2 Early Greek Mathematics Flashcards

Question

Euclid’s Elements | BOOK 1: STRUCTURE

Answer 1

* Definitions * Opens abruptly with twenty-three definitions from plane geometry * then follow 5 postulates for his geometry (without proof) geometric in nature * 5 axioms- *48 PROPOSITIONS 34 proofs theorems and 14 constructions *47 is pythagoras' theorem so is 48 *doesn't explain, was supposed to be a textbook for students in Alexandria

Answer 2

three definitions from plane geometry 1. A point is that which has no part 2. A line is breadthless length 12. An acute angle is an angle less than a right angle 23. Parallel lines straight lines in same plane that being produced indefinitely in both directions, do not meet.

Answer 3

five postulates (self-evident truths) for his geometry- geometrical assumptions (unproven) 1. A straight line may be drawn from any point to any other 2. A straight line segment can be produced to any length 3. A circle may be described with any centre and radius 4. All right angles are equal 5. Parallel Postulate: If a straight line falls on two straight lines, making the interior angles on the same side less than two right angles, the two straight lines, if indefinitely produced, meet on the side on which the angles are less than two right angles. (5 is the most important: if less than 180 THEN they will meet) postulate that is it cannot be proven

Answer 4

``` A X I O M S assumptions/logical about logic rather than geometry Now follow five axioms (self-evident GENERAL truths) ``` 1. Things equal to the same thing are equal to one another 2. If equals be added to equals, the wholes are equal 3. If equals be subtracted from equals, the remainders are equal 4. Things which coincide with one another are equal 5. The whole is greater than the part

Answer 5

I had been told Euclid proved things, and was disappointed he started with axioms. At first, I refused to accept them unless my brother could offer me a reason for doing so, but he said “if you don’t accept them, we cannot go on” As I wished to go on, I reluctantly admitted them. Bertrand Russell His demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative form when no figure is drawn, but many of Euclid's earlier proofs fail before this test. HE USED DIAGRAMs

Answer 6

STATEMENT always uses a diagram in his proof diagram QEF --- Compass & straightedge construction only, equilateral triangle * Uses Definition 15 (circle), Postulates 1, 3 and Axiom 1 * Tacit assumption (reasoning from diagram) that two circles intersect - not justifiable from basic assumptions- HE TOOK THIS FOR GRANTED; using a diagram as proof not from axioms.he wanted to prove everything from first principles however he assumed something not from his axioms * QEF ~ quod erat faciendum. That which was to be done. Indicates Proposition I is a construction not a theorem.

Answer 7

In a right-angled triangle, the square on the side subtending the right angle is equal to the sum of the squares on the sides containing the right angle Proof diagram QED * Pythagoras’ Theorem * Euclid’s own proof (?) we think * Square on hypotenuse is divided into two rectangles, having areas, equal in turn, to the squares on the other two sides. * QED ~ quod erat demonstrandum. That which was to be demonstrated. PROVED Indicates Proposition 47 is a theorem not a construction. uses definition 22 on a circle postulates 1 and 4 axiom 2 and previous propositions 4 14 31 41 46

Answer 8

(1628) He was 40 years old before he looked on Geometry, which happened accidently. Being in a Gentlemen’s Library, Euclid’s Elements lay open, and ‘twas 47 El. libri I. He read the Proposition. By G---, sayd he (he would now and then sweare an emphaticall Oath by way of emphasis) this is imposssible. So he reads the Demonstration of it, which referred him back to such a Proposition; which he read. That referred him back to another, which he also read. Et sic deinceps [and so on] that at last he was demonstratively convinced of that trueth. This made him in love with Geometry. (Brief Lives by John Aubrey) read book 1 by referring proposition to proposition referring back

Answer 9

If the square on one side of a triangle equals the squares on the remaining two, then the angle contained by the remaining sides is right Proof diagram QED *CONVERSE of Pythagoras’ Theorem * Final proposition in ELEMENTS I * Proof uses Proposition 47 & Proposition 8. * QED indicates Proposition 48 is theorem NOT construction * Pythagoras’ Theorem & Converse thrilling climax to Book I user's previous propositions 3 8 and 11 47

Answer 10

