the appropriate square cells of the grid as two-digit
entries. (Thus compute 3 × 2 as 06, for example.)
Add the entries in each diagonal starting with the
bottom right diagonal. Write down the units figure and
carry any tens figures that appear to the next diagonal.
The answer, 6,831, now appears along the left column
and bottom row.
This procedure works for multidigit multiplications
of any size. Its success relies on the DISTRIBUTIVE PROPERTY of arithmetic and the process of EXPANDING
BRACKETS. In our example,
253 × 27 = (2 × 10
2 + 5 × 10 + 3) × (2 × 10 + 7)
= (2 × 2) × 10
3 + (2 × 7) × 10
2 + (3 × 2) × 10
+ (5 × 2) × 10
2 + (5 × 7) × 10
+ 3 × 7
Each diagonal corresponds to a powers-of-10 place,
with entries placed in an upper portion of a square cell
corresponding to carried figures to the next powers-of10 position. Try computing a multiplication problem
both the Elizabethan way and the usual way, side-byside, to see that the two methods do not differ.
See also EGYPTIAN MULTIPLICATION; FINGER MULTIPLICATION; MULTIPLICATION; NAPIER’S BONES; RUSSIAN
MULTIPLICATION.
ellipse As one of the CONIC SECTIONS, an ellipse is
the plane curve consisting of all points P whose distances from two given points F 1 and F 2 in the plane
have a constant sum. The two fixed points F 1 and F 2
are called the foci of the ellipse. An ellipse also arises as
the curve produced by the intersection of a plane with
a single nappe of a right circular CONE.
Using the notation |PF 1 | and |PF 2 | for the lengths of
the line segments connecting P to F 1 and F 2 , respectively,
the defining condition of an ellipse can be written:
|PF 1 | + |PF 2 | = d
where d denotes the constant sum.
The equation of an ellipse can be found by introducing a coordinate system in which the foci are
located at positions F 1 = (–c,0) and F 2 = (c, 0), for some
positive number c. It is convenient to write d = 2a, for
some a > 0. If P = (x,y) is an arbitrary point on the
ellipse, then the defining condition states:
+
= 2a
Moving the second radical to the right-hand side,
squaring, and simplifying yields the equation:
Squaring and simplifying again yields:
+
Noting that a is greater than c, we can set the positive
quantity a
2 – c
2 as equal to b
2
, for some positive number b. Thus the equation of the ellipse is:
+
= 1
y
2
–
b
2
x
2
–
a
2
= 1
y
2
––
a
2 – c
2
x
2
–
a
2
(
)
x c
y
a
c
a
x
−
+
= −
2
2
√(x – c)
2 + y
2
√(x + c)
2 + y
2
160 ellipse
Elizabethan multiplication
Ellipse
entries. (Thus compute 3 × 2 as 06, for example.)
Add the entries in each diagonal starting with the
bottom right diagonal. Write down the units figure and
carry any tens figures that appear to the next diagonal.
The answer, 6,831, now appears along the left column
and bottom row.
This procedure works for multidigit multiplications
of any size. Its success relies on the DISTRIBUTIVE PROPERTY of arithmetic and the process of EXPANDING
BRACKETS. In our example,
253 × 27 = (2 × 10
2 + 5 × 10 + 3) × (2 × 10 + 7)
= (2 × 2) × 10
3 + (2 × 7) × 10
2 + (3 × 2) × 10
+ (5 × 2) × 10
2 + (5 × 7) × 10
+ 3 × 7
Each diagonal corresponds to a powers-of-10 place,
with entries placed in an upper portion of a square cell
corresponding to carried figures to the next powers-of10 position. Try computing a multiplication problem
both the Elizabethan way and the usual way, side-byside, to see that the two methods do not differ.
See also EGYPTIAN MULTIPLICATION; FINGER MULTIPLICATION; MULTIPLICATION; NAPIER’S BONES; RUSSIAN
MULTIPLICATION.
ellipse As one of the CONIC SECTIONS, an ellipse is
the plane curve consisting of all points P whose distances from two given points F 1 and F 2 in the plane
have a constant sum. The two fixed points F 1 and F 2
are called the foci of the ellipse. An ellipse also arises as
the curve produced by the intersection of a plane with
a single nappe of a right circular CONE.
Using the notation |PF 1 | and |PF 2 | for the lengths of
the line segments connecting P to F 1 and F 2 , respectively,
the defining condition of an ellipse can be written:
|PF 1 | + |PF 2 | = d
where d denotes the constant sum.
The equation of an ellipse can be found by introducing a coordinate system in which the foci are
located at positions F 1 = (–c,0) and F 2 = (c, 0), for some
positive number c. It is convenient to write d = 2a, for
some a > 0. If P = (x,y) is an arbitrary point on the
ellipse, then the defining condition states:
+
= 2a
Moving the second radical to the right-hand side,
squaring, and simplifying yields the equation:
Squaring and simplifying again yields:
+
Noting that a is greater than c, we can set the positive
quantity a
2 – c
2 as equal to b
2
, for some positive number b. Thus the equation of the ellipse is:
+
= 1
y
2
–
b
2
x
2
–
a
2
= 1
y
2
––
a
2 – c
2
x
2
–
a
2
(
)
x c
y
a
c
a
x
−
+
= −
2
2
√(x – c)
2 + y
2
√(x + c)
2 + y
2
160 ellipse
Elizabethan multiplication
Ellipse