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Ancient and medieval China Юшкевич А.П. О достижениях китайских ученых в области математики // ИМИ, 1955. № 8. С. 539–572. Математика в девяти книгах / Перевод Э.И. Березкиной. // ИМИ, 1957. № 10. С. 439–513. Раик А.Е. Очерки по истории математики в древности. – Саранск: Мордовское книжное изд-во, 1967. Хуан Т. О древнекитайском трактате “Математика в девяти книгах” в русском переводе”// УМН, 1958, т.13, в.5. С.235–237 Mathematics in China emerged independently by the 11th century BC. The Chinese independently developed very large and negative numbers, decimals, a decimal system, a binary system, algebra, geometry, trigonometry People in China were using written numbers by about 1500 BC, in the Shang Dynasty. Sometime before 190 AD, people in Han Dynasty China began to use the abacus. Mathematics was one of the Six Arts, students were required to master during the Zhou Dynasty (1122 BC - 256 BC). Six Arts have their roots in the Confucian philosophy. http://aleph0.clarku.edu/~djoyce/ma105/china.html Basic mathematical treatises Chu Chang Suan Shu, Nine Chapters on the Mathematical Art (II c. BC) Chou Pei Suan Ching (II c. BC)) The Arithmetical Classic of the Gnomon and the Circular Paths of Heaven Ts'e-yuan hai-ching, or Sea-Mirror of the Circle Measurements,(Чжу ши Цзе, 1303) - fan fa, or Horner's method, to solve equations of degree as high as six, although he did not describe his method of solving equations Shu-shu chiu-chang, or Mathematical Treatise in Nine Sections (ca. 1202 - ca. 1261 A.D.) - a method of solving simultaneous congruences, it marks the high point in Chinese indeterminate analysis. 235 Nine Chapters 方田 Фан тянь (Измерение полей) 粟米 Су ми (Соотношение между различными видами зерновых культур) 衰分 Шуай фэнь (Деление по ступеням) 少廣 Шао гуан 商功 Шан гун (Оценка работ) 均輸 Цзюнь шу (Пропорциональное распределение) 盈不足 Ин бу цзу (Избыток-недостаток) 方程 Фан чэн . 勾股 Гоу гу in Russian (R. I. Berezkina 1957) in German (K. Vogel 1968) in English (D. B. Wagner 1978) This book contains a total of 246 questions. For each question in the book, there is only answer given. The method of solving the question is omitted. Nine Chapters arithmetic Nine Chapters, two false positions 9 x 11y 8 x y 13 10 y x c1 x2 c2 x1 x c1 c2 c1 y 2 c2 y1 y c1 c2 Nine Chapters, two false positions 1 цзинь=16 лан 1 лан = 24 чжоу 9 x 11y 8 x y 13 10 y x x Избыток: x1=2 цзиня, y1=18/11 цзиня, Недостаток: x2=3 цзиня, y2=27/11 цзиня, с1 16 18 13 180 13 8 3 3 14 16 19 11 15 2 16 14 2 1 1 11 16 11 16 11 11 16 11 16 11 16 с2 24 27 13 270 13 1 10 3 21 16 35 11 49 3 24 22 3 1 2 11 16 11 16 11 11 16 11 16 11 16 3 15 49 2 143 15 15 3 2 цзиня 2 цзиня лана 2 цзиня 3 лана 2 цзиня 3 лана 18 чжу 64 64 64 4 4 y 143 / 64 13 9 117 53 53 1 1 цзиня 1цзинь лана 1цзинь 13 ланов 1цзинь 13 ланов 6 чжу 11 / 9 64 64 64 4 4 方程 Фан чэн Geometry Чжан Хэн (II в.н.э.) 10 3,16227 Лю Хуэй (III в.н.э.) Liu Hui =157/50 Цзу Чун-чжи (430-501) Zu Chongzhi 3,1415926<<3,1415927, =355/113 Лю Хуэй «Mathematics of sea island» "There is a pond with a side of 1 zhang = 10 chi. At its center is growing cane that protrudes above the water at 1 chi. If you pull the reeds to the shore, he just touches it. The question is: what is the depth of the water and what is the length of the cane? ". Trigonometry The embryonic state of trigonometry in China slowly began to change and advance during the Song Dynasty (960-1279), where Chinese mathematicians began to express greater emphasis for the need of spherical trigonometry in calendarical science and astronomical calculations. The polymath Chinese scientist, mathematician and official Shen Kuo (1031-1095) used trigonometric functions to solve mathematical problems of chords and arcs. He’s work in the lengths of arcs of circles provided the basis for spherical trigonometry developed in the 13th century by the mathematician