4

I have dipped into the Metaphysics pretty extensively, and I know that he uses a number of fascinating methods for decomposing arguments in that book. No wonder St. Thomas Aquinas read it while he was imprisoned. The book is a cerebral tour de force.

One of the above mentioned methods for decomposing arguments is geometrical. He draws analogies between the way three-dimensional figures are drawn and the way propositions about Plato's forms constructed, in order to shed a dubious light on the latter. I have to say that while I love his thinking style in that book, he's preaching to the choir really. I have never really liked Plato's forms, so I tend to tune out there and tune back in when he brings up the likes of Anaxagoras and Democritus.

So I'd like to know where else I can read that kind of 'shapely' argumentation, particularly where (else) Aristotle makes the most references to Euclidean geometry.

4

Henry Mendell discusses Greek mathematics in Aristotle's Works in SEP:

Here are twenty-five of his favorite propositions (the list is not exhaustive). Where a proposition occurs in Euclid's Elements, the number is given, * indicates that we can reconstruct from what Aristotle says a proof different from that found in Euclid). Where the attribution is in doubt, I cite the scholar who endorses it. In many cases, the theorem is inferred from the context.

1. In a given circle equal chords form equal angles with the circumference 
    of the circle (Prior Analytics i.24; not at all Euclidean in conception)
2. The angles at the base of an isosceles triangle are equal (Prior 
    Analytics i.24; Eucl. i.5*).
3. The angles about a point are two right angles (Metaphysics ix 9; Eucl. 
    follows from i def. 10).
4. If two straight-lines are parallel and a straight-line intersects them, 
   the interior angle is equal to the exterior angle (Prior Analytics ii.17; 
   Eucl. i 29, cf. 28).
5. If two straight-lines are parallel and a straight-line intersects them,the 
   alternate angles are equal (possibly, but not likely Prior
   Analytics ii.17 (Heiberg), although the theorem is about parallel
   lines and uses (7); Eucl. i 29*).
6. If a straight-line intersects two straight-lines and makes interior or 
    exterior angles equal to two right angles on the same side
   with each, then the lines are parallel (possibly Posterior Analytics
   i.5 (Blancanus, Heath), but it is possibly the weaker theorem that
   each angle formed by the intersecting line is right, rather than their
   sum equals two right angles); Eucl. i 28*)
7. The internal angles of a triangle are equal to two right angles (frequent, 
   but cf. Prior Analytics i.35, Metaphysics ix.9; Eucl.
   corollary to i.32*)
8. The angle in a semicircle is a right angle (Posterior Analytics i.1, ii.11, 
    Metaphysics ix.9; Eucl. iii.31*)
9. In a right triangle the squares on the legs are equal to the square on the 
   hypotenuse (De incessu animalium 9 (Heath); Eucl. i.47).
10. To find the mean proportion of two lines (De anima ii.2, Metaphysics 
   iii.2; Eucl. vi.13, cf. ii.14)

Complete list is here.

0

Thomas Aquinas's works, including his commentaries on Aristotle, contain many references to mathematics, all of which are documented here (& ff.).

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