**Archimedes of Syracuse (287 – 212 B.C.)**

Archimedes was one of the three greatest mathematicians of all time – the other two being Newton and Gauss. The son of an astronomer, Archimedes had an appreciation for both mathematics and science and made major contributions to both. He gave accurate estimations to π, developed much of solid geometry, and anticipated the theory of integration well before it was discovered by others many years later. He was also a brilliant engineer who developed many weapons of war which were used to defend his home city of Syracuse while under siege by the Romans. Rumors have it that Archimedes also developed a parabolic mirror which could direct sunlight to Roman ships to set them on fire! This is most likely not true — but it sure makes a good story!

(a 1600 painting by Parigi showing Archimedes burning ships with his parabolic mirror.)

Fortunately much of Archimedes’ written work survives and it regarded as brilliant for what is discovers as well for its exposition. His writings are considered works of art. T. L. Heath, a scholar and translator of Archimedes’ work stated

*The treatises are, without exception, monuments of mathematical exposition; the gradual revelation of the plan of attack, the masterly ordering of the propositions, the stern elimination of everything not immediately relevant to the purpose, the finish of the whole, are so impressive in their perfection as to create a feeling akin to awe in the mind of the reader.*

Check out Heath’s translation of Archimedes as well as a brief online biography.

We will discuss two theorems from Archimedes. They both deal with the number π.

**Theorem:** *The ratio of any circle to its diameter is constant.*

**Proof:** We want to show that for any two circles with circumference C_1 and radius r_1, respectively C_2 and r_2, that C_1/2r_1 = C_2/2r_2. To start, we notice that one of these circles will have larger circumference. Let’s say it is the first one. Place the two circles concentric to each other as follows.

Within each circle, inscribe a polygon with n sides as follows:

Let A_n be the length of one of the outer sides of this n-sided polygon and B_n be the length of one of the sides of the inner polygon:

Observe that as n increases that the polygons become more and more like circles. A more precise way of saying this is that n A_n -> C_1 as n -> ∞ and n B_n -> C_2 as n -> ∞.

Look at the above drawing and you will see two similar triangles, one with base of length B_n and one with base of length A_n. By Thales’ theorem we have A_n/B_n = r_1/r_2. Rearranging the terms of this expression we get A_n = B_n (r_1/r_2). Putting this all together, we get

C_1/2r_1 = lim n A_n/2r_1 = lim n B_n(r_1/r_2)(1/2r_1) = lim n B_n/2r_2 = C_2/2r_2.

Thus C_1/2r_1 = C_2/2r_2. **QED**.

I’m sure you all know that this number, the ratio of any circumference to its diameter, has a name — π. How do we estimate the value of π? Here is this wonderful construction of Archimedes.

**Theorem (Archimedes):*** 3 10/71 < π < 3 1/7.*

The proof uses the “method of exhaustion” by getting lower bounds of π by using perimeters of inscribed regular polygons and upper bounds using perimeters of circumscribed polygons. Here is the main idea:

Using polygons as above with 96 sides, along with some very beautiful geometry (which shows his true genius), and a very clever approximation of √3 > 265/153, he gets his estimate. In our class discussion, will not use the original proof of Archimedes but a more modern proof. The original proof, though absolutely wonderful, can be difficult to read since it uses notation the modern students will not recognize. Here is a link to a treatment of Archimedes’ original proof which contains more of the details.

Notice the upper bound 3 1/7 = 22/7 which many people think is equal to π. It is not! We will prove in a later on that π is irrational and thus is not equal to the rational number 22/7. As you can imagine, there is a lot to be said about π. We mention a few facts here,

**Some facts about π:**

- π is irrational and
*not*equal to 22/7! (though many people believe this) - There are some really cool infinite series formulas involving π due to the Indian mathematician S. Ramanujan. We will mention him later in connection to G. H. Hardy.
- π is ubiquitous in mathematics, physics, engineering, statistics, etc.
- An exact value of π is unknown and one can only approximate it. Here are the first 1120 digits of π

3.1415926535 8979323846 2643383279 5028841971 69399375105820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 3460348610 4543266482 1339360726 0249141273 7245870066 0631558817 4881520920 9628292540 9171536436 7892590360 0113305305 4882046652 1384146951 9415116094 3305727036 5759591953 0921861173 8193261179 3105118548 0744623799 6274956735 1885752724 8912279381 8301194912 9833673362 4406566430 8602139494 6395224737 1907021798 6094370277 0539217176 2931767523 8467481846 7669405132 0005681271 4526356082 7785771342 7577896091 7363717872 1468440901 2249534301 4654958537 1050792279 6892589235 4201995611 2129021960 8640344181 5981362977 4771309960 5187072113 4999999837 2978049951 0597317328 1609631859 5024459455 3469083026 4252230825 3344685035 2619311881 7101000313 7838752886 5875332083 8142061717 7669147303 5982534904 2875546873 1159562863 8823537875 9375195778 1857780532 1712268066 1300192787 6611195909 2164201989 3809525720 1065485863 2788659361 5338182796 8230301952 0353018529 6899577362 2599413891 2497217752 8347913151 5574857242 4541506959…..

According to Plutarch, Archimedes favorite result was the fact that the surface area of a sphere is 2/3 the volume of the circumscribed cylinder. The same is true for the volumes. Can you prove this?

In class we will say a few words about “The Method” which Archimedes used to prove this result. He essentially equates volume and mass and shows that if you have a scale, and place a cylinder of height 2r and radius 2r (the cylinder which circumscribes the sphere but with twice the radius) on one side of the scale and place two cones (with height 2r and radius 2 r) and two spheres (of radius r) on the other side of the scale, it will be balanced.

The volume of a cylinder was known by Archimedes (π r^2 h) as was the volume of a cone (1/3 π r^2 h – shown by Euclid). A little algebra now yields the result. Showing the balancing is an ingenious trick of Archimedes.

In class we will also cover some of the details of the quadrature of a parabola problem:

**Theorem:** The area under the parabola is 4/3 the area of the triangle.

Archimedes was killed by a Roman soldier, supposedly while taking a bath and considering mathematics, at the battle of Syracuse. Legend says that Archimedes requested that the above figure of a sphere inscribed in a cylinder be inscribed on his tomb.

The above painting, *Cicero Discovering the Tomb of Archimedes*, was painted in 1797 by the American painter Benjamin West.

The British philosopher Alfred North Whitehead has the following stinging quote about the Romans who killed Archimedes

The death of Archimedes by the hands of a Roman soldier is symbolical of a world-change of the first magnitude: the Greeks, with their love of abstract science, were superseded in the leadership of the European world by the practical Romans. The Romans were a great race, but they were cursed with the sterility which waits upon practicality. They did not improve upon the knowledge of their forefathers, and all their advances were confined to the minor technical details of engineering. They were not dreamers enough to arrive at new points of view, which could give a more fundamental control over the forces of nature. No Roman lost his life because he was absorbed in the contemplation of a mathematical diagram.One last tidbit about Archimedes: There is no Nobel prize in mathematics — long story. The highest prize in mathematics is the Fields Medal. The image of the actual medal is one of Archimedes

The inscription reads *Transire suum pectus mundoque potiri* (Rise above oneself and grasp the world).