The Helenistic Period of Greek Mathematics

Aristarchus of Samos

(ca. 310-230 BC)

He was very knowledgeable in all sciences, especially astronomy and mathematics.

He discovered an improved sundial, with a concave hemispherical circle.

He was the first to formulate the Copernican hypotheses and is sometimes called the Ancient Copernican He countered the nonparallax objection by asserting that the stars to be so far distant that parallax was not measurable.

Wrote On the Sizes and Distances of the Sun and Moon. In it he observed that when the moon is half full, the angle between the lines of sight to the sun and the moon is less than a right angle by 1/30 of a quadrant. From this he concluded that the distance from the earth to the sun is more than 18 but less than 20 times the distance from the earth to the moon. (Actual ). Without trigonometry he was aware of and used the fact that

He also made other trigonometic estimates -- without trigonometry.

Apollonius of Perga
(ca 262 BC - 190 BC)

Apollonius was born in Perga in Pamphilia (now Turkey), but was possibly educated in Alexandria where he spent some time teaching. Very little is known of his life. He seems to have felt himself a rival of Archimedes. In any event he worked on similar problems.

He was known as the ``great geometer" because of his work on conics.

Apollonius wrote many books. All but one are lost. Among those we know he wrote are:

Quick Delivery

Cutting-off of a Ratio

Cutting-off of an Area

On Determinate Section

Tangencies

Vergings (Inclinations)

Plane Loci

Apollonius was 25 years younger than Archimedes, and they together with Euclid stood well above all other mathematicians of the first century of this period. Because of them, this period is sometimes called the ``golden age" of Greek mathematics.

In his book Quick Delivery (lost), he gives the approximation to as 3.1416. We do not know his method.

His only known work is On Conics - 8 Books only 4 survive. Features:

Using the double oblique cone he constructs the conics parabola, ellipse, hyperbola, whose names he fixed for all time.

He made use of the idea of Symptoms which were similar to equations, there results an analytic-like geometry - but without coordinates!

Proposition I-33. If AC is constructed, where |AE| = |ED|, then AC is tangent to the parabola.

Proof. Assume AC cuts the parabola at K. Then CK lies within the parabola. Pick F on that segment and construct a to AD at B and G on the curve. Then

Since G and C lie on the curve, the symptom shows that and implies . This implies from (*) that

Thus . Since |AE| = |DE| it follows that and since |BE|>|EA| it follows that . (Why?) Since AB=EA+EB, this is a contradiction.

Proposition I-34. (ellipse) Choose A so that

Then AC is tangent to the ellipse at C.

The Apollonius model for the sun used epicyles. The sun rotates about a circle whose center revolves about the earth. The directions of rotation are opposed. The ratio

There was also the eccenter model, designed to explain the seasons better.

He improved upon the numbering system of Archimedes by using the base .

In the works of Apollonius, Greek mathematics reached its zenith. Without his predecessors, the foremost being Euclid, Apollonius never would have reached this height. Together, they dominated geometry for two thousand years. With the works of Apollonius, mathematics was now well beyond the reaches of the dedicated amateur. Only a professional would be able to advance this theory further.

Hipparchus
(fl 140 B.C.)

Hipparchus of Nicaea was a scientist of the first rank. So carefully accurate were his observations and calculations that he is known in antiquity as the ``lover of truth." He worked in nearly every field of astronomy and his reckonings were canon for 17 centuries.

Only one work of his remains, a commentary on Phainomena of Eudoxus and Aratus of Soli. We know him, however, from Ptolemy's The Almagest. Indeed the ``Ptolemaic Theory" should be called Hippachian.

His mathematical studies of astronomical models required a computation of a table of sines. He constructed a table of chords for astronomical use. Here a chord was given by Crd

in modern notation.

Hipparchus knew the half angle formula as well.

He could compute the chord of every angle from to 180

Using no doubt Babylonnian observations, he improved the lunar, solar and sidereal years. He reckoned the solar year at 365 days, less 4 minutes, 48 seconds -- an error of 6 minutes from current calculations. He computed the lunar month at 29 days, 12 hours, 44 minutes, 2 seconds -- less than one second off. He also computed the synodic periods of the planets with astonishing accuracy.

He estimated the earth-moon distance at 250,000 miles, less than 5 percent off.

Hipparchus almost concluded the orbit of the earth about the sun to be elliptic through his theory of ``eccentrics" to account for orbital irregularities.

In about 129 B.C., he made a catalogue of 1080 known fixed stars in terms of celestial longitude and latitude. Comparison of his chart with that of Timochares from 166 years earlier, he made his most brilliant observation. Noting a shift in the apparent position of the stars he predicted the precession of the equinoxes -- the advance, day by day, of the moment when the equinoctial points come to the meridian. He calculated the precession to be 36 seconds/year -- 14 seconds slower than the current estimate of 50 seconds.

