(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 different 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
EUCLID may have drawn on the work of Hippocrates
when he described and proved this result in his famous
text THE ELEMENTS.
histogram See DISTRIBUTION; STATISTICS: DESCRIPTIVE.
homogeneous A POLYNOMIAL in several variables
p(x,y,z,…) is called homogeneous if all terms in the polynomial have the same total degree. For example, each
term of the polynomial x
3 + 5x
2
y – xy
2
, 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,…) = k
n f(x,y,z,…)
For example, the function
is
homogeneous.
Identifying homogeneous functions can be helpful
in solving DIFFERENTIAL EQUATIONs. Any formula that
represents the MEAN of a set of numbers is required to
be homogeneous.
In physics, the term homogeneous describes a substance 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 f between two sets S and T,
each equipped with some kind of algebraic structure, is
called a homomorphism (Greek, homos, “same”; morphé, “form” or “shape”) if it preserves the algebraic
relations within the sets. For example, if the algebraic
operation is “multiplication” in the set S and “addition” in the set T, and if z is the product of two elements 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 f is the doubling function, f(x) = 2x, on real
numbers, then f is 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) = x
2 preserves multiplication but
not addition.
A map between two GROUPs is called a homomorphism if it preserves the algebraic operations of the
groups. A map f : S → T between two RINGs 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 x and y of the ring S.
See also ISOMORPHISM.
Hypatia of Alexandria (ca. 370–415 C.E.) Greek Philosophy, Commentary Born about 370 in Alexandria,
Egypt, Hypatia is remembered as the first woman in the
history of mathematics to have made significant contributions 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 commentaries on her father’s work and also on the work of
APOLLONIUS OF PERGA (ca. 263–190 B.C.E.) and DIOPHANTUS OF ALEXANDRIA (ca. 200–284 C.E.).
Little is known of Hypatia’s life. She likely received
instruction in mathematics from her father, and certainly maintained a close working relationship with
him throughout her life. It is known that she assisted
x
x y
y
xy
y
x
2
2
2
sin(
)
cos( )
+ +
252 histogram
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 different 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
EUCLID may have drawn on the work of Hippocrates
when he described and proved this result in his famous
text THE ELEMENTS.
histogram See DISTRIBUTION; STATISTICS: DESCRIPTIVE.
homogeneous A POLYNOMIAL in several variables
p(x,y,z,…) is called homogeneous if all terms in the polynomial have the same total degree. For example, each
term of the polynomial x
3 + 5x
2
y – xy
2
, 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,…) = k
n f(x,y,z,…)
For example, the function
is
homogeneous.
Identifying homogeneous functions can be helpful
in solving DIFFERENTIAL EQUATIONs. Any formula that
represents the MEAN of a set of numbers is required to
be homogeneous.
In physics, the term homogeneous describes a substance 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 f between two sets S and T,
each equipped with some kind of algebraic structure, is
called a homomorphism (Greek, homos, “same”; morphé, “form” or “shape”) if it preserves the algebraic
relations within the sets. For example, if the algebraic
operation is “multiplication” in the set S and “addition” in the set T, and if z is the product of two elements 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 f is the doubling function, f(x) = 2x, on real
numbers, then f is 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) = x
2 preserves multiplication but
not addition.
A map between two GROUPs is called a homomorphism if it preserves the algebraic operations of the
groups. A map f : S → T between two RINGs 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 x and y of the ring S.
See also ISOMORPHISM.
Hypatia of Alexandria (ca. 370–415 C.E.) Greek Philosophy, Commentary Born about 370 in Alexandria,
Egypt, Hypatia is remembered as the first woman in the
history of mathematics to have made significant contributions 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 commentaries on her father’s work and also on the work of
APOLLONIUS OF PERGA (ca. 263–190 B.C.E.) and DIOPHANTUS OF ALEXANDRIA (ca. 200–284 C.E.).
Little is known of Hypatia’s life. She likely received
instruction in mathematics from her father, and certainly maintained a close working relationship with
him throughout her life. It is known that she assisted
x
x y
y
xy
y
x
2
2
2
sin(
)
cos( )
+ +
252 histogram