Mersenne, Marin 337

Mercator’s expansion (Mercator’s series) The T

AY

-

LOR SERIES

expansion of the natural

LOGARITHMIC

FUNCTION

is given as follows:

It is valid for –1 < x≤1. This series is sometimes called

Mercator’s expansion in honor of Danish mathemati-

cian Nicolaus Mercator (ca. 1619–87) who, in 1668,

was the first to publish this expansion.

The Mercator expansion has the curious of prop-

erty of apparently proving the absurd statement that 1

equals 2. Setting x= 1 yields:

and so

This paradox alerted mathematicians to the fact

that it is not always permissible to rearrange the terms

of a series. In 1837 P

ETER

G

USTAV

L

EJEUNE

D

IRICHLET

(1815–59) proved that such an operation is valid if the

series is absolutely convergent. Unfortunately, the alter-

nating

HARMONIC SERIES

expressed above is not.

See also

ABSOLUTE CONVERGENCE

.

Mercator’s projection It is not possible to make a flat

map of the world without incorporating some kind of

distortion. In the mid-1600s, Flemish cartographer Ger-

hard Kremer (1512–94), known as Mercator, devised a

method for mapping points from the surface of the Earth

onto a planar surface in such a way that all compass

directions, at least, are preserved. Although the distances

and areas are distorted under this

PROJECTION

, the gen-

eral shapes of small regions, such as small countries and

small bodies of water, are reasonably well preserved.

The mathematical construct of Mercator’s projec-

tion is obtained by imagining a cylinder placed around

the sphere of the Earth tangent to the equator and par-

allel to the axis of the Earth. A point on the surface of

the Earth is mapped to a point on this cylinder by

drawing a line from the center of the Earth and

through this point until it cuts the cylinder. (The North

and South Poles are not mapped.) The cylinder is then

cut and unrolled to form a flat surface.

In Mercator’s projection, lines of longitude are the

same distance apart, but lines of latitude get farther

apart from the equator. Mercator adjusted the vertical

spacing of the lines of latitude on his flat map to com-

pensate for this distortion and to make the shapes of

countries resemble more closely their true shape as they

appear on the globe.

Mercator’s projection can be given by mathematical

formulae. Under his mapping, a point on the Earth’s sur-

face at an angle αlatitude and angle βlongitude has

C

ARTESIAN COORDINATES

xand yon the plane given by:

for some constant k. The angles between lines on the

surface of the sphere (away from the poles) are pre-

served under Mercator’s projection and so this map-

ping is an example of a

CONFORMAL MAPPING

.

See also

STEREOGRAPHIC PROJECTION

.

Mersenne, Marin (1588–1648) French Number the-

ory, Theology Born on September 8, 1588, in Oize,

France, Marin Mersenne is remembered for the list of

PRIME

numbers that bear his name. These primes are

intimately connected with the formulation of even

PER

-

FECT NUMBER

s.

Mersenne studied theology as a teenager and at age

23 joined the Minims, a religious order devoted to

prayer and scholarship. Throughout his life Mersenne

pursued interests in

NUMBER THEORY

, mechanics, and

acoustics. He defended the work of

GALILEO GALILEI

(1564–1642) and R

ENÉ

D

ESCARTES

(1596–1650) against

theological criticism, and took on the task of translating

many of Galileo’s texts into French. Historians believe