Medal from the R
OYAL
S
OCIETY
of London in 1925,
and the Gold Medal from the Royal Astronomical
Society in 1926.
With the rise of anti-Semitism in Europe, Einstein
accepted a position at the Institute of Advanced Study
at Princeton, N.J., in 1933. He stayed there until his
death on April 18, 1955.
One cannot exaggerate the effect that Einstein’s
work has had on modern physics. One of his principal
goals was to unite the discrete description of particles
and matter with the continuous description of electro-
magnetic radiation and develop a single unified theory
of the two. The result is quantum mechanics. Intense
work continues today to incorporate other physical
forces, such as gravity, into a grand unified theory.
elementary matrix See G
AUSSIAN ELIMINATION
.
The Elements No doubt the most influential mathe-
matics text of all time, The Elements, written by E
UCLID
(ca. 300–260
B
.
C
.
E
.), provided the model for all of math-
ematical thinking for the two millennia that followed it.
Mathematicians agree that this work defines what the
pursuit of
PURE MATHEMATICS
is all about. More than
2,000 editions of The Elements have been printed since
the production of its first typeset version in 1482.
Written in 13 volumes (called “books”) The Ele-
ments represents a compilation of all the mathematics
that was known at the time. Organized in a strict logi-
cal structure, Euclid begins the work with a set of basic
definitions, “common notions,” and axioms (E
UCLID
’
S
POSTULATES
), and deduces from them, by the process of
pure logical reasoning, some 465 propositions (
THEO
-
REM
s) on the topics of plane geometry, number theory
(typically presented in terms of geometry), and solid
geometry. The work is revered for its clarity, precision,
and rigor.
The work is extremely terse in its presentation.
There is no discussion or motivation, and results are
simply stated and proved, often referring to a figure
accompanying the statement. Each proof ends with a
restatement of the proposition studied along with the
words, “which was to be proved.” The Latin transla-
tion of this phrase is quod erat demonstrandum, and
many mathematicians today still like to end a formal
proof with the abbreviation Q.E.D.
Although it is generally believed that no result pre-
sented in The Elements was first proved by Euclid, the
organization of the material and the logical develop-
ment presented is original. Euclid’s choice of beginning
postulates shows remarkable insight and a deep wis-
dom of the subject. His recognition of the need to for-
mulate the controversial
PARALLEL POSTULATE
, for
instance, shows a level of genius beyond all of those
who tried to prove his choice irrelevant during the two
millennia that followed. (It was not until the 19th cen-
tury, with the development of
NON
-E
UCLIDEAN GEOME
-
TRY
, did mathematicians realize that the parallel
postulate was an essential assumption in the develop-
ment of standard planar geometry.)
The first six books of The Elements deal with the
topic of plane geometry. In particular, Books I and II
establish basic properties of triangles, parallel lines,
parallelograms, rectangles, and squares, and Books III
and IV examine properties of circles. In Book V, Euclid
examines properties of magnitudes and ratios, and
applies these results back to plane geometry in Book
VI. Euclid presents a proof of P
YTHAGORAS
’
S THEOREM
in Book I.
Books VII to X deal with
NUMBER THEORY
. The
famous E
UCLIDEAN ALGORITHM
appears in book VII,
and E
UCLID
’
S PROOF OF THE INFINITUDE OF PRIMES
in
book IX. Book X deals with the theory of irrational
numbers, and Euclid actually proves the existence of
these numbers in this work.
The final three volumes of The Elements explore
three-dimensional geometry. The work culminates with
a discussion of the properties of each P
LATONIC SOLID
and proof that there are precisely five such polyhedra.
elimination method for simultaneous linear
equations Another name for G
AUSSIAN ELIMINATION
.
Elizabethan multiplication Also known as the gal-
ley method and the lattice method, this multiplication
technique was taught to students of mathematics in
Elizabethan England. To multiply 253 and 27, for
example, draw a 2 ×3 grid of squares. Write the first
number along the top, and the second number down
the right side. Divide each cell of the grid diagonally.
Multiply the digits of the top row, in turn, with each of
the digits of the right column, writing the products in
Elizabethan multiplication 159