ois the center of the Earth, Pis a point on a

circle of latitude, P′is the point on the equator

directly south of P, and αis the angle POP′, then the

circle of latitude containing Pis labeled α. The lati-

tude of any point can thus be 0°(on the equator) up

to 90°north (the North Pole) or 90°south (the South

Pole). New York City, for example, has a latitude of

40°45′06′′ N.

Conveniently, the North Star lies directly above the

North Pole. By measuring the angle of elevation of the

North Star (90°at the North Pole, 0°at the equator),

one can quickly determine the latitude of any location

in the Northern Hemisphere. Using sextants to measure

elevation, sailors of the past relied on the North Star to

help determine their locations. The meridians, or lines

of longitude, run from the North Pole to the South

Pole, perpendicular to the circles of latitude. Each

meridian is a semicircle. The equator is divided into

360°, with the meridian passing through Greenwich,

England (called the prime meridian), deemed angle

zero, and the angle of any other meridian ranges from

180°east to 180°west. Stated precisely, the longitude

of a point Pon the sphere is given by the angle P′OG′,

where P′is the point on the equator directly south or

p

–

q!