History of Mathematics Week 8 Notes Chapter 14: Europe in the Middle Ages • Author chooses to mark end of medieval period in the year 1436, which is probably when al-Kashi (Arabian mathematician who calculated π to 14 decimal places) died and Regiomontanus (Johann M¨uller) was born. • Alcuin of York was hired by Charlemagne to revitalize education in France. He explained that creation took 6 days because 6 is a perfect num- ber. • Early Christian attitudes to scholarship were that it was unnecessary because Jesus had told them everything they needed. All scientific knowl- edge, then, was held by Moors, Arabs, and Jews. • Europeans begin a massive program of translating classics, including Elements, Almagest, and works of al-Khwarizmi. • The word \sine" comes from a mistranslation of \jiba" as \jaib". • Fibonacci's Liber Abaci (13th century) is about methods of computa- tion. It uses Hindu-Arabic numerals, but avoids decimal fractions in favor of unit and sexigesimal fractions. • The Fibonacci sequence first appears in Liber abaci 3 2 • Fibonacci studied the cubic equation x +p 2x + 10x = 2 and proved no rational solutions or solutions of the form a + b. He found an approximate solution, presumably using Horner's method. • Jordanus Nemorarius (likely of the century) was the first to give the law of the inclined plane (F = W sin(θ)), so was an Aristotelian in his physics. Jordanus wrote Arithmetica, which was a common university text until 16th century. He used letters for numbers, and so could describe mathematics ab- stractly. He proved theorems such as factors of perfect numbers are deficient. He also is the first to state the quadratic formula in full generality. • Campanus and Jordanus trisected an angle • Thomas Bradwardine and Nicole Oresme develop theory of exponents, in particular what exponents can be used. Bradwardine showed that corre- lations need not be linear in physics (for example, inverse-square laws are possible), and Oresme develops ideasp leading toward any real number being a possible exponent (gets close to x 2). • Oresme developed the latitude of forms, which showed that an object accelerating linearly covers the same distance as an object with no accelera- tion travelling at the speed of the first half-way. 1 P1 n • Richard Suiseth showed that n=1 2n = 2. Oresme showed that the harmonic series diverges. • By the fifteenth century, England and France were devastated by the Black Death and the Hundred Years War, and so Italian, German, and Polish scholars ascended to prominence. Chapter 15: The Renaissance • Regiomontanus was the most influential mathematician of the 15th cen- tury. He gives tan(90◦) is infinite. His geometric methods were algebraic: in making geometric constructions, he would use algebra to determine certain lengths, for example. He planned to reform astronomy (perhaps championing a heliocentric system), but he died prematurely • Nicolas Chuquet wrote Triparty en la science des nombres in 1484, which remained a manuscript until the 19th century. Chuquet finally gave a notation that made exponents clear. He would write axb as :a:b and ax−b as :a:b:m. He also deals with isolated negative numbers 4x = −2, but generally does not accept 0 as a solution. • The Italian friar Luca Pagioli wrote Summa de arithmetica, geometrica, proportionali et proportionalita in 1487, which collected knowledge in arith- metic, algebra, and geometry. It was a compendium of general mathematical knowledge, lacking in deep substance and modern notation. However, some symbols had been introduced to make ideas clearer. • In the 16th century, Germans took the lead in algebra, to the extent that algebra became known as \the cossic art". They introduced the mod- ern + and - symbols (replacing p. and m.). They also freely acknowledged negative coefficients, although they were not sure what to do with negative or irrational solutions. • In the 1540's, the solution of the cubic and quartic become generally known, marking the beginning of the modern period of mathematics. To solve x3 + px = q, let u and v be such that uv = p=3 and let x = u − v. Then one gets u3 − 3u2v + 3uv2 − v3 + p(u − v) = q u3 − 3(u − v)uv − v3 + p(u − v) = q u3 − v3 = q; which gives a quadratic in u3 (when v is eliminated). This led to solutions involving square roots of negatives that simplify to real solutions, which Car- dan did not understand. To solve quartic x4 + 6x2 + 36 = 60x: (1) add squares to make left-hand side a perfect square: x4 + 12x2 + 36 = 60x + 6x2 (2) add a variable y so that LHS remains square: (x2 + 6 + y)2 = 60x + 6x2 + y2 + 2x2y + 12y (3) choose y so that RHS is a perfect square 2 (4) step 3 results in solving a cubic (5) substitute back • Algebra was king during the 16th century, and advances in geometry were at the hands of artists like Werner and D¨urerin the theory of perspective • The Melancholia of D¨urerwas dated (1514) using a 4 × 4 magic square 3.