and astronomer Guo Shoujing (1231-1316).. «Не надо беспокоиться о своем низком социальном уровне, а надо беспокоиться о своем низком уровне морали» (Чжан Хэн, эпоха династии Хань) Facsimile of Zhu Shijie's Jade Mirror of Four Unknowns Yang Hui, Qin Jiushao, Zhu Shijie all used the Horner-Ruffini method to solve certain types of simultaneous equations, roots, quadratic, cubic,and quartic equations. Yang Hui was also the first person in history to discover and prove "Pascal's Triangle", along with its binomial proof Yang Hui triangle (Pascal's triangle) using rod numerals, as depicted in a publication of Zhu Shijie in 1303 AD There are many summation series equations given without proof in the Precious mirror Chinese literature dating from as early as 650 BC tells the legend of Lo Shu or "scroll of the river Lo".According to the legend, there was at one time in ancient China a huge flood. While the great king Yu (禹) was trying to channel the water out to sea, a turtle emerged from it with a curious figure / pattern on its shell: a 3×3 grid in which circular dots of numbers were arranged, such that the sum of the numbers in each row, column and diagonal was the same: 15, which is also the number of days in each of the 24 cycles of the Chinese solar year. According to the legend, thereafter people were able to use this pattern in a certain way to control the river and protect themselves from floods. Ancient Indian Mathematics Middle of the III millennium BC. Slaveholding states - I millennium BC, the power struggle between warriors (Kshatriyas) and the priests (Brahmins). IX с BC - Communication with Babylon. VI с. BC - The northern part of India captures the Persian king Darius V century BC - Buddhism in the IV. BC comes Alexander of Macedon The IV. BC - The northern and central India unites Gupta dynasty. The I millennium BC - the first sacred books of brahmen («Vedas.») VII-V сс. BC– Sulbasutras IV c. – Siddhanthams 499 – «Aryabhatiya» VI-VIII cc. – the anonymous manuscript on arithmetics and algebra 628 – «Brahmasphutasiddhanta» by Brahmagupta 850 – «Ganita Sara Samgraha» by Mahavira XI c. – «Arithmetic course» by Sridhara XII c. – «Crown of science» by Bhaskara II Aryabhata (476 - 550) Bhaskara I (VII в.) Brahmagupta (598 - 670) Sridhara (VIII-IX сс.) Mahavira (Mahaviracharya) (814/815-880) Aryabhata II (X с.) Bhaskara II (1114 – 1185) Madhava (XIV-XV вв.) Aryabhata (476 – ок.550) π= =62832/20000= = 3.1416 «Aryabhatiya» Дашагитика – introduction of 10 verses Ганитапада – mathematics (33 verses giving 66 mathematical rules without proof) Калакриапада – 5 verses on the reckoning of time and planetary models Голапада – 50 verses being on the sphere and eclipse Brahmagupta (ок. 598 – 670) «Brahmasphutasiddhanta» (628) A debt minus zero is a debt. fortunes - positive numbers A fortune minus zero is a fortune. Zero minus zero is a zero. debts - negative numbers A debt subtracted from zero is a fortune. A fortune subtracted from zero is a debt. The product of zero multiplied by a debt or fortune is zero. The product of zero multipliedby zero is zero. The product or quotient of two fortunes is one fortune. The product or quotient of two debts is one fortune. The product or quotient of a debt and a fortune is a debt. The product or quotient of a fortune and a debt is a debt. "Khandakhadyaka " (665.), fundamental work on astronomy Brahmagupta (ок. 598 – 670) 10 Brahmagupta's Theorem In a cyclic quadrilateral having perpendicular diagonals, the perpendicular to a side from the point of intersection of the diagonals always bisects the opposite side. Brahmagupta (ок. 598 – 670) ax2 + bx = c, where a > 0, b и c – arbitrary 4a2x2 + 4abx = 4ac, 4a2x2 + 4abx + b2 = 4ac + b Brahmagupta also solves quadratic indeterminate equations of the type ax2 + c = y2 and ax2 - c = y2. For example he solves 8x2 + 1 = y2 obtaining the solutions (x, y) = (1, 3), (6, 17), (35, 