History of Math
Key anecdote with Archimedes
"Eureka, I found it!"
Gnomon
"one that knows or examines",and is the part of a sundial that casts the shadow
An octahedron can be constructed by:
( 1/3√2a3), where a is the edge length
Who was Nicole Oresme?
(1323-1383) ▪ geometric techniques ▪ Configuration: geometric figure with perpendicular lines drawn over a base (like distance = rate*time) ▪ Later influenced Galileo
Ptolomy 1
(323-283 BCE) was a great king / pharaoh of Egypt at the time, and funded Elements
Euclid of Megara
(435-365), also an important philosopher of logic and ethics
Aryabhata was known for?
(476-550 CE) - Sine tables; mathematical methods ◦ Main work: Aryabhatiya - 499 CE ▪ on mathematics and astronomy ◦ Estimates П ≈ 62832/20000 = 3.1416 ◦ Geometric progressions, quadratics, simultaneous and indeterminate equations ◦ Defines the trig functions ◦ gives rules for solving quadratics ◦ uses a more practical alphabetic numeric system (Āryabhaṭa numeration)
Who was Brahmagupta?
(598-c.670 CE) - Indeterminate equations; discovers zero and makes rules for it ◦ Main work: Brahma Sphuta Siddhanta (Extensive Treatise of Brahma ← various translations) ▪ Covers extensively mathematics and astronomy ▪ points out quadratics can have two solutions, one of which would be negative ▪ Also solves quadratics as system of coupled equations, not later seen again until Fermat in around 1657. ◦ Establishes rules for 0, like 1+0 = 1, 1-0 = 1, 1x0 =0, etc. ◦ Generalizes Heron / Qin Juishao's formula to quadrilateral inscribed in a circle ▪ Called a "Cyclic Quadrilateral" ◦ Observes quadratics have two solutions, possibly negative
The Chinese square root algorithm is based on which algebraic formula?
(x + y)^2 = x^2 + 2xy + y^2
The Greeks used the cycle of the ____ for ____ while the cycle of the ____ was for _____ .
,,,
Rhind Mathematical Papyrus
- Egyptian mathematics -It is named after Alexander Henry Rhind, a Scottish antiquarian, who purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum
Hippocrates of Chios
- the first to write a systematically organized geometry textbook called Elements -was an ancient Greek mathematician, geometer, and astronomer, who lived c. 470 - c. 410 BCE.
Ahmose
- wrote the Rhind Mathematical Papyrus - he is the earliest contributor to mathematics whose name is known
What are the five theorems of elementary geometry Thales is credited with?
-A circle is bisected by any diameter. -The base angles of an isosceles triangle are equal. -The angles between two intersecting straight lines are equal. -Two triangles are congruent if they have two angles and one side equal. -An angle in a semicircle is a right angle.
Base 10
-Base 10 refers to the numbering system in common use. -used by Babylonians
important ancient works of Hellenistic Period
-Euclid's Elements - 13 books on Geometry (300 BCE) -Ptolemy's Almagest - 13 books on Astronomy, including trigonometry
What is the work of Heron?
-Hero published a well recognized description of a steam-powered device called an aeolipile (sometimes called a "Hero engine"). -Among his most famous inventions was a windwheel, constituting the earliest instance of wind harnessing on land. -The first vending machine was also one of his constructions, when a coin was introduced via a slot on the top of the machine, a set amount of holy water was dispensed. -Hero also invented many mechanisms for the Greek theater, including an entirely mechanical play almost ten minutes in length, powered by a binary-like system of ropes, knots, and simple machines operated by a rotating cylindrical cogwheel. The sound of thunder was produced by the mechanically-timed dropping of metal balls onto a hidden drum. -The force pump was widely used in the Roman world, and one application was in a fire-engine. -A syringe-like device was described by Heron to control the delivery of air or liquids. -In optics, Hero formulated the Principle of the Shortest Path of Light: If a ray of light propagates from point A to point B within the same medium, the path-length followed is the shortest possible. It was nearly 1000 years later that Alhacen expanded the principle to both reflection and refraction, and the principle was later stated in this form by Pierre de Fermat in 1662; the most modern form is that the path is at an extremum. -A standalone fountain that operates under self-contained hydrostatic energy. (Heron's fountain) -A programmable cart that was powered by a falling weight. The "program" consisted of strings wrapped around the drive axle.
According to Hipparchus' findings, in a circle of radius r, the measure of an ______ is equal to its ______ measure.
-If the radius of the circle is denoted by R, then the chord is related to the sine by the equations -Because the angle or arc was to be measured in degrees and minutes, Hipparchus decided to use the same measure for the radius of the circle
Hellenistic Period
-Like an ancient Renaissance -Starts about the time of Aristotle's death in 322 BCE, ends with the foundation of the Roman Empire at Batlle of Actium
Transmission of ideas to / from China
-Not much is known about the possible transmission of mathematical ideas between China and other cultures before the sixteenth century. All that is known is that there are certain similarities in techniques in the mathematics of China, India, Europe, and the Islamic world. -At the end of the sixteenth century, the Jesuit priest Mateo Ricci (1552-1610) came to China (Fig. 7.20). Ricci and one of his Chinese students, Xu Guangqi (1562-1633), translated the first six books of Euclid's Elements into Chinese in 1607. And although it took many years for the Chinese to understand that the form and content of Euclidean geometry were inseparable (to Western minds, at least), nevertheless from this time period forward, Western mathematics began to enter China and the indigenous mathematics began to disappear.
What rules for numbers did Brahma Sphuta Siddhanta have?
-The sum of two positive quantities is positive -The sum of two negative quantities is negative -The sum of zero and a negative number is negative -The sum of zero and a positive number is positive -The sum of zero and zero is zero -The sum of a positive and a negative is their difference; or, if they are equal, zero -In subtraction, the less is to be taken from the greater, positive from positive -In subtraction, the less is to be taken from the greater, negative from negative -When the greater however, is subtracted from the less, the difference is reversed -When positive is to be subtracted from negative, and negative from positive, they must be added together -The product of a negative quantity and a positive quantity is negative -The product of two negative quantities is positive -The product of two positive quantities is positive -Positive divided by positive or negative by negative is positive -Positive divided by negative is negative. Negative divided by positive is negative -A positive or negative number when divided by zero is a fraction with the zero as denominator -Zero divided by a negative or positive number is either zero or is expressed as a fraction with zero as numerator and the finite quantity as denominator -Zero divided by zero is zero
Hieratic
-a cursive writing system used in the provenance of the pharaohs in Egypt and Nubia -It was primarily written in ink with a reed brush on papyrus, allowing scribes to write quickly without resorting to the time-consuming hieroglyphs.
