Contribution to the Theory of Evolution,” that, over
their 21-year span, outlined the principles of statistical
analysis. His work, however, was not without controversy. While other statisticians at the time, including SIR
RONALD AYLMER FISHER (1890–1962) and WILLIAM
SEALY GOSSET (1876–1937), emphasized the role of
causes rather than correlation, and worked with small
sample sizes in deducing reliable information, Pearson
analyzed large sample sizes and explored overall trends.
This difference in philosophy led to a bitter dispute
between Pearson and Fisher, sufficiently vehement to
cause Fisher to turn down a prestigious position at the
college so as to simply avoid working with Pearson.
Nonetheless, Pearson’s work was duly recognized as significant, and Pearson was awarded the position as the
first Galton Professor of Eugenics at the Galton
National Laboratory of Eugenics. He was chair there
from 1911 to 1933.
Apart from making significant contributions to
the field of statistics, Pearson also practiced law, was
active in politics, and wrote literary works. In 1892
he published The Grammar of Science, a philosophical text that attempted to extend the approach and
methods of science to a wide range of general pursuits. This piece even anticipated some of the ideas of
relativity theory.
Pearson founded the journal Biometrics and was
the editor of Annals of Eugenics. Because of his fundamental work in the development of modern statistics,
many scholars today regard Pearson as the founder of
20th-century statistics. He died in Coldharbour, England, on April 27, 1936.
See also SIR FRANCIS GALTON, HISTORY OF PROBABILITY AND STATISTICS (essay); STATISTICS: INFERENTIAL.
pedal triangle Given a point P in a triangle ABC, the
pedal triangle with respect to P is formed by dropping
perpendicular lines from P to each of the three sides of
the triangle. If X, Y, and Z are the locations at which
these lines meet the sides of the triangle, then triangle
XYZ is the pedal triangle for P. The pedal triangle may
extend outside the triangle if ABC is an obtuse triangle.
The three altitudes of any triangle meet at a common point O called the orthocenter of the triangle. (See
TRIANGLE.) The pedal triangle with respect to O, with
vertices the feet of the altitudes of the triangle, is often
referred to as the pedal triangle of a given triangle. It is
also called the orthic triangle. Geometers have shown
that if one constructs a nested sequence of three orthic
triangles, then the third orthic triangle is similar to the
original triangle.
The pedal circle of a triangle with respect to a
point P is the circle that passes through all three vertices of the pedal triangle with respect to P. The pedal
circle with respect to the orthocenter O is the inscribed
circle of the triangle.
In 1775 Italian mathematician Giovanni Fagnano
proposed the following problem:
Of all triangles inscribed in an acute triangle,
which has the least perimeter?
Using the techniques of CALCULUS, Fagnano was able to
prove that the orthic triangle is the desired inscribed
triangle. (This can also be proved by pure geometric
means.) This observation leads to an interesting result:
if XYZ is the inscribed triangle of least perimeter, then
the two sides X to Y and Y to Z are a solution to the
famous path-walking problem in OPTIMIZATION theory.
Consequently these two sides intercept the “wall” of
the given triangle at equal angles, and so represent the
path a ball would follow when thrown at that side of
the triangle. As the same argument applies to any pair
of sides in the orthic triangle, we have:
The orthic triangle represents the closed path
of a ball bouncing inside an acute triangle.
Mathematicians have proved that this is the only
closed-circuit path a ball could follow inside an acute
triangle. For obtuse triangles there is no inscribed triangle of least perimeter.
pentagram (pentacle, pentalpha, pentangle) The starshaped figure formed by the five diagonals of a regular
pentagon (or equivalently, formed by extending all sides
of a regular pentagon to meet in pairs) is called a pentagram. The Pythagorean Order, a group of followers of
PYTHAGORAS, bestowed deep significance to various
numbers and geometric shapes, including the pentagram. The figure is often referred to as the “Pentagram
of Pythagoras.” Its geometric properties were considered divine.
