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Euclid – geometry

Euclid

Euclid

Postulate 1 ‘[It is possible] to draw a straight line from any point to any point’.

Postulate 2 ‘[It is possible] to produce a finite straight line continuously in a straight line’.

Postulate 3 ‘[It is possible] to describe a circle with any centre and diameter’.

Geometry

Euclid’s theorems are still true and his methods are still admired. For millenia his books have been studied and referenced, though they are no longer used as a school text-book.53 He entitled his principal work Elements, and it was intended to be a foundational work in the subject, a starting point. The same Greek word (stoikheia) also means the letters in the alphabet, and Euclid’s elements are to geometry what letters are to language: the building blocks or basic components.

One of the most outstanding features of Euclid’s work is its structure: the first book contains a number of definitions, postulates and common notions, and the following twelve books endeavour to introduce or assume no extraneous material as they progress, but only to construct from definitions and propositions already done. Thus, for any proposition one can trace back the reasoning for a particular result through earlier propositions until one comes back ultimately to the original postulates and common notions.

This trace can be illustrated by drawing a proof tree, of which an example is given below in Figure 3, to illustrate the reasoning for Pythagoras’ Theorem. Of course Euclid was not infallible, and there are occasionally holes in the arguments, but these should not be allowed to detract from the overall aim and success of his method. Another outstanding feature is the thoroughness with which propositions are proved, as will become apparent in the example given below. Let us first review the Elements.

Book 1 builds from twenty-three definitions, five postulates, and nine common notions.54 The definitions explain the basic terms of geometry, what is meant by words such as ‘point’ or ‘line’. The common notions are axioms or self-evident truths; statements that any sensible person would take as true, although it is not possible to prove them. For example, Common Notion 1 is ‘Things which are equal to the same thing are also equal to one another’. The postulates are unproved assertions about geometry. The first three postulates are assertions that amount to the possibility of doing geometry.

Postulate 1 ‘[It is possible] to draw a straight line from any point to any point’.

Postulate 2 ‘[It is possible] to produce a finite straight line continuously in a straight line’.

Postulate 3 ‘[It is possible] to describe a circle with any centre and diameter’.

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Sacred Geometry – levels

Plato considered geometry and number as the most reduced and essential, and therefore the ideal, philosophical language. But it is only by virtue of functioning at a certain ‘level’ of reality  that geometry and number can become a vehicle for philosophic contemplation. Greek philosophy  defined this notion of levels, so useful in our thinking,  distinguishing the ‘typal‘ and the ‘archtypal‘.  Following the indication given by Egyptian wall reliefs, which are laid out in three registers, an upper, a middle and a lower, we can define a third level, the ‘ectypal‘, situated between the archtypal and typal.

To see how these operate, let us take an example of a tangible thing, such as the bridle of a horse. This bridal can have a number of forms, materials, sizes, colours, uses, all of which are bridals. The bridal considered in this way, is typal; it is existing, diverse and variable. But on another level there is the idea or the form of the bridal, the guiding model of all bridals. This is an unmanifest, pure, formal idea and its level is ectypal. But yet above this there is an archtypal level which is that of the principal or power-activity, that is a process which the ectypal form and typal example of the bridal only represent. The archtypal is concerned with universal processes or dynamic patterns which can be considered independently of any structure or material form

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Greek Columns

Parthenon

Greek Columns

Greek Columns

Three Greek columns; Ionic, Corinthian and Doric made up of the capital, shaft and base. Of the three columns found in Greece, Doric columns are the simplest. They have a capital (the top, or crown) made of a circle topped by a square. The shaft (the tall part of the column) is plain and has 20 sides.

There is no base in the Doric order. The Doric order is very plain, but powerful-looking in its design. Doric, like most Greek styles, works well horizontally on buildings, that’s why it was so good with the long rectangular buildings made by the Greeks. The area above the column, called the frieze [pronounced “freeze”], had simple patterns.

Above the columns are the metopes and triglyphs. The metope [pronounced “met-o-pee”] is a plain, smooth stone section between triglyphs. Sometimes the metopes had statues of heroes or gods on them. The triglyphs are a pattern of 3 vertical lines between the metopes.

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