Sexagesimal (base 60)
-a numeral system with sixty as its base -it was passed down to the ancient Babylonians, and it is still used—in a modified form—for measuring time, angles, and geographic coordinates
What is Elements?
-a systematically organized geometry textbook that is, basic theorems, or building blocks of mathematical theory -13 volumes, building on Aristotle, Eudoxus, Plato, Pythagoras, Thales, and others, almost all of which was already known to earlier mathematicians (mainly his contribution is compiling them into a single source, and including rigorous proofs)
Eratosthenes of Cyrene
-circumference of the Earth along the great circle through Alexandria and Syene -Sieve of Eratosthenes
Thales
-the first known Greek philosopher, scientist and mathematician although his occupation was that of an engineer -He is believed to have been the teacher of Anaximander -he was the first natural philosopher in the Milesian School -none of his writing survives so it is difficult to determine his views or to be certain about his mathematical discoveries
In Ancient Chinese geometry, the relation between a circle's diameter and circumference was taken to be:
-the number used for the ratio of circumference to diameter of a circle was always taken as 3, the same value used by theBabylonians -the Chinese scribe stated not one but four separate formulas by which the calculation of area could be made 1. The rule is: Half of the circumference and half of the diameter are multiplied together to give the area. 2. Another rule is: The circumference and the diameter are multiplied together, then the result is divided by 4. 3. Another rule is: The diameter is multiplied by itself. Multiply the result by 3 and then divide by 4. 4. Another rule is: The circumference is multiplied by itself. Then divide the result by 12.
magnitude
-the size of a mathematical object, a property by which the object can be compared as larger or smaller than other objects of the same kind. More formally, an object's magnitude is an ordering (or ranking) of the class of objects to which it belongs -The Greeks distinguished between several types of magnitude, including: ~Positive fractions ~Line segments (ordered by length) ~Plane figures (ordered by area) ~Solids (ordered by volume) ~Angles (ordered by angular magnitude)
Law of the lever
1. Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline toward the weight that is at the greater distance. 2. If, when weights at certain distances are in equilibrium, something is added to one of the weights, they are not in equilibrium but incline toward the weight to which the addition was made. 3. Similarly, if anything is taken away from one of the weights, they are not in equilibrium but incline toward the weight from which nothing was taken. 6. If magnitudes at certain distances are in equilibrium, other magnitudes equal to them will also be in equilibrium at the same distances.
What approximation for sqrt(2) did the Babylonians use, and why was their "less accurate" approximation not an issue?
1.41421356
Indian Mathematics had numbers up to _ and larger?
10^53 and larger
How many problems are in the 13 books "Arithmetica"?
130 Problems
When was the downfall of indigenous Chinese mathematics?
1368 (fall of Yuan Dynasty, rise of the Ming Dynasty)
How many problems are in Ceyuan haijing?
170 Problems
When was Suan Shu Shu discovered?
1984
How many chapters did Suanxue qimeng consist of?
20 Chapters
How man bamboo strips was the Suan Shu Shu written on?
200
What estimated time period was Suan Shu Shu written?
200 BC
How many sine differences were used in Indian sine tables?
24
How many problems did Suanxue qimeng consist of?
259 Problems
How many problems are in Jade Mirror of the Four Unknowns?
288 Problems
How many volumes did Suanxue qimeng consist of?
3 Volumes
When was the Counting Rod System used?
475 BCE - 1500 CE (2000 years)
What value did Ptolemy assign to the size of the earth?
500 stadia
What was the Classical period of Indian mathematics called?
500CE-1200 CE - "Golden Age"
Hipparchus' development of a half-angle formula allowed him to calculate chord lengths of angles, up to increments of ____, and also chord lengths of _____
7.5 degrees, 60 degrees
How old was Diophantus of Alexandria when he died?
84 years old
In a book Liu Hui commented on and added the problem of calculating height of a distant object, it has ____ chapters
9
What is the Zhoubi suanjing?
A book on Astronomy and practical applications
How was Hypatia of Alexandria murdered?
A mob of Christians gathered, led by a reader (i.e., a minor cleric) named Peter. They kidnapped Hypatia on her way home and took her to the "Church called Caesareum. They then completely stripped her, and then murdered her with tiles." She was still alive though, and Hypatia's flesh was torn off using oyster shells . Afterward, the men proceeded to mutilate her and, finally, burn her limbs.
posteriori
A posteriori knowledge or justification is dependent on experience or empirical evidence (for example "Some bachelors I have met are very happy").
priori
A priori knowledge or justification is independent of experience (for example "All bachelors are unmarried"). Galen Strawson has stated that an a priori argument is one in which "you can see that it is true just lying on your couch. You don't have to get up off your couch and go outside and examine the way things are in the physical world. You don't have to do any science."
Fibonacci numbers (Fibonacci sequence)
A sequence of numbers introduced by Fibonacci
What equation was developed by Brahmagupta?
A= sqrt( (s-a) (s-b) (s-c) (s-d) )
What was Liu Hui know for?
Adding a 10th chapter to Jiuzhang suanshu (Nine Chapters on the Mathematical Art)
Archimedes did what of Exodus?
Advanced the limit method of Exodus
Who gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials?
Al-Karaji
Who was known for binomial theorem and Pascal's triangle?
Al-Karaji
1st systematized study of Algebra
Al-Khwarizmi
Who gave an exhaustive explanation for the algebraic solution of quadratic equations with positive roots?
Al-Khwarizmi
Who popularized Hindu-Arabic numerals including zero?
Al-Khwarizmi
Who had the first systematized study of algebra?
Al-Kwarizmi
Who was known for the law of exponents?
Al-Samaw'al
Rhind Mathematical Papyrus was named after...?
Alexander Henry Rhind
How was the Rhind Mathematical Papyrus found?
Alexander Henry Rhind, a Scottish antiquarian, purchased the papyrus in 1858 in Luxor, Egypt; it was apparently found during illegal excavations in or near the Ramesseum
Ptolemy's work on astronomy was called (the):
Algamest
Which book is Ptolemy most famous for?
Algamest
What did Li Ye like connecting?
Algebra and Geometry
Extreme and mean ratio
Also known as the golden ratio, if their ratio is the same as the ratio of their sum to the larger of the two quantities
Egyptian Fractions
Although they had a notation for 1/2 and 1/3 and 1/4 and so on (these are called reciprocals or unit fractions since they are 1/n for some number n), their notation did not allow them to write 2/5 or 3/4 or 4/7 as we would today.
Put all these philosophers in order: Apollonius, Anaximander, Ptolemy, Eudoxus, Hipparchus
Anaximander, Eudoxus, Apollonius, Hipparchus and Ptolemy
Who was famous for Conics?