99), (204, 577), (1189, 3363), ... For the equation 11x2 + 1 = y2 Brahmagupta obtained the solutions (x, y) = (3, 10), (161/5, 534/5), ... He also solves 61x2 + 1 = y2 which is particularly elegant having x = 226153980, y = 1766319049 as its smallest solution. Bhaskara II (1114 – 1185) Six works by Bhaskaracharya are known : Lilavati (The Beautiful) which is on mathematics; Bijaganita (Seed Counting or Root Extraction) which is on algebra; the Siddhantasiromani which is in two parts, the first on mathematical astronomy with the second part on the sphere; t he Vasanabhasya of Mitaksara which is Bhaskaracharya's own commentary on the Siddhantasiromani ; the Karanakutuhala (Calculation of Astronomical Wonders) or Brahmatulya which is a simplified version of the Siddhantasiromani ; the Vivarana which is a commentary on the Shishyadhividdhidatantra of Lalla. y2 = ax2 +1 y2 = ax2 +b a a2 b a a2 b a b 2 2 Bhaskara is finding integer solution to 195x = 221y + 65. He obtains the solutions (x, y) = (6, 5) or (23, 20) or (40, 35) and so on a b a b 2 ab x2 x 12 0 64 «На две партии разбившись, Забавлялись обезьяны, Часть восьмая их в квадрате В роще весело резвилась. Криком радостным двенадцать Воздух свежий оглашали. Вместе сколько, ты мне скажешь, Обезьян там было в роще?» Доказать: 16 120 72 60 48 40 24 2 3 5 6 10 24 40 60 2 3 5 5 24 2 3 9 54 450 75 3 2 3 5 3 Geometry The area of a circle is equal to the area of the rectangle, which party – a semi-circle and semi-diameter. 2 1 2 2 x x 2 2 1 x 4 4 Над озером тихим, с полфута размером, Высился лотоса цвет. 3 Он рос одиноко. И ветер порывом ) фута ( 3 x Отнес его в сторону. Нет 4 Больше цветка над водой, Нашел же рыбак его ранней весной В двух футах от места, где рос. Итак, предложу я вопрос: Как озера вода Здесь глубока? Geometry Аryabhata 1. Square and cube definition, area and volume 2. Triangle and cube areas, π 3. Inexact formula of volume of a sphere 4. Pythagorean theorem 5. Gnomon theory Brahmagupta 1. Brakhmagupta's theorem of the area of a quadrangle 2. Theorems of chords and semi-chords 3. Measurement of a prism, pyramid, approximate formulas for other bodies 4. Tasks about a gnomon Trigonometry sin2 α + sin vers2 α = 4 sin2 (α/2) sin2 (α/2) = (sin vers α) / 2 полухорда ардхаджива джива джайб (впадина) синус Sinus totus Complimenti sinus (cosinus) – синус дополнения, 1620, Гюнтер Sinus versus джива – хорда, ватар «зилл ма'кус» umbra versa тангенс (1583, Финке) «зилл мустав» umbra recta котангенс (Гюнтер, 1620) Π =3,14159265359 3 4 4 4 4 ... 2 3 4 4 5 6 6 7 8 8 9 10 Nilakantha Somayaji (1444-ок.1501) Madhava (1340 - 1425) Paramesvara (ок.1370 - ок.1460) Islamic Mathematics and Mathematicians Baghdad (IX в) Bukhara, Khoresm, Cairo Gazna (X cent), Isfahan (XI cent) Марага (XIII in) Samarkand (XV cent) Scientific centers Abu Ja'far Muhammad ibn Musa AlKhwarizmi (about 790 - about 850) The book about the construction of the astrolabe The book about the actions of the astrolabe The book about the sundial A Treatise on the definition of the era of the Jews and their holidays history Book Algoritmi de numero Indorum in English AlKhwarizmi on the Hindu Art of Reckoning Principle of record of numbers Ways of calculation, elements of "the Indian arithmetics" – addition and subtraction, "doubling" and "bifurcation" operations, multiplication and division. The rule of check by means of the nine Arithmetics of fractions (six-denary and usual) Extraction of a square root 5+9+6+3=23; мерило 23-9-9=5 3+4+1+9=17; мерило 17-9=8 8+5=13; мерило 13-9=4 9+3+8+2=22; мерило 22-9-9=4 5963+3419=9382 Hisab al-jabr w’al-muqabala «Я составил это небольшое сочинение из наиболее легкого и полезного в науке счисления и притом такого, что требуется постоянно людям, в делах о наследовании, наследственных пошлинах, при разделе имущества, в судебных процессах, в торговле и во всех деловых взаимоотношениях, случаях измерения земель, проведения каналов, в геометрических вычислениях и других предметах различного рода и сорта...» ... what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned. «Ты разделил 10 дирхемов на две части, затем умножил каждую часть на себя, затем сложил их и прибавил к ним разность между частями до умножения и в сумме получил 54» 2 2 (10 x) x (10 x) x 54 «После восполнения и противопоставления ты скажешь: 110 дирхемов и два квадрата равны 54 дирхемам и двадцати двум вещам» 2 110 2x 54 22 x «Приведи два квадрата к одному, т.