After stating that "the extremities of a line are points- and postulating that it is possible "to draw a straight line from any point to any point-, he opened the doors to a realm of perfect forms, or figures, that can only exist in our minds. Figures we can only see through the eyes of reason.
Although I think that we should never renounce reality in our search for philosophical answers, I'm persuaded that it is in a universe similar in nature to the world of geometry that Euclid opens up for us, where Beauty herself exists. Using reason as our guide, we should look here for her ei]doj. Great dangers await one who ventures into a realm of forms without well examined hypotheses and rigorously followed logical procedures. So, lest we end up being "robbed of the truth and knowledge of the things that are- like those who "trust arguments to be true without the art of arguments,"" we ought to learn from Euclid and follow his lead into the cosmos of geometry.
A good way to start, then, is perhaps by taking a look at the masterpiece of his first book: The 47th proposition. In a rigorous, fast, and easily understandable way, XLVII proves that in a right-angled triangle the square of the hypotenuse, or diagonal, is equal to the sum of the squares on the remaining sides. This theorem is named after its first observer, Pythagoras, but it was Euclid's insight what made it clear enough to become the basis for a great number of further mathematical discoveries. .
"In right-angled triangles-, the Alexandrian mathematician tells us, "the square on the side subtending the right angle is equal to the squares on the sides containing the right angle-. We set out on our inquiry armed with twenty-three definitions, five postulates, an equal number of common notions, and forty-six propositions.
