In Part 1 we introduced concepts from Tony Crilly's wonderful grand introduction to mathematics book (essentially Chapters 1 -14), and here we continue with Chapters 15 through 19, the timelines that appear at the bottom of each 4-page chapter.

**The additions are in bold.**

Since there are so few (Part 3 will be Chapters 20-50), we may as well list them:

15. Euclid's algorithm

16.

*Logic*

17. Proof

18. Sets

19. Calculus

And folks, Calculus is VERY important.

Think of Calculus as the broadest "dividing line" in Mathematics. Think of it as the "fulcrum."

**Calculus I**is the single most important dividing line in undergraduate Science education, because the REALLY GOOD STUFF in any Science requires you master that one-semester subject. The most concise book on this subject is Schaum's Outline of Calculus (although nothing beats a good Teacher and textbook).

But the most IMPORTANT subject in undergrad education is Calculus IV, that is to say:

**Differential Equations**, with the boring Calc II and Calc III multi-dimensional stuff in between.

Master THAT topic, Diff-E-Q as we called it, and you will recognize at least the Maths parts of the vast majority of the scientific papers that are published.

It's important, and that's why we are exposing this stuff, in this way, and in this order.

Enjoy:

30,000 BC - Paleolithic peoples in Europe make number marks on bones (2)

2000 BC - The Babylonians use symbols for numbers (2)

2000 BC - The Babylonians observe pi is roughly 3 (5)

1950 BC - The Babylonians work with quadratic equations (14)

1800 BC - Fractions are used in Babylonian cultures (3)

1750 BC - The Babylonians compile tables of square roots (4)

1650 BC - The Egyptians make use of unit fractions (3)

700 BC - The Babylonians use zero as a placeholder in their number system (1)

525 BC - The Pythagoreans study geometrically arranged square numbers (4)

525 BC - The Pythagoreans are associated with both perfect and abundant numbers (10)

c. 500 BC - Fragmentary evidence exists for Pascal's triangle in Sanskrit (13)

**c. 450 BC - Zeno ridicules infinitesimals with a paradox (19)**

350 BC - Aristotle rejects an actual infinite (7)

**c. 335 BC - Aristotle formalizes the logic of the the syllogism (16)**

c. 300 BC - The theory of the irrational numbers by Eudoxus is published in Book 5 of Euclid's

*Elements*(4)

300 BC - Euclid's

*Elements*gives a proof that there are infinitely many prime numbers (9)

300 BC - Book 9 of Euclid's

*Elements*discusses perfect numbers (10)

c. 300 BC - The extreme and mean ratio is published in Euclid's

*Elements*(12)

**c. 300 BC - Euclid's**

*Elements*provides the model for mathematical proof (17)**c. 300 BC - Euclid's algorithm is published in Book 7 of**

*Elements*(15)250 BC - Archimedes gives the close approximation of pi of 22/7 (5)

230 BC - Eratosthenes of Cyrene describes a method for sieving out prime numbers from the whole numbers (9)

100 - The Chinese devise a system for calculating with fractions (3)

100 - Nicomachus of Gerasa gives a classification of numbers based on perfect numbers (10)

250 - Diophantus of Alexandria publishes

*Arithmetica*(14)

**300 - Sun Tzu discovers the Chinese Algorithm (15)**

600 - The forerunner of our modern decimal notation is used in India (2)

628 - Brahmagupta uses zero and states rules for its use with other numerals (1)

630 - Brahmagupta gives methods for computing square roots (4)

**810 - Al-Khwarizmi gives the word "algorithm" to mathematics (15)**

825 - Derived from "al-jabr" Al-Khwarizmi gives the word "algebra" to mathematics (14)

830 - Mahavira has ideas on how zero interacts with other numerals (1)

c. 1070 - Omar Khayyam discovers Pascal's triangle, which in some countries is named after him (13)

1100 - Bhaskara uses zero as a symbol in algebra and attempts to show how it is manipulated (1)

1200 - The Hindu-Arabic system of writing numerals 1,...,9, and a zero, spreads (2)

1202 - Fibonacci uses the extra symbol 0 added to the Hindu-Arabic system of numerals 1,...,9 but not as a number on par with them (1)

1202 - Fibonacci publishes the Liber Abaci and Fibonacci numbers (11)

1202 - Fibonacci popularizes the bar notation of fractions (3)

**1202 - Fibonacci publishes work on congruences in**

*Liber Abaci*(15)1303 - Zhu Shijie defines Pascal's triangle and shows how to sum certain sequences (13)

1509 - Paciola publishes The Divine Proportion (12)

1550 - The square root symbol is introduced (4)

1572 - Rafael Bombelli calculates with imaginary numbers (8)

