Archimedes: Numerical Analyst

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 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.

Discoveries and inventions

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.

  • His invention of the water-screw, still in use in Egypt, for irrigation, draining marshy land and pumping out water from the bilges of ships.

  • His invention of various devices used in defending Syracuse when it was besieged by the Romans. These include powerful catapults, the burning-mirror and systems of pulleys. It was his pride in what he could lift with the aid of pulleys and levers which provoked his glorious hyperbole "Give me a place to stand and I will move the earth". (This saying of Archimedes is even more grandly laconic in Greek, in which it transliterates as the eight word sentence "dos moi pou stó kai kinó tén; gén;". [1]

  • His discovery of the hydrostatic principle that a body immersed in a fluid is subject to an upthrust equal to the weight of fluid displaced by the body. This discovery is said to have inspired his famous cry "Eureka" ("I have found it").

Mathematical achievements

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.

  • He computed the area of a segment of a parabola. He used a most ingenious argument involving the construction of an infinite number of inscribed triangles which "exhausted" the area of the parabolic segment. This is a most beautiful piece of mathematics.

  • He computed the area of an ellipse by essentially "squashing" a circle.
    He found the volume and surface area of a sphere. Archimedes gave instructions that his tombstone should have displayed on it a diagram consisting of a sphere with a circumscribing cylinder. C H Edwards [2] writes how Cicero, while serving as quaestor in Sicily, had Archimedes' tombstone restored, and adds
    The Romans had so little interest in pure mathematics that this action by Cicero was probably the greatest single contribution of any Roman to the history of mathematics.

  • He discussed properties of the "Archimedean spiral", which is defined as follows : the distance from a fixed point OO of any point PP on the spiral is proportional to the angle between OPOP and a fixed line through OO. In his evaluation of areas involving the spiral he anticipated methods of the calculus which were not developed until the seventeenth century AD.

  • He found the volumes of various "solids of revolution" obtained by rotating a curve about a fixed straight line.

Archimedes and π\pi

The following three propositions are contained in Archimedes' book Measurement of the Circle.

  1. 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.

  2. 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 πis close to the fraction 227\large\frac{22}{7}\normalsize.)

  3. The circumference of a circle is less than 3173\large\frac{1}{7}\normalsize its diameter but more than 310713\large\frac{10}{71}\normalsize 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 96 sides.

Archimedes the Numerical Analyst

Here we summarize the main points in the paper [3].

  1. Let p(n)p(n) and P(n)P(n) respectively denote half the perimeter of the inscribed and circumscribed regular nn-gons of the unit circle. It can be shown that p(2n)p(2n) and P(2n)P(2n) are simply related to p(n)p(n) and P(n)P(n). The most obvious way is to express p(2n)p(2n) as a function of p(n)p(n) only, and to express P(2n)P(2n) solely in terms of P(n)P(n). However, there is a more elegant way of proceeding, in which both P(n)P(n) and p(n)p(n) are used to compute each of P(2n)P(2n) and p(2n)p(2n) : it happens that P(2n)P(2n) is the harmonic mean of P(n)P(n) and p(n)p(n), and p(2n)p(2n) is the geometric mean of P(n+1)P(n+1) and p(n)p(n).

  2. It is easily verified that P(3)P(3) is 333√3, and p(3)p(3) one half of this. We can compute P(6)P(6) and p(6)p(6) from the "double mean" process defined above, then P(12)P(12) and p(12)p(12), and so on. After five applications of the double mean process, we find that P(96)=3.1427P(96) = 3.1427 and p(96)=3.1410p(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))/3u(n)=(2p(n) + P(n))/3 is a closer approximation to π\pi than either of P(n)P(n) or p(n)p(n).

  4. It can be shown similarly that the quantity v(n)=(4p(2n)p(n))/3v(n) = (4p(2n) - p(n))/3 is a closer approximation to π than either of p(n)p(n) or p(2n)p(2n). The approximation p(n)p(n) differs from π\pi by a quantity which tends to zero like one-over-n-squared; however, v(n)v(n) differs from π\pi by a quantity which tends to zero like one-over-n-to-the-power-4. The same remarks hold if we replace "pp" by "PP" in the relation for v(n)v(n). The process used in computing v(n)v(n) is called extrapolation to the limit, also known as Richardson extrapolation.

  5. In fact, the error between p(n)p(n) or P(n)P(n) and is a power series in the variable xx = one-over-n-squared, beginning with the term in xx. The computation of v(n)v(n) is defined so that the error between v(n)v(n) and π\pi has a series in xx beginning with the term in xx-squared.

  6. To produce still faster convergence to , we "remove" the term in xx-squared, by computing w(n)=(16v(2n)v(n))/15w(n) = (16v(2n) - v(n))/15. We can extend this process further, using w(2n)w(2n) and w(n)w(n) to remove the term in xx-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: [5] say.

  7. Beginning with p(3),p(6),p(12),p(24),p(48)p(3), p(6), p(12), p(24), p(48) and p(96)p(96), the "raw material" computed by Archimedes, we can extrapolate five times. The final number in this calculation differs from π\pi by less than one unit in the eighteenth decimal place. (We obtain a siP(3) = (a-1)/(a+1)milar result if we use the "PP" sequence in place of the "pp" sequence.)

  8. It is geometrically obvious that the two sequences P(n)P(n) and p(n)p(n) converge to the common limit . What if we assign arbitrary positive values to P(3)P(3) and p(3)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)p(3) is less than or greater than P(3)P(3).

  9. In particular, let us choose any value of t>1t > 1 and write a=t2a = t^{2}. Then let us choose and p(3)=(a1)/(2t)p(3) = (a-1)/(2t), and compute the "PP" and "pp" sequences using the "double mean" process defined in paragraph 1 above. Note that P(3)<p(3)P(3) < p(3). In this case the two sequences converge to the common limit logt\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 [4].

"Double mean" processes

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.

Foster and Phillips define an "Archimedean" process where, given positive real numbers x(0)x(0) and y(0)y(0), the sequences (x(n))(x(n)) and (y(n))(y(n)) are computed recursively from
x(n+1)=M(x(n),y(n))x(n+1) = M(x(n),y(n))
y(n+1)=N(x(n+1),y(n))y(n+1) = N(x(n+1),y(n))
for n=0,1,...n = 0, 1, ... , where MM and NN are any two means belonging to the class mentioned above.
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 MM and NN 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 MM and NN 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 454^{5} is approximately 1000.

Foster and Phillips consider also the superficially similar "Gaussian" process where, again given positive real numbers x(0)x(0) and y(0)y(0), the sequences (x(n))(x(n)) and (y(n))(y(n)) are computed recursively from
x(n+1)=M(x(n),y(n))x(n+1) = M(x(n),y(n))
y(n+1)=N(x(n),y(n))y(n+1) = N(x(n),y(n))
for n=0,1,...n = 0, 1, ... , where MM and NN are again any two of the means mentioned above.

This generalizes the famous "arithmetic-geometric mean" process which Gauss used to compute an elliptic integral. In this classical case, MM and NN 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)x(n+1) behaves asymptotically as a multiple of the square of the error in x(n)x(n), with the errors in y(n+1)y(n+1) and y(n)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 MM and NN possess continuity of their derivatives up to second order.

Written by G M Phillips