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Ptolemy, Hipparchus, Diophantus, Pappus, Hypatia

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Ptolemy, Hipparchus, Diophantus, Pappus, Hypatia Ptolemy, Hipparchus, Diophantus, Pappus, Hypatia Chapters 5 & 6 Astronomy • We talked a little about astronomy and calendaring in Egypt and Babylon. • In Greece, there began to be models of the visible universe, most based on Plato’s injunction to “save the appearances presented by the planets” by admitting as hypotheses “uniform motions, uniform and perfectly regular.” By this was meant circular motion, so… • Most of these models were based on circles or spheres. Eudoxus • We have already mentioned Eudoxus of Cnidos in the context of 1) the method of exhaustion, which he developed, and 2) a theory of ratios presented in Euclid’s Elements. He also developed astronomy into a mathematical science, and put forward a model of concentric spheres, centered at the earth, which rotated about different axes to account for the various motions of the sun, moon, stars, and planets. The axis of rotation of each sphere was not fixed in space but, for most spheres, this axis was itself rotating as it was determined by points fixed on another rotating sphere. Eudoxus • The axis of rotation of each sphere was not fixed in space but, for most spheres, this axis was itself rotating as it was determined by points fixed on another rotating sphere. • The combination allowed for modeling retrograde motion of the planets, but didn’t explain changes in brightness. Apollonius • We have already talked about Apollonius in conjunction with his (yawn) Conics. However, he also developed a model of the visible universe that included several innovations: • He placed the center of the sun’s orbit not at the earth, but at a point called the eccenter some distance from the earth. Apollonius • The sun, moon, and planets moved in small circles, called epicycles, whose centers themselves followed the circular orbits (deferent circles) about the earth or about an eccenter. • By combining these ideas – Explained planetary motion – Explained changes in brightness Moving to Prediction • Apollonius’ model allowed for calculations of ‐ ‐ and thus predictions of ‐ planetary motion, but these calculations were based on the solution of triangles. Thus the need for a practical method for finding these solutions was born: • Trigonometry – literally, the measure of tri‐ gons, or three‐sided figures. Hipparchus • Hipparchus developed a table of chords; that is, a table that gave the length of a chord subtended by a given angle in a circle of fixed radius. • The fixed radius was 3438, so chosen so that the chord subtended by a angle was equal to the radius. Hipparchus • He created a table of chords for angles in ଵ increments of . This angle was arrived at ଶ essentially by using what we would call “half‐ angle” formulas, beginning with a angle. Ptolemy and the Almagest • Ptolemy wrote Mathematical Collection, a work in 13 volumes. It was the astronomical version of Euclid’s Elements, and replaced all earlier works on astronomy. It was the word on astronomy up to the time of Copernicus. It came to be known as the greatest collection, or megisti syntaxis, or, in Arabic, the al‐ magisti, which became Almagest. Ptolemy and the Almagest • Ptolemy, like Hipparchus, created a chord table in order to do the calculations necessary for astronomical predictions that tested the model. • Unlike Hipparchus, his was given in increments ଵ of . ଶ Ptolemy and the Almagest • Ptolemy calculated chords for angles of and degrees (using equilateral triangles and pentagons respectively), then developed what we would call “difference formulas” in order to get chords of , then half‐angle ଵ ଷ formulas for , and . ଶ ସ Ptolemy and the Almagest • He then used something like linear interpolation to get to chords for angles of ଵ and . As mentioned in the book, the tables ଶ have an accuracy equivalent to about 5 decimal places. • He used a circle of fixed radius 60. • Both he and Hipparchus did their calculations and produced their tables using Babylonian sexagecimal numbers. Ptolemy and the Almagest • Two things to note about Ptolemy’s trigonometry: – Using chords instead of ratios of sides, and using a radius of 60 instead of 1, made calculations more difficult. – Nevertheless, Ptolemy had access to many of the fundamental formulas of modern trigonometry, adapted to chords – including the law of sines, the law of cosines, half‐angle formulas, and addition and subtractions formulas. Heron and Practical Mathematics • As the book notes, from about the second century BCE, the eastern Mediterranean slowly became part of the Roman Empire. The Romans were a practical people, concerned with building and with conquest, but not so