Practical Geometry - Drawing a Square with a Compass, Part 2

 4 more ways to draw a square with a compass.

For Part 1 see: https://www.jgrarchitect.com/2019/12/practical-geometry-drawing-square-with.html


How to draw a square with a compass  #3
Peter Nicholson wrote about Practical Geometry in 1793.  His first plates are introductions to the first rules of geometry: using a compass to bisect a line,

My blog post about him is : https://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by_16.html
It includes images of Plate 2 and Plate 3.

Here I have copied just the image of a square. Nicholson includes instructions for finding the square 'abcd' by dividing the arc a-e (the black spot) in half then adding that half to a-e and b-e find d and c.

Asher Benjamin and Owen Biddle in their pattern books copy Nicholson.
They do change the order of the letters which makes the steps easier to follow: a and b are 2 corners of the square. The arcs of a and b create c. Half of arc a-c is d. Add the length b-c to the arcs of a-c and b-c to find e and f: the square has its 4 corners.




How to draw a square with a compass, #4


A 3/4/5 triangle always has a right angle (90*) where the lengths 3 and 4 meet.
2 3/4/5 triangles are a rectangle which is 3/4 of a square.
I have drawn this on graph paper for clarity.

When carpenter squares became widely available and accurate, the square corners were easy to establish. The compass was only needed to lay out the length.

Before that - before about 1830 - the carpenter could have laid out his square like this:

His length is laid out in 4 units.
He knows approximately where the 2 sides will be. He does not know if his angle is 90*.

Here I have drawn the arc of the length of 4 units - on the right side. Then the arc of 5 units with its center at 3 units  on the left side. where they meet will be the 3/4/5 triangle.











The carpenter did not need to layout the full arcs as I have drawn them.
If he held his Line at the right lengths he could have marked a bit of both arcs where he thought they cross, and then placed a peg where they did cross. He would have checked his square by matching diagonals.

The relationship between the 3/4/5 triangle and the square is good to recognize. However, the 3/4/5 triangle is usually the only geometry. Layout by a carpenter square, widely available in the 1840's, was simpler and took less training than using a compass.





This small, simple house, built c. 1840 for a cobbler, was probably laid out using a carpenter square. I've tried other geometries which almost fit. The 3/4/5 triangle does.


I wrote the original post in 2014. It's time to revisit and review.
Here's the link to the post:
  https://www.jgrarchitect.com/2014/10/the-cobblers-house-c-1840.html



How to draw a square with a compass, #5

 Lay out a perpendicular through a line. Draw a circle with its center where the lines cross.
Draw lines - here dash/dot lines - between the points where the circle crosses the lines.

This square, as a diamond, was often used by finish carpenters because it easily evolves into more complex layouts. 

Below is the entrance porch for Gunston Hall, designed by William Buckland, c. 1761. The rotated squares determine the size of the porch. They also locate the floor, the pediment, the roof pitch, the size of the arch, the height of the rail.
 My post on Gunston Hall is: https://www.jgrarchitect.com/2014/05/gunston-hall-ason-neck-virginia.html

Here the glass facade of
 the Mass. Ave. entrance to MIT. For more, see:











https://www.jgrarchitect.com/2018/04/a-little-bit-of-geometry-of-mit.html



 How to draw a square with a compass, #6

 On a line select a length - see the dots .
Using the length as the radius draw a circle using one dot as the center.
Now there are 3 dots. Draw 3 circles using all 3 dots as centers.
Drop a perpendicular line at the first circle's center.
Now there are 2 new dots for centers of more circles.
Connect the petals where the 4 circles cross.
A square.




This modest farm house, c. 1840, used the square crossed as the squares above are for the Gunston Hall porch.

https://www.jgrarchitect.com/2014/09/how-to-construct-square.html

One last note: the circle to square diagram #6 can also become the diagram for #5. 

Each master builder probably had his preferred way of using his compass, even when he practiced within a tradition.
Still, just as a 3/4/5 triangle is part of a square, these diagrams are also simply different choices, different perceptions of the same geometry.

  

Portraits of Master Builders with Their Compasses, Part I

This past year I have given 8 presentations on Practical Geometry, the last 3 at the International Preservation Trades Workshop (IPTW) in Stirling, Scotland. My workshop begins with a power point introduction about how geometry was used in construction. Then we practice using compasses, straightedges, pencils, and twine (chalk lines, anyone?) to layout and design frames and buildings. 
 
 
The portraits here of master builders holding a compass, the symbol of their profession,  are part of those presentations.

