354 Neyman, Jerzy

≈

and so on. Hence, to four decimal places, we see √

–

2 =

1.4142.

If one initially makes two guesses x0and x1to the

root of an equation f(x) = 0, then one can use the

formula to approximate the slope

of the curve at position x= x1. An improved approxi-

mation for the root is thus given by

. Repeated application leads to

the secant method for approximating roots: given two

initial guesses x0and x1set:

Neyman, Jerzy (1894–1981) Moldavian Statistics

Born on April 16, 1894, in Bendery, Moldavia, statis-

tician Jerzy Neyman is remembered for his funda-

mental 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 Univ-

ersity and received a doctorate in mathematics and

statistics in 1924 from the University of Warsaw.

Collaborating with K

ARL

P

EARSON

’

S

son, E. S. Pear-

son, 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, Berke-

ley, 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 popula-

tions led to a new field of statistical theory. The struc-

ture 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 MA, MB, and MCbe the

MIDPOINT

s 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

HA, HB, and HCbe 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 Hcalled the orthocenter of the triangle. (See

ALTI

-

TUDE

.) Let PA, PB, and PCbe the midpoints of the line

segments connecting Hto each vertex A, B, and C,

respectively. We now have nine points associated with

the triangle: MA, MB, MC, HA, HB, HC, PA, PB, and PC.

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 mathemati-

cian Karl Wilhelm Feuerbach (1800–34). The center of

the circle turns out to lie on the E

ULER LINE

of the trian-

gle. In fact, if Ois the circumcenter of the triangle, that

is, the center of the

CIRCUMCIRCLE

of the triangle, then

the center of the nine-point circle Qis the midpoint of

the line segment connecting Oto H. (With this known,

one can then prove that Qis 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 Abs-

tract algebra Born on March 23, 1882, in Erlangen,

Bavaria, mathematician Emmy Noether is remembered