Palladio Discusses Geometry

This is part of Plate 1, The Second Book of Andrea Palladio's ARCHITECTURE.

Palladio's 4 books of Architecture were published in`1570 in Venice.  They were meant for Italians. He wrote in terms of the local climate, and materials he used in and around Venice. He also shares ways of  working that  today we would agree are 'best practice".
Here is a small example.
He explains in Book I, Chap. II, On Timber
that timber should be felled in autumn or winter, "in the wane of the moon". This may be because of the gravitational pull of the full moon on sap.
Palladio adds that the timber must be "laid in a proper place,.. shelter'd from the south sun, high winds, and rain."  
This final sentence would delight all modern contractors: "Those therefore who are about to build, ought to be inform'd from men thoroughly acquainted with the nature of timber, that they may know which is fit for such and such uses, and which not."

Many Northern Europeans traveled to Italy as part of their education. The 4 Books were admired and translated. Inigo Jones, English Master Builder, brought his own copy back when he returned from Italy in 1614. He marked it up. He used it. Today his copy is kept at Worcester College, Oxford.

The first Book is about construction and materials and the all important Roman columns; the second Book discusses houses; the third Book reviews public places and spaces; the fourth Book concludes with Roman temples.
Note: the word 'fabric" means something fabricated, constructed, a building.

He starts Book I, Chapter I :  "...three things, according to Vitruvius, ought to be considered in every fabric*...and these are utility, or convenience, duration, and beauty."...
"Beauty will result from the form and correspondence of the whole, with respect to the several parts, of the parts with regard to each other, and of these again to the whole; that the structure may appear an entire and complete body, wherein each member agrees with the other, and all necessary to compose what you intend to form."
This is what  we called Practical Geometry 200 years later.

Book I has beautiful drawings of the five orders - the proper proportions and ornaments for columns.
In the introduction, Chapter VIII, Palladio writes:
"... in the dividing and measuring of the said orders, I would not make use of any certain or determinate measure particular to any city, as a cubit, foot, or palm knowing that these several members differ as much as the cities and countries; but imitating Vitruvius, who divides the Dorick order with a measure taken from the thickness or diameter of the columns, common to all, and by him called a module. I shall therefore make use of the same measure in all the orders."




Here is Plate XII, Book I. The Dorick Order

Note that he has given dimensions: the column width is 2 modules ( MO.2). the space between the columns is MO. 5 .1/2. tThe  height of the columns is MO. 15.


Palladio includes half of his  'Venetian foot' in his 4 Books. explaining that he divides his whole foot into 12 inches and then each inch into 4 minutes.  He notes that the numbers on his fabrics are dimensions based on this particular foot.




First Book, Chapter XXI, Of the loggias, entries, halls, rooms and of their form 

The last paragraph:
The most beautiful and proportionable manners of rooms, and which succeed best, are seven, because they are either made round (tho' but seldom) or square, or their length will be the diagonal of the square, or of a square and a third, or of one square and a half, or of 1 square and two thirds, or of 2 squares.




Here are the diagram for the floor plan house above based on Palladio's list of beautiful proportions.

I have copied from the book. It is not quite square. Therefore my analysis is general.

I have used the division of the square into halves and quarters, and the division of the square into thirds, both ways of lay out that could have been drawn on a board or on plaster and laid out with a rope.


The plan is a square. The square divided into 4 equal rectangles marks the position of the interior walls from top to bottom. They are thicker than the other interior walls, so probably structural.  The line established is on the outside of the walls which would be appropriate for building a wall which would sit on a foundation and support the building.
Palladio may be showing off what a good designer he is, but his drawings are also construction drawings meant to convey to his workmen where to build the walls and the columns.  

When the diagonals of the square are added, corner to corner, the interior walls (left to right) at the edge of the stairs are determined.

The placement of the columns in the grand hall are also noted.

Both points divide the square into thirds.





The loggia is 2 squares - one of Palladio's 7 favorite room shapes.

The side rooms are also his "beautiful proportions". Upper right: the square with its width the diagonal of the square. Middle right: the square and a half. Lower right: a square and two thirds.

The main hall is different, not only grand with columns, but centered on the whole house
The upper columns' location comes from the lines. They are at one of the thirds of the square floor plan. The lower columns are then placed the same distance from the wall for symmetry. The diagonals on each side position the columns more directly.
The squares overlap - not in a proportion that Palladio admired. However the overlap is the width of the doors which run from one end of the house to the other.


