I. Ancient Period
A. Number systems and arithmetic
1. Many different numeration systems
2. Arithmetic performed with calculating devices, e.g. the abacus
1. Creation of units of measure --- ways to quantify distance, area, volume, time
2. Simple forms of trigonometry and other rules of geometry used for indirect measurement
C. Practical problems
1. Art and architecture: temples, pyramids, geometric patterns
2. Engineering: canals, aqueducts, ship building
3. Economics and government: census, taxes, land surveys
II. Greek Period ( 600 B.C. to 400 A.D.)
A. Greek logic and philosophy
1. First schools of logic, science and mathematics.
2. Development of philosophical systems that valued logic and a rational explanation of natural phenomena.
3. Mathematics viewed as more than a tool to solve practical problems; it was seen as a means to understand divine laws.
B. Euclidean geometry
1. First mathematical system based on the ideas of axioms, postulates theorems and proofs.
2. Basic formulas for the area and volume of geometric shapes and solids.
3. First "textbook" of geometry --- Euclid's Elements
C. First mathematicians known by name.
1. Pythagoras (540 B.C.) – founded the Pythagorean school, discovered the Pythagorean theorem
2. Euclid (300 B.C.) – wrote the “Elements” and established geometry as a logical system based on axioms, postulates, theorems and proof
3. Archimedes (225 B.C.) – discovered fundamental laws of physics, area and volume formulas
4. Eratosthenes (200 B.C.) – sieve for prime numbers, estimated the size of the earth
5. Ptolemy (125) – sphere model of the universe with the earth at the center
III. Hindu-Arabian Period ( 200 B.C. to 1200 A.D. )
A. Development and spread of the Hindu-Arabic number system
1. Numeration system using base 10, positional notation and the zero symbol developed in Hindu India (approx. 200 B.C. to 800 A.D.).
2. The Hindu system is adopted by the Arabs and spread throughout their sphere of influence (approx. 700 A.D. to 1200 A.D.). Later adopted by Europeans to become the system used world-wide today.
B. Preservation of Greek Mathematics
1. After the fall of the Greco-Roman civilization Arab scholars copied and studied Greek mathematical works, principally in Baghdad. These copies were spread throughout the Arab empire.
C. Development of Algebra and Trigonometry
1. Study of cubic and higher degree polynomial equations
2. The English word “algebra” derived from the title of an Arabic textbook
3. Hindu trigonometry, especially sine tables, developed by Arab mathematicians
IV. Period of Transmission ( 1000 – 1500)
A. Discovery of Greek and Hindu-Arab mathematics
1. Greek mathematics texts translated from Arabic into Latin. Greek ideas about logic, geometrical reasoning, and a rational view of the world are re-discovered.
2. Arab works on algebra and trigonometry are also translated into Latin and disseminated throughout Europe.
B. Spread of the Hindu-Arabic numeration system
1. Hindu-Arabic numerals slowly spread over Europe
2. Arithmetic algorithms replace the use the abacus (e.g. long division )
V. Early Modern Period ( 1500 – 1800 )
1. First European text on trigonometry by Regiomontanus (1464)
2. Development of precise trigonometry tables, surveying methods using trigonometry, and mathematical analysis of trigonometric relationships (approx. 1530 – 1600)
1. Logarithms introduced by Napier (1614)
2. Logarithmic tables and the slide rule developed as calculation tools (1620 – 1625)
C. Analytic Geometry
1. Cartesian coordinate system developed by Rene Descartes and Pierre Fermat (1630 – 1640)
2. Synthesis of algebra and geometry
D. Creation of the calculus
1. Calculus co-invented by Isaac Newton and Gottfried Leibniz. Central ideas of the calculus expanded and refined by other mathematicians, especially the Bernoulli family and Leonhard Euler. (approx. 1660 – 1750).
2. Powerful tool to solve scientific and engineering problems, it opened the door to a scientific and mathematical revolution
V. Modern Period ( 1800 – present)
A. Non-Euclidean geometry
1. Gauss, Lobachevsky, and Riemann develop an alternative to Euclidean geometry in the 19th century.
2. The new geometries inspire modern theories of gravitation, space curvature and the physics of the molecular world, e.g. Einstein’s theory of relativity.
B. Statistics and probability
1. Theories of probability and statistics developed by many mathematicians during the 19th and 20th centuries have ever-greater significance in the complex, computerized modern world. Numerous examples include weather prediction, polls and surveys, insurance tables, medical studies, etc.
1. Development of electronic computer hardware and software (1940 – present) allow mathematicians to solve many previously unsolvable problems and opens up new fields of mathematical research.
D. Mathematics as a world-wide language
1. The Hindu-Arabic numeration system and a common set of mathematical symbols are used and understood throughout the world.
2. Mathematics expands into many branches and is created and shared world-wide at an ever-expanding pace.
3. Modern mathematics is now too large a subject to be mastered by any one mathematician