The GOLDEN RATIO appears a number of times in a
pentagram. For instance, point C divides the length AD
pentagram 387
their 21-year span, outlined the principles of statistical
analysis. His work, however, was not without controversy. While other statisticians at the time, including SIR
RONALD AYLMER FISHER (1890–1962) and WILLIAM
SEALY GOSSET (1876–1937), emphasized the role of
causes rather than correlation, and worked with small
sample sizes in deducing reliable information, Pearson
analyzed large sample sizes and explored overall trends.
This difference in philosophy led to a bitter dispute
between Pearson and Fisher, sufficiently vehement to
cause Fisher to turn down a prestigious position at the
college so as to simply avoid working with Pearson.
Nonetheless, Pearson’s work was duly recognized as significant, and Pearson was awarded the position as the
first Galton Professor of Eugenics at the Galton
National Laboratory of Eugenics. He was chair there
from 1911 to 1933.
Apart from making significant contributions to
the field of statistics, Pearson also practiced law, was
active in politics, and wrote literary works. In 1892
he published The Grammar of Science, a philosophical text that attempted to extend the approach and
methods of science to a wide range of general pursuits. This piece even anticipated some of the ideas of
relativity theory.
Pearson founded the journal Biometrics and was
the editor of Annals of Eugenics. Because of his fundamental work in the development of modern statistics,
many scholars today regard Pearson as the founder of
20th-century statistics. He died in Coldharbour, England, on April 27, 1936.
See also SIR FRANCIS GALTON, HISTORY OF PROBABILITY AND STATISTICS (essay); STATISTICS: INFERENTIAL.
pedal triangle Given a point P in a triangle ABC, the
pedal triangle with respect to P is formed by dropping
perpendicular lines from P to each of the three sides of
the triangle. If X, Y, and Z are the locations at which
these lines meet the sides of the triangle, then triangle
XYZ is the pedal triangle for P. The pedal triangle may
extend outside the triangle if ABC is an obtuse triangle.
The three altitudes of any triangle meet at a common point O called the orthocenter of the triangle. (See
TRIANGLE.) The pedal triangle with respect to O, with
vertices the feet of the altitudes of the triangle, is often
referred to as the pedal triangle of a given triangle. It is
also called the orthic triangle. Geometers have shown
that if one constructs a nested sequence of three orthic
triangles, then the third orthic triangle is similar to the
original triangle.
The pedal circle of a triangle with respect to a
point P is the circle that passes through all three vertices of the pedal triangle with respect to P. The pedal
circle with respect to the orthocenter O is the inscribed
circle of the triangle.
In 1775 Italian mathematician Giovanni Fagnano
proposed the following problem:
Of all triangles inscribed in an acute triangle,
which has the least perimeter?
Using the techniques of CALCULUS, Fagnano was able to
prove that the orthic triangle is the desired inscribed
triangle. (This can also be proved by pure geometric
means.) This observation leads to an interesting result:
if XYZ is the inscribed triangle of least perimeter, then
the two sides X to Y and Y to Z are a solution to the
famous path-walking problem in OPTIMIZATION theory.
Consequently these two sides intercept the “wall” of
the given triangle at equal angles, and so represent the
path a ball would follow when thrown at that side of
the triangle. As the same argument applies to any pair
of sides in the orthic triangle, we have:
The orthic triangle represents the closed path
of a ball bouncing inside an acute triangle.
Mathematicians have proved that this is the only
closed-circuit path a ball could follow inside an acute
triangle. For obtuse triangles there is no inscribed triangle of least perimeter.
pentagram (pentacle, pentalpha, pentangle) The starshaped figure formed by the five diagonals of a regular
pentagon (or equivalently, formed by extending all sides
of a regular pentagon to meet in pairs) is called a pentagram. The Pythagorean Order, a group of followers of
PYTHAGORAS, bestowed deep significance to various
numbers and geometric shapes, including the pentagram. The figure is often referred to as the “Pentagram
of Pythagoras.” Its geometric properties were considered divine.
The GOLDEN RATIO appears a number of times in a
pentagram. For instance, point C divides the length AD
pentagram 387