MATH 147 Part I Outline
Plimpton 322

Mathematics is as old as humanity;
some of the earliest written artifacts are mathematical.

Plimpton 322 was created by an unknown
Babylonian Scribe around the year 1850 BC.

Babylonian mathematics employed a sexagesimal
numeration system, a positional notation based on powers of 60.

Babylonians knew the content of the
Pythagorean Theorem 1300 years before Pythagoras lived.
Know how to

recognize Babylonian numerals.

convert whole numbers and fractions
from sexagesimal into decimal and back (with fractions in lowest terms).
Pythagoras

Founder of a 5th century, BCE, school
of philosophy that flourished for many centuries; it believed that everything
reduced to number.
Zeno

Heraclitus, Parmenides and Zeno were
among the earliest of Greek philosophers, flourishing in the 6th century,
BC.

A proof by contradiction, or reductio
ad absurdum, proves that something true by assuming its falseness then
deducing something known to be false from this assumption.

Zeno formulated four paradoxes, discussed
by Aristotle in his Physics about two hundred years later, the Dichotomy,
the Achilles, the Arrow, and the Stadium. Each provides a different argument
to discredit the Heraclitean philosophy that all things are in motion.
The Dichotomy and Achilles arguments assume that spacetime is continuous
while the Arrow and Stadium assume that it is discrete. Be able to
explain the content of each of the paradoxes.

An infinite series converges to a sum
S if the sequence of its partial sums gets closer and closer to
S as more and more terms are added in.

The geometric series formula:
a + ar + ar^{2} + ^{...} = a/(1
 r), valid when r < 1.
Know how to

tell when an argument represents a
proof by contradiction.

form the sequence of partial sums for
an infinite series.

sum a geometric series, in particular,
those which correspond to infinite repeating decimal numbers.
Hippocrates of Chios

The area, or quadrature, problem: Given
a plane figure, find a square of the same area or determine how many times
bigger the figure is than a given square.

Three straightedgeandcompass problems
from ancient geometry went unsolved for centuries, until they were proven
to be impossible in the 19th century: the quadrature of the circle, the
trisection of angles, the duplication of the cube.

The quadrature problem for polygons
was easily handled by ancient mathematicians, but the quadrature problem
for the circle ultimately requires the determination of p.
Ancient geometers devised various approximation methods for this.

Hippocrates of Chios lived in the fifth
century, BC; unsuccessful as a merchant, he became a geometer/philosopher
in Athens and was the first to determine the exact quadrature of a figure
with curved sides, a type of lune. His work is known only through references
in other later works.

A lune is the figure bounded by two
intersecting circles that lies outside one circle but inside the other.
Know how to

find the areas of simple polygonal
regions, and how to recognize these regions using integral notation.
The integral
represents the area of the region
above the xaxis, between the vertical lines x = a
and x = b, and below the curve with equation y
= f(x).

find the area of Hippocrates' lune.
Eudoxus

Eudoxus of Cnidos lived in the 4th
century, BC, was a student of Plato in Athens, and developed the method
of exhaustion for approximating the area of the circle aribtrarily closely
with inscribed polygons.

A double reductio ad absurdum
argument is used to show that X = Y by showing instead that X
> Y and X < Y are both impossible.

Eudoxus' proof that circles are as
the squares on their diameters appears as Proposition xii.2 in Euclid's
Elements.

Euclid's Elements, written in
the 4th century, was a compendium of Greek mathematics in 13 books that
proved to be the most influential mathematical work of all time.
It organized its results in a very spare literary form: definitions of
terms, axioms taken as first principles and needing no justification, then
proposition after proposition, stated and proved. It became the model
for mathematical writing to the present day.
Know how to

recognize the format of a double reductio
ad absurdum proof.

recognize an application of the method
of exhaustion.

use proportional ratios to compare
areas of similar figures.
Eratosthenes

Stoic philosopher of the 3rd century,
BCE, who became one of the Librarians at the Alexandrian Museum, an important
center of learning for the Hellenistic world.

Eratosthenes used a gnomon (a shadowcasting
stick) to conduct a simple measurement of the radius of the earth at the
time of the summer solstice in Alexandria; the angle of the shadow and
the known distance to the town of Syene on the equator allowed him to apply
similar triangles to achieve remarkable accuracy for so crude a measurement.
Know how to

measure an angle in radians.
Aristarchus

A 3rd century, BCE, mathematician and
astronomer, author of a work called On the Sizes and Distances of the Sun
and Moon, in which he applies simple geometry of similar triangles to certain
astronomical observations derived from the behavior of eclipses to measure
the distances to the sun and moon, as well as the relative sizes of these
bodies compared to the earth.
Archimedes

Greatest mathematician of the ancient
world.

The son of an astronomer, he lived
in the 3rd century, BCE, and traveled widely across the Greek world, from
his birthplace in Syracuse to Alexandria and elsewhere to study with other
philosophers and mathematicians.

His fame as an engineer was heralded
by ancient historians; he is legendary for having successfully thwarted
the Roman naval siege of Syracuse for many months during the Second Punic
War by inventing many military devices for the defense against the Roman
legions.

The problem of the duplication of the
cube inspired Menaechmus to use conic section curves, perhaps for the first
time in history.

Archimedes quoted wellknown facts
about the conic sections and the properties of lines tangent to the curve
from a lost work of Euclid.

The Quadrature of the Parabola
describes this result of Archimedes: the area of a segment of a parabola
is equal to 4/3 the area of the triangle inscribed in the segment with
the same base and same height.

He discovered the result by means of
a mechanical thought experiment in which he balanced the weight of the
parabolic segment with that of the triangle, but he proved this result
by a double reductio ad absurdum employing Eudoxus' method of exhaustion.
Know how to

recognize the types of conic section
(parabola, ellipse and hyperbola) and how they are found by cutting a cone
with a certain vertex angle by a plane perpendicular to the side of the
cone

identify the elements of a parabola
(axis/diameter, vertex, base of segment, vertex of segment)

use Archimedes proposition on the area
of a parabolic segment to calculate integrals of quadratic polynomials
Apollonius

Lived in the third century BCE and
died in Alexandria.

Like Euclid's Elements, Apollonius'
Conics was a very influential work in Greek mathematics; it displaced
all earlier treatises in the subject by presenting a unified theory of
conic sections.

Apollonius showed how all conic curves
can be formed from sections of the same cone; earlier theorists had required
different cones for different curves. He coined the terms parabola,
hyperbola and ellipse, whose origins come from the nature of the symptoms
of the curves they represent.
Know how to

identify equations (symptoms) of the
various conic sections and their graphs:

given a point on a parabola with given
equation, find the tangent line to a parabola (by Apollonius' Proposition
i.33) and its equation.
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last modified 9/29/02