and so on. Hence, to four decimal places, we see √
–
2 =
1.4142.
If one initially makes two guesses x 0 and x 1 to the
root of an equation f(x) = 0, then one can use the
formula
to approximate the slope
of the curve at position x = x 1 . An improved approximation for the root is thus given by
. Repeated application leads to
the secant method for approximating roots: given two
initial guesses x 0 and x 1 set:
Neyman, Jerzy (1894–1981) Moldavian Statistics
Born on April 16, 1894, in Bendery, Moldavia, statistician Jerzy Neyman is remembered for his fundamental work in the theory of inferential statistics. He
made significant contributions to the practice of
hypothesis testing and estimation, and was the first to
introduce the notion of a confidence interval.
Neyman studied mathematics at Kharkov University and received a doctorate in mathematics and
statistics in 1924 from the University of Warsaw.
Collaborating with KARL PEARSON’S son, E. S. Pearson, Neyman wrote a number of influential papers.
The two most prominent pieces “On the Problem of
the Most Efficient Tests of Statistical Hypotheses” and
“The Testing of Statistical Hypotheses in Relation to
Probabilities A Priori” were both published in 1933.
In 1938 Neyman emigrated to the United States,
where he worked at the University of California, Berkeley, for the remainder of his life. He died in Oakland,
California, on August 5, 1981.
His work in statistics, with the introduction of the
confidence interval, revolutionized sampling techniques
and practices in agriculture, biology, medicine, and the
physical sciences, and his study of stratified populations led to a new field of statistical theory. The structure of the Gallup poll used today, for instance, is
based on this work.
See also STATISTICS: INFERENTIAL.
nine-point circle Consider a triangle with vertices A,
B, and C. Let M A , M B , and M C be the MIDPOINTs of
each of its three sides. Consider too the three altitudes
of the triangle, lines that each pass through one vertex of
the triangle to meet the opposite side at a right angle. Let
H A , H B , and H C be the feet of the three altitudes, one on
each side of the triangle. A study of EQUIDISTANT points
shows that the three altitudes pass through a common
point H called the orthocenter of the triangle. (See ALTITUDE.) Let P A , P B , and P C be the midpoints of the line
segments connecting H to each vertex A, B, and C,
respectively. We now have nine points associated with
the triangle: M A , M B , M C , H A , H B , H C , P A , P B , and P C .
Surprisingly, these nine points will always lie on a circle.
This circle is called the nine-point circle of the triangle.
This result was discovered by German mathematician Karl Wilhelm Feuerbach (1800–34). The center of
the circle turns out to lie on the EULER LINE of the triangle. In fact, if O is the circumcenter of the triangle, that
is, the center of the CIRCUMCIRCLE of the triangle, then
the center of the nine-point circle Q is the midpoint of
the line segment connecting O to H. (With this known,
one can then prove that Q is indeed the same distance
from each of the nine points listed, thereby establishing
the claim that all nine points lie on a circle.)
Feuerbach also proved that the nine-point circle
and the INCIRCLE of the triangle touch at just one point
and, moreover, that the nine-point circle is TANGENT to
each of the three excircles of the triangle.
Given four arbitrary points in the plane, one can
construct a nine-point circle for any triangle formed by
a set of three of those points. As there are four sets of
three points, this yields four nine-point circles in all.
Surprisingly, these four nine-point circles all pass
through a single common point.
Noether, Amalie (Emmy) (1882–1935) German Abstract algebra Born on March 23, 1882, in Erlangen,
Bavaria, mathematician Emmy Noether is remembered
x
x f x
x x
f x
f x
n
n
n
n
n
n
n
+
−
−
=
−
−
−
1
1
1
( ) ( ) (
)
x f x
x x
f x
f x
1
1
1
0
1
0
−
−
−
( ) ( ) ( )
x
x
f x
f x
2
1
1
1
= − ′
≈
( )
( )
′
≈
−
−
f x
f x
f x
x x
( )
( ) ( )
1
1
0
1
0
x
x
x
2
3
4
17
12
1 416667
99
70
1 414216
577
408
1 414213
=
=
=
=
=
=
.
.
.
