Department of Mathematics Education

Dr. J. Wilson, EMAT 6690

with a close look at

The Method of Exhaustion

by David Wise

Euclid’s *Elements*, by far his most famous and important
work, is a comprehensive collection of the mathematical knowledge
discovered by the classical Greeks, and thus represents a mathematical
history of the age just prior to Euclid and the development of
a subject, i.e. Euclidean Geometry. There is question as to whether
the *Elements* was meant to be a treatise for mathematics
scholars or a text for students (**Kline 57**). Regardless
of the original purpose, the thirteen books that comprise the
*Elements* became the "centre of mathematical teaching
for 2000 years" (**Euclid
of Alexandria ****3**).

Relatively little is known about the classical period, but
historians are certain that Euclid did not discover most of the
results in the Elements. However, Euclid’s brilliance as
a mathematician is evident in that he chose the axioms, arranged
the theorems, and applied a new level of rigor to proofs. It is
believed that Euclid acquired a great deal of the material for
the *Elements* from the Platonists, Eudoxus, Theaetetus,
and others. In fact, Eudoxus’ work is the basis for Books
V and XII of the *Elements* (**Kline 57**).

Book V is considered by most authorities to be the greatest
achievement of Euclidean Geometry because it extended the Pythagorean
theory of proportion. The Pythagorean theory of proportion only
dealt with commensurable magnitudes, which are magnitudes whose
ratio is restricted to those that can be expressed by a ratio
of whole numbers. Euclid relied upon Eudoxus’ work with proportions
to develop Book V, which extended the theory of proportions to
include incommensurable ratios and still avoided irrational numbers.
Book V was crucial in the development of the remaining books of
the Elements because it provided the theory for all kinds of magnitudes.
In fact, Book V played a key role in the subsequent history of
mathematics in that Eudoxus’ theory of proportion required
continuous quantities to be treated entirely on a geometric basis.
It was not until about 1800 that a theory of rational numbers
was developed to provide a foundation for irrational numbers (**Kline
69**).

- In Book XII, Euclid proves eighteen propositions on areas
and volumes bounded by curves. Eudoxus’ theory of proportion
provides a necessary foundation, but it is Euclid’s use
of Eudoxus’ method of exhaustion that is the key element
to providing rigorous proofs. The method of exhaustion is a modern
term that came into use during the seventeen century and refers
to the approximation of a figure using a sequence of inscribed
figures within it. The successive inscribed figures "exhaust"
the original figure. The important feature of this principle
is that the sequence of approximations is made so that the difference
between the original figure and the inscribed figure decreases
by at least half at each step of the sequence. The method of
exhaustion was essential in proving propositions 2, 5, 10, 11,
12, and 18 of Book XII (
**Kline 83**).

Perhaps two of the most easily recognized propositions from Book XII by anyone that has taken high school geometry are propositions 2 and 18:

- Proposition 2. Circles are to one another as the squares on the diameters.
- Proposition 18. Spheres are to one another in triplicate
ratio of their diameters.

(**Euclid****Elements**)*toc,12*1

Proposition 2 is stating that circles are proportional to the
squares of their diameters (C1/C2 = (d1)^{2}/(d2)^{2
}), while proposition 18 is stating that circles are proportional
to the cubes of their diameters (C1/C2 = (d1)^{3}/(d2)^{3
}). While these propositions are routinely shrugged at by
our students as being simplistic, known facts, Euclid’s proofs
are quite rigorous. High school students can benefit greatly by
studying Eudoxus’ method of exhaustion. We will examine the
proof of proposition 2 in detail, paying particular attention
to the use of the method of exhaustion.

In Heath’s translation of the *Elements*, Euclid’s
proof of proposition 2 is as follows. At the end of each appropriate
paragraph, a reference to the proposition(s) used is provided:

(diagram provided by **Joyce**)

* *Let *ABCD*, *EFGH* be circles, and
*BD*, *FH* their diameters; I say that, as the circle
*ABCD* is to the circle *EFGH*, so is the square on
*BD* to the square on *FH*.

For, if the square on *BD* is not to the square on *FH*
as the circle *ABCD* is to the circle *EFGH*, then,
as the square on *BD* is to the square on *FH*, so
will the circle *ABCD* be either to some less area than
the circle *EFGH*, or to a greater.

First, let it be in that ratio to a less area *S*.

