1

–

x

Ptolemy, Claudius 421

Claudius Ptolemy, an eminent scholar of the second century,

wrote the Almagest, the most influential work on the topic of

trigonometry of ancient times. (Photo courtesy of the Science

Museum, London/Topham-HIP/The Image Works)

120

If a quantity yis proportional to the inverse of a

variable x, that is, y∝, then we say that yand xare

inversely proportional or that yvaries inversely as x.

For example, the gas laws in physics assert that gas

pressure is inversely proportional to volume.

In algebra, terms in a statement about the ratios of

quantities are sometimes called proportionals. For

example, in the expression:

a: b= c: d

the quantity cis the third proportional.

proposition In mathematics, any statement for

which a proof is required or has been provided is called

a proposition. In

FORMAL LOGIC

, any statement that

has a truth-value of either true or false (but not both) is

called a proposition.

See also

THEOREM

.

Ptolemy, Claudius (ca. 85–165

C

.

E

.) Greek Geome-

try, Trigonometry, Astronomy Born in Egypt, around

85

C

.

E

., scholar Claudius Ptolemy is remembered for

his epic piece Syntaxis mathematica (Mathematical col-

lection), also known as the Almagest (The greatest),

often deemed as the most significant work in

TRIGONOMETRY

of ancient times. The work contains

accurate tables of “chords” (equivalent to a modern

table of sine values), as well as a clear description as to

how that table was constructed. (He used a result

today known as P

TOLEMY

’

S THEOREM

, the geometric

equivalent of the trigonometric

ADDITION

formulae we

use today.) It is known that Ptolemy also attempted to

prove E

UCLID

’s famous

PARALLEL POSTULATE

.

The exact location of Ptolemy’s birth is unknown.

Although born in Egypt, he is referred to as a classical

Greek scholar because he followed the scholarly tradi-

tions of the Greeks and wrote in that language. His last

name, Ptolemy, is Greek, but his first name, Claudius,

is Roman. This suggests that he was also considered a

Roman citizen.

Accurate translations of all of Ptolemy’s works sur-

vive today. His most noted work, Syntaxis mathemat-

ica, comes in 13 volumes, and much of the

mathematics developed in it is motivated by concerns

of astronomy. In particular, Ptolemy worked to give

detailed descriptions of the motions of the Sun, Moon,

and the known planets at the time. He believed that the

Earth lay at the center of the solar system.

In Books 1 and 2 of Syntaxis mathematica, Ptolemy

develops a mathematical theory of compound circular

motions (motion in

EPICYCLE

s) as a means to explain

the observed motion of the planets. This motivated his

need for his accurate table of chords. It is also worth

mentioning that in this work Ptolemy used a 360-sided

polygon inscribed in a circle to find the following

approximation for πcorrect to three decimal places:

Book 3 of Syntaxis mathematica is concerned with

the motion of the Sun. By carefully analyzing the