Sebastiano Serlio, 1475-1554, master builder and author of    'On Architectura'
 posthumous portrait by Bastolomeo Passerotti c. 1575, 
now in the Martin von Wagner Museum,  University of Wurtzburg.

I wrote an introduction to Serlio here: https://www.jgrarchitect.com/2017/04/serlio-writes-about-practical-geometry.html

 Andrea Palladio,  1508-1580
architect and author of The 4 Books of Architecture, 1570

The engraving and the painting it came from may have been made 100 years after Palladio died. 




https://www.jgrarchitect.com/2017/04/palladio-discusses-geometry.html

James Gibbs, born in Scotland, died in England: 1682 - 1754. Architect of St. Martin's in the Field church, 1722. author of the Book of Architecture. 1728. This book of engravings of his buildings is known to have been in the Colonies. It influenced a great many designs.

Portrait in the Scottish National Portrait Gallery

William Buckland - 1734-1774
 
indentured to  George Mason of Gunston Hall, 1755 

Master Builder in Virginia and Maryland

Note his compass on the table below his hand 

I write about him here: https://www.jgrarchitect.com/2014/05/gunston-hall-ason-neck-virginia.html







Peter Nicholson  - 1765 -1844  
Author of texts for master builders, architects and engineers, and mathematicians beginning in 1793 , extending through the 1840's. His books were in print many years after he died and went through many editions both in London and in the States. Asher Benjamin and Owen Biddle credit him in their pattern books. Minard Lefever says (here I paraphrase):  "Refer to Nicholson . He's the master!"

For more on Peter Nicholson see: https://www.jgrarchitect.com/2018/09/practical-geometry-what-our-ancestors.htm
 

Much of what I taught this year is not yet on this blog.  It should be, so this is a beginning.

Practical Geometry - what our ancestors called this geometry

Practical Geometry.
It's what our ancestors called these diagrams I draw.


 Here is Peter Nicholson who wrote about Practical Geometry. His writings make clear that geometry was once an expected and necessary part of construction, used both by the designer and the artisan.

His first book, The Carpenter's New Guide, published in London in 1792.

He begins with a Preface, some of which I quoted in an earlier post: http://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by_16.html


Page 2 is copied here.

The use of geometry in construction was so accepted that Peter Nicholson waits until his third paragraph before he shares that geometry is useful in mathematics and science too.



By the time of his death in 1844, Nicholson had published 27 books in London, New York City, and Philadelphia.  More than 10 years later his books were still in print.

This portrait is in his updated book The New and Improved Practical Builder, published in 1837.

This time he writes a whole paragraph explaining Practical Geometry. Has he been asked to be more thorough? Have the new uses of geometry in science changed the perception of what geometry is? Have men become carpenters by necessity - especially in the New World - rather than by apprenticeship, and thus desire to educate themselves?




Here is the Introductory Chapter. 

The second paragraph describes the 2 branches of Geometry: Theoretical and Practical.
Now  the Theory of  Geometry is carefully described, including a reference to Euclid, but it is still one of the 2 branches of Geometry.
The other branch, Practical Geometry,
 allows "the architect to regulate his designs and the artisan to construct his lines".



Later, on page vii, he writes, "There is no mechanical profession that does not derive considerable advantage from it."


first portrait: by James Green, 1816, now in the National Portrait Gallery, London
second portrait: the frontispiece of The New and Improved Practical Builder. Don't miss his compass.

Practical Geometry - as described by those who used it, Part 2

The last post  discussed how Asher Benjamin and Owen Biddle presented Practical Geometry in their pattern books in 1805 and 1806.
This post focuses on Minard Lefever, and finally Peter Nicholson, who inspired them all.

Minard Lefever ( 1798-1854) wrote 5 pattern books between 1829 and 1856.
The Modern Builder's Guide was published in September 1833, in New York.
In his Preface Lefever says "...it will be proper to specify the authors whom I have either consulted or made extractions from,..."
One of these was Peter Nicholson.  Because Lefever copies Nicholson's drawings  directly I will post only the latter's introductory geometry.

Lefever writes 35 pages of  descriptions for 21 plates on "Geometry Adapted to Practical Carpentry".
Here are Plate 8  and Plate 20.


















Minard Lefever, The Modern Builder's Guide, NY, 1833, reprint by Dover Publications, NY, 1969.






Peter Nicholson (1765-1844) practiced architecture, mathematics, and engineering in Scotland.  He taught and wrote 27 books.  The Carpenter's New Guide was first  published in 1792 in Great Britain. His books were regularly reprinted in the States.