Let the square *EFGH* be inscribed in the circle *EFGH*;
then the inscribed square is greater than the half of the circle
*EFGH*, inasmuch as, if through the points *E*, *F*,
*G*, *H* we draw tangents to the circle, the square
*EFGH* is half the square circumscribed about the circle,
and the circle is less than the circumscribed square; hence the
inscribed square *EFGH* is greater than the half of the
circle *EFGH*. [**IV.
6 and III. 17**]

Let the circumferences *EF*, *FG*, *GH*, *HE*
be bisected at the points *K*, *L*, *M*, *N*,
and let *EK*, *KF*, *FL*, *LG*, *GM*,
*MH*, *HN*, *NE* be joined; therefore each of
the triangles *EKF*, *FLG*, *GMH*, *HNE*
is also greater than the half of the segment of the circle about
it, inasmuch as, if through the points *K*, *L*, *M*,
*N* we draw tangents to the circle and complete the parallelograms
on the straight lines *EF*, *FG*, *GH*, *HE*,
each of the triangles *EKF*, *FLG*, *GMH*, *HNE*
will be half of the parallelogram about it, while the segment
about it is less than the parallelogram; hence each of the triangles
*EKF*, *FLG*, *GMH*, *HNE* is greater than
the half of the segment of the circle about it. [**III.
17**]

Thus, by bisecting the remaining circumferences and joining
straight lines, and by doing this continually, we shall leave
some segments of the circle which will be less than the excess
by which the circle *EFGH* exceeds the area *S*.

For it was proved in the first theorem of the tenth book that,
if two unequal magnitudes be set out, and if from the greater
there be subtracted a magnitude greater than the half, and from
that which is left a greater than the half, and if this be done
continually, there will be left some magnitude which will be
less than the lesser magnitude set out. [**X.
1**]

Let segments be left such as described, and let the segments
of the circle *EFGH* on *EK*, *KF*, *FL*,
*LG*, *GM*, *MH*, *HN*, *NE* be less
than the excess by which the circle *EFGH* exceeds the area
*S*.

Therefore the remainder, the polygon *EKFLGMHN*, is greater
than the area *S*.

Let there be inscribed, also, in the circle *ABCD* the
polygon *AOBPCQDR* similar to the polygon *EKFLGMHN*;
therefore, as the square on *BD* is to the square on *FH*,
so is the polygon *AOBPCQDR* to the polygon *EKFLGMHN*.
[**XII.
1**]

But, as the square on *BD* is to the square on *FH*,
so also is the circle *ABCD* to the area *S*; therefore
also, as the circle *ABCD* is to the area *S*, so is
the polygon *AOBPCQDR* to the polygon *EKFLGMHN*; therefore,
alternately, as the circle *ABCD* is to the polygon inscribed
in it, so is the area *S* to the polygon *EKFLGMHN*.
[**V.
11 and V. 16**]

But the circle *ABCD* is greater than the polygon inscribed
in it; therefore the area *S* is also greater than the polygon
*EKFLGMHN*.

But it is also less: which is impossible.

Therefore, as the square on *BD* is to the square on
*FH*, so is not the circle *ABCD* to any area less
than the circle *EFGH*.

Similarly we can prove that neither is the circle *EFGH*
to any area less than the circle *ABCD* as the square on
*FH* is to the square on *BD*.

I say next that neither is the circle *ABCD* to any area
greater than the circle *EFGH* as the square on *BD*
is to the square on *FH*.

For, if possible, let it be in that ratio to a greater area
*S*.

Therefore, inversely, as the square on *FH* is to the
square on *DB*, so is the area *S* to the circle *ABCD*.

But, as the area *S* is to the circle *ABCD*, so
is the circle *EFGH* to some area less than the circle *ABCD*;
therefore also, as the square on *FH* is to the square on
*BD*, so is the circle *EFGH* to some area less than
the circle *ABCD*: which was proved impossible. [**Lemma
and V. 11**]

Therefore, as the square on *BD* is to the square on
*FH*, so is not the circle *ABCD* to any area greater
than the circle *EFGH*.

And it was proved that neither is it in that ratio to any
area less than the circle *EFGH*; therefore, as the square
on *BD* is to the square on *FH*, so is the circle
*ABCD* to the circle *EFGH*.

Therefore etc. Q. E. D.

I say that, the area *S* being greater than the circle
*EFGH*, as the area *S* is to the circle *ABCD*,
so is the circle *EFGH* to some area less than the circle
*ABCD*.

For let it be contrived that, as the area *S* is to the
circle *ABCD*, so is the circle *EFGH* to the area
*T*.

I say that the area *T* is less than the circle *ABCD*.

