What is the role of deductive reasoning in Euclid’s method of proceeding? Give an example.
Euclid used deductive reasoning to prove his Optics theorems. Deductive reasoning is basically using many statements is a sort of 'top-down' logic to prove theorems.
Now what Euclid did in Opticswas that he used many postulates and definitions to prove the theorems about vision.
Some of the postulates in Optics are:
Let it be assumed :
1. That rectilinear rays proceeding from the eye diverge indefinitely;
2. That the figure contained by a set of visual rays is a cone of which the vertex is at the eye and the base at the surface of the objects seen;
3. That those things are seen upon which visuals rays fall and those things are not seen upon which visual rays do not fall;
4. That things seen under a larger angle appear larger, those under a smaller angle appear smaller, and those under equal angles appear equal;
5. That things seen by higher visual rays appear higher, and things seen by lower visual rays appear lower;
6. That, similarly, things seen by rays further to the right appear further to the right, and things seen by rays further to the left appear further to the left;
7. That thing seen under more angles are seen more clearly.
Using these postulates, Euclid proved his theorems about optics. This methodology is called Deductive Reasoning.