484 Steiner, Jakob

2

This is known today as Student’s t-distribution (with

N– 1 degrees of freedom). Student’s t-test is used to

test whether any difference in the means of two differ-

ent samples is statistically meaningful. For example, a

study might indicate that the mean height of 100 ran-

domly selected basketball players is 8 in. higher than

the mean height of 100 baseball players. Student’s t-

test would test the hypothesis, “Both samples were

drawn from the same normal population.” If the value

8 in. is deemed too high, the hypothesis would be

rejected and the difference in the means would be con-

sidered significant.

The F-test tests whether or not two samples come

from the same population by focusing instead on the

VARIANCE

of each sample. If two samples of sizes N1and

N2come from the same normal population, then the

ratio of their variances should be approximately

equal to 1. The F-distribution tabulates values of these

ratios, and the F-test determines whether the observed

ratio for two particular samples has an acceptable value.

See also

CHI

-

SQUARE TEST

;

CORRELATION COEFFI

-

CIENT

.

Steiner, Jakob (1796–1863) Swiss Geometry Born on

March 18, 1796, in Utzenstorf, Switzerland, mathemati-

cian Jakob Steiner is remembered for his fundamental

contributions to the study of

PROJECTIVE GEOMETRY

and

for his work on the solution of the famous

ISOPERIMET

-

RIC PROBLEM

. A collection of geometric points are today

named in his honor, S

TEINER POINT

s, to acknowledge his

work in geometric

OPTIMIZATION

.

Steiner had no early formal education. It is said

that he did not begin to read and write until he was 14,

and did not attend any kind of school until age 18.

During the latter part of his teen years, however,

Steiner demonstrated a talent for mathematics, which

earned him admission to the Johann Heinrich

Pestalozzi school in Yverdon, Switzerland. Just 2 years

after entering the school as a student, he was hired as a

teacher of mathematics at the school.

Steiner entered the University of Heidelberg in

1818 and transferred to the University of Berlin 3 years

later to pursue a research career in mathematics. His

innovative work in geometry was duly noted by the

mathematicians of the time. Steiner was awarded an

honorary doctoral degree from the University of

Königsberg and the position as chair of mathematics at

the University of Berlin in 1834. He held that post for

the remainder of his life. He died in Berlin on April 1,

1863.

A number of finite configurations in projective

geometry are named in his honor, as well as a geomet-

ric surface, the Steiner surface, which has the property

that each of its tangent planes slices the surface in a

pair of

CONIC SECTIONS

. His influence on the develop-

ment of geometric optimization was profound.

Steiner point Given three points A, B, and Cin the

plane, a Steiner point for that system is a point Pwhose

sum of distances AP + BP + CP is at a minimum. That

such a point always exists was first established by Swiss

mathematician J

AKOB

S

TEINER

(1796–1863). He proved

that this point occurs at the location where the angle

between each of the line segments AP, BP, and CP is

120°. To see this, first suppose that the point Pis

placed at a fixed distance from Cso that it lies on a cir-

cle with Cas its center. Then a study of

OPTIMIZATION

shows that the location on the circle that minimizes the

sum AP + BP + CP occurs at the point Pwhere the

lines AP and BP make equal angles to the line tangent

to the circle at P. Thus the solution to this restricted

version of the problem occurs when the angles between

line segments AP, BP, and CP are equal. The same

occurs for a solution with Pa fixed distance from Aor

Pa fixed distance from B. As the solution to the gen-

eral problem simultaneously solves each restricted ver-

sion of the problem, all three angles must be equal.

Since the three angles sum to 360°, each must therefore

equal 120°.

The Steiner point solves the following road-build-

ing problem:

What design of a road system connects three

towns using the minimal total length of road?

The solution is a design that connects each town with a

straight segment of road directly to the Steiner point of

the three towns.

Steiner also analyzed road-building problems that

involve more than three towns. He showed that given

Ntowns, N≥3, it is necessary to introduce N– 1 spe-

cial points between the towns and draw straight-line

segments between these points and the towns in such a

way that: