History of Mathematics Week 6 Notes Chapter 10: Greek Trigonometry and Mensuration

• Law of Cosine appeared in two forms for obtuse and acute angles in Elements of Euclid. Otherwise there was not much organized trigonometry, even though theorems of a trigonometric flavor (dealing with ratios of sides of triangles) had been known to a variety of mathematicians.

• Aristarchus proposed a heliocentric system, but his proposal was lost.

• Aristarchus wrote On the Sizes and Distances of the Sun and Moon, which uses a geocentric universe. He used the “fact” that during a half- moon, the angle sun-earth-moon is 87 degrees, so the distance from the sun to the earth is 18-20 times the distance from the earth to the moon. In real- ity, it is 400 times the distance, but that is because at half-moon, the angle sun-earth-moon is closer to 89.8 degrees.

• Eratosthenes estimated the circumference of the earth at about 25000 miles (actually 24,901 miles at the equator). The original work is lost, but survives through Heron and Ptolemy.

• Eratosthenes is best known for developing the “Sieve of Eratosthenes” for computing prime numbers.

• Hipparchus earned title “father of trigonometry” for computing a table of arcs and chords for angles from 0 to 180 degrees. The work was intended for use in astronomy.

• Menelaus wrote about trigonometry in the plane and on the sphere, showing that spherical triangles are equal if their angles are.

• Ptolemy, who lived in the second century A.D., wrote Mathematical Syntaxis, containing 13 books. Since it was considered “the greatest” math- ematical text, it became known as Almagest.

• The Almagest contained trigonometric tables of chords. Ptolemy mea- sured angles with 360 degrees (as had been customary in Greece before him) and split degrees into minutes and seconds following Babylonian numeration. His tables measured chords of angles from 0 to 180 degrees by every half de- gree.

• The Almagest presented a geocentric system (which is defended by lack of stellar parallax). He moved the earth from the center of the orbit, though, to more accurately match observations. He also used epicycles in his astron- omy.

• Ptolemy wrote an authoritative Geography which introduced latitude and longitude. He proposed orthographic and stereographic projections.

1 • Ptolemy also wrote Optics, where angles of incidence and reflection were studied but incorrectly given.

• Ptolemy wrote Tetrabiblos, which was essentially an astrology book, using the stars to describe a pseudoreligion.

• Heron wrote Metrica where he studied areas of polygons. However, he did not distinguish between exact and approximate answers.

• In Heron’s Geometrica, he asks for the perimeter, diameter, and area of a circle if the sum is given. Even though this is adding unlike quantities (of unlike dimensions), he solves the problem.

• Heron “solved” indeterminate problems, like solving a right triangle given the sum of area and perimeter is 280.

• Heron, in Catoptrics, develops the law of reflection using a least-distance argument.

• He developed a primitive steam engine in his Pneumatics and described the simple machine of the inclined plane (although did not correctly develop its properties)

Revival and Decline of Greek Mathematics

• Perhaps there were always two levels of Greek mathematics, but few works of the lower level survived.

• Rome contributed little to mathematics. The Roman author Vitruvius wrote that the√ three best mathematical discoveries were the 3-4-5 right tri- angle, that 2 is irrational, and Archimedes’s discovery of the fraudulent 1 crown. He also gave the equivalent of π = 3 8 , which is worse an approxima- tion than even Archimedes knew.

• Romans were chiefly interested in engineering and architecture, not pure mathematics.

• Diophantus wrote Arithmetica, a text in the theory of numbers in 13 books. He included a multiplication table up to 10 times 10. Nicomachus wrote Introductio arithmeticae, which reiterated much of Pythagorean sci- ence and math.

• Diophantus did not rely on geometry as a basis for number. He required exact solutions to his problems, even though they were sometimes indeter- minate.

• Diophantus would solve a problem like giving two numbers with sum 20 and product 208 by first calling the numbers 10 − x and 10 + x and then multiplying.

2 • Diophantus would solve a problem like finding a and b such that a2 + b and b2 + a are perfect squares by letting b = 2a + 1 so that automatically a2 + b is a square, even though this does not give all solutions. He then only looked for when b2 +a was of the form (2a−2)2, which was certainly arbitrary.

• Diophantus would obtain exact rational solutions to his problems.

• Pappus studied isoperimetric problems, and as such admired bees for their economy.

• Pappus declared the Problem of Apollonius open and solved it. He went on to generalize it into what became known as the six lines problem.

• In his Collection, Pappus first proved Pappus’s Theorem from calculus.

• Rome was not a scholar-friendly society, and many chose to leave by about 500 A.D. for the near east.

3