е. возьми половину всего, что у тебя» 55 x 27 11x 2 После противопоставления 28 x 2 11x Al-Khwārizmī's Zīj al-Sindhind Al-Khwārizmī's Geometry 22 3,1428 7 10 3,16227 62832 3,1416 20000 Area of circle sector Volume of the parallelepiped of a circular cylinder, prism, cone, pyramid, truncated square pyramid Measurement of triangles and rectangles Classification of rectangles (square, rectangle, rhombus, rhomboid, rectangle with different angles and sides) Abu Kamil Shuja ibn Aslam ibn Muhammad (ab. 850 – 930). He worked on integer solutions of equations. He also gave the solution of a fourth degree equation and of a quadratic equations with irrational coefficients. Abu Kamil's work was the basis of Fibonacci's books. He lived later than AlKhwarizmi; his biggest advance was in the use of irrational coefficients Mohammad Abu'l-Wafa al'Buzjani (940-998) Kitab fima yahtaj ilayh al-sani ’min al-a’mal al-Handasiyha (”Book on What Is Necessary from Geometric Construction for the Artisan”). Abu'l-Wafa translated and wrote commentaries on the works of Euclid, Diophantus and Al-Khwarizmi. He is best known for the first use of the tangent function and compiling tables of sines and tangents at 15' intervals. This work was done as part of an investigation into the orbit of the Moon. Trigonometric tables! «Kitab fima yahtaj ilayh al-kuttab wa al-ummal min ’ilm al-hisab» («Book on What Is Necessary from the Science of Arithmetic for Scribes and Businessmen») Abu Bakr Muhammad al-Khasan-al-Karaji (early 11 century) 3 k k 1 1 n n 2 n 2 1 k k 3 k 3 1 1 n 2 Abu al-Hasan ibn Marwan al-Sabi Thabit ibn Qurra (836-901) He generalizes Pythagoras’s theorem to an arbitrary triangle, as did Pappus He considers parabolas, angle trisection and magic squares He was regarded as Arabic equivalent of Pappus, the commentator on higher mathematics He was also founder of the school that translated works by Euclid, Archimedes, Ptolemy, and Eutocius contribution to amicable numbers Kitab fi'l-qarastun (The book on the beam balance) Abu Ali al-Husain ibn Abdallah ibn Sina (Avicenna) (980-1037) «Kitab an-Najat (Book of Safety)», «Danish Nameh Alali (Book of knowledge dedicated to Alai Dawlah)», «Remarks and Admonitions Part One: On Logic» «Kitab ash-Shifa (Book of Healing)» «Canon of Medicine (al-Qanun fi al-tibb)» Ibn Sina's wrote about 450 works, of which around 240 have survived. Of the surviving works, 150 are on philosophy while 40 are devoted to medicine, the two fields in which he contributed most. He also wrote on psychology, geology, mathematics, astronomy, and logic. His most important work as far as mathematics is concerned, however, is his immense encyclopaedic work, The Book of Healing. One of the four parts of this work is devoted to mathematics and ibn Sina includes astronomy and music as branches of mathematics within the encyclopaedia. Abu Ali al-Husain ibn Abdallah ibn Sina (Avicenna) (980-1037) X 9k 8 X 2 81k 2 144k 64 81k 2 144k 63 1 Mechanics, Optics Astronomy and astrology Logic از قعر گل سیاه تا اوج زحل کردم همه مشکالت گیتی را حل بیرون جستم زقید هر مکر و حیل هر بند گشاده شد مگر بند اجل Up from Earth's Centre through the Seventh Gate, I rose, and on the Throne of Saturn sate, And many Knots unravel'd by the Road, But not the Master-Knot of Human Fate. Correspondence between Ibn Sina (with his student Ahmad ibn 'Ali al-Ma'sumi) and Abū Rayhān al-Bīrūnī has survived