Transcript Лекция 1.

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