Euclid of Alexandria

1. Euclid of Alexandria Born: ~ 325 BC Died: ~ 265 BC

2. During the last part of the fourth century B.C. Alexander the Great set out from Macedonia to conquer the world. Among his many conquests was Egypt where in 332 B.C. Alexander the Great founded the city of Alexandria at the mouth of the Nile River. The city rapidly grew to a population of over a half million in three decades supporting the Library that supplanted the Academy of Alexandria. This was the intellectual focus of the Mediterranean world through both the Greek and Roman periods. [1 p 29] Euclid set up the school of mathematics at Alexandria sometime around 300 B.C. as one of its acting scholars. There is not much known for sure about Euclid before or after his arrival on the African coast, but it is clear he received his schooling in the academy from Plato.[1 p 30] Plato was a student of Socrates who traveled meeting great thinkers to enable him to formulate his own philosophy. In 387 B.C. he returned to Athens where he founded an academy. The most highly regarded subject at the academy was mathematics. The entry to the Academy  was engraved with the words "Let no man ignorant of geometry enter here." Plato basically established geometry as an entrance requirement of his school.  Plato does not have many original mathematical ideas attributed to him, but by mathematical knowledge being requisite forentry and study, the Academy cultivated many outstanding mathematicians. [1 p 28] Preceding Euclid at the Academy was an outstanding mathematician Eudoxus who came to the academy at about the time of its creation and attended Plato's lectures. He traveled to Egypt gathering the discoveries of science. Interested in science he developed an idea for lunar and planetary motion that was felt till the Copernican era. In mathematics Eudoxus is most remembered for two things: one is his theory of proportion and the second is his method of exhaustion. Proportion had been an idea that the Greeks knew was correct but could not establish logically. This left logical holes in many theories and ideas. Eudoxus work filled in many of those holes in Book V of Euclid's Elements.

3. Eudoxus work filled in many of those holes in Book V of Euclid's Elements. In the method of exhaustion Eudoxus was one of the first people to develop a calculus concept when it came to determining the area and volume of geometrical figures. This method used better approximations to determine these values similar to taking a limit.[1 pp 28-29] • Greek geometry is one of the major achievements of the human intellect for both mathematical and historical as well as practical and aesthetic reasons. The ancient Greek mathematicians developed geometry from a practical method of measuring land to a body of knowledge with abstract theorems, constructions, computations tied together by the rules of logic. Greek geometers shared a general philosophy and style such as: • 1. Logical rigor in proving theorems 2. The geometric as opposed to the numerical nature of their mathematics 3. Skillful organization in presentation 4. Axiomatic minimization • There is no question that certain geometric ideas predated classical Greek thought, but the Greeks were the first we look to for geometric theorems proven with logical rigor. [2 p 76] • Although responsible for a number of mathematical treatises Euclid is best remembered for the Elements, a systematic development of Greek Mathematics up to that point in history. This work is divided into 13 books and contains some 465 propositions on plane and solid geometry as well as number theory. Many people refer to it as the greatest mathematics textbook of all time. [2 p 77] Today it is generally agreed that relatively few of the theorems in the book were Euclid's own innovation, but Euclid created a treatise that was so superbly organized it obliterated all other works of its kind.

4. Euclid's genius was not in creating new mathematics but in presenting old mathematics in a clear, organized, and logical fashion. Euclid gave us an axiomatic development of mathematics which was a critical distinction. He began the Elements with 23 definitions, 5 postulates and 5 common notions. These were the foundations or "givens" of his system. This established a clear ordering for what could and could not be used in a proof. This enabled Euclid to not just give proofs but also the axiomatic framework on which they are based. This avoided circular reasoning and filling in gaps with unestablished facts.[1 p 31] • The Elements has been one of the most widely read treatises in its over 2000 year history. It has been read by leaders of the world in their education. Abraham Lincoln recounts how as a young lawyer trying to sharpen his reasoning skills • ... bought the Elements of Euclid, a book twenty-three centuries old... [It] went into his carpetbag as he went out on the circuit. At night... he read Euclid by the light of a candle after other had dropped off to sleep.[1 p 30] • As mentioned earlier Euclid's axiomatic system was an extremely important innovation as to how mathematics developed. His book started with 23 definitions. Here are some examples: • Definition 1: A point is that which has no part • Definition 2: A line is breathless in length • Definition 4: A straight line is a line that lies evenly with itself • Definition 15: A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another • Definition 23: Parallel straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.

