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Early Greeks & Aristotle

II. The Method of Exhaustion (Eudoxus and Archimedes)

Euclidean Geometry: Two notions of the infinite:

(a) infinite divisibility of line segments
(b) infinite extendability of line segments

lead to paradoxes of infinitely small and infinitely big


BUT: Early Greeks tended to avoid talk of the infinite. In geometry, all objects are really

finite (like natural numbers: any one is finite; together are all infinite)

Example: Method of Exhaustion (Eudoxus 408-355 BC) as used by
Archimedes to prove area of circle = πr2.

Let C be a circle with radius r.

For each natural number n, let be a regular polygon inscribed in C.
n equal sides and
n equal angles
Divide into n congruent triangles  

= base of triangle
= height of triangle

THEN: area of triangle =
AND: area of 

Now visualize C as -- a polygon with infinitely many infinitely small sides.

SO: When n = ∞:

The height of each (infinitely thin!) triangle in is
identified with the radius of C (and the base of each
triangle in is very, very small... infinitely small!).

SO: area of C = area of
, when n = ∞


(1) What does it mean to multiply by an infinitely small amount ( when
n = ∞)? (Can’t be same as multiplying by 0!)

(2) What is a polygon with infinitely many infinitely small sides?

(3) As n goes to infinity, approximates C, but also C*:

What does it mean to say C is what
is tending towards and not C*?

Archimedes’ Solution

Proved 2 claims:

Claim I: There is a regular polygon as close in area to C as you care to specify.

i.e., For any arbitrary small area ε, there is always a number n such that differs
in area from C by less than ε.) 

not true for C*

Consequence: The area of C is at most πr2.

Claim II: The area of C is at least πr2.

Consequence of I and II: The area of C is exactly πr2

Significance of Archimedes’ Solution: No mention of infinity!

III. Aristotle

  only world there
is for Aristotle
Empiricist: Platonic Forms are in the physical world.
Relevant Question: Is anything in nature infinite?

Aristotle’s Answer:

The infinite exists potentially and not actually.

actual infinite: that whose infinitude exists at some point in time
potential infinite: that whose infinitude exists over time (not wholly present)

Note: For A., this is literally the distinction: Time is infinite, but not space

Aristotle’s Response to Zeno’s Paradoxes:

Achilles and the Tortoise

The distance between Achilles and the Tortoise is only potentially infinitely divisible; it is not
actually infinitely divisible. And there is no contradiction in claiming that a finite length is
potentially infinitely divisible.


To travel a potentially infinitely divisible distance, Achilles needs a potentially infinite time. And
there is nothing wrong in claiming he has such a time available.

Aristotle’s infinite: “the untraversible”

Something is infinite if, taking it quantity by
quantity, we can always take something outside.”

“It is not what has no part outside it that is infinite,
but what always has some part outside it.”

Under Moore’s reading, Aristotle rejects the “metaphysically infinite” and adopts the “mathematically infinite”.

Problem for Aristotle: What about the infinite past?
  Already traversed?
So actually infinite?

Wittgenstein’s Story:
Suppose we come across a man saying “... 5, 1, 4, 3.”, who then proceeds to tell
us that he has just finished reciting π backwards for all past eternity. Why does
this strike us as impossible, whereas someone who just starts reciting π forwards
and will continue for all future eternity does not (given that we concede the
possibility of living forever).