For since, as the area *S* is to the circle *ABCD*,
so is the circle *EFGH* to the area *T*, therefore,
alternately, as the area *S* is to the circle *EFGH*,
so is the circle *ABCD* to the area *T*. [**V.
16**]

But the area *S* is greater than the circle *EFGH*;
therefore the circle *ABCD* is also greater than the area
*T*.

Hence, as the area *S* is to the circle *ABCD*,
so is the circle *EFGH* to some area less than the circle
*ABCD*. Q. E. D." (**Euclid’s
Elements 12.2**** 1**).

- Euclid deals with three cases when comparing the ratio of
the squares of
*BD:FH*to the ratio of the circles*ABCD:EFGH*. The first case is that the ratio of the squares*BD:FH*equals*ABCD:S*where*S*is some area less than circle*EFGH*. Euclid uses most of the proof to refute this case by employing Eudoxus’ method of exhaustion. In order to approximate the circles by successive inscribed polygons, the square*EFGH*is inscribed in the circle*EFGH*, and it is shown that the remainder is less than half the circle. The circumferences are bisected to construct an octagon*EKFLGMHN*, and again the remainder of the circle is shown to be less than half the previous remainder. This process can be continued as much as one wishes, constructing polygons with 16, 32, 64, etc., sides which increasingly approximate the circumscribed circle. Each successive polygon leaves a remainder less than half of the previous remainder. At some stage of this process, the circle*EFGH*exceeds the area*S*by some finite amount, and using proposition X. 1, the remainder will be less than the excess of circle*EFGH*over*S*. Euclid uses the octagon*EKFLGMHN*stage throughout the rest of the proof. More specifically,

- At this step in the proof a similar octagon
*AOBPCQDR*is inscribed in circle*ABCD*. Thus,

- This last statement contradicts the prior statement that
area
*S*< octagon*EKFLGMHN*(**Joyce 1**). - The second case is that the ratio of the squares
*BD:FH*equals*ABCD:S*, where*S*is some area greater than circle*EFGH*. Euclid inverts this statement to the statement that the ratio of the squares*FB:BD*equals*EFGH*to some area less than circle*ABCD*, which is the first case that has been contradicted. Thus, by proving a contradiction in the first two cases, Euclid proves the third case of the ratios of the squares*BD:FH*equals the ratio of the circles*ABCD:EFGH*. As a note, modern mathematicians point out that there is a gap in Euclid’s last step of this proof, in that he never showed that the three cases he dealt with were the only three cases (**Joyce 1**). - Euclid relied on the method of exhaustion in much the same
way for proposition XII. 18, that spheres are to one another
in triplicate ratio of their diameters, but instead of inscribing
polygons to exhaust circles, polyhedra were inscribed to exhaust
spheres. In Book XII of the
*Elements*, Euclid demonstrates the rigor, the power, and the beauty of Eudoxus’ method of exhaustion. This method provided the ability to determine areas and volumes bounded by curves without the use of limits and is considered to be the predecessor of integral calculus (**Aulie 1**). Eudoxus, through his work in extending the theory of proportion and inventing the method of exhaustion, played a significant role in Euclid’s development of the*Elements*, considered to be one of the greatest works in history.

Aulie, Richard Paul. (Feb. 1, 2000). *Eudoxus Of Cnidus,
Method of exhaustion. *Britannica.com. **http://www.britannica.com/bcom/eb/article/6/0,5716,33776+3,00.html**.

Heath, Thomas L. *The Thirteen Books of Euclid’s *Elements,
3 vols., Dover (reprint), 1956.

Joyce, D.E. (Feb. 2, 2000). *Euclid’s Elements Book
XII Proposition 2*. (1996). Clark University. **http://aleph0.clarku.edu/~djoyce/java/elements/bookXII/propXII2.html**.

Kline, Morris. *Mathematical Thought from Ancient to Modern
Times*. New York: Oxford University Press, 1972.

No author cited. (Feb. 2, 2000). *Euclid of Alexandria*.
School of Mathematics and Statistics, University of St. Andrews,
Scotland. (January 1999). **http://www-history.mcs.st-andrews.ac.uk/history/Mathematicians/Euclid.html**.

No author cited. (Feb. 1, 2000). *Euclid *Elements*
toc,12*** **1. Perseus Project, Tufts University. **http://www.perseus.tufts.edu/cgi-bin/text?lookup=euc.+toc,12&vers=english;heath**.

If you have any comments that would be useful, especially for
use at the high school level, please send e-mail to **esiwdivad@yahoo.com**.

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