Wednesday, April 12, 2017

The Bible and Vitruvius knew about Practical Geometry; Plato did too.


Practical Geometry  -  A lecture for SAH, Latrobe Chapter , May 9, 2017
I will be in Washington, DC, speaking to architectural historians

The lecture will be copiously illustrated, but not hands-on. Unlike the IPTN Workshops no one will learn to use a compass.

Preparing a talk always requires that I do more research, more than I can share in one talk. So here is some of what I will paraphrase, starting with what was written at least 2600 years ago.

Compasses,  basic tools in geometry, have been standard equipment for builders since early times.In the Bible, the 6th c, BCE, the prophet Isaiah describes the work of woodsmen, blacksmiths, and carpenters as he deplores the creation of graven images:
Isaiah 44: The carpenter stretches out his rule: he marks it out with a line; he fits it with planes, and he marks it out with the compass.

  Vitruvius,1st. c. BCE, does not write easily. He works hard to find the right phrase. However, he is so present, so involved, that I enjoy his work. I feel as if  he is here, intently explaining an idea. I wish I could have discussed Vitruvius with the translator of my edition, Morris Hickey Morgan,

Vitruvius,  The Ten Books of Architecture,
Book I, Chapter I, The Education of the Architect

1: Theory ... is the ability to demonstrate and explain ... the principals of proportion.

3:  Neither natural ability without instruction nor instruction without natural ability can make the perfect artist. Let him be educated, skillful with the pencil, instructed in geometry,know much history,  have followed the philosophers with attention, understand music...  

He then elaborates (I have left some of it out but it is well worth reading ):
...he must have knowledge of drawing s that he can readily make sketches to show the appearance of the work which he proposes. Geometry, also, is of much assistance in architecture, and in particular it teaches us the use of the rule and compasses, by which especially we acquire  readiness for making plans for buildings in their grounds, and rightly apply the square, the level and the plummet. .. It is true that it is by arithmetic that the total of buildings is calculated and measurements are computed, but difficult questions involving symmetry are solved by means of geometrical theories and methods.

Chapter II, The Fundamental Principles of Architecture

1: Architecture depends on Order, Arrangement, Eurythmy, Symmetry, Propriety, and Economy.   
Vitruvius then writes a paragraph for each idea. Again I am quoting pieces, and suggest you enjoy reading the whole
2: Order gives due measure to the members of a work considered separately, and symmetrical agreement to the proportions of the whole... selection of the modules from the members of the work itself, and starting from these individual parts of members, constructing the whole work to correspond.
3. Eurythmy is beauty and fitness in the adjustments of the members. This is found when the members of a work are at a height suited to their breadth, of a breadth suited to their length, and in a word that they all correspond symmetrically.
4. Symmetry is a proper agreement between the members of the work itself, and relationship between the different parts and the whole general scheme, in accordance to a certain part selected as a standard.

Vitruvius then  mentions how in the human body there is a kind of symmetrical harmony -   which becomes in the Renaissance the Vitruvian Man.

Chapter III.
Buildings must be ... built with due reference to durability, convenience and beauty. Beauty is when the appearance of the work is pleasing and in good taste, and when its members are in due proportion according to correct principles of geometry.

Book IX, Introduction
Vitruvius praises the Greek writers, specifically Aristotle, Democritus, Plato and Pythagoras. He specifically discusses Plato and Pythagoras.

3. ... Of their many discoveries that have been useful for the development of humans life, I will site a few examples.
4. First of all, among the many very useful theorems of Plato, I will cite one as demonstrated by him.
Paragraphs 4 and 5 are examples of  Plato's teachings of geometry.Paraphrasing, a square field needs to be doubled in size, and still be square. Vitruvius says finding the side of the new square cannot be done with arithmetic and describes this :

A-B-C-D is a square. A-C is its diagonal. The triangle A-B-C is the same size as A-C-D. Using A-C as the side of the new square , see that A-C-E-F is made up of 4 triangles, each the size of the original 2 in A-B-C-D.
Look at that! A-C-E- F is twice as big!

 Paragraphs 6,7, and 8 describe Pythagoras'  knowledge of the 3/4/5 right triangle:
7. ...When Pythagoras discovered this fact, he had no doubt that the Muses had guided him in the discovery, and it is said that he very gratefully offered sacrifice to them. 

Book IX goes on to discuss the zodiac, planets, astrology, phases of the moon and sundials.


I have not yet read Plato or Pythagoras on geometry.
Next post will be a brief review of  the use of geometry in Medieval Europe.

Vitruvius, The Ten Books on Architecture, translated by Morris Hicky Morgan, Harvard University, 1916, reprinted by Dover Publications, Inc. 1960

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