Apollonius
Who was more famous for Eccenters, and Epicycles?
Apollonius
What technique did Diophantus use that was similar to another ancient civilization?
Arabian mathematics
Who was the Father of integral calculus?
Archimedes
Archimedes Spiral
Archimedes needed the last inequality to determine the area bounded by one turn of the "Archimedean spiral," the curve given in modern polar coordinates by the equation r = aθ
What are Archimedes credited inventions?
Archimedes' screw, Claw of Archimedes, Heat ray, Mirrors, improving catapults accuracy and power, block-and-tackle pulley systems
Who wrote the book "Mechanics?"
Aristotle
What does Zhoubi suanjing in English?
Arithmetica Classic of the Gnomon and the Circular Paths of Heaven
Who computed first approximation of pi = 3.1416?
Aryabhata
Who computed size of the moon and the distance it was away from Earth?
Aryabhata
Who created first notion of sine tables?
Aryabhata
Who defined trig functions?
Aryabhata
Who discovered that Earth rotates on its axis by looking at the daily (nightly) movement of the stars?
Aryabhata
Who estimated pi to 3.1216?
Aryabhata
Who explained lunar/solar eclipses?
Aryabhata
Who gave rules for solving quadratics?
Aryabhata
Who wrote aryabhatiya and surya-siddhanta?
Aryabhata
Who wrote aryabhatiya?
Aryabhata
Who wrote surya-siddhanta?
Aryabhata
Who wrote Āryabhaṭīya?
Aryabhata
This (Indian Mathematician) had the most accurate approximation of this (trigonometric function).
Aryabhata, sine
What is the story of the fall of mathematics in Islamic Empire?
As early as the tenth or eleventh century, the Abbasid empire began to factionalize and fragment due to increased provincial autonomy and frequent uprisings. By 1258, the little that was left of the Abbasid state was swept away by the Mongol invasion.
Who used Rhetorical Algebra?
Babylonians "I found a stone and did not weigh it..." etc., al-Khwarizmi
As early as our records show, the Chinese used what number system?
Base 10
The Counting Rod System used which base system?
Base 10
What base-number system did the Babylonians use?
Base 60
What base and system did Indian Mathematics use?
Base-10, decimal place value system
Order these Indian Mathematicians from oldest to newest: Aryabhata, Bhāskara, Mahviria, Birahmagupta
Bhāskara, Birahmagupta, Aryabhata, Mahāvīra
Which book / proposition of Elements the Pythagorean theorem in?
Book 1 Prop 47
Which book of Elements you will find the 5 platonic solids in?
Book 13
Which book of Pappus's Collection was the most influential and contains the method of analysis from Greek times?
Book 7
Which book of Elements you will find the proposition there are infinitely many primes?
Book 9
What contained the first clear description of the quadratic formula (the solution of the quadratic equation)
Brahma Sphuta Siddhanta
0 was discovered by which famous Indian Mathematician?
Brahmagupta
Which mathematician developed a formula for the area of a quadrilateral similar to Heron's formula for the area of a triangle?
Brahmagupta
Who discovered zero and made a rule for it?
Brahmagupta
Who wrote ?
Brahmagupta
What is the order of countries developing the Decimal Place System?
China or India → Islam (Baghdad, c. 700) → Europe (via Spain and Italy c. 1000-1100)
According to Katz, who was responsible for spreading rumors of Hypatia's Sorcery that eventually resulted in her horrific death?
Cyril
Describe the decimal system we use today?
Decimal notation often refers to a base-10 positional notation such as the Hindu-Arabic numeral system or rod calculus; however, it can also be used more generally to refer to non-positional systems such as Roman or Chinese numerals which are also based on powers of ten.
Which of the late Greek Mathematicians was the first to use symbolism in his equations?
Diophantus
Who wrote Arithmetica?
Diophantus
Who is the Father of Number Theory?
Diophantus of Alexandria
Who wrote the 13 books "Arithmetica"?
Diophantus of Alexandria - translated in Fermat's time, and was influential then
Who studied equations with rational solutions?
Diophantus of Alexandria ~(on study notes)
Who introduced unknown "x", the algebraic symbolism?
Diophantus of Alexandria ◦ Syncopated Algebra
Who is the Father of Algebra?
Diophantus of Alexandria (debatable with al-Khwarizmi)
Who used Syncopated Algebra?
Diophantus, Brahmagupta (mix of words and symbols)
What was significant about the Counting Rod System?
Dot or blank used for zero and had negative numbers (red rods)
1600 BCE Ancient and Medieval China
Early evidence of number system (carved on bones)
The Greeks believed the universe to be centered around the _______.
Earth
What did Plato conclude about the Earth?
Earth and universe are spheres and move on circular paths, Earth is the center
How did Diophantus' method of false position differ from that of the Egyptians?
Egyptians used the method of false position to solve relatively simple algebraic equations, while Diophantus was more complex
What is Rhetorical Algebra?
Equations are written in full sentences. For example, the rhetorical form of x + 1 = 2 is "The thing plus one equals two" or possibly "The thing plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century.
Who was the father of Geography?
Eratosthenes of Cyrene
List these Helenic (Classical Period) in order, oldest to youngest: Apollonius, Euclid, Eratosthenes, Hipparchus
Euclid, Apollonius, Hipparchus, Eratosthenes
What was the standardized testing for competence to serve in pubic offices in Ancient and Medieval China based off of?
Examination system based on memorization and solving related problems to those in the text of the Ten Mathematical Classics
While working on trigonometry, Ptolemy said nothing about spheres or 3D space. True or False?
False
What is Symbolic Algebra?
Full symbolism is used. Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna and al-Qalasadi, though fully symbolic algebra has been developed by François Viète. Later, René Descartes has introduced the modern notation, and shown that the problems occurring in geometry may be expressed and (hopefully) solved in terms of algebra (Cartesian geometry).
What is the Cyclic Quadrilateral?
Generalizes Heron / Qin Juishao's formula to quadrilateral inscribed in a circle
Hypatias death, or murder some might argue, was the end of ______ mathematics?
Greek
What is the 10th chapter to Jiuzhang suanshu called in Chinese?
Haidao suanjing
Which dynasty is Zhoubi suanjing from?
Han Dynasty (202 BC-AD 220)
Archimedes wanted what on his tombstone?
He had requested that his tomb include a cylinder circumscribing a sphere together with an inscription of what he evidently thought one of his most important theorems, that a cylinder whose base is a great circle in the sphere with height equal to the diameter is 3/2 of the sphere in volume and also has surface area 3/2 of the surface area of the sphere
What did Qin Jiushao (1202-1261) contribute to mathematics?