much with debate and mathematics. • Practical mathematics involved with such things as architecture, surveying, and feeding troops were more the norm for the Romans. Heron and Practical Mathematics • Heron (sometimes called Hero) was an Alexandrian mathematician who wrote works on surveying (Dioptra), simple mechanics (Mechanics), and measurements of areas and volumes (Metrica). • Included in Metrica is Heron’s famous formula for the area of a triangle in terms of its sides. Heron’s Formula • For a triangle with sides a, b, c: ଵ Let . Then the area of the ଶ triangle is given by: To find the square root, it looks like he used something very like the recursive procedure we sometime use today. An Algorithm for Square Roots An Algorithm for Square Roots Nicomachus • His Introduction to Arithmetic is of some interest, but mainly serves as an example of how much of Greek mathematics was lost during the middle ages in Europe. Diophantus • In many ways, Diophantus can be considered the Greek “Father of Algebra.” His major work, Arithmetica, was divided into 13 volumes of which 10 survive, 6 in Greek and 4 in Arabic. • A set of “indeterminant” problems together with solutions, and a more symbolic way of dealing with them. Diophantus’ Symbolism • Here is an example of one of Diophantus’ equations: • The Greek letters with bars over the top are of course numbers; in this case, 1, 10, 2, and 5. • The ‘ισ is short for ,ίσος, with means “equals” • The inverted trident indicates subtraction of what follows it. Diophantus’ Symbolism • Thus, becomes: This leaves four symbols to explain, and they all deal with the unknown in the equation. Diophantus’ Symbolism The is the unknown, while ஌ stands for the cube of the unknown (from the Greek word κύβος, “cube”). ஌ is the square of the unknown, and M is the “zeroeth” power of the unknown. Thus a literal translation of the above equation becomes: ଷ ଶ ଴ ଴ Diophantus’ Symbolism From ଷ ଶ ଴ ଴ , we have only to put in some addition signs and some parentheses: ଷ ଶ ଴ ଴ Finally, writing coefficients before the variable, letting ଴ , and writing things in decreasing order of degree, we have: ଷ ଶ Diophantus • In his writings, he solved third‐ and fourth‐ degree equations in one unknown, systems of equations in two, three, and four unknowns, and a problem that is equivalent to solving a system of eight equations with twelve unknowns. Diophantus • In calculating what amounts to the product of binomials such as , he knew how to handle the minus signs: – “Wanting [i.e., negativity] multiplied by wanting yields forthcoming [i.e. positivity]; wanting multiplied by forthcoming equals wanting.” • This is evidence that even though Diophantus did not recognize what we would call a negative integer as a number, he did nevertheless know how to deal with them in some sense, during calculations. Diophantus • Another indication that this is so is his ability to subtract, for example, from ଶ to obtain ଶ . The ‐6 didn’t make sense as a number, but it was clear to Diophantus that subtracting 6 in the expression was the correct thing to do. • Further, he showed evidence of combining like terms to simplify, and moving a term from one side of an equation to the other, changing the sign appropriately. Pappus • Wrote a Mathematical Collection, a consolidation of all the geometric knowledge of the time. • The 7th book in the collection, Domain of Analysis, was an attempt to help others gain the understanding they needed from older works, to perform the “analysis” in solving problems. Theon and Hypatia • The last of the Alexandrian mathematicians. • Theon wrote mostly commentary (like Pappus). We have little evidence of original work. • His daughter, Hypatia, is the first woman mathematician for which we have any details of her life. She wrote commentaries, much like her father. Hypatia – Taught philosophy and astronomy as head of the Platonist School in Alexandria. A sought‐after teacher by both Christian and “pagan” scholars. Well respected, and held up as a symbol of virtue by later Christian writers. – Hypatia was a political backer of Orestes, an Imperial Prefect in Alexandria, who opposed the “ecclesiastical encroachment” of Bishop Cyril into governmental affairs. A certain faction was anxious for the two leaders to be “reconciled” (i.e., Orestes would lose, Cyril would win), and Hypatia came to be seen as one impediment to this reconciliation. – In March 415, her chariot was attacked on her way home by a Christian mob, who stripped her and dragged her through the streets to the Caesareum church, where she was brutally killed. Some reports suggest she was flayed with ostraca (potshards) and set ablaze while still alive, though other accounts suggest those actions happened after her death. .
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