1585 - Simon Stevin sets out a theory of decimal fractions (3)

1591 - Francois Viete writes a mathematical text in terms of letters for knowns and unknowns (14)

1600 - The symbols of the decimal system take their recognizable modern forms (2)

1603 - Pietro Cataldi finds the 6th and 7th perfect numbers, 2^16(2^17 -1) = 8,589,869,056 and 2^18(2^19 - 1) = 137,438,691,328 (10)

1618 - John Napier encounters a constant, e, in connection with logarithms (6)

1639 - Girard Desargues introduces the concept of infinity into geometry (7)

1655 - John Wallis is credited with being the first to use the "love knot" symbol for infinity (7)

1664 - Pascal's paper on the properties of Pascal's triangle is published posthumously (13)

**1660's-1670's - Newton and Leibniz take the first steps in Calculus (19)**

**1687 - Descartes promotes mathematical rigour as a model in his**

*Discourse on Method*(17)1700 - The fractional line "-" is in general use (as is %) (3)

1706 - William Jones introduces the pi symbol (5)

1714 - Leibniz discusses the harmonic triangle (13)

1724 - Daniel Bernoulli expresses the numbers of the Fibonacci sequence in terms of the golden ratio (11)

1727 - Euler uses the notation e in connection with the theory of logarithms; it is sometimes called Euler's number (6)

**1734 - Berkeley draws attention to foundational weaknesses in Calculus (19)**

1742 - Goldbach speculates that every even number (more than 2) is a sum of two primes (9)

1748 - Euler calculates e to 23 digits; he is given the credit for the discovery of the famous formula e^i(pi) + 1 = 0 around this time (6)

1777 - Euler first uses the symbol i to represent the square root of -1 (8)

1761 - Lambert proves that pi is irrational (5)

1806 - Argand's diagrammatic representation leads to the name "Argand diagram" (8)

1811 - Carl Friedrich Gauss works with functions of complex number variables (8)

**1820 - Cauchy formalizes calculus in a rigourous way (19)**

1837 - William R. Hamilton treats complex numbers as ordered pairs of real numbers (8)

**1838 - De Morgan introduces the term "Mathematical Induction" (17)**

**1847 - Boole publishes**

*The Mathematical Analysis of Logic*(16)**1854 - Riemann introduces the Reimann integral (19)**

1872 - Richard Dedekind sets out a theory of irrational numbers (4)

**1872 - Cantor takes a tentative step in the creation of set theory (18)**

1874 - Cantor treats the notion of infinity rigorously, specifying different orders of infinity (7)

1876 - Fechner writes on psychological experiments to determine the proportions of the most "aesthetic" rectangle

1873 - Hermite proves e is a transcendental (6)

**1881 - Venn produces "Venn Diagrams" for sets (18)**

1882 - Lindemann proves that pi is transcendental (5)

1896 - The prime number theorem on the distribution of primes is proved (9)

**1902 - Lebesgue sets out the theory of the Lebesgue integral (19)**

**1910 - Russell and Whitehead attempt to reduce mathematics to logic (16)**

1920's - Emmy Noether publishes papers in the development of modern abstract algebra (14)

1923 - Bartok composes his "Dance Suite", believed to be inspired by the Fibonacci numbers (11)

1930 - Bartel van der Waerden publishes his famous

*Moderne Algebra*(14)

**1931 - Godel proves that any formal axiomatic mathematical system contains undecidable statements (18)**

**1939 - The pseudonym Bourbaki is first used by French mathematicians (18)**

1960s - Abraham Robinson devises a non-standard arithmetic based on the notion of the infinitesimal (7)

1963 - The Fibonacci Quarterly, a journal devoted to the number theory of the Fibonacci sequence, is founded (11)

**1964 - Cohen proves the independence of the continuum hypothesis (18)**

**1965 - Lofti Zadeh develops fuzzy logic (16)**

**1967 - Bishop proves results exclusively by constructive methods (17)**

1966 - Chen Jingrun almost confirms the Goldbach conjecture (9)

**1970's - The Chinese remainder theorem is applied to message encryption (15)**

1975 - The International Organization for Standardization (ISO) defines the A paper size (12)

**1987 - The underground train system in Japan is based on fuzzy logic (16)**

**1976 - Imre Lakatos publishes the influential**

*Proofs and Refutations*(17)2006 - The great prime search project finds the 44th Mersenne prime (with almost ten million digits) and yet another new perfect number can be generated (10)

2007 - e is calculated to 10^11 digits (6)

2007 - Sculptor Peter Randall-Page creates the 70 tonne sculpture "Seed" based on the Fibonacci sequence for the Eden Project in Cornwall, UK (11)

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