Geometry and the Quadrivium

Geometry and the Quadrivium

Whist sitting in school, slaving away with compasses and a ruler, one hardly remembers that geometry is the study of the measurement of the earth. Earth. The thing we sit on, utilize, and finally rest in when this is all over. The geometry in schools today looks nothing like the geometry of 3000 years ago. It is difficult to divorce geometry from the other liberal arts when we take into consideration the scale to while discoveries are interconnected. Geometry arose from the needs of agriculture, civilization, and war. For so much of this, we can thank Archimedes of Syracuse. A student of Euclid in the 3rd c. BCE, his advances in the field of geometry furthered irrigation (Archimedes’ Screw), astronomy (the first planetarium), and weights & measures (Archimedes’ Principle). The most interesting, to me, is The Method of Exhaustion (remember Dialectica) also known as “The Method” or “Archimedes’ Method.”

“…, to estimate the area of a circle, he constructed a larger polygon outside the circle and a smaller one inside it. He first enclosed the circle in a triangle, then in a square, pentagon, hexagon, etc, etc, each time approximating the area of the circle more closely. By this archimedes_circleso-called ‘method of exhaustion’ (or simply ‘Archimedes’ Method’), he effectively homed in on a value for one of the most important numbers in all of mathematics, π.” 1

Linked together with this Method is the “Method of Mechanical Theorems.” Proofs are everything to the mathematician, and in his Method of Mechanical Theorems, Archimedes had none that would be accepted. He set out using Eudoxus’ The Method of Exhaustion to prove what he knew to be true. In a letter to Eratosthenes, in manuscripts discovered in 1906, Archimedes outlines his thought processes. This document is known as the Archimedes Palimpsest.

Certain theorems first became clear to me by means of a mechanical method. Then, however, they had to be proved geometrically since the method provided no real proof. It is obviously easier to find a proof when we have already learned something about the question by means of the method than it is to find one without such advance knowledge.

The importance of these discoveries and the methods by which Archimedes came to them may be obvious – who doesn’t need π? However, it is also fascinating to peer inside the mathematician’s mind and view it with a Freemason’s perspective. Here was a man who could see the Plan, understand the Plan, and only needed to bring it to life: a divine spark of wisdom, the will to discover, and beauty in its presentation.

For an interesting and short expose on The Method and the “Archimedes Palmipsest,” whence this Method is documented, review  “The Illustrated Method of Archimedes” by  Andre Koch Torres Assis and Ceno Pietro Magnaghi. The PDF can be found here.

Additionally, the originally translated letter from Archimedes to Eratosthenes can be downloaded here. (Thank you, JSTOR.)


Just a note (1): The Story of Mathematics, Luke Mastin – http://www.storyofmathematics.com/hellenistic_archimedes.html – I’ve done my best to verify statements here, and so should you.

Trivium: Logic or Dialectica

Trivium: Logic or Dialectica

Today’s theme is Logic, or as seen the picture here, Dialectica. As the New Catholic Encyclopedia states, “Logic is the science and art which so directs the mind in the process of reasoning and subsidiary processes as to enable it to attain clearness, consistency, and validity in those processes. The aim of logic is to secure clearness in the definition and arrangement of our ideas and other mental images, consistency in our judgments, and validity in our processes of inference.”

trivio7

Aristotle is generally considered the “Founder of Logic,” although many others before him put themselves to the task of thinking about how we think. One of these, Zeno of Elea, was considered to have developed reductio ad absurdum, or the method of indirect proof. If something cannot be both true and false, then an argument can be made from reducing the statement to the absurd.

For example, “The earth is round. The earth is not flat. If it were flat, people would fall off the edge.” Since the earth cannot be both round and flat, the statement is true.

Another good example, from Wikipedia (Yes, I know. Don’t judge.), reads:

xenophanes1The ‘reduction to the absurd’ technique is used throughout Greek philosophy, beginning with Presocratic philosophers. The earliest Greek example of a reductio argument is supposedly in fragments of a satirical poem attributed to Xenophanes of Colophon (c.570 – c.475 BC). Criticizing Homer’s attribution of human faults to the gods, he says that humans also believe that the gods’ bodies have human form. But if horses and oxen could draw, they would draw the gods with horse and oxen bodies. The gods can’t have both forms, so this is a contradiction. Therefore, the attribution of other human characteristics to the gods, such as human faults, is also false.

logic2Logic is mental training: once the words and language have been developed, we can think through situations, problems, and reason our way to clear conclusions that work in conjunction with the natural world. For example, if we seekers of Truth are to grow and understand how a symbol might be applied to our everyday lives, we need to understand not only what the symbol is, but how it works in the world around us, how nature employs it.

Logic utilizes the senses but the connection must be made in the mind to form usable conclusions. Logic is, to me, a fundamental aspect of any human being’s career, if one expects to progress through life and learn. We can learn Logic in the modern age via University, but this really teaches us about Logic, not how to employ our logical mind. It seems that only through discourse, or dialectica, are we able to truly develop logical thought processes and reasoning at a higher level. Masonic Philosophical Society, anyone?

 


As a side note, the Catholic Encyclopedia on newadvent.org has a very good article on Logic and its history. It’s concise and certainly doesn’t include manuscripts; I would encourage anyone with a keen interest in Logic or Dialectica to read Aristotle but also some of the pre-Socractic philosophers, whence a great deal of our modern ideas of logic come.