* fabric is an archaic word for a building, My copy of Palladio was translated into English around 1738.

Andrea Palladio, The Four Books of Architecture, published by Isaac Ware, London, 1738, reprinted by Dover Publications, Inc. NY, 1965.

I will add the practical geometry of the facade next. 

Serlio writes about Practical Geometry

The  hypothetical church facade - below -  drawn by Sebastiano Serlio, 1475-1554, was one of the illustrations in his  'Architectura', published in 5 volumes, beginning in 1537. It was translated from Italian into Dutch, then into English in 1611. Christopher Wren and Inigo Jones both had copies.
In 1996 it was translated again by Vaughan Hart and Peter Hicks.

For me a drawing of a building is as good evidence of someone's thinking as his/her words in a book. Here I have both.

This image is informed by geometry.
The extravagance of details - plinth blocks, double columns, not one but three oriel window, stone worked in patterns, is startling. Nothing here is backdrop; all is to be seen. Serlio holds it all together with geometry.


I've put the diagrams for the image at the end of this post because I want to first show a few explanations he gave about the use of geometry.


Serlio's Book I, On Geometry, is not very long, 13 pages.

Here are 2 illustrations.

He presents a problem to the reader on the last page: how to design a door for an existing church.
The answer is, if the church has pillars to take the width between the columns as the dimension of a square, "as high as broad," adding
"the diagonal lines,and the other two cross  cutting lines" will not only give you "the width of the door but also the places and points of the ornaments of the same door as you may see here in this figure."  


The diagram is part of what today we call The Rule of Thirds.  It can regularly be seen in  Pre-WWI American construction in layout and design.


Serlio also shows 3 ways to lay out an ellipse. Here are two.

Draw a circle and add one on each side. The left and right sides of the outside circles are the ends of the ellipse.
He does not comment about the need to set the circles on a line - which will establish the point on the circumference where the next circle is added, nor that a line perpendicular to the first line is also necessary to find the points T and K. These are details an experienced mason or carpenter would have known. The
T and K become the centers for radii to draw arcs of the upper and lower sections of the ellipse. "Then placing one point of the compass at K you must " draw "a line with the other point from the figure of 1 to 2.... This figure is very like the form of an Egge."    

The second example begins with 2 squares and their diagonals. The radius for the ends of the ellipse is half the diagonal, g to 1 for example. The radius for the top and bottom of the ellipse is the whole diagonal, f to 2.

Here is the diagram  Owen Biddle of Philadelphia added to his instructions for drawing an ellipse in his pattern book published in 1805.

He did not seem to know of Serlio's layout which looks to be simpler.

Practical Geometry:
The main floor is made up of 2 squares, or perhaps 1 square in the middle and a half square on each side.
The upper floor is also 2 squares if its arcs are included.
I drew in red the diagonals of the squares on the right side. I also extended the main floor diagonal with a dotted line on the upper floor to show how one part is an extension of the other.

On the left side I drew the diagram shown above that Serlio drew for finding a door. Here it indicates the location of the oriel window and the edges of the rounded pediment over the window.

This engraving's  proportions are a little off. Is it because it's just hypothetical, or because it has been reprinted so many times?


Footnotes: I am reading Serlio's  Architettura, Book I, 1611, English translation,  on-line, printing parts of it and then enlarging the page. The English is archaic. So is the spelling. The printing blurred. If I read it out loud it is easier, much like reading hand written property deeds from the 1830's and '40's. The diagrams are clear!

2 books on early American libraries do not mention Serlio.
American Architects and Their Books to 1848, edited by Kenneth Hafertepe and James F. O'Gorman, 2001, U. Mass. Press, Amherst. MA.
Architectural Books in Early America, Janice G. Schimmelman, 1999, Oak Knoll Press, New Castle, DE.

January, 2018. I ordered the 1996 translation of Serlio in the fall. It had to be specially printed at my request - not cheap! Easier to read with illustrations
The translators - Vaughan Hart and Peter Hicks - have not heard of  Practical Geometry and think Serlio invented 'Lines' ( see Glossary, p. 458, Linee occulte. 
They also seem to imply that the tools illustrated on the cover ( see above) are symbols, not the actual tools masons used in construction. They do not seem to view  this book as a manual for practitioners, but as a philosophical treatise.
Still I'm glad they translated it.

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.

Vitruvius refers to illustrations. The lone surviving copy of his book was discovered in a Swiss monastery in medieval times. It had no illustrations and none have yet been discovered. 


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