north of P, and G′is the point on the equator south of
Greenwich. New York City, for example, has longitude
73°59′39′′ W.
The longitude of a point Pon the Earth’s surface
can be measured by identifying the time difference
between high noon at Pand high noon at Greenwich.
As the day is divided into 24 1-hour periods, each
delay of 1 hour corresponds to 1/24 of a full turn
about the Earth’s circumference. Thus, at 360/24 = 15°
longitude west, for instance, the Sun reaches its highest
point in the sky 1 hour later than it does at Greenwich;
at 30°longitude west, it occurs 2 hours later, and so
forth. After about 1735, sailors were able to carry
accurate chronometers to keep track of Greenwich
time. By measuring the time of high noon at any given
location, sailors could accurately determine the longi-
tude of that location.
See also
OBLATE
/
PROLATE
.
eccentricity For any
CONIC SECTION
, the ratio of the
distance of a point on the curve from a fixed point, the
focus, to its distance from a fixed line, the directrix, is
constant. The value of this ratio, denoted e, is called
the eccentricity of the curve, and it gives a measure of
the curve’s shape. For a
PARABOLA
, eequals one. If e
lies between zero and one, then the curve is an
ELLIPSE
.
If eis greater than one, then the curve is a
HYPERBOLA
.
The eccentricity of a
CIRCLE
is defined to be zero. (In
this context, the eccentricity eis not to be confused
with L
EONHARD
E
ULER
’s number e.)
Egyptian fractions Any fraction with unit numerator,
such as , , and , is called an Egyptian fraction.
The Egyptians of 4,000 years ago expressed all frac-
tional quantities as sums of distinct Egyptian fractions.
For example, was written + , and as +
+ .
In 1202, F
IBONACCI
began his own investigation of
Egyptian fractions and was the first to prove that every
fraction can indeed be expressed as a finite sum of dis-
tinct Egyptian fractions. (It is not clear whether the
ancient Egyptians ever questioned this.) He showed
that subtracting a quantity of the form 1/n, with nas
small as possible, from a given fraction always pro-
duces a new fraction with a smaller numerator. Thus
repeated application of this procedure must eventually
produce a fraction with unit numerator itself.
As an example, for the fraction 5/17 we have:
– = ( is too large a quantity to subtract.)
and
– = ( is too large a quantity to subtract.)
giving:
= + +
Such representations need not be unique. For example,
equals both + + and + .
See also E
GYPTIAN MATHEMATICS
.
Egyptian mathematics Our knowledge of ancient
Egyptian mathematics from around 2000
B
.
C
.
E
. comes
chiefly from the R
HIND PAPYRUS
. There we learn, for
example, that the Egyptians followed a very natural sys-
tem for denoting numerals: 1 was a vertical stroke |, 2
was two of them ||, 3 was |||, and 4 was ||||, and separate
symbols were used for 5, 6, 7, 8, and 9, and for 10,
20,…, 100, 200,…, 1000, and so on. All other numbers
were represented as groups of these symbols, usually
arranged in order from largest to smallest. Like the
R
OMAN NUMERAL
system, the Egyptian system did not
use a
PLACE
-
VALUE SYSTEM
(the symbol for 5, for exam-
ple, denoted “5” no matter where it appeared in the
number). It is very difficult to do pencil-and-paper cal-
culations without place-value notation, but the Egyp-
tians always used a calculating board, much like an
ABACUS
, to perform arithmetic calculations, and needed
only to record the results. They were therefore not hin-
dered by their cumbersome numerical system. The
ancient Egyptians were adept at multiplication, using a
method of successive doubling to calculate products.
This method is today called E
GYPTIAN MULTIPLICATION
.
Division problems lead to
FRACTION
s. It did not
occur to the ancient Egyptians to express fractions with
numerators and denominators. In the Rhind papyrus,
the mathematician Ahmes simply placed a dot over a
number to indicate its reciprocal, except in the case of
the fractions , , , and , each of which had its
1
–
4
2
–
3
1
–
3
1
–
2
1
–
20
1
–
4
1
–
30
1
–
15
1
–
5
3
–
10
1
–—
1564
1
–
23
1
–
4
5
–
17
1
–
22
1
–—
1564
1
–
23
3
–
68
1
–
3
3
–
68
1
–
4
5
–
17
1
–
30
1
–
15
1
–
5
3
–
10
1
–
18
1
–
6
2
–
9
1
–
598
1
–
45
1
–
2
Egyptian mathematics 155