He introduced techniques for solving certain types of algebraic equations using a numerical algorithm and for finding sums of arithmetic series. He also introduced the use of the zero symbol into written Chinese mathematics. -Qin also recorded the earliest explanation of how Chinese calendar experts calculated astronomical data according to the timing of the winter solstice.
What is remarkable about Qin Jiushao in mathematics?
He is regarded as one of the greatest mathematicians in Chinese history, but Qin did not devote his life to mathematics, instead was accomplished in many other fields and held a series of bureaucratic positions in several Chinese provinces
Who was Li Ye?
He was a Chinese mathematician and scholar, who published and improved the tian yuan shu method for solving polynomial equations of one variable
Greeks (Hellenistic Times) call their land
Hellas
Greeks (Hellenistic Times) call themselves
Hellenes
Greeks (Hellenistic Times) call their society
Hellenic
Who invented a standalone fountain that operates under self-contained hydrostatic energy?
Heron of Alexandria 10 - 70 CE
Who invented a syringe-like device to control the delivery of air or liquids?
Heron of Alexandria 10 - 70 CE
Who invented many mechanisms for the Greek theater, including an entirely mechanical play almost ten minutes in length, powered by a binary-like system of ropes, knots, and simple machines operated by a rotating cylindrical cogwheel?
Heron of Alexandria 10 - 70 CE
Who invented the first vending machine?
Heron of Alexandria 10 - 70 CE
Who invented the force pump was widely used in the Roman world, and one application was in a fire-engine?
Heron of Alexandria 10 - 70 CE
Who invented the windwheel?
Heron of Alexandria 10 - 70 CE
Hieratic vs. Hieroglyphic
Hieratic is much more cursive, having large numbers of ligatures and signs unique to hieratic. However, there is, as might be expected, a certain degree of influence from hieratic in the visual appearance of some signs. One significant difference is that the orientation of cursive hieroglyphs is not constant, reading right to left or left to right depending on the context, whereas hieratic is always read right to left.
Where was our modern day decimal place system introduced?
Hindu-Arabic numeral system, our modern day decimal system, was introduced in India
Who created the first interpolation for finding values of sine?
Hipparchus
Who definitely influenced the Indian development of trigonometry
Hipparchus
Who developed "Half-Angle" formula?
Hipparchus of Bithynia
Who was the Father of Trigonometry?
Hipparchus of Bithynia
Who was the first to write a systematically organized geometry textbook?
Hippocrates of Chios
Who wrote Elements?
Hippocrates of Chios
What famous mathematician's death marked the end of the Greek mathematical tradition in Alexandria?
Hypatia
Who was the first known woman mathematician?
Hypatia of Alexandria
Who was the president of the largest "university" in the known world?
Hypatia of Alexandria
Who was one the first women in math/science to become someone of knowledge?
Hypatia of Alexandria (CE 350 - 415)
How does Hypatia of Alexandria murder fit into history at her time period?
Hypatia of Alexandria, being a women, had a seat of power, and was viewed as a threat to other men, more so than others Some were very angry that she was even allowed to have such a level of power and killed her for being a women and having her position
Who used Symbolic Algebra?
Ibn al-Banna (c. 1300) and al-Qalasadi (c. 1450), but most notably: François Viète. and René Descartes. This is our modern algebra
Modus tollens
If p, then q. If not p, therefore not q.
Modus Ponens
If p, then q. If p, therefore q.
Hypothetical syllogism
If p, then q. If q, then r. If p, therefore r.
Buoyancy was used to determine?
If the crown was made of solid gold
Hexagon construction, dividing a segment into equal parts
In a regular hexagon, the side length is equal to the distance from the center to a vertex, so we use this fact to set the compass to the proper side length, then step around the circle marking off the vertices
What is Fibonacci sequence?
In mathematics, the Fibonacci numbers or Fibonacci sequence are the numbers in the following integer sequence
Islam mathematics was built on what?
Indian mathematics (Aryabhata, Brahmagupta) and Greek mathematics (Euclid, Archimedes, Apollonius)
What could Indians represent in terms of numbers?
Indians could represent very large numbers (base 10) and had numbers up to 10^53 and larger
Nicomachu is best known for?
Introduction to Arithmetic
The original Chinese approximation of 3 for (pi) is similar to what other civilization?
Islamic
Who specifically studied extensive geometry and trigonometry?
Islamic Empire Mathematicians
What was the Treatise on Demonstration of Problems of Algebra?
It laid down the principles of algebra, part of the body of Islamic Mathematics that was eventually transmitted to Europe. In particular, he derived general methods for solving cubic equations and even some higher orders. ▪ One of the greatest achievements of the Arab mathematicians! ▪ This work improves on the earlier Greek mathematical work of Hippocrates and Menaechmus ▪ Solution of cubic equations by intersection of conic sections ▪ solves all 14 varieties of cubic equations ▪ geometric solutions, not trigonometric or algebraic solutions which would come much later
What book reduced a system of equations using techniques with counting rods?
Jade Mirror of the Four Unknowns
Who first discovered Yang Hui triangle?
Jia Xian (1010-1070 AD)
Who was Fibonacci also known as?
Leonardo de Pisa, or Leonardo Bonacci
Liber abbaci (Book of Calculation) was written by who?
Leonardo de Pisa, or Leonardo Bonacci, also known as Fibonacci (1170-1250)
Who solved x^3 + 2x^2 + 10x = 20?
Leonardo de Pisa, or Leonardo Bonacci, also known as Fibonacci (1170-1250)
Fibonacci sequence
Leonardo of Pisa
Who popularized the Hindu-Arabic numerals throughout Europe in the 13th century, (at time still used roman numerals which were cumbersome)?
Leonardo of Pisa
Who were the three contemporaries that made significant contributions to the mathematics in Qin Jiushao?
Li Yhe, Yang Hui, and Zhu Shijie
What did Li Ye tell his son to do?
Li told his son to burn all of his books except for Sea mirror of circle measurements
Most of his achievements and work was devoted to solving area and volume problems.
Liu Hui
The method of double differences is attributed to which mathematician?
Liu Hui
Who added a 10th chapter called Sea Island Mathematical Manual?
Liu Hui
Who created the most accurate estimation of pi in ancient mathematics?
Liu Hui
Who developed the method of double differences?
Liu Hui
Who edited and published the surviving edition of Jiuzhang suanshu?
Liu Hui
Who edited and published the surviving edition of Nine Chapters on the Mathematical art?
Liu Hui
Who used the concept of a decimal number for the first time?
Liu Hui
Who was credited for discovering the method of double differences?
Liu Hui
Liu Hui used what method to show that the value of (pi) was not 3?
