1. The geometer
Hippocrates of Chios is sometimes confused with a contemporary of his,
the famous physician
Hippocrates
of Cos, for whom the Hippocratic Oath is named. Not much is known
about him past what is read here. He was an accomplished geometer,
but was thought to have been otherwise simple-minded. A more detailed
biography can be found here.
2. John Philoponus, also called Grammaticus, was a sixth century (AD) scholar of philosophy (and a Christian theologian) who studied the texts of Greek philosophers, especially Aristotle, and wrote commentaries on them. A native of Alexandria in Egypt, he comes at the end of the period of Greek progress in the sciences.
3. A lune is a figure bounded by two circular arcs. The term is very descriptive of the resulting shape.
4. We will analyze in detail Hippocrates' quadrature of a lune below. It is enough now to understand the sentiment alluded to in this portion of the text. For the first time, someone had determined the area of a figure with curved sides--in fact, circular sides--and it was thought that the techniques used for the quadrature of the lune might lead to the quadrature of the circle.
5. Simplicius was another sixth century commentator on early Greek texts, notably on the work of Aristotle and Euclid. He was born in Cilicia, a Roman province in modern-day Turkey, studied in Athens at the Academy that Plato had instituted centuries earlier, and served for a time in the court of the Persian king.
6. Eudemus of Rhodes (350?BCE--290?BCE) was the first historian of mathematics. He was a fellow student with Aristotle in Athens and authored histories of geometry, arithmetic and astronomy, none of which survive today. What we know of his work is based on references and quotes in other works.
7. We no longer have access to what Hippocrates' general quadrature of lunes may have been, but we do have an account, quoted below from Eudemus, of the quadrature of the lune "on the side of a square". Simplicius believes that Hippocrates had claimed to have squared the circle as a result of his work on lunes, but the argument he gives contains an obvious fallacy, and it is not clear that Simplicius is to be trusted on this account. We will study this argument in the exercises.
8. Recall what we discussed earlier about the terse style of writing adopted by mathematicians. This style persists today; just open any upper level mathematics book or research journal and see.
9. Mathematical prose labels assertions as one of a small number of types: Axioms and postulates are self-evidently true. They are stated without proof and form the first principles of the theory that follows. Propositions and theorems are results that require proof, and these proofs may cite the axioms and postulates or previously proven propositions and theorems in their arguments. Theorems are usually more important results than propositions. Lemmas are results whose assertions are important only because they are needed in the service of the proof of a proposition or theorem that is to follow, and corollaries are results that are immediate consequences of other propositions or theorems. The result cited here is a perfect example of a lemma: the statement about segments of circles is "useful to his purpose".
A segment of a circle is any portion of the circle cut off by a line. Segments are similar if the angles they subtend at the centers of the circles are equal.
Thus, to say that "similar segments of circles have the same ratios as the squares on their bases" means that the ratio of the areas of the segments is the same as the ratio of the squares of the lengths of their bases. In symbols, if segment 1 has area s and base b and segment 2 area S and base B, then
10. The lemma he asserts in the previous sentence is a corollary to the result he cites here, that the ratio of the areas of two circles is the same as the ratio of the areas of the squares whose sides are the diameters of the two circles. In symbols, if circle 1 has area a and diameter d and segment 2 area A and diameter D, then
This result is the focus of the article on Eudoxus which we will read next.
11. The figure described here is
12. This is deduced by combining the lemma on areas of circular segments (see the end of note 9) with the Pythagorean theorem:
The Pythagorean theorem says that
By the lemma, we can transform this into a relationship between the areas of the segments:
This shows that "the segment about the base is equal to the sum of those about the sides".
13. The lune
contains both small segments (see the diagram in note
12 again); if we replace both with the single larger segment whose
area is the same, we get the triangle. Thus, the area of the lune is equal
to that of the triangle.
Read the scholium
on the integral
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