1, 2, and 5 of E
UCLID
’
S POSTULATES
remain valid in
affine geometry.
affine transformation Any map from the
PLANE
to
itself that transforms straight
LINES
into straight lines
and preserves
RATIO
s of distances between
POINT
s (so
that the midpoint of a line segment, for instance,
remains the midpoint after the transformation) is called
an affine transformation. One can prove that any affine
transformation must be a
LINEAR TRANSFORMATION
followed by a translation. Thus an affine transforma-
tion Tis completely specified by a
MATRIX
Aand a
VECTOR
bso that T(x) = Ax+ bfor any vector xrepre-
senting a point in the plane.
An affine transformation Tsatisfies the relation:
T(sx+ ty) = sT(x) + tT(y)
for any two vectors xand yand any two real numbers
sand tsuch that s+ t= 1. This is sometimes taken as
the definition of what it means for a transformation to
be affine.
Affine transformations generally do not preserve
the lengths of line segments nor the measure of
ANGLE
s between segments. It is possible to transform
a
CIRCLE
into an
ELLIPSE
, for instance, via an affine
transformation.
See also
AFFINE GEOMETRY
.
Agnesi, Maria Gaëtana (1718–1799) Italian Calcu-
lus Born on May 16, 1718, to a wealthy family of silk
merchants, Maria Agnesi is best remembered for her
influential expository text outlining the methods and
techniques of the newly invented
CALCULUS
. Written
with such clarity and precision, Istituzioni analitiche
(Analytical institutions) garnered her international
fame. Agnesi is considered the first major female math-
ematician of modern times, and she holds the distinc-
tion of being the first woman to be awarded a
professorship of mathematics on a university faculty.
Agnesi demonstrated remarkable academic talents
as a young child. By age 13 she had mastered many
languages and had published translations of academic
essays. Although little consideration was given to edu-
cating women at the beginning of the 18th century,
Agnesi’s father encouraged her intellectual develop-
ment, provided tutors of the highest quality, and pro-
vided forums for her to display her talents to Italian
society. In preparation for these events, Agnesi had pre-
pared discourses on a wide variety of topics in science
and philosophy, which she published as a collection of
190 essays at age 20.
After the death of her mother, Agnesi undertook
the task of instructing her younger brothers in the sub-
ject of mathematics. In 1738 she began preparing a
textbook for their use, and found the topic so com-
pelling that she devoted her complete intellectual atten-
tion to mathematics. Ten years later, her famous
two-volume text Istituzioni analitiche was published.
The work was the first comprehensive overview of
the subject of calculus. Although designed for young
students beginning their studies of the subject, Agnesi’s
work was recognized as providing hitherto unnoticed
connections between the different approaches of S
IR
I
SAAC
N
EWTON
(1642–1727) and G
OTTFRIED
W
IL
-
HELM
L
EIBNIZ
(1646–1716), independent coinventors
of the subject. Her piece also provided, for the first
time, clear explanations of previously confusing issues
in the topic. Her text collated and explained the work
of other contributors to the subject from several differ-
ent countries, a task no doubt facilitated by her skills in
translation. Her talents, not just as an expository
writer, but also as a great scholar in mathematics, were
apparent. Mathematicians at the time recognized her
text as a significant contribution to the further develop-
ment of the topic of calculus. In 1750 Agnesi was
appointed the chair of mathematics at the University of
Bologna in recognition of her great accomplishment.
Curiously, she never officially accepted or rejected the
faculty position. It is known that she never visited the
city of Bologna, even though her name appears on uni-
versity records over a span of 45 years.
After the death of her father in 1752, Agnesi with-
drew from mathematics and devoted her life to charita-
ble work, helping the sick and the poor. In 1783 she
was made director of a women’s poorhouse, where she
remained the rest of her life. Having given all her money
and possessions away, Agnesi died on January 9, 1799,
with no money of her own and was buried in a pauper’s
grave. The city of Milan, where she had lived all her
life, today publicly honors her gravesite.
John Colson, Lucasian professor of mathematics at
Cambridge, published an English translation of Isti-
tuzioni analitiche in 1801. He said that he wanted to
Agnesi, Maria Gaëtana 7