By Frater Apollonius 4°=7□



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Of Euclid

Euclid's residence in Alexandria is reported by Pappus (320 AD), who observes that Apollonius spent many years studying with Euclid's pupils in that city. Writing more than a century later, Proclus, after a brief synopsis of the Platonic school, which included Hippocrates, Eudoxus, Menaechmus and Theaetetus; says of Euclid:

"Those who compiled histories carry the development of this science up to this point. Not much younger than these is Euclid, who put together the Elements, arranging in order many of Eudoxus' theorems, perfecting many of Theaetetus', and also bringing to irrefutable demonstration the things which had been only loosely proved by his predecessors. This man lived in the time of the first Ptolemy; for Archimedes, who came immediately after the first Ptolemy, makes mention of Euclid; and further they say that Ptolemy once asked him if there was in geometry a way shorter than that of the elements; he replied that there was no royal road to geometry. He is therefore younger that the pupils of Plato, but older than Eratosthemnes and Archimedes. For these men were contemporaries, as Eratosthenes somewhere says. In his aim he was a Platonist, being in sympathy with this philosophy, whence it comes that he made the end of the whole Elements the construction of the so-called Platonic figures. There are many other mathematical writings by this man, wonderful in their accuracy and replete with scientific investigations. Such are the Optics and Catoptrics, and the Elements of Music, and again the book on Divisions. He deserves admiration pre-eminently in the compilation of his Elements of Geometry on account of the order and of the selection both of the theorems and of the problems made with a view to the elements. For he included not everything which he could have said, but only such things a he could set down as elements. And he used all the various forms of syllogisms , some getting their plausibility from the fist principles, some setting out from demonstrative proofs, all being irrefutable and accurate and in harmony with science."

There is an aesthetic, minimalistic simplicity to Euclid. Starting with three simple statements - there is point, there is line, there is plane - the Elements advance step by step like a Jacob's ladder from the geometry of the triangle, to that of the circle, the parallelogram and the formation of areas. Propelled by the logic of ratios and the theory of irrational lines, Euclid arrives finally at the geometry of the sphere and the five Platonic solids that inhabit three-dimensional space. Sometimes plodding, at times repetitious, each wrung is anchored in an unshakable logic that rises from the elementary to the highly complex, at certain points taking astonishing turns and transformations that flip human perception inside-out and upside down.

One can argue that Euclid is a prime example of intellectual abstraction. In fact, many have contended that the Greeks were less concerned with resolving concrete problems than with the advancement of pure logic. This argument is often employed to explain Archimedes' works and attitude. Yet history attests that Archimedes addressed a great many practical problems involving warfare and weaponry. At the same time, while Euclidean mathematics and geometry may indeed epitomize pure abstraction, the last three books of the Elements lay the groundwork for modern physics. Even though the Greeks fashioned mathematics as an abstract construct, it would become a very potent tool for resolving pragmatic problems.

With Euclid, of course, there are no short-cuts. One must take the argument step by step, wrung by wrung, and climb the ladder slowly. Having reached the summit, one finally comprehends why people have scaled this structure for more than twenty centuries. One marvels at the way it affords such certainty in a uncertain universe where the gods are often fickle.

Permeating the structure, moreover, are two fundamental mathematical constructs, to which we have already been introduced. The combination of these would prove vital to the development of trigonometry - the point at which geometry becomes physics.

If the Pythagoreans had provided a general rule for the sides and hypotenuse of a triangle, it would be left to Euclid to present a final proof. The proof comes at the end of Book I, under the form of Preposition 47. Since Book I is comprised of 48 prepositions, the proof appears as a crowning moment in the gradual progression of the Elements of Book I. Later, in Book VI, the theorem is represented in a version of that has wider applications, through an argument that is entirely based on proportion. "If we listen," says Proclus, "to those who wish to recount ancient history, we may find some of them referring this theorem to Pythagoras...But for my part... I marvel more at the writer of the Elements, not only because he made it fast by a most lucid demonstration, but because he compelled assent to the still more general theorem by the irrefragable arguments of science in Book VI."



The general theorem in question is Proposition 31, Book VI, which states: "In right-angled triangles the figure on the side subtending the right angle is equal to the similar and similarly described figures on the sides of the right angle. As the figure shows, the construction for the proof is a fusion of the geometrical mean and the Pythagoras Theorem. (fig. 31)





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