in which they debated Aristotelian natural philosophy and the Peripatetic school. Abu Rayhan began by asking Avicenna eighteen questions, ten of which were criticisms of Aristotle's On the Heavens. Ты спрашиваешь меня — храни тебя Аллах невредимым! — о вопросах, которые ты находишь достойными порицания в словах Аристотеля, в его произведении, известном (под названием) книги «О небе и вселенной»; из них я выбрал те, которые более всего тебя затруднили, и отвечаю на них. Я постарался истолковать и объяснить эти (вопросы) кратко и сжато, ибо кое-какие неожиданные занятия мешают мне подробнее остановиться на каждом из них и ответить на них так, как они того заслуживают. Кроме того, мое послание не запоздало бы до сего времени, если бы законовед Ма'суми мог (своевременно) его переписать ради тебя и включить в свое письмо к тебе. (ибн Сина) Abu Arrayhan Muhammad ibn Ahmad al-Biruni (973-1048) Mathematics and astronomy. Geography. Pharmacology and mineralogy. History and chronology. History of religions. Physics (hydrostatics and very accurate measurements of specific weight) Bīrūnī’s major contribution to astronomy is al-Qānūn almasʿūdī fi’l-hayʾa wa’l-nojūm (Masʿudic canon of astronomy), covering the same ground as Ptolemy’s Almagest but introducing new material. http://www.iranicaonline.org/articles/biruniabu-rayhan-index Masudic canon He concentrated mainly on the applications of spherical trigonometry in astronomy and provided a detailed classification of spherical triangles and their solutions He propounded trigonometric theorems equivalent to those related to the sums and differences of angles He developed his solution to the algebraic equation of the third degree as part of an attempt to compute the sine of 1°; not only defined all the trigonometric functions used today but also discussed methods of computing them from a circle with radius R = 1 (still used for this purpose). «The Book of Instruction in the Elements of the Art of Astrology» Классификация натуральных чисел Четные Нечетные Четно-четные (2n, n>1) Четно-нечетные (2(2n+1)) Четно-четно-нечетные (2n(2k+1)) Нечетно-нечетные ((2k+1)(2m+1)) In mathematical geography Biruni developed a new technique for measuring the difference in longitude between two given cities "Geometry is the science of quantities and amounts with respect to each other, learning about the properties of shapes and figures inherent in the body. It makes the science of numbers from private to general and translates astronomy from the realm of conjecture and assumptions on the ground of the truth" An illustration from al-Biruni's astronomical works, explains the different phases of the moon Абу Али Хасан ал-Хайсам алБасри (Альгазен) (965-1039) He seems to have written around 92 works of which, remarkably, over 55 have survived. The main topics on which he wrote were optics, including a theory of light and a theory of vision, astronomy, and mathematics, including geometry and number theory. Ibn al-Haytham's answer to a geometrical question addressed to him in Baghdad Book of Optics. Ghiyath al-Din Abu'l-Fath Umar ibn Ibrahim Al-Nisaburi al-Khayyami(1048-1131) Khayyam, who stitched the tents of science, Has fallen in grief's furnace and been suddenly burned, The shears of Fate have cut the tent ropes of his life, And the broker of Hope has sold him for nothing! Алгебра как наука о решении уравнений Геометрические построения корней Арифметические идеи, связанные с биномом Теория параллельных прямых, полемика с АлХайсамом, неприемлемость введения движения Развитие теории отношений, стирание грани между числами и иррациональными величинами Календарная реформа Khayyam measured the length of the year as 365.24219858156 days. Two comments on this result. Firstly it shows an incredible confidence to attempt to give the result to this degree of accuracy. We know now that the length of the year is changing in the sixth decimal place