Liu Hui used only one inscribed 96-gon to obtain his π inequality, and his results were a bit more accurate than Archimedes'
How did Archimedes die?
Marcellus at bay for months during the siege of Syracuse. Finally,however, probably through treachery, the Romans were able to enter the city. Marcellus gave explicit orders that Archimedes be spared, but Plutarch relates that, "as fate would have it, he was intent on working out some problem with a diagram and, having fixed his mind and his eyes alike on his investigation, he never noticed the incursion of the Romans nor the capture of the city. And when a soldier came up to him suddenly and bade him follow to Marcellus, he refused to do so until he had worked out his problem to a demonstration; where at the soldier was so enraged that he drew his sword and slew him."
What is Shushu Jiuzhang in english?
Mathematical Treatise in Nine Sections
To which subject did Diophantus contribute?
Mathematics (Algebra?)
What is closely related to the mean value theorem from modern calculus?
Mean Speed Rule
List these Post-Helenic (Classical Period) in order, oldest to youngest: Heron, Menelaus, Ptolemy
Menelaus, Ptolemy, Heron
What did Jade Mirror of the Four Unknowns modernize?
Modernized Chinese algebra
As the Greeks studied astronomy, what was the most prominent feature of the moon's appearance in the sky?
Moon's phase
What is the most prominent feature of the moon's appearance in the sky?
Moon's phase
What is Yigu yanduan in English?
New steps in computation
Who later influenced Galileo?
Nicole Oresme
Who wrote Introduction to Arithmetic and Manual of Harmonics?
Nicomachus of Gerasa
Who explained Pythagorean Number Philosophy?
Nicomachus of Gerasa ◦ One of the best and only surviving sources on Pythagorean philosophy ◦ Euclid's books VII-IX in Elements is only other treatise on number theory so far (Diophantus comes later)
What does Jiuzhang suanshu in English?
Nine Chapters on the Mathematical Art
Archimedes was known for...
On plane equilibriums, Stomachion, Quadrature of the parabola, On the sphere and cylinder, On spirals, On conoids and spheroids, On floating bodies, Measurement of a circle, The Sandreckoner, On the method of mechanical problems
Who was Liu Hui?
One of the greatest mathematicians of ancient China
Calculation of pi technique
PROPOSITION 3 The ratio of the circumference of any circle to its diameter is less than 3 1/7 but greater than 3 10/71 Archimedes began with regular hexagons, the ratios of whose perimeters to the diameter of the circle are known from elementary geometry. He then in effect used the following lemmas to calculate, in turn, the ratios to the diameter of the perimeters of regular polygons with 12, 24, 48, and 96 sides
Who wrote Collection?
Pappus of Alexandria
What were 3 Medieval Universities?
Paris, Oxford, Bologna
Yang Hui's diagram is similar to what other figure that many people know of today?
Pascal's Triangle
Dioptra (The Mirror) is written by:
Philippos Monotropos
Ptolemy's work and writing of the planetary movements is called:
Planetary Hypotheses
Ptolemy used _____ Theorem to improve the table of chords from ____ increments to _____ increments in ____ (this last blank being the reference):
Ptolemy's,7.5 degree,1/16 degree,
Who is the creator of what is now known as the Algamest (previously known as the "mathematical collection"?
Ptolomy
Pythagoras
Pythagoras of Samos is often described as the first pure mathematician. He is an extremely important figure in the development of mathematics yet we know relatively little about his mathematical achievements. Unlike many later Greek mathematicians, where at least we have some of the books which they wrote, we have nothing of Pythagoras's writings. The society which he led, half religious and half scientific, followed a code of secrecy which certainly means that today Pythagoras is a mysterious figure.
Who introduced the use of the zero symbol into written Chinese mathematics?
Qin Jiushao
Who wrote Shushu Jiuzhang?
Qin Jiushao (1202-1261)
Who were the four outstanding mathematicians during the 13th century in Ancient and Medieval China?
Qin Jiushao, Li Yhe, Yang Hui, and Zhu Shijie
This (Chinese mathematician) was first to publish a general method of solving ____, using what he described as the ____ rule.
Qin Jiushao, systems of linear congruences, Ta-Yen
What equations form was designed by Diophantous for solving equations?
Quadratic Equations: a^2x+c=bx a^2x=bx+c a^2+bx=c
What is the 10th chapter to Jiuzhang suanshu called in English?
Sea Island Mathematical Manual
What is Ceyuan haijing in English?
Sea mirror of circle measurements
Which Chinese mathematician used trigonometric functions to solve mathematical problems of chords and arcs?
Shen Kuo
Jade Mirror of the Four Unknowns showed what with variables?
Shows how to solve systems of equations up to 4 variables
Indian Mathematics was the first to develop what trigonometry functions?
Sine, Cosine, Tangent, Inverse Sine
What is Syncopated Algebra?
Some symbolism is used but which does not contain all of the characteristic of symbolic algebra. For instance, there may be a restriction that subtraction may be used only once within one side of an equation, which is not the case with symbolic algebra. Syncopated algebraic expression first appeared in Diophantus' Arithmetica, followed by Brahmagupta's Brahma Sphuta Siddhanta.
Which period did Qin Jiushao lived?
Song dynasty
What was Shen Kuo important for?
Spherical Trigonometry and Astronomy
What was the earliest chinese math book?
Suan Shu Shu (book of numbers and computation)
What book had an important influence on the development of mathematics in Japan?
Suanxue qimeng
What was Problem 1 of Liu Hui's the Sea Island Mathematical Manual?
Survey of the sea island Q: Now surveying a sea island, set up two three zhang poles at one thousand steps apart, let the two poles and the island in a straight line. Step back from the front post 123 steps, with eye on ground level, the tip of the pole is on a straight line with the peak of island. Step back 127 steps from the rear pole, eye on ground level also aligns with the tip of pole and tip of island. What is the height of the island, and what is the distance to the pole ? A: The height of the island is four li and 55 steps, and it is 120 li and 50 steps from the pole.
How would the number 6 be represented in ancient China?
T
When was the Ten Mathematical Classics compiled?
Tang dynasty (AD 618-907)
None of _____ writing survives so it is difficult to determine his views or to be certain about his mathematical discoveries.
Thales
Who is believed to have been the teacher of Anaximander?
Thales
List these Pre-Helenic (Classical Period) in order, oldest to youngest: Plato, Thales, Aristotle, Socrates, Eudoxus, Pythagoras
Thales, Pythagoras, Socrates, Plato, Eudoxus, Aristotle
Put all these philosophers in order: Plato, Thales, Euclid, Eratosthenes, Hipparchus, Aristotle, Socrates, Eudoxus, Heron, Menelaus, Ptolemy, Pythagoras, Apollonius
Thales, Pythagoras, Socrates, Plato, Eudoxus, Aristotle, Euclid, Apollonius, Hipparchus, Eratosthenes, Menelaus, Ptolemy, Heron
Aristarchus of Samos concluded what?