5. An important aspect of Greek geometrical thought was that of constructing figures with a straight edge and compass. These constructions constituted the reasoning for many of their results. The postulates that Euclid put forth concerned how such constructions could be done or the results of such constructions. • Postulate 1: It is possible to draw a straight line from any point to any point. • Postulate 2: It is possible to produce a finite straight line continuously in a straight line. • Postulate 3: It is possible to describe a circle with any center and distance (i.e. radius). • Postulate 4: All right angles are equal to each other. • Postulate 5: If a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on the that side on which are the angles less than the two right angles. • The fifth postulate is of course key in defining the essence of Euclidean geometry. Later people questioned this and developed non-Euclidean geometries. • The ideas Euclid proposed in the common notions were used to address the lack of algebraic sophistication of the time. • Common Notion 1: Things which are equal to the same thing are also equal. • Common Notion 2: If equals be added to equals, the wholes are equal. • Common Notion 3: If equals be subtracted from equals, the remainders are equal. • Common Notion 4: Things which coincide with one another are equal. • Common Notion 5: The whole is greater than the part.

6. The Geometry of Euclid and Greeks in general is best characterized by Plato in the Republic: they are not thinking about these figures but of those things which the figures represent; thus it is the square in itself and the diameter in itself which are the matter of their arguments, not that which they draw; similarly, when they model or draw objects, which may themselves have images in shadows or in water, they use them in turn as images, endeavoring to see those objects which cannot be seen otherwise than by thought.[2 p. 76] [1] Dunham, William. Journey Through Genius The Great Theorems of Mathematics. New York: John Wiley & Sons 1990. [2] Dunham, William. The Mathematical Universe. New York: John Wiley & Sons 1994.

7. Euclid was aquatinted with much of Pythagoras's work both in terms of Geometry and number theory. The Pythagoreans classified numbers into 3 types according to the sum of it proper divisors. A number was called abundant if its proper divisors summed to more than the number. For example 12 is abundantsince its proper divisors are 1,2,3,4,6 and 1+2+3+4+6=16>12. A number was called deficient if its proper divisors summed to less than the number. For example 10 isdeficientsince its proper divisors are 1,2,5 and 1+2+5=8<10. A number was called perfect if its proper divisors sum is equal to the number. For example 6 is perfect since its proper divisors are 1,2,3 and 1+2+3=6. Euclid was one of the first to explore the idea of Perfect numbers. He was able to establish a way of classifying perfect numbers. Before we proceed we need to remember a method known for summing a geometric series. A finite geometric series is a series for the form: rn+rn-1+rn-2+…+r2+r+1. This is all of the powers of r up to n. The method used to sum this can is demonstrated below. Let Multiply by r Subtract the expressions Solve for S. In the special case when r=2 we get the following:

8. Euclid's theorem on perfect numbers: If 2n-1 is prime and N=2n-1(2n-1) then N is perfect. Proof: Let p=2n-1 be a prime number other than the number 2. Then the sum of proper divisors of a number N of the form 2n-1p is:

9. This is a direct translation of Euclid’s proof that there exist infinitely many primes. It is a good illustration as to how the ancient Greeks thought of numbers being related to the lengths of line segments. Also notice there is no algebraic expressions for sums and products.