Elements textbook covering most elementary mathematics Books I - IV: rectilinear figures, circles. Book V: Eudoxus’ theory of proportion. Book VI: similarity. Books VII - IX: number theory. Book X: incommensurables. Books X1-XIII: solid geometry. 1. Plane geometry: Propositions 47, 48 - Pythagoras, Converse 2. Geometrical Algebra: Propositions 12, 13 - Cosine Rule 3. Circles: Proposition 31 - Angle in semicircle is right 4. Construction of regular polygons: 3, 4, 5, 6, 15 sides 5. Theory of proportion: EUDOXUS 6. Similarity: Book V in action 7. Elementary properties of numbers: Euclid’s algorithm 8. Continued proportion: geometric progressions 9. Number theory: infinity of primes, formula for perfects 10. Incommensurables: basis for method of exhaustion 11. Lines, planes in space 12. Areas, volumes: circles, cones, pyramids 13. Regular polyhedra: only five regular polyhedra

Answer 11

Elements IX, Proposition 20: Prime numbers are more than any assigned multitude of prime numbers. (a) Assertion: Given any (finite) set of primes, there is always another prime not in the set. Modern terminology: there are infinitely many primes (b) Euclid only shows: given any three primes, a fourth can be found. Assumes his method generalizes to any finite set of primes. (c) Numbers represented by line segments: UNIT NOT thought a number. (generates the others) (d) References to previously established results: Book VII, 31. (another number theory book) (e) Euclid’s proof is by contradiction (reductio ad absurdum). (f) QED ~ quod erat demonstrandum, it is a theorem being proved PAGE 9 problem studies *sheet handed out look at numbers and generate a prime number by multiplying together and adding 1 (last years exam question)

Answer 12

If numbers are in continued proportion, and there be subtracted from the second and last numbers equal to the first, then as the excess of the second is to the first, so will be the excess of the last to those before it. ``` Numbers in continued proportion: a, ar, . . . , ar^{n−1}, ar^n Interpretation: (ar − a)/a = (ar^n − a)/(a + ar + · · · + arn−1) ``` whence a + ar + · · · + ar^{n−1} = a(r^n − 1)/(r − 1) Tacit assumptions: a not equal to 0, r not equal to 1. Example: 1 + 2 + 2^2 + · · · + 2^{n−1} = 2^{n} − 1.

Answer 13

If numbers beginning with a unit be set out in double proportion, until their sum becomes prime, and if the sum is multiplied into the last, the product will be perfect. Interpretation: (1 + 2 + · · · + 2^{n−1})2^{n−1} perfect when 1 + 2 + · · · + 2n−1 prime, i.e. (2^n − 1)2^{n−1} perfect when 2^{n} − 1 prime. (by previous result we can write and simplify) *** If we have a number n st 2^{n-1} is a prime we get a perfect number if we multiply by the last term in the progression 2^{n-1} *comment on the understanding and importance IN EXAM

Answer 14

*RESEARCH MATHEMATICIAN: not rigourous in SOME places, sometimes assumed things from diagrams which isnt a proper proof. Although the results are very interesting * FOR STUDENTS: no motivating examples, no introduction, not a student friendly book * HISTORIANS OF MATHEMATICS: no dates, no names of people, no attributed people to any of the results.

Answer 15

*unchanged is Mesopotamia, Early Greek, Euclid, Greek calc cubics

Answer 16

Russell Criticizes Euclid's Basics His demonstrations require many axioms of which he is quite unconscious. A valid proof retains its demonstrative form when no figure is drawn, but many of Euclid's earlier proofs fail before this test On the structure: I had been told Euclid proved things, and was disappointed he started with axioms. At first, I refused to accept them unless my brother could offer me a reason for doing so, but he said “if you don’t accept them, we cannot go on” As I wished to go on, I reluctantly admitted them. ‘At eleven, I began Euclid, one of the great events of my life, dazzling as first love. I had not imagined there was anything so delicious in the world.’ Bertrand Russell

Answer 17

doesn't explain, was supposed to be a textbook for students in Alexandria Begins abruptly not very reader friendly begins with 2 useless definitions, what is a part what is a line? NO STUDENT FRIENDLY BORING FOR STUDENTS no motivation for students

Answer 18

48 propositions each of which is meticulously proved by back reference to the initial definitions, postulates, axioms and propositions already established. Propositions themselves comprise 14 straight edge and Compass constructions and 34 theorems the proofs of the former and with qts and Latter qet. Most proofs are illustrated by diagrams. Book one concern plane rectilinear geometry, points, lines, lengths, perpendiculars, parallel, angles, congruences, triangles, quadrilaterals, parallelograms, areas etc. Examples of a construction and 0 are respectively proposition 1 and proposition 47