That Earth goes around the fixed sun in a circular orbit
What speculation about Shulba Sutras was there?
That Pythagoreans learned some key principles from this text
What is the Chinese Algorithm for calculating square roots?
The Chinese algorithm for calculating square roots is similar to one that was taught in schools in recent years: (x + y)^2 = x^2 + 2xy + y^2
Archimedes Palimpsest
The Method, which was unexpectedly discovered in 1899 in a Greek monastery library in Constantinople. The manuscript, containing several other works of Archimedes as well, is the oldest extant manuscript of Archimedes. It dates from the tenth century, but the writing was partially washed out in the thirteenth century and the parchment reused for a religious work
What is interesting about Diophantus of Alexandria death?
The age of his death is precisely known from epitaph ◦ "In this tomb lies Diophantus. 1/6 marked his childhood, 1/12 marked his adolescense, out of the 7 part of his life, one part went on before he got married (1/7 before marriage), then 5 years later, his life gave him a beloved son, his child passed away at half of his age. He survived this disaster for 4 more years, then passed away. So tell me at what age he passed away?"
What is interesting about Brahma Sphuta Siddhanta?
The book was written completely in verse and does not contain any kind of mathematical notation.
Where did the earliest records of combinatorial rules come from?
The earliest known connection to combinatorics comes from the Rhind papyrus, problem 79, for the implementation of a geometric series.
Describe the methods which Thales might have used to calculate the distance to a ship at sea.
The first method was to have an instrument consisting of two sticks nailed into a cross so that they could be rotated about the nail. An observer then went to the top of a tower, positioned one stick vertically (using say a plumb line) and then rotating the second stick about the nail until it point at the ship. Then the observer rotates the instrument, keeping it fixed and vertical, until the movable stick points at a suitable point on the land. The distance of this point from the base of the tower is equal to the distance to the ship.
What is Qin Jiushao's formula it used for and is it the same as?
The formula finding the area of a triangle from the given lengths of three sides; this formula is the same as Heron's formula, proved by Heron of Alexandria about 60 BCE, though knowledge of the formula may go back to Archimedes
What is the incident of the gold crown and Buoyancy Principle?
The incident of the gold crown and the bath led Archimedes to the study of an entirely new subject, that of hydrostatics, in which he discovered its basic law, that a solid heavier than a fluid will, when weighed in the fluid, be lighter than its true weight by the weight of the fluid displaced. As in his study of levers, Archimedes began the mathematical development of hydrostatics, in his treatise On Floating Bodies, by giving a simplifying postulate. He was then able to show, among other results, that the surface of any fluid at rest is the surface of a sphere whose center is the same as that of the earth. He could then deal with solids floating or sinking in fluids by assuming that the fluid was part of a sphere. Archimedes was able to solve the crown problem by using the basic law, proved as Proposition 7.
Solve 16x^2 + 192*x - 1863.2 = 0 using Qin Jiushao's procedures.
The initial steps in solving such an equation are the same as those in the solution of the pure equation, x^n = b, namely, first, determine the number of decimal digits in the answer and, second, guess the appropriate first digit.
What is Āryabhaṭīya?
The oldest extant Indian work with alphabet numerals. That is, he used letters of the alphabet to form words with consonants giving digits and vowels denoting place value. This innovation allows for advanced arithmetical computations which would have been considerably more difficult without it.
What is the Introduction to Arithmetic?
The only extant work on mathematics by Nicomachus (60-120 AD). It contains both philosophical prose and basic mathematical ideas. Nicomachus refers to Plato quite often, and writes that philosophy can only be possible if one knows enough about mathematics. Nicomachus also describes how natural numbers and basic mathematical ideas are eternal and unchanging, and in an abstract realm. It consists of two books, twenty-three and twenty-nine chapters, respectively.
What is so important about Hypatia of Alexandria that is relevant even today?
The prejudice and bias against women in the sciences is an extremely important and relevant issue even today, with bitter and deep historical roots.
Chinese square root algorithm
The square root algorithm is based on the algebraic formula (x + y)^2 = x^2 + 2xy + y^2
What is geometry?
The study of static objects and space, what their properties are and relations to other objects
What is The Brāhmasphuṭasiddhānta ("Correctly Established Doctrine of Brahma", abbreviated BSS), also translated into Brahma Sphuta Siddhanta?
The text is notable for its mathematical content, as it contains ideas including a good understanding of role of zero, rules for manipulating both negative and positive numbers, a method for computing square roots, methods of solving linear and quadratic equations, and rules for summing series, Brahmagupta's identity, and Brahmagupta's theorem.
What is Āryabhaṭīya significance?
The treatise uses a geocentric model of the solar system, in which the Sun and Moon are each carried by epicycles which in turn revolve around the Earth
Where does "Sine" come from?
The word "sine" comes from a Latin mistranslation of the Arabic jiba, which is a transliteration of the Sanskrit word for half the chord, jya-ardha.
What about their decimal system was significant in Islam mathematics?
They included decimal fractions
To help one read the numbers easily, the Chinese did what to two consecutive arrangements of rods?
They were placed either horizontally or vertically
What did Shushu Jiuzhang cover?
This treatise covered a variety of topics including indeterminate equations and the numerical solution of certain polynomial equations up to 10th order, as well as discussions on military matters and surveying
How did Chinese mathematicians represent numbers greater than 10?
To avoid confusion, vertical and horizontal forms are alternately used. Generally, vertical rod numbers are used for the position for the units, hundreds, ten thousands, etc.
PTOLEMY'S THEOREM
To prove that AC * BD = AB * CD + AD * BC in quadrilateral ABCD, choose E on AC so that ABE = DBC Then ABD = EBC Also BDA = BCA since they both subtend the same arc Therefore, ABD is similar to EBC Hence, BD : AD = BC : EC or AD * BC = BD * EC. Similarly, since BAC = BDC, ABE is similar to DBC Hence, AB : AE = BD :CD or AB * CD = BD * AE Adding equals to equals gives AB * CD + AD * BC = BD * AE + BD * EC = BD(AE + EC) = BD * AC, and the theorem is proved.
What solved all 14 varieties of cubic equations?
Treatise on Demonstration of Problems of Algebra
What was one of the greatest achievements of the Arab mathematicians
Treatise on Demonstration of Problems of Algebra
How did Fibonacci solve x^3 + 2x^2 + 10x = 20?