over a person's lifetime. Secondly it is outstandingly accurate. For comparison the length of the year at the end of the 19th century was 365.242196 days, while today it is 365.242190 days. I was unable to devote myself to the learning of this algebra and the continued concentration upon it, because of obstacles in the vagaries of time which hindered me; for we have been deprived of all the people of knowledge save for a group, small in number, with many troubles, whose concern in life is to snatch the opportunity, when time is asleep, to devote themselves meanwhile to the investigation and perfection of a science; for the majority of people who imitate philosophers confuse the true with the false, and they do nothing but deceive and pretend knowledge, and they do not use what they know of the sciences except for base and material purposes; and if they see a certain person seeking for the right and preferring the truth, doing his best to refute the false and untrue and leaving aside hypocrisy and deceit, they make a fool of him and mock him. Khayyám works on problems of geometric algebra. First is the problem of "finding a point on a quadrant of a circle. Again in solving this problem, he reduces it to another geometric problem: "find a right triangle having the property that the hypotenuse equals the sum of one leg (i.e. side) plus the altitude on the hypotenuse ".To solve this geometric problem, he specializes a parameter and reaches the cubic equation x3 + 200x = 20x2 + 2000. Indeed, he finds a positive root for this equation by intersecting a hyperbola with a circle. In the Treatise, he wrote on the triangular array of binomial coefficients «Explanations of the difficulties in the postulates in Euclid's Elements» (1077) ОСНОВА – ПРИНЦИПЫ АРИСТОТЕЛЯ - Величины можно делить до бесконечности - Прямую линию можно продолжать до бесконечности - Всякие две пересекающиеся прямые линии раскрываются и расходятся по мере удаления от вершины угла пересечения - Две сходящиеся линии пересекаются, и невозможно, чтобы две сходящиеся прямые линии расходились в направлении схождения - Из двух нервных ограниченных величин меньшую можно взять с такой кратностью, что она превзойдет большую The book consists of several sections on the parallel postulate (Book I), on the Euclidean definition of ratios and the Anthyphairetic ratio (modern continued fractions) (Book II), and on the multiplication of ratios (Book III). Nasir al-Din Muhammad ibn Muhammad ibn al-Hasan al-Tusi (1201-1274) «The collection on arithmetics using the boards and dust» n an r a 6 r (a 1) n a n 244140626 25 1 26 6 256 25 1 64775151 n n 1 n n 2 2 (a b) a a b a b ... bn 1 2 n n n n 1 n 1 m m 1 m He wrote many commentaries on Greek texts. These included revised Arabic versions of works by Aristarchus, Euclid, Apollonius, Archimedes, Hypsicles, Menelaus and Ptolemy. One of al-Tusi's most important mathematical contributions was the creation of trigonometry as a mathematical. In Treatise on the quadrilateral al-Tusi gave the first extant exposition of the whole system of plane and spherical trigonometry. In logic al-Tusi followed the teachings of ibn Sina (five works). Tusi shows how to choose the proper disjunction relative to the terms in the disjuncts. He also discusses the disjunctive propositions which follow from a conditional proposition. Mīrzā Muhammad Tāraghay bin Shāhrukh Ulugh Beg (1394-1449) Ghiyath al'Din Jamshid Mas'ud al'Kashi (1390, in Kashan, Iran – 1450, in Samarkand, Uzbek) «The Reckoners’ Key» (1427) The arithmetic of integers The arithmetic of fractions The calculus of the astronomers on the measurement On finding the unknown al’Kashi applied the method now known as fixed-point iteration to solve a cubic equation having sin10 as a root. «The treatise about a circle» (1424) He calculated to 16 decimal places and considered himself the inventor of decimal fraction π=3,14159265358979325