Mathematical Legacy of Archimedes By Alex Martirosyan and Jenia Tevelev 1
ARCHIMEDES LULLABY (EXCERPT) Gjertrud Schnackenberg A visit to the shores of lullabies, Where Archimedes, counting grains of sand, Is seated in his half-filled universe And sorting out the grains by shape and size. Above his head a water-ceiling sways, Beneath his feet the ancient magma-flows Of metamorphic, underearth plateaus Are moving in slow motion, all in play, And all is give-and-take, all comes and goes, And hush now, all is well now, close your eyes, Distant ocean-engines pulverize Their underwater mountains, coarse to fine, In granite-crumbs and flakes of mica gold And particles of ancient olivine; And water waves sweep back and forth again, Materialize, and dematerialize, Retrieving counted grains and dropping more Uncounted grains in heaps along a shore Of granite-particled infinities, Amassing shores for drawing diagrams. Behind him, on the shores of Sicily, His legendary works accumulate: 2 Discarded toys, forgotten thought-machines, And wonder-works, dismantled on the sand: A ship, reduced to ashes by a mirror; A planetarium in hammered bronze Whose heaven rotates, taking its own measure; The fragment of a marble monument A sphere inscribed within a cylinder Forgotten, overgrown with stems and leaves; A vessel, filled with water to the brim To weigh Hiero's golden diadem, But emptied on its side now, overturned; And numbers fading in papyrus scrolls He sent by ship to Alexandria: Approximated ratios glimpsed within The wondrously unlocked square root of 3; And 3.141... : a treasure-store Marcellus cannot plunder; cannot use; And 1.618... : the weightless gold No scales are needed for, no lock and key, Ratio divine, untouchable in war; And block-and-tackle pulleys; water-screws And other spirals, angles, cubes, and spheres; The iron lever rusting at his feet A relic from the time he told the King Assembled with the court: Give me a place Whereon to stand, and I will move the earth
Archimedes of Syracuse Archimedes is regarded as one of the greatest mathematicians of all times Archimedes was born in the city of Syracuse during ~287 BC. He studied mathematics in Alexandria before returning to Syracuse for a life of inventions and mathematical discoveries. Legend has it that he took it upon himself to solve problems for King Hiero - can you name a few? 3
The Screw King Hiero wanted to empty water from the hull of one of his ships Archimedes took on the challenge created an Archimedes screw and The machine is a hollow tube with a spiral on the inside and operated by a handle. When the handle is turned water is carried up the tube and out of the area This method is still used to this day as an irrigation technique 4
Claw of Archimedes King Hiero anticipated an attack f r o m R o m e a n d e n t r u s t e d Archimedes with defending the city Legend has it that Archimedes designed a weapon that was able to lift ships out of the water The machine was claw like and was dropped onto oncoming enemy ships 5
Death Ray According to several ancient historians, Archimedes constructed the burning mirror to set ablaze ships Today we know that this was was not realistically possible These legends highlight how highly many thought of Archimedes 6
Death of Archimedes I should say that Archimedes diligence also bore fruit if it had not both given him life and taken it away. At the capture of Syracuse Marcellus had been aware that his victory had been held up much and long by Archimedes machines. However, pleased with the man s exceptional skill, he gave out that his life was to be spared, putting almost as much glory in saving Archimedes as in crushing Syracuse. But as Archimedes was drawing diagrams with mind and eyes fixed on the ground, a soldier who had broken into the house in quest of loot with sword drawn over his head asked him who he was. Too much absorbed in tracking down his objective, Archimedes could not give his name but said, protecting the dust with his hands, I beg you, don t disturb this, and was slaughtered as neglectful of the victor s command; with his blood he confused the lines of his art. So it fell out that he was first granted his life and then stripped of it by reason of the same pursuit. -Valerius Maximus 7
On the Equilibrium of Planes Archimedes used the word equilibrium which is now commonly thought of as center of gravity or center of mass The treatise contains propositions and postulates that are used to establish the law of the lever. Postulate 1: Equal weights at equal distances are in equilibrium, and equal weights at unequal distances are not in equilibrium but incline toward the weight that is at the greater distance Postulate 6: If magnitudes at certain distances are in equilibrium, other magnitudes equal to them will also be in equilibrium at the same distances. 8