10. Euclid's theorem on the infinitude of the prime numbers(Modern Interpretation): There exist infinitely many prime numbers. Proof (by contradiction): Assume there exists only a finite number of prime numbers (exactly n of them). Call the finite number of prime numbers p1, p2, p3, ..., pn. Let the number A=p1·p2·p3·...·pn+1 Since A is larger than the product of all of the primes by 1, A can not equal any of the prime numbers. This means A is not a prime number. The number A must be a composite number and hence must have a prime factor that is one of the numbers p1,p2,p3,...,pn. But each of these numbers will leave a remainder of 1 when it is divided into A. This is a contradiction. Therefore, there must exist an infinite number of primes. QED

11. Euclid considered Pythagoras's result of having an infinite number of Pythagorean triple. Like much of Euclid's work he was able to take a result refine it and make it much more concise and clear. Euclid's Theorem on Pythagorean Triples: There exist an infinite number of Pythagorean triples. Proof: Consider the following differences: The difference of the squares of two consecutive numbers is an odd number. Since the are an infinite number of odd numbers that are perfect squares (i.e. 12, 32, 52,72,...) the number of Pythagorean triples is infinite. QED

12. As mentioned above Greek Geometry was one of the most important advancements in the history of mathematics for it use of rigorous axiomatic logic. Euclid's Elements is the most famous example of this. I will show you the development of his first and third Propositions. Both of these are concerned with constructions. Proposition 3 is the equivalence of a collapsible compass and a ridged compass. Proposition 1  An equilateral triangle can be constructed upon any line segment. (A given line segment is the side of an equilateral triangle) Proof: Given segment AB construct circle with center A and radius AB as permitted by postulate 3. Similarly construct a circle with center B and radius AB. Let C be the point of intersection (Euclid realized the existence of such a point was a logical gap). The segments AB and AC are equivalent since they are both radii of the same circle. The segments AB and BC are equivalent since they are both radii of the same circle. By postulate 1 AC is equivalent to BC. Thus the triangle ABC is equilateral. QED

13. Proposition 3 It is possible to transfer a segment of given length onto a second segment using a collapsible compass. Proof: Given segments AB and CD. Construct segment AC. Construct equilateral triangle ACE as in proposition 1. Construct a circle with center A and radius AB. This circle intersects AE at a point F. Construct a circle with center E of radius EF. This circle intersects EC at a point G. Construct a circle at point C of radius CG. This circle intersects CD at a point H. Segment pairs AB and AF, CG and CH, EF and EG are all congruent since all pairs are radii of a circle. The segments AE and CE are congruent since triangle ACE is equilateral. The segments AF and CG are equal by common notion 3 if equals EF and EG are subtracted from equals AE and CE the results AF and CG are equal. The segments AB and CH are equal by common notion 1.

14. One of the main purposes of what Euclid was doing was to establish the Pythagorean theorem using logical rigor and axiomatic ideas. The algebra known at the time was not sophisticated enough to classify the conventional arguments as logically rigorous. Euclid set up a series of propositions to establish the Pythagorean theorem in his work the elements was Proposition 47. Aside from the two propositions above Euclid proved propositions about angle congruency, triangle congruency (SSS,ASA,SAS and AAS), parallel lines and areas. These preceded Proposition 47. The proof shows how a square can be dissected to form the areas of two other squares. It uses as a guide one of the most famous figures in geometry. Proposition 47 In right-angled triangles the square on the side subtending the right angle is equal to the sides containing the right angle. I will not give the complete proof because we did not examine all of the propositions. I will give an outline. He used construction techniques he established to construct squares to each side of right triangle ABC with right angle A. Euclid first established that CA and AG lie on the same line as constructed. This is because two right angles form a straight line.  FBC is congruent to ABD because both are a right angle together with ABC. The triangles ABD and FBC are congruent by SAS an earlier result. This produces the following using his propositions Area(∆ ABD) = Area(∆ FBC) 2 Area(∆ ABD) = 2 Area(∆ FBC)

15. A result proven earlier by Euclid stated if the base of a triangle is incident with the base of a rectangle and the remaining vertex of the triangle is on the line passing through the opposite side of the rectangle then the area of the rectangle was equal to the area of two triangles. This is a result he makes use of several times. Area(ABFG) = 2 Area(∆ FBC) and Area(BMLD) = 2 Area(∆ ABD) Thus he concludes: Area( ABFG) = Area(BMLD) He gave a similar argument to prove that: Area( ACHK) = Area(CMLE) thus, Area( CBDE) = Area(BMLD) + Area(CMLE) = Area( ABFG) + Area( ACKH) QED