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Apollonius of Perga (~200BCE) Today’S Plan Studied Conic Sections • Apollonius and Conic Sections • Ptolemy and His Table of Chords

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Apollonius of Perga (~200BCE) Today’S Plan Studied Conic Sections • Apollonius and Conic Sections • Ptolemy and His Table of Chords Apollonius of Perga (~200BCE) Today’s plan Studied conic sections • Apollonius and conic sections • Ptolemy and his table of chords. • Erathostenes and the measurement of the Earth • Start with Ancient and Medieval Chinese Mathematics. MAT 336 Hellenic (click on the link Mathematics After on the class website Activity 2) Euclid http://jwilson.coe.uga.edu/emt725/Conics%20Images/Conics.html Apollonius of Perga (~200BCE) Apollonius of Perga (~200BCE) Studied parabolas. Current applications of properties parabolas Alexandria, worked out more fully and https://www.geogebra.org/m/pgu4vvtj generally than in the writings of (click on the link on the class website Activity 2) others A parabolic dish is a surface with a studied and later taught under cross-sectional shape of a parabola - the followers of Euclid paraboloid- used to direct light or sound waves. This concept is used by Conics - 8 books • Radio telescope antennas “the most and prettiest of these theorems are new, and it was • satellite dishes their discovery which made me aware that Euclid did not work out the syntheses of the locus with respect to three and four lines, but only a • parabolic reflector, chance portion of it, and that not successfully; for it was not possible for the said synthesis to be completed without the aid of the • car's headlights and in spotlights. additional theorems discovered by me.” https://mathbitsnotebook.com/Algebra2/Quadratics/QDParabolaApplied.html Claudius Ptolemy ~100AD Claudius Ptolemy Table of Chords Mathematical Collection (Almagest) - Mathematical Astronomy The idea of angles seems to have appeared in Astronomer and physicist (hence, mathematician of his Greece between 6th and 5th century BCE. era) Trigonometry - Table of chords Trigonometry is the branch of mathematics that studies relationships between side lengths Solved plane triangles and angles of triangles. Solved spherical triangles Ptolemy made a table of correspondence chord of angle α between angles and their chords. denoted by chord(α) R α Claudius Ptolemy ~100AD Claudius Ptolemy ~100AD • Worked with a circle of radius 60 (because he did In the Almagest, Ptolemy announces that he is going to calculations in base 60) Question: Why base 60? employ in general the • Draw AE so angles BAE and DAC sexagesimal system, in order • Computed chord(36°). How? to avoid "the embarrassment are equal. of the fractions (referring to he visualizes the fractional Main idea: computed chord tables of certain angles using system of Egyptian origin, • Then triangles BAE and CAD are which was at his time in • propositions from Euclid’s elements common use) similar. • square roots (or approximations) • Also, triangles DAE and CAB are Example: E bisects DC similar. F is chosen so EF=EB Then (prove it!) DF is the side of a decagon inscribed on a circle with radius DC. He computed DF=chord(36°): DF2=(EF-ED)2=(EB-ED)2=((BD2+ED2)1/2 - ED)2= ((602+302)1/2 - 30)2=(37;4;55)60 Claudius Ptolemy ~100AD Claudius Ptolemy ~100AD • Use to “Ptolemy’s Theorem” to compute the chord of the difference of two arcs and the chord Ptolemy's world map, reconstituted from Ptolemy's Geography (circa 150) in the 15th century, indicating "Sinae" (China) at the extreme right, beyond the island of "Taprobane" (Ceylon or Sri Lanka, oversized) and the of half an arc. "Aurea Chersonesus" (Southeast Asian peninsula). • Using the construction of the regular dodecagon and regular decagon, he could compute the chords of 36° and 30° . • The difference of the chords of 36° and 30° gave him the chord of 6° • Bisecting he got the chord of 3°, 1° 30’ and 45’. • Then he set up upper and lower values of the chord of 1°. • Finally, he set out a table of 360 entires, giving the chord at half-degree increments up to 180°. Beginning of Ptolemy table of chords one of the first scholars to look at the problem of representing large portions of the Earth's surface on a https://www.wilbourhall.org/pdfs/HeibergAlmagestComplete.pdf flat map. Diophantus - (possibly ~250AD) - Alexandria Arithmetica “The Greeks drew upon reasoning, Major advance in the solution of equation: Introduction to symbolism. proof and definition in their Linear and quadratic equations mathematics; and, to a degree Higher degree equations unique among ancient cultures,….” The method of false position The Rainbow of Mathematics: A History of the Mathematical Problem II.8 in Science- Ivor Grattan-Guinnes the Arithmetica (edition of 1670), annotated with ALGEBRA Fermat's comment which became Fermat's Last Theorem. Wikipedia Difficulties Studying Math History of China, India, and Islamic World • Large (in space and time) • Not homogeneous (for instance, different languages in each one) • Western centered approach Ancient Chinese • tendency to see them as alien or exotic • accounts from historians or writers with a variety of agendas. Mathematics- • Lack of documents (destroyed by climate, wars, fires, and Magic Squares people) Reasons to do mathematics How mathematics was done in China, India, and Islamic World China, India, and Islamic World • Astronomy - including computations of the calendar, • How numbers were written: astrology and cosmology.) • Zero • Calculating the direction of Mecca for the Islamic world. • Fraction • Negative numbers • Measuring time. • Very large numbers • Land surveying • Calculation with numbers • Estimating areas and volumes • Counting rods in China • Rules and algorithms (long multiplication • Taxation and division of states. and division, root extraction, etc) • Teaching numeracy to an elite. • Geometry, trigonometry. • Math for the sake of it. • Algebra Counting boards Why mathematics was accepted? • in use by 400 BCE, • made of polished wood with rulings that formed a grid of square cells History of “European” • numerals 1 through 9, were laid out from right to left, from the lowest to the highest power of 10. mathematics • positional stystem • distinction between geometry • Alternating vertical and horizontal forms (concerned with proof) and • Rods with a red dot for positive numbers, and with a black dot algebra (concerned with for negative numbers. calculation, possibly with a set of China, India, and Islamic • To “write” a number on the counting board, its digit characters rules forming an algorithm.) World Mathematics could be placed, one per cell, on one row of the grid. (A blank cell stood for zero.) • 17th century: algebra itself • greater emphasis on started to be considered as • To do an arithmetic problem, two or more numbers could be Paper facsimile is shown above were known to have been number, calculation and used in Edo Period Japan (1603-1867) - No old drawings of placed on neighboring rows, and the results calculated on counting boards survive from China. capable of “proof”. algorithms successive rows of the board, much as we might do today in https://www.maa.org/press/periodicals/convergence/a- working a lengthy addition or multiplication on paper. classic-from-china-the-nine-chapters-numbers-and-units • debate among historians • uniquely Chinese, do not appear to have been used by any about to what extent other civilization. 537 = understanding of • highly developed set of algorithms for multiplication, division, algorithms implies computation of square and cubic roots. existence of proof. • Suggested introduction of negative numbers and flexible mean of solving equations linear and higher degrees with multiples unknowns. Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD ‹ ‹ https://www.dkfindout.com/us/history/ancient-china/chinese-paper-making/.
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