• Ptolemy was an astronomer/mathematician
• He wrote
• The Geography-- a compilation of places in the known world along with their geographic coordinates.
• Mathematical Collection, later called The Almagist, for the Arabic `al-magisti'.
• Mathematical Collection - 13 books, which contains
1. table of chords - 180 intervals -- R=60, in sexigesimal.
2. He solve triangles, planes and spherical triangles.
3. The book was very algorithmic - this was a text/tutorial not a research monograph.

4. We also find the well known Ptolemy's Theorem.

Theorem.

• The Almagest was the standard of astronomers until the time of Copernicus.

Heron of Alexandria
1 century A.D.

Very little is known of Heron's life. However we do have his book: Metrica which was more of a handbook. In it we find the famous Heron's formula. For any triangle of sides and c, and with perimeter s= a+b+c, the areas is given by

He also gives formulas for the area of regular polygons of n sides, each of length a:

Nicomachus of Gerasa

Nicomachus was probably a neo-Pythagorean as he wrote on numbers and music. The period from 30BC to 641 AD is sometimes called the Second Alexandrian School.

He studied in Alexandria.

Only two of his books are extant:

Introduction to Arithmetic

Introduction to Harmonics

A third book on geometry is lost.

Introduction to Arithmetic - in two books

Book I is a classification of integers no proofs For example we have

Classification of ratios of numbers. Assume a/b is completely reduced form of A/B. (i.e. a and b are relative prime)

Book II discusses plane and solid numbers but with . (Were the proofs removed in the many translations?) He studies the very Pythagorean:ß

triangular numbers square numbers pentagonal numbers hexagonal numbers heptagonal numbers

He notes an interesting result about cubes:

This should be compared with the summation of odd numbers to achieve squares. (Recall, the square numbers of Pythagoras.)

The other known work, Introduction to Arithmetic was a ``handbook'' designed for students, primarily. It was written at a much lower level than Euclid's Elements but was studied intensively in Europe and the Arabic World throughout the early Middle Ages.

Other Great Geometers

Hypsicles of Alexandria (fl. 175 BC) added a fourteenth book to the Elements on regular solids. In short, it concerns the comparison of the volumes of the icosahedron and the dodecahedron inscribed in the same sphere.

In another work, Risings, we find for the first time in Greek mathematics the right angle divided in Babylonian manner into 90 degrees. He does not use exact trigonometry calculations, but only a rough approximation. For example, he uses as data the times of rising for the signs from Aries to Virgo as an arithmetic progression.

He also studied the polygonal numbers, the one of which is given by

Diocles of Carystus, fl 180 BC, invented the cissoid, or ivy shaped curve. It was used for the duplication problem. In modern terms it's equation can be written as

Diocles also solved an open problem posed by Archimedes, that of dividing a sphere into two parts whose volumes have a prescribed ratio.

He was able to solve certain cubics by intersecting an ellipse and a hyperbola.

He studied refraction and reflection in the book On Burning Mirrors.

Nicomedes,(fl. 260 BC), wrote Introduction Arithmetica. He discovered the conchoid and used it for angle trisection and finding two mean proportionals. The conchoid is given by

Eratosthenes of Cyrene, (c. 276 - c. 195 BC)

Eratosthenes achieved distinction in many fields. and rnaked second only to the best in each. His admirers call him the second Plato and some called him beta, indicating that he was the second of the wise men of antiquity. By the age of 40, his distinction was so great that Ptomemy III made him head of the Alexandrian Library.

He wrote a volume of verse and a history of comedy.

He wrote mathematical monographs and devised mechanical means of finding mean proportions in continued proportion between two straight lines. He also invented the sieve for determining primes.

In a remarkable achievement he attempted the measurement of the earth's circumference, and hence diameter. Using a deep well in Syene (nowadays Aswan) and an Obelisk in Alexandria, he measured the angle cast by the sun at noonday in midsummer at both places. He measured the sun to be vertical in Syene and making an angle equal to 1/50 of a circle at Alexandria. With this data he measured the circumference of the earth to be 25,000 miles. Remember, this measurement of the radius of the earth was made in 250 B.C. Here's the diagram:

In modern terms, we have distance = radius angle. Using the angle to be 1/50 of a circle, we obtain

= 25,000 miles

He also calculated the distance between the tropics as of the circumference, which make the obliquity of the ecliptic , an error of 1/2 of one percent.

Having measured the earth he set out to describe it, compiling accounts of all voyagers. He advocated regarding each person not as a Greek of Babylonnian, but as an individual with individual merits.

It is said that in his old age he became blind and committed suicide by starvation.

Perseus (75 BC) discovered spiric sections, curves obtained by cutting solids, not cones. In modern terms a spire is given by

Zenodorus (fl 180 BC) - isoperimetric figures (same perimeter different shape)

Of all regular polygons of equal perimeter, that is greatest in areas has the most angles.

A circle is greater than any regular polygon of equal contour.

Of all polygons of the same number of sides and equal perimeter the equilateral and equiangular polygon is the greatest in area.