Eudoxus of Cnidus

Eudoxus (c. 400 B.C.) is the greatest of the ancient mathematicians, surpassed only by Archimedes -- but later. Biographical highlights:

• Eudoxus  was born in Cnidos, on the Black Sea.
• He studied mathematics with Archytus in Tarentum.
• He studied medicine with Philistium on Sicily.
• At 23 years he went to Plato's academy in Athens to study philosophy and rhetoric.
• Some time later he went to Egypt to learn astronomy at Helopolis.
• He established a school at Cyzicus on the sea of Marmora and had many pupils.
• In 365 B. C. he returned to Athens with his pupils. He became a colleague of Plato.
• At the age of 53 he died in Cnidos, highly honored as a lawgiver.
• He was the leading mathematician and astronomer of his day.

Eudoxus was the most reknown astronomer and mathematician of his day. In astronomy devised an ingenious planetary system based on spheres.

The spherical earth is at rest at the center.

Around this center, 27 concentric spheres rotate.

The exterior one caries the fixed stars,

The others account for the sun, moon, and five planets.

Each planet requires four spheres, the sun and moon, three each.

Consider the moon.

• The outer sphere rotates in one day as the sphere of the stars and with axis perpendicular to the zodiac circle. One period is 24 hours.
• The next middle sphere rotates on an circle at an angle to the plane of zodiac circle, and from east to west. One period is 223 lunations. From this sphere the ``recession of the nodes" is realized.
• The inner sphere rotates about an axis inclined to the axis of the second at an angle equal to the highest latitude attained by the moon, and from west to east. The draconitic month, the period of this sphere, is 27 days, 5 hours, 5 minutes. The moon is fixed on the great circle at this angle.

Homocentric spheres for the moon

The description of the motion of the planets is more clever still.

This model was improved by Callippus by adding more spheres and by Aristotle added to this certain `retrograde" spheres. But all the emendations never accounted for variation of luminosity, which had been observed.

Eudoxus  also described the constellations and the rising and setting of the fixed stars.

However, within 50 years the whole theory had to be abandoned.

Eudoxus 's contributions to mathematics include:

• A theory of proportion; this allowed the study of irrationals (incommensurables).
• The concept of magnitude, as not a number but stood for such as line segments, angles, areas, etc, and which could vary continuously. Magnitudes were opposed to numbers, which could change discontinuously. This avoided giving numerical values to lengths, areas, etc. Consequently great advances in geometry were made.
• The method of exhaustion.
• Establishing rigorous methods for finding areas and volumes of curvilinear figures (e.g. cones and spheres).
• A profound influence in the establishment of deductive organization of proof on the basis of explicit axioms.

There is little question that Eudoxus added to the body of geometric knowledge. Details are scant, but probably his main contributions can be found in Euclid, Books V, VI, and XII.

The Theory of Proportion of Eudoxus  is found as Definition 5 of Euclid, Book V.

Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively tken in corresponding order.

In modern terms: a/b=c/d if and only if, for all integers m and n , whenever ma<nb then mc<nd, and so on for > and =.

This is tantamount to an infinite process. But it was needed to deal with incommensurables.

The Method of Exhaustion unquestionably helped resolve number of loose ends then extant. It contained as Proposition 1 of Book X.

Two unequal magnitudes being set out, if from the greater there is subtracted a magnitude greater than its half, and from that which is left a magnitude greater than its half, and if this process be repeated continually, there will be left some magnitude which will be less than the lesser magnitude set out.

Let be given. Let , and . Let and . Continue this process, generating the sequence , we eventually have .

How does this differ from our limit concept today?

With this result, Eudoxus  was able to establish following:

Proposition 2. (Book XII) Circles are to one another as the squares on the diameters.

This was proved on the basis of the previous proposition.

Proposition 1. (Book XII) Similar polygons inscribed in circles are to one another as the squares on the diameters.

To prove the Proposition 2, polygonal figures, of indefinitely increasing numbers of sides, are both inscribed and circumscribed in the circle. Assuming Proposition 2 does not hold will lead to the contratiction that the result must be false for the polygons also.

Proof of Proposition 2. Let a and A, d and D be the repectively diameters of the circles. Suppose that

Then there is an a'<a so that

Set . Let ( resp. ) be the inscribed regular polygons of n sides in circle a (resp A). Then

By the method of exhaustion it follows that for large enough n

which implies that

We know that

Thus, since , it follows that . But this is impossible, and we have a contradiction.

To complete the proof, it must now be shown that

is also impossible.

This is a double reductio ad absurdum argument, a requirement of this method.

Eudoxus  also demonstrated that the ratios of the volumes of two spheres is as the cube of their radii.

On pyramids Eudoxus proved

Proposition 5. (Book XII) Pyramids which are of the same height and have triangular bases are to each other as their bases.

Other propositions are more famous:

Proposition. The volume of every pyramid is one third of the prism of on the same base and with the same height.

Proposition. The volume of every cone is one third of the cylinder on the same base and with the same height.

Curiously, the proof is by the method of slabs, familiar to all freshmen.

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