## 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 space-time 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 + ar2 + ...  =  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 straightedge-and-compass 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 x-axis, 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 shadow-casting 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 well-known 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.