252 histogram

(One can check this algebraically by labeling the

radius of the larger circular arc r and computing the

side-length of the large square, the area of the sector

DAC, the area of the semicircle ABC, and consequently

the area of the lune.)

In applying this type of analysis to a number of dif-

ferent lunes, Hippocrates managed to prove that the

ratio of the areas of two different circles is the same as

the ratio of their radii squared. This new result was a

significant achievement. Some historians believe that

E

UCLID

may have drawn on the work of Hippocrates

when he described and proved this result in his famous

text T

HE

E

LEMENTS

.

histogram See

DISTRIBUTION

;

STATISTICS

:

DESCRIPTIVE

.

homogeneous A

POLYNOMIAL

in several variables

p(x,y,z,…) is called homogeneous if all terms in the poly-

nomial have the same total degree. For example, each

term of the polynomial x3+ 5x2y– xy2, has total degree

three, and so represents a homogeneous polynomial.

More generally, a function of several variables

f(x,y,z,…) is homogeneous if there is a natural number n

such that the following relation holds for any nonzero

constant k:

f(kx,ky,kz,…) = knf(x,y,z,…)

For example, the function is

homogeneous.

Identifying homogeneous functions can be helpful

in solving

DIFFERENTIAL EQUATION

s. Any formula that

represents the

MEAN

of a set of numbers is required to

be homogeneous.

In physics, the term homogeneous describes a sub-

stance or an object whose properties do not vary with

position. For example, an object of uniform density is

sometimes described as homogeneous. In cooking,

when creaming butter and sugar, for instance, one aims

to produce a homogeneous mixture.

homomorphism A map fbetween two sets Sand T,

each equipped with some kind of algebraic structure, is

called a homomorphism (Greek, homos, “same”; mor-

phé, “form” or “shape”) if it preserves the algebraic

relations within the sets. For example, if the algebraic

operation is “multiplication” in the set Sand “addi-

tion” in the set T, and if zis the product of two ele-

ments in S, z = x×y, then f(z) should correspond to the

sum of the two corresponding elements in T:

f(x×y) = f(x) + f(y)

The

LOGARITHM

function, for example, is a

homomorphism from the set of positive real numbers

under multiplication to the set of all real numbers

under addition.

If fis the doubling function, f(x) = 2x, on real

numbers, then fis a homomorphism under addition,

but not under multiplication:

f(x+ y) = 2(x+ y) = 2x+ 2y= f(x) + f(y)

f(x×y) = 2 ×x×y≠f(x) ×f(y)

The squaring map f(x) = x2preserves multiplication but

not addition.

A map between two

GROUP

s is called a homomor-

phism if it preserves the algebraic operations of the

groups. A map f: S→Tbetween two

RING

s is called a

homomorphism if it preserves both the additions and

multiplications of the system:

f(x+ y) = f(x) + f(y)

f(x×y) = f(x) ×f(y)

for all elements xand yof the ring S.

See also

ISOMORPHISM

.

Hypatia of Alexandria (ca. 370–415

C

.

E

.) Greek Phi-

losophy, Commentary Born about 370 in Alexandria,

Egypt, Hypatia is remembered as the first woman in the

history of mathematics to have made significant contri-

butions to the field, both as a scholar and as a teacher.

She was the daughter of mathematician and astronomer

Theon of Alexandria. Hypatia wrote influential com-

mentaries on her father’s work and also on the work of

A

POLLONIUS OF

P

ERGA

(ca. 263–190

B

.

C

.

E

.) and D

IO

-

PHANTUS OF

A

LEXANDRIA

(ca. 200–284

C

.

E

.).

Little is known of Hypatia’s life. She likely received

instruction in mathematics from her father, and cer-

tainly maintained a close working relationship with

him throughout her life. It is known that she assisted