Using sexagesimal numerals, he computes the answer correct to 6.72e-10 error x = 1,22,7,42,33,4,40 = 1 + 22/60 + 7/60^2 + 42/60^3 + 33/60^4 + 4/60^5 + 40/60^6
deferent circle
When the sun moves uniformly around the new circle
How did Diophantus name exponents?
Words: a sixfold number increased by twelve, which is divided by the difference by which the square of the number exceeds three
What was Yang Hui's Triangle?
Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian who expounded it around 1100 AD, about 500 years before Pascal.
Who wrote Suanxue qimeng and Jade Mirror of the Four Unknowns?
Zhu Shijie
Who was Émilie du Châtelet?
a French mathematician, physicist, and author during the Age of Enlightenment
What is the Ceyuan haijing?
a collection of 170 problems, all related to the same example of a circular city wall inscribed in a right triangle and a square. They often involve two people who walk on straight lines until they can see each other, meet or reach a tree in a certain spot. It is an algebraic geometry book, the purpose of book is to study intricated geometrical relations by algebra.
What is trigonometry?
a general theory for solving geometry problems: measurements are taken and calculations are made
Liber Abaci
a historic book on arithmetic by Leonardo of Pisa, known later by his nickname Fibonacci.
Number
a mathematical object used to count, measure, and label
eccentricity
a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.
apogee
a point of greatest distance of a body in an elliptic orbit about a larger body
perigee
a point of least distance of a body in an elliptic orbit about a larger body
Palimpsests
a reused parchment
Logarithmic Spiral
a self-similar spiral curve which often appears in nature. The logarithmic spiral was first described by Descartes and later extensively investigated by Jacob Bernoulli, who called it Spira mirabilis, "the marvelous spiral".
What area today was known as Oikoumene?
a term originally used in the Greco-Roman world to refer to the inhabited universe (or at least the known part of it).
Hieroglyphic
a variety of Egyptian hieroglyphs commonly used for religious documents written on papyrus, such as the Book of the Dead
What is the Yigu yanduan?
a work of more basic mathematics written soon after Li Ye completed Ceyuan haijing, and was probably written to help students who could not understand Sea mirror of circle measurements. Yigu yanduan consists of three volumes dedicated to solving geometrical problems on two tracks, through Tian yuan shu and geometry.
Archimedes was...
an applied mathematician and inventor of numerous mechanical devices, known and feared for his weapons of war
Katz speculates that the Babylonian study of problems involving quadratic equations was primarily a method for what?
area of a rectangle
Nicomachus separated the integers into which three classes?
arithmetic, geometric, and harmonic
What does Suan Shu Shu in English?
book of numbers and computation
What did the Liber abbaci (Book of Calculation) contain?
contains much of the Arab knowledge of Algebra and Arithmetic, even copying some problems verbatim
What is Liber abbaci (Book of Calculation)?
contains much of the Arab knowledge of Algebra and Arithmetic, even copying some problems verbatim ◦ enlarging, reorganizing and compiling a vast amount of the known mathematics ◦ Profound impact on development of mathematics in Europe ◦ First Western work to use Hindu-Arabic numerals ▪ For centuries afterward many people still used Roman numerals and the abacus anyway ▪ Uses 0-9, negative numbers
What did Shulba Sutras contain?
contains non-axiomatic description of Pythagorean Theorem (i.e. triples for right triangles)
Who was Anaximander?
cosmologist, first Multiverse Theory, first to theorize evolution (biology)
The perpendicular coordinate, measured north and south from the equator, is called?
declination δ
What did Nicole Oresme distinguish?
distinguished the intensio (the degree of heat at each point) and the extensio (as the length of the heated rod). These two terms were often replaced by latitudo and longitudo
When was the key to transmission of mathematics to Europe in Islamic Mathematics?
during the 10th - 12th centuries
What was the Suan Shu Shu?
earliest chinese math book
What did Ancient Babylonians have relating to Astronomy?
extensive astronomical database, later stolen by Alexander the Great during his conquest and subsequently given to his teacher Aristotle
Diophantus used what method similar to the Egyptians?
false position
In book IV, ____ was used by Diophantus, a technique with which he solved many problems.
false position
It is said that over the entrance to Plato's Academy, this Greek phrase was inscribed. "Let no one ignorant of ____ enter here."
geometry
What is in Collection?
geometry, recreational mathematics, doubling the cube, polygons and polyhedra
The sine table was incremented by ____ arcs.
half-chord
What was Pappus of Alexandria part of?
he was part of the Alexandrian school ◦ Katz associates him uncommon clarity in describing mathematical problems and solutions ▪ Basically he is remembered for being a good teacher
What was Émilie du Châtelet known for?
her translation and commentary on Isaac Newton's work Principia Mathematica. The translation, published posthumously in 1759, is still considered the standard French translation
Babylonian Fractions
i) Each figure has a symbol which isn't like the value it represents ii) The value of the figure depends on the position of it within the entire number iii) A zero is needed to mean nothing and also to fill the place of units that are missing
600 BCE Ancient and Medieval China
intellectual flowering
Minutiae
means small
What did red rods mean in the Counting Rod System?
negative numbers
What is highly unusual about Diophantus of Alexandria's life?
nothing much is known about his life, which is unusual for someone so famous
Arithmetica
one of the earliest treatises on algebra by Diophantus
Who was Zhu Shijie, and whay Dynasty did he live in?
one of the greatest Chinese mathematicians living during the Yuan Dynasty
Alternative syllogism
p or q. If not p, therefore q.
Elements books 1-4
plane geometry
Who was Shen Kuo?
polymath, scientist, Chinese mathematician
What does Suan Shu Shu consist of?
problems and solutions
Elements Books 5-10
ratios and proportions
The coordinate along the equator, also measured counterclockwise from the vernal point is know as?
right ascension α
Ptolemy's chord table used a ___ system measuring angles by ____.
sexagesimal, sixteenths
What did Suanxue qimeng show?
showed how to measure different two-dimensional shapes and three-dimensional solids
Elements Books 11-13
spatial geometry
What number system did Indian Mathematics influence?
the Arabic number system development (pre-dates it)
The "eccenter" is defined as:
the center of the sun's orbit at a point
the lune quadrature
the lune of Hippocrates, named after Hippocrates of Chios, is a lune bounded by arcs of two circles, the smaller of which has as its diameter a chord spanning a right angle on the larger circle. Equivalently, it is a non-convex plane region bounded by one 180-degree circular arc and one 90-degree circular arc. It is the first curved figure to have its exact area calculated mathematically.
