Archimedes - Computations of Areas and Volumes

     Glenn Mason-Riseborough (31/10/1997)

Introduction
Archimedes (287-212 BCE) is often considered to be the 
greatest mathematician of antiquity (Gow, 1884) and one of the 
three greatest mathematicians of all time (along with Isaac 
Newton (1642-1727) and Carl Gauss(1777-1855)).  Plinus called 
him ‘the god of mathematics,’ and the Romans who were 
besieging his home city of Syracuse called him ‘the mathematical 
devil’ because of his military constructions (Lanczos, 1974).  The 
range and variety of his work is especially remarkable—there are 
very few fields of mathematics in which Archimedes did not 
contribute an original piece of work.  Especially remarkable was 
his work in higher mathematics which was only fully appreciated 
from the nineteenth century.  Many of Archimedes’ solutions to 
his problems used techniques that we would call calculus today, 
thus in a sense he is one of the creators of that branch of 
mathematics.  Archimedes considered that his greatest discovery 
was the proof that the surface of a sphere is equal to the surface 
of the circumscribed cylinder, while its volume is two thirds of 
the volume of the circumscribed cylinder (Lanczos, 1974).
In this essay, we will discuss Archimedes’ computations of areas 
and volumes.  After a brief overview of the life of Archimedes, 
we will examine his proofs and methods for finding the areas and 
volumes of various polygons and polyhedrons.

Archimedes’ life
As with most of the ancient Greeks, very little is known about 
the life of Archimedes.  It is known that he studied in Alexandria 
but spent most of his life in Syracuse where he was a friend (and 
may even have been related) to the ruling family at the time 
(Aaboe, 1964).  There are many stories and legends which tell of 
Archimedes solving one problem or another, his single-
mindedness in carrying out his speculations and problem-solving, 
and his consequent absent-mindedness in everyday activities.  
One classic story (which may or may not be true) is of when he 
discovered the law of buoyancy while taking a bath.  The story 
tells us that he was so elated by his insight, that he jumped out of 
the bath and (without putting on any clothes) ran through the 
streets of Syracuse shouting Eureka! (I have found it!) (Anglin, 
1994).  Aaboe (1964) suggests that the popular image of 
Archimedes was such that he was regarded as the epitome of a 
learned man in the same way that Albert Einstein (1879-1955) is 
regarded today.
Many written works of Archimedes are known to us, some of 
which have survived in the original Greek texts (Archimedes 
wrote in the Doric dialect of Syracuse, although many books are 
written the Attic dialect (Gow, 1884)), while many others may 
have been destroyed.  Some texts (such as the Book of Lemmas) 
have been preserved in Arabic translations (albeit in a corrupted 
form) while still others are only known to us because they are 
referred to in other writings (e.g. On Sphere-making) (Aaboe, 
1964).
Archimedes was killed during the sacking of Syracuse in 212 
BCE by the Romans; Tzetzes reports that he was 75 at the time 
(Gow, 1884).  There are many conflicting stories about how he 
was killed but most accounts suggest that he was so engrossed in 
his calculations that he was not aware that the city had been 
taken by the Romans and was run through by a soldier who 
unexpectedly found him.  Some stories say that this was because 
he refused to move until he finished the problem he was working 
on, others say that it was because the mathematical equipment he 
was carrying at the time was mistaken for gold.  All variants tell 
us that Marcellus, the Roman leader was so enraged by the death 
of Archimedes that he regarded the soldier who killed him as a 
murderer, and that he sought out Archimedes family to favour 
them (Aaboe, 1964).

Area of a circle
In The Measurement of a Circle, Archimedes proved that the 
area A of a circle of radius r is equal to that of a triangle whose 
base is equal to the circumference C of the circle and whose 
height is r (Aaboe, 1964).  In modern notation this is:
 .
Tee (1996) gives the following proof (by reductio ad absurdum) 
that Archimedes used:
Let ABCD be a circle and K a triangle with the requirements as 
above, then if the area of the circle is not equal to K it must be 
either greater or less (see Fig. 1).

1. If the circle is greater than K:
A square ABCD is inscribed on the circle; the arcs AB, BC, 
CD, DA are bisected then bisected repeatedly until the sides 
of the inscribed polygon whose angular points are the points 
of division subtend segments whose sum is less than the 
excess of the area of the circle over K.  Thus the area of the 
polygon is greater than K.  Let AE be any side of it and ON 
the perpendicular on AE from the centre O.  ON is less than 
the radius of the circle and thus less than the sides about the 
right angle in K.  The perimeter is less than the 
circumference of the circle and thus less than the other side 
of the right angle in K.  Therefore the area of the polygon is 
less than K; this is inconsistent so the area of the circle is not 
greater than K.
2.	If the circle is less than K:
A square is circumscribed, and two adjacent sides, touching 
the circle in E, H, meet at T.  The arcs between adjacent 
points of contact are bisected and tangents are drawn at the 
points of bisection.  Let A be the middle point of the arc EH, 
and FAG the tangent at A.  Thus, the angle TAG is a right 
angle and TG > GA
>
 
G
H
Thus, the triangle FTG is greater than half the area TEAH.  
Similarly, if the arc AH is bisected and the tangent at the 
point of bisection is drawn, it will cut off from the area GAH 
more than one half.  Repeating this process sufficiently many 
times, we will get a circumscribed polygon such that the 
spaces intercepted between it and the circle are together less 
than the excess of K over the area of the circle.  Thus, the 
area of the polygon will be less than K.  Since the 
perpendicular from O on any side of the polygon is equal to 
the radius of the circle, while the perimeter of the polygon is 
greater than the circumference of the circle, then the area of 
the polygon must be greater than K, which is a 
contradiction, so the area of the circle is not less than K.
Thus, the area of the circle must be equal to K since it is not 
greater than or less than it.