354 Neyman, Jerzy
–
2 =
1.4142.
If one initially makes two guesses x 0 and x 1 to the
root of an equation f(x) = 0, then one can use the
formula
to approximate the slope
of the curve at position x = x 1 . An improved approximation for the root is thus given by
. Repeated application leads to
the secant method for approximating roots: given two
initial guesses x 0 and x 1 set:
Neyman, Jerzy (1894–1981) Moldavian Statistics
Born on April 16, 1894, in Bendery, Moldavia, statistician Jerzy Neyman is remembered for his fundamental work in the theory of inferential statistics. He
made significant contributions to the practice of
hypothesis testing and estimation, and was the first to
introduce the notion of a confidence interval.
Neyman studied mathematics at Kharkov University and received a doctorate in mathematics and
statistics in 1924 from the University of Warsaw.
Collaborating with KARL PEARSON’S son, E. S. Pearson, Neyman wrote a number of influential papers.
The two most prominent pieces “On the Problem of
the Most Efficient Tests of Statistical Hypotheses” and
“The Testing of Statistical Hypotheses in Relation to
Probabilities A Priori” were both published in 1933.
In 1938 Neyman emigrated to the United States,
where he worked at the University of California, Berkeley, for the remainder of his life. He died in Oakland,
California, on August 5, 1981.
His work in statistics, with the introduction of the
confidence interval, revolutionized sampling techniques
and practices in agriculture, biology, medicine, and the
physical sciences, and his study of stratified populations led to a new field of statistical theory. The structure of the Gallup poll used today, for instance, is
based on this work.
See also STATISTICS: INFERENTIAL.
nine-point circle Consider a triangle with vertices A,
B, and C. Let M A , M B , and M C be the MIDPOINTs of
each of its three sides. Consider too the three altitudes
of the triangle, lines that each pass through one vertex of
the triangle to meet the opposite side at a right angle. Let
H A , H B , and H C be the feet of the three altitudes, one on
each side of the triangle. A study of EQUIDISTANT points
shows that the three altitudes pass through a common
point H called the orthocenter of the triangle. (See ALTITUDE.) Let P A , P B , and P C be the midpoints of the line
segments connecting H to each vertex A, B, and C,
respectively. We now have nine points associated with
the triangle: M A , M B , M C , H A , H B , H C , P A , P B , and P C .
Surprisingly, these nine points will always lie on a circle.
This circle is called the nine-point circle of the triangle.
This result was discovered by German mathematician Karl Wilhelm Feuerbach (1800–34). The center of
the circle turns out to lie on the EULER LINE of the triangle. In fact, if O is the circumcenter of the triangle, that
is, the center of the CIRCUMCIRCLE of the triangle, then
the center of the nine-point circle Q is the midpoint of
the line segment connecting O to H. (With this known,
one can then prove that Q is indeed the same distance
from each of the nine points listed, thereby establishing
the claim that all nine points lie on a circle.)
Feuerbach also proved that the nine-point circle
and the INCIRCLE of the triangle touch at just one point
and, moreover, that the nine-point circle is TANGENT to
each of the three excircles of the triangle.
Given four arbitrary points in the plane, one can
construct a nine-point circle for any triangle formed by
a set of three of those points. As there are four sets of
three points, this yields four nine-point circles in all.
Surprisingly, these four nine-point circles all pass
through a single common point.
Noether, Amalie (Emmy) (1882–1935) German Abstract algebra Born on March 23, 1882, in Erlangen,
Bavaria, mathematician Emmy Noether is remembered
x
x f x
x x
f x
f x
n
n
n
n
n
n
n
+
−
−
=
−
−
−
1
1
1
( ) ( ) (
)
x f x
x x
f x
f x
1
1
1
0
1
0
−
−
−
( ) ( ) ( )
x
x
f x
f x
2
1
1
1
= − ′
≈
( )
( )
′
≈
−
−
f x
f x
f x
x x
( )
( ) ( )
1
1
0
1
0
x
x
x
2
3
4
17
12
1 416667
99
70
1 414216
577
408
1 414213
=
=
=
=
=
=
.
.
.
354 Neyman, Jerzy