Eudoxus used...
the method of exhaustion, which is like limits to calculate areas and volumes
geometric
the only one in the strict sense of the word to be called a proportion, the greatest term is to the next greatest as that one is to the next; for example, 3, 9, 27, are in geometric proportion
What form did Diophantus strive to put his solutions into?
the solution of each problem began with the assumption that the answer x, for example, had been found
method of double differences
two differences in Problem 1 of Liu Hui's the Sea Island Mathematical Manual are used in the solution procedure
Who was Mateo Ricci?
was an Italian Jesuit priest and one of the founding figures of the Jesuit China missions; his 1602 map of the world in Chinese characters introduced the findings of European exploration to East Asia
arithmetic
which each consecutive pair of terms differs by the same quantity. For example, 3, 7, 11, are in arithmetic proportion. Among the properties of such a proportion are that the product of the extremes is smaller than the square of the mean by the square of the difference
harmonic
which the greatest term is to the smallest as the difference between the greatest and mean terms is to the difference between the mean and the smallest terms. For example, 3, 4, 6, are in harmonic proportion because 6 : 3 = (6 − 4) :(4 − 3)
What method did the Babylonians use to solve quadratic equations?
x^2 + bx = c and x^2 - bx = c
What did a blank mean in the Counting Rod System?
zero
What did a dot mean in the Counting Rod System?
zero
This famous French mathematician worked closely with Voltaire on a translation of Newton's Principle:
Émilie du Châtelet
What are some important points about Ptolemy?
• Combined and extended the work of Hipparchus and Apollonius ◦ ½ degree chord table ◦ Used circle of radius 60 ◦ used Sexagesimal number system • "Mathematical Collection" in 13 volumes WAS Astronomy until Copernicus ◦ It is the greatest collection = the Almagest (Arabic) • Had a lot of trigonometry ◦ Law of sines ◦ Law of cosines ◦ Half-Angle Formulas ◦ Addition and subtraction formulas for sine and cosine
Number system development
• Decimal place value system: ◦ China or India → Islam (Baghdad, c. 700) → Europe (via Spain and Italy c. 1000-1100) ◦ India, China and Islamic Empire used negative numbers ▪ Chinese had red sticks (Rod system) for negative numbers and did arithmetic with them for commercial and government business as early as 200 BCE! ▪ Brahmagupta gave rules for dealing with negative numbers, also in the context of debts and business c. 620 CE ▪ al-Khwarizmi balances equation with negative numbers, but won't allow negative answers to his algebra problems. He does use negative numbers in a treatise on debts though, as did other Islamic mathematicians. ▪ Greeks were geometrically inclined and thought negative length / area / volume absurd ▪ An argument can be made that negative exponents were in use from Babylonian times, but not for negative numbers as solutions to algebraic equations • "Hindu-Arabic numerals" is the modern term ◦ Arab numerals are the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. ◦ Most common today
What are some important points about Hipparchus of Bithynia?
• Measurement of Trigons (3-sided figures) made somewhat precise • Chord - subtended by a given angle in a circle • 7.5 degree increments, • Fixed radius of 3438, so a 60 degree angle has chord 3438
What are some important points about Apollonius?
• Not spheres, but circular orbits • Earth not exact center of universe anymore • Can account for retrograde motion • CAN account for the seasons • CAN account for relative brightness of the planets throughout the year • Really needs trigonometry though to make predictions
How did Eudoxus of Cnidus contribute to Astronomy?
• Spheres inside spheres, centered at the Earth • Axis of rotation not fixed • Can account for retrograde motion (quite remarkable) • CANNOT account for the seasons • CANNOT account for relative brightness of the planets throughout the year • Similar to Anaximander's Universe (c. 610-546 Bc)
What comprised of the Ten Mathematical Classics?
◦ Arithmetica Classic of the Gnomon ◦ Nine Chapters on the Mathematical Art ◦ Liu Hui's Sea Island Mathematical Manual ◦ Mathematical Classic of Master Sun (4th Century) ◦ Mathematical Classic of Zhang Qiujuan
What is important to note about Indian Mathematics and trigonometry functions?
◦ Basically had "rational approximations" to these "transcendental functions" ◦ Hipparchus definitely influenced the Indian development of trigonometry ◦ Essentially they used crude tables to approximate these functions, but used first (linear) and second (quadratic) order interpolation schemes to correct these estimates ◦ 3.75 degree arcs - very imprecise compared to Ptolemy (0.5 degrees) for example, but better than Hipparchus
Shulba Sutras contains what two important things?
◦ Contains notion of square root 2 ◦ Contains geometric constructions
Euclid is also credited with writing a number of other important works such as...?
◦ Data - plane geometry, studies nature of given information ◦ On Divisions of Figures - geometric divisions ◦ Optics - treatise on perspective ◦ Phenomena - spherical astronomy ◦ Some other works are loosely attributed to Euclid, but it is not sure he is the author
Path of transfer of knowledge in Indian Mathematics?
◦ Greek → India → Arabs → Europeans ◦ India → Greek → India → Arabs → Europeans (speculation, paths of transfer of knowledge is unclear in ancient history)
Who was Al-Kwarizmi?
◦ His work on the 6 quadratic forms and solutions by completing the sqaure ◦ Rhetorical Algebra, but al-Khwarizmi balances equation with negative numbers, but won't allow negative answers to his algebra problems ▪ al-jabr, al-muqabala, algorismi ← Algebra, Balancing, Algorithm ◦ He does use negative numbers in a treatise on debts though, as did other Islamic mathematicians.
What is the Mean Speed Rule?
◦ Is closely related to the mean value theorem from modern calculus. ◦ Sometimes called the Merton mean speed theorem (Work done by the Oxford Calculators) ◦ Stated by Heytesbury in Rules for Solving Sophisms ◦ "When any mobile body is uniformly accelerated from rest to some given degree, it will in that time traverse one-half the distance that it would traverse if, in that same time it were moved uniformly at the final degree of velocity. For that motion, as a while, will correspond to precisely one-half that degree which is its terminal velocity." p. 356
What notable research took place in Ancient and Medieval China?
◦ Numerical calculations ◦ Geometry ◦ Equation solving and solution of linear congruences
What are some important facts on the 13 books "Arithmetica"?
◦ Solving quadratic and cubic equations based on geometric formulations ◦ Solved non-geometric, higher order equations
Who was Al-Karaji?
◦ known for binomial theorem and Pascal's triangle ◦ Also regarded as further free algebra from geoemetry ▪ gave the rules for arithmetic operations for adding, subtracting and multiplying polynomials ▪ dividing polynomials by monomials
Who was Omar Khayyam?
◦ poet and mathematician ◦ known for Rubaiyat (which is a work of poetry) ◦ Treatise on Demonstration of Problems of Algebra, finds general geometric solutions by intersecting conic sections