Archimedes never offers a definition of the center of gravity. Instead he offers axioms which can be justified by our intuition about levers or using a principle in philosophy known as Principle of Insufficient Reason For example, the first axiom (equal weights at equal distances balance the lever) is correct because there is no good reason to make another guess 9
Law of the Lever Proposition 6-7: Two magnitudes, whether commensurable or incommensurable, balance at distances inversely proportional to the magnitudes 10
Given two unequal weights such as 3 and 5 set on a lever. Replace each weight with an equivalent number of unit weights such that this new subset preserves the center of gravity. For instance, for weight 3 place one unit at the original position and the two other units on the left and right sides. We now have a new lever comprised of units that is easier to balance. The total length is 8 units and if we find the balance to this lever of 8 units then we know how to balance the original weights. 11
Give me a place to stand and I will move the Earth Archimedes 12
Theorem. Medians of a triangle intersect in one point. Can we prove this using center of gravity? Porism. The intersection point of medians cuts each of them in 1:2 ratio. Porism is a word used by Euclid for a corollary of the proof of the theorem rather than the statement of the theorem. Nowadays we usually say Corollary of the proof. 13
By placing unequal weights in the vertices of the triangle, one can prove Ceva s Theorem: In a triangle ABC, three lines AD, BE, and CF intersect at a single point if and only if: 14
Measurement of the Circle Proposition 1: The area A of any circle is equal to the area of a right triangle in which one of the legs is equal to radius of the circumference. In modern terms, the proposition says that the area of the circle is equal to pi multiplied by r 2 15
Archimedes offered a mathematically rigorous proof by using what is now known as Eudoxus exhaustion argument Eudoxus of Cnidus (390-340BCE) was a Greek mathematician and astronomer. Although none of his writings survive, his contributions are known from many discussions throughout antiquity. Eudoxus s theory of proportions (equal ratios) and incommensurable magnitudes (magnitudes lacking a common measure) influenced several books of the Elements. His method of exhaustion was explained in several writings of Archimedes who highly praised it. 16
Let K be the area of the given triangle and let A be the area of the circle. Begin by supposing that A>K and inscribe in the circle a polygon of area P such that A-P<A-K. This is possible because A-K is a positive quantity and we can make the difference A-P as little as we want by approximating the circle by the polygon better and better. 17
Since A-P<A-K, we know that P>K. We are going to show that this is impossible, and thus derive a contradiction. The perpendicular h from the center of the circle to the side of the inscribed polygon is less than the radius r of the circle. h Also, the perimeter of the polygon is less than the circumference of the circle. Therefore, 18
A contradiction with P>K. Applying similar logic to the case A<K, we find another contradiction This completes the proof because we have shown that both A>K and A<K are impossible, thus A=K In many ways, the exhaustion argument is a precursor to calculus. 19
Proposition 3: The ratio of the circumference of any circle to its diameter is less than 22/7 (=3.142 ) but greater than 223/71 (=3.140 ) This gives us a lower bound and an upper bound for the value of pi (=3.141 ) Recall that the Babylonians used approximation 3, so a huge improvement The idea is to circumscribe a polygon round a circle and by doubling the number of its sides get a tighter and tighter approximation. The circumference of the circle is approximated by the perimeter of the polygon. A hexagon and 12-gon circumscribed around a circle 20
Lemma of Archimedes: How to prove it? (Hint: OD is a bisector) 21
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Now let s double the hexagon From the Hexagon: We have CA = We have OA = 1 We have CO = DA = 23
Now we can compute the perimeter of the 12-gon to get a closer approximation of pi We know what DA is so we compute the perimeter by multiplying DA by the number of sides If we now double the sides of the 12-gon to get a 24-gon we can get even closer to pi. This is done by now finding a new DA recursively. The previously found DA now becomes CA and the new CO can be found by the pythagorean theorem 24
Thus, we can compute the perimeter of the 24-gon to get a new pi approximation. This method converges to pi: Archimedes probably got his approximations by using two more iterations. Notice that to apply this method we also need to know how to approximate the square root of 3 The wondrously unlocked square root of 3 25