The Greek Legacy Flashcards
(23 cards)
What mathematical contributions did the Greeks make?
- Geometry
- Deductive reasoning
- Geometric algebra
- Ratios
- Trigonometry
- Estimation of Earth’s size
- Statics (systems in equilibria)
- Number theory
- Results in quadrature, maxima and minima
What mathematical concepts did the Greeks lack?
- Negative numbers
- Fractional numbers
- Symbolic algebra
- Coordinate geometry
- Functions
- Infinitesimal calculus
Why is Ancient Greece significant in the history of mathematics?
It introduced:
- rigorous proofs
- general statements
- individual attribution
- advanced results beyond geometry
How do we know about Greek mathematics?
Most original writings are lost due to perishable materials and re-use but cross-references in later books preserve knowledge.
What century is Thales of Miletus from? What is he known for?
600 BCE
- Measurement of pyramid height
- Prediction of a solar eclipse
- Proving the triangle in a semicircle right-angle theorem.
What century is Pythagoras from? What is he famous for?
500 BCE
- Pythagorean triples
- Summation of series
- Discovery of incommensurability of square roots
- The philosophy that “all is number”
What century was Zeno of Elea from? What is he known for?
400 BCE
- Paradoxes about continuity and motion (e.g., Achilles and the tortoise)
- Illustrating problems between discrete and continuous concepts
What century are Plato and Aristotle from? How did they influence mathematics?
300 BCE
- Plato’s academy hosted mathematicians
- Aristotle emphasised logical arguments and syllogisms (e.g., all animals have legs, all dogs are animals, so all dogs have legs)
- Both distinguished number from magnitude
What century is Euclid from? What is he famous for?
300 BCE
His work “The Elements” which compiled and extended mathematical knowledge using axioms definitions and rigorous proofs.
What are the three bases in the Elements?
- Definitions
- Common Notions
- Postulates/Axioms
What are the postulates in The Elements?
- To draw a straight line from any point to any point
- To produce a finite straight line continuously in a straight line
- To describe a circle with any center and radius
- That all right angles equal one another
- That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles are less than the two right angles
What are the common notions in The Elements?
- Things which equal the same thing also equal one another
- If equals are added to equals, then the wholes are equal
- If equals are subtracted from equals, then the remainders are equal
- Things which coincide with one another equal one another
- The whole is greater than the part
Worth noting:
no.5 is not always accurate in mathematics (e.g., negative numbers)
What is interesting about the common notions from The Element’s?
These seem to be real life common notions, applied to mathematics. For example, no.5 (the whole is greater than the part) is always true in life but not always true in modern mathematics.
What are the definitions in The Elements?
- A point is that which has no part
- A line is a breadthless length
- The ends of a line are points
- A straight line is a line which lies evenly with the points on itself
Which postulate in The Elements is the odd one out, and why?
- That, if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles
Number 5 is the odd one out, quite obviously from appearance. It is essentially the “parallel line” postulate, but caused quite a bit of controversy.
What century is Archimedes from? What mathematical contributions did he make?
200 BCE
- Quadrature and cubature
- Curved line geometry
- Estimates of pi
- Practical engineering applications
What century is Apollonius from? What did he contribute to mathematics?
200 BCE
- Unified study of conic sections
- Early ideas approaching coordinate geometry influenced by astronomy
What century is Hipparchus from? What is he known for?
200 BCE
- Trigonometry and spherical trigonometry
- Catalogue of stars
- Estimate of earth-moon distance
What century is Ptolemy from? What is his major work?
100 BCE
Almagest - which describes a geocentric model of the universe using Babylonian arithmetic traditions.
This work influenced Galileo and Newton
What century is Pappus from? What contribution did he make?
300 CE
- Generalisations of previous geometric problems
- Three and four line problems.
What century are Theon and Hypatia from? What mathematical developments did they contribute?
400 CE
- Theon edited Euclid’s Elements
- Hypatia helped preserve mathematical knowledge
What century is Diophantus from? What is he known for?
200 CE
- Developing early symbolic algebra
- Studying indeterminate equations which inspired modern algebra
What is interesting about the definitions in The Elements?
That they are new concepts which don’t necessarily exist in real life. For example, the concept of “having no part” does not come from the real world. The rise of philosophy at the same time could explain this.