Calculation of ?
From the result above of
 ,
it follows that the ratio of the area of a circle to the square of its 
radius is the same as the ratio of its circumference to its 
diameter.  In modern notation, this ratio is called ? (1).  By applying 
the method of exhaustion of Eudoxus using inscribed and 
circumscribed regular polygons of 6, 12, 24, 48 and 96 sides, 
Archimedes calculated upper and lower bounds of ? (Bunt, et 
al., 1976).  He did this by showing that the circumference of the 
circle is greater than any of the inscribed polygons and less than 
any of the circumscribed polygons.  He also showed that as the 
number of sides of the polygons increase the inscribed polygon 
increases and the circumscribed polygon decreases towards the 
circumference of the circle.  Thus, for a polygon of 96 sides the 
upper and lower bounds for ? are: 
 .
In decimal notation this is:
3.1408 < ? < 3.1429.
This is accurate to three significant figures since ? = 
3.1415926536...

Volume of a Sphere
A manuscript of hymns and prayers written by a monk in 
Jerusalem around the thirteenth century was discovered in 1906.  
When it was examined, it was found to be written over the top of 
a tenth century manuscript of the works of Archimedes.  This 
manuscript included the previously unknown book On 
Mechanical Theorems, Method (usually referred to as the 
Method) (Lanczos, 1970).  This book was written in the form of 
a letter to his friend Eratosthenes, and in it Archimedes attempts 
to explain how he forms his ideas before establishing a formal 
proof for his theorems.  As an example, Archimedes includes a 
non-rigorous demonstration of his theorem of  the areas, 
volumes, et cetera of objects.  This argument used the law of the 
lever such that given two objects A and B hanging in equilibrium 
from a balance with a distance a and b respectively from the 
fulcrum (see Fig. 2) then:
 

Objects A and B may be considered as pairs of areas, volumes, 
moments or anything in which we desire to discover their 
equality.  One object, A of which we know its area, is suspended 
at its centre of gravity and placed such as to maintain equilibrium 
of the lever.  At each distance a, a line l is drawn vertically on A 
and a line l’ is drawn at some distance b from the fulcrum on B 
such that al equals bl’.  If some infinite number of these lines are 
drawn, then we may consider objects A and B to be composed 
entirely of lines such as l and l’ respectively.  Thus, if all the lines 
l’ of B are hung at distance b from the fulcrum then b times the 
area of B equals the area of A times a where all lines l of A are at 
distance a.  Thus, we can find the area of B (equals area of A 
times a divided by b).

Conclusions
Archimedes was one of the greatest mathematicians in recorded 
history, and accounts of his many original ideas in various fields 
of mathematics and the sciences attest to this.  Many people 
consider that Archimedes was one of the creators of the branch 
of mathematics called calculus.  The techniques that he used to 
find the areas and volumes of various polygons and polyhedrons 
were far ahead of his time.  It was not for almost another 2000 
years before Newton developed his theory of fluxions and fluids 
and Leibniz developed the differential and integral calculus.  
Archimedes calculation of the area of a circle was obtained by 
successively getting closer and closer approximations to it by 
using circumscribed and inscribed polygons with successively 
greater numbers of sides.  This limit process was a powerful way 
of arriving at results but required previous knowledge of the 
desired results.  From the result of the area of circles, 
Archimedes was able to calculate upper and lower bounds to the 
ratio of the circumference of a circle to its diameter.  In modern 
terms this is the calculation of ? and Archimedes was able to 
achieve results with an accuracy of 0.002 between the upper and 
lower bounds.  Archimedes’ considered his method for finding 
areas, volumes, et cetera as described in The Method to be 
informal and as such only useful for suggesting results which 
could then be proved using the method of exhaustion.

References:
Aaboe, A. (1964). Episodes from the early history of 
mathematics. Yale: Random House.

Anglin, W. S. (1994). Mathematics: A concise history and 
philosophy. New York: Springer-Verlag.

Bunt, L. N. H., Jones, P. S., & Bedient, J. D. (1976). The 
historical roots of elementary mathematics. Englewood 
Cliffs, NJ: Prentice-Hall.

Gow, J. (1884). A short history of Greek mathematics. 
Cambridge: Cambridge University Press.

Lanczos, C. (1970). Space through the ages: The evolution of 
geometrical ideas from Pythagoras to Hilbert and Einstein. 
London: Academic Press.

Tee, G. J. (1996). History of the differential and integral 
calculus. Unpublished Manuscript, University of Auckland.

Endnotes:
1 This ratio was named ? by William Jones in 1706