In the very long line of Greek mathematicians from Thales of Miletus and Pythagoras of Samos in the 6th century BC to Pappus of Alexandria in the 4th century AD, Archimedes of Syracuse (287 - 212 BC) is the undisputed leading figure. His pre-eminence is the more remarkable when we consider that this dazzling millenium of mathematics contains so many illustrious names, including Anaxagoras, Zeno, Hippocrates, Theodorus, Eudoxus, Euclid, Eratosthenes, Apollonius, Hipparchus, Heron, Menelaus, Ptolemy and Diophantus. (See the St Andrews archive History of Mathematics for information about these mathematicians.)
Although his main claim to fame is as a mathematician, Archimedes is also known for his many discoveries and inventions in physics and engineering, which include the following.
Before discussing the work covered in his book Measurement of the Circle , we mention briefly a few of the other significant contributions which Archimedes made to mathematics.
1. C H Edwards. The Historical Development of the Calculus , Springer-Verlag, New York, 1979.
2. T L Heath. A History of Greek Mathematics (2 vols.) , Dover, New York, 1981.
The following three propositions are contained in Archimedes' book Measurement of the Circle .
(i) The area of a circle is equal to that of a right-angled triangle where the sides including the right angle are respectively equal to the radius and circumference of the circle.
(ii) The ratio of the area of a circle to that of a square with side equal to the circle's diameter is close to 11:14. (This is of course equivalent to saying that is close to the fraction 22 over 7.)
(iii) The circumference of a circle is less than three and one-seventh times its diameter but more than three and ten-over-seventy-one times the diameter. He obtained these inequalities by considering the circle with radius unity and estimating the perimeters of inscribed and circumscribed regular polygons of ninety-six sides.
Here we summarize the main points in the paper by Phillips with the above title.
1. Let p(n) and P(n) respectively denote half the perimeter of the inscribed and circumscribed regular n-gons of the unit circle. It can be shown that p(2n) and P(2n) are simply related to p(n) and P(n). The most obvious way is to express p(2n) as a function of p(n) only, and to express P(2n) solely in terms of P(n). However, there is a more elegant way of proceeding, in which both P(n) and p(n) are used to compute each of P(2n) and p(2n) : it happens that P(2n) is the harmonic mean of P(n) and p(n), and p(2n) is the geometric mean of P(n+1) and p(n). (Click on means for definitions of the harmonic and geometric mean.)
2. It is easily verified that P(3) is 3 times the square root of 3, and p(3) one half of this. We can compute P(6) and p(6) from the "double mean" process defined above, then P(12) and p(12), and so on. After five applications of the double mean process, we find that P(96) = 3.1427 and p(96) = 3.1410 to 4 decimal places. This is consistent with Archimedes' proposition (iii) above.
3. Some analysis shows that the quantity u(n)=(2p(n) + P(n))/3 is a closer approximation to than either of P(n) or p(n).
4. It can be shown similarly that the quantity v(n) = (4p(2n) - p(n))/3 is a closer approximation to than either of p(n) or p(2n). The approximation p(n) differs from by a quantity which tends to zero like one-over-n-squared; however, v(n) differs from by a quantity which tends to zero like one-over-n-to-the-power-4. The same remarks hold if we replace "p" by "P" in the relation for v(n). The process used in computing v(n) is called extrapolation to the limit, also known as Richardson extrapolation.
5. In fact, the error between p(n) or P(n) and is a power series in the variable x = one-over-n-squared, beginning with the term in x. The computation of v(n) is defined so that the error between v(n) and has a series in x beginning with the term in x-squared.
6. To produce still faster convergence to , we "remove" the term in x-squared, by computing w(n) = (16v(2n) - v(n))/15. We can extend this process further, using w(2n) and w(n) to remove the term in x-cubed from the error, and so on. This is called repeated extrapolation. For more information on extrapolation to the limit, see any text on numerical analysis. (Phillips and Taylor, Theory and Applications of Numerical Analysis , is one that comes to mind.)
7. Beginning with p(3), p(6), p(12), p(24), p(48) and p(96), the "raw material" computed by Archimedes, we can extrapolate five times. The final number in this calculation differs from by less than one unit in the eighteenth decimal place. (We obtain a similar result if we use the "P" sequence in place of the "p" sequence.)
8. It is geometrically obvious that the two sequences P(n) and p(n) converge to the common limit . What if we assign arbitrary positive values to P(3) and p(3)? It can be shown that the two sequences converge to a common limit which can be expressed as a multiple of an inverse cosine or an inverse hyperbolic cosine (cosh), depending on whether p(3) is less than or greater than P(3).
9. In particular, let us choose any value of t > 1 and write a = t-squared. Then let us choose P(3) = (a-1)/(a+1) and p(3) = (a-1)/(2t), and compute the "P" and "p" sequences using the "double mean" process defined in paragraph 1 above. Note that P(3) < p(3). In this case the two sequences converge to the common limit log t.The above method of the logarithm should not be regarded as a serious practical algorithm, since there are faster methods available. See, for example, Borwein and Borwein, Pi and the Arithmetic-Geometric Mean .
In the first paragraph of the previous section, we combined the harmonic and geometric means in a "double mean" process. In order to generalize this, Foster and Phillips used of a class of abstract means. For a definition of this class and associated material, click on means.
Foster and Phillips define an "Archimedean" process where, given positive real numbers x(0) and y(0), the sequences (x(n)) and (y(n)) are computed recursively from
This generalizes the process defined in paragraph 1 of the previous section. It can be shown that the two sequences defined by this "Archimedean" process converge monotonically to a common limit. Moreover, assuming that the functions M and N possess continuity of their derivatives up to second order, the two sequences converge to zero in a first order manner, meaning that the errors tend to zero asymptotically like a geometric sequence. Somewhat surprisingly, no matter what means M and N we choose, with the properties specified above, the common ratio of this geometric sequence is always one-quarter. Thus about three further decimal digits of accuracy are gained for every five iterations carried out in an "Archimedean" process, since 4 to the power 5 (1024) is approximately 1000.
Foster and Phillips consider also the superficially similar "Gaussian" process where, again given positive real numbers x(0) and y(0), the sequences (x(n)) and (y(n)) are computed recursively from
This generalizes the famous "arithmetic-geometric mean" process which Gauss used to compute an elliptic integral. In this classical case, M and N are (obviously from its name) the arithmetic and geometric means, and it is well known that the "arithmetic-geometric mean" process converges quadratically . This means that the error in x(n+1) behaves asymptotically as a multiple of the square of the error in x(n), with the errors in y(n+1) and y(n) being related similarly.
Foster and Phillips show that this quadratic convergence carries over to the general "Gaussian" process defined above provided that the means M and N possess continuity of their derivatives up to second order.