Discoveries and inventions

• 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

• 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 $O$ of any point $P$ on the spiral is proportional to the angle between $OP$ and a fixed line through $O$. 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 $\pi$

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 $\large\frac{22}{7}\normalsize$.)
   
   
3. The circumference of a circle is less than $3\large\frac{1}{7}\normalsize$ its diameter but more than $3\large\frac{10}{71}\normalsize$ 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

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 $p(n)$.
   
   
2. It is easily verified that $P(3)$ is $3√3$, 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 $\pi$ 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 π than either of $p(n)$ or $p(2n)$. The approximation $p(n)$ differs from $\pi$ by a quantity which tends to zero like one-over-n-squared; however, $v(n)$ differs from $\pi$ 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 $\pi$ 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: [5] say.
   
   
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 $\pi$ 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 "$P$" 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 $a = t^{2}$. Then let us choose 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 $\log t$.

"Double mean" processes