The book reproduced here was printed in Philadelphia in 1830, his 10th Edition with, he writes,"6 new Plates".  The book is 121 pages long not including the Index.
27 of those pages are of - as his title page says - Practical Geometry for Carpentry and Joinery, "the whole founded on the geometric principals; the theory and practice well explained and fully exemplified" on 10 copper-plates.

In the Preface he says, "...it is Geometry which lays down all the first principals of building, measures lines, angles, and solids, and gives rules for describing the various kinds of figures used in buildings; therefore, as a necessary introduction to the art treated of, I have first laid down, and explained in the terms of workmen, such problems of Geometry as are absolutely prerequisite to the well understanding and putting into practice the necessary lines for Carpentry."

His introductory geometry plates match those of Asher Benjamin, Owen Biddle and Minard Lefever, all of whom acknowledge him in their prefaces.

Nicholson's Plate 10 is Lefever's Plate 8.

I will bring this book to the 2016 IPTN Workshops in September. It is fragile.

If you would like to read the titles of Peter Nicholson's books, they are listed at the end of his Wikipedia biography.

Other architectural historians must have looked at the first pages of these books. Everyone cannot have just turned to the illustrations of mantles and window casings, building plans and elevations and ignored the plates on geometry. Why hasn't someone else wondered out loud why so many pages on geometry were included in a book about construction?

Someone must have considered that if Nicholson's The Carpenter's New Guide went through 10 editions and was published in the States as well as Great Britain - as well as being directly copied - that carpenters were reading it, using it, that his information was useful, that maybe we should understand what he wrote.

The builders who came before us used geometry to design and build. The knowledge was taught to the next generation hands-on. Books were not needed.
Boys were 'apprenticed', learned their craft and became 'journeymen', traveling to sites to earn and learn. Eventually these men became full carpenters, 'masters', and were admitted to a guild. The guild system was not always possible in the States. Men quit their apprenticeships. moved west or into cities. The skills and knowledge that masters were expected to impart had to be taught in other ways. Asher Benjamin and others set up a school in Boston. The pattern book was another solution - a way for 'young carpenters'  (to quote Owen Biddle) to teach themselves the necessary construction skills, beginning with geometry.

Part 1 can be read here: http://www.jgrarchitect.com/2016/08/practical-geometry-as-described-by.html

Learning from a Workshop

The IPN Workshops at the Shelburne Farms Coach Barn were superb.
The barn is magnificent. To be able to be in and around it for 4 days was true luxury.

Above left is the main entrance from the court yard. Right is the dormer for the hay loft door above the stable. Below is half of one barn door showing its hinges and brick work.



The food was plentiful and excellent - local and fresh.
The company and the workshops couldn't be beat.
Of course I plan to go to the 20th annual IPT Workshop to be held in Virginia, autumn of 2016 . www.ptn.org


My presentation was almost derailed by the cheap school compasses I brought. The compasses did not hold their angle, so the diagrams we drew weren't true. I had not anticipated that the participants might not know how to draft: they needed basic instructions and better tools.

Luckily people bore with me and I presented twice. Many people talked with me about geometry between sessions.




Here is what worked best:

The daisy wheel: As people found the rectangle created by 4 points  they easily understood the geometry of the Old First Church in Bennington.








Making a square:  beginning with a line and a circle.

The hardest part for people to figure out was how to draw the arcs for the vertical line. I felt very successful when I heard one person explaining how to do it to another.










The 1830's farmer's cottage pleased everyone. They could see how to use what they had drawn.


A few people were able to rotate the square 45* to complete the diagram as shown

I brought the pictures from my post on  Asher Benjamin, Owen Biddle and Peter Nicholson. http://www.jgrarchitect.com/2015/07/geometry-of-cobb-hepburn-house-part-2.html

 We were to draw squares based on their diagrams  - as shown here:
This is where, especially, the compasses were not up to the task. The squares were not true;they were cock-eyed. I loaned my good compass out - so much easier to draw with good tools!
So, I explained and demonstrated. People practiced.
The pattern books' first pages of geometry turned out to be an adventure.


I thought to show 4 different ways to grow a layout from one dimension.
Instead I used the different buildings and diagrams as illustrations as people asked questions,
Good, thoughtful questions.

What a good time we had!

Geometry of the Cobb-Hepburn House, an aside for skeptics

For previous posts on this house please read
http://www.jgrarchitect.com/2015/02/baring-bones-of-house.html
http://www.jgrarchitect.com/2015/06/the-cobb-hepburn-house-frame-tinmouth-vt.html
http://www.jgrarchitect.com/2015/06/geometry-for-cobb-hepburn-house-part-1.html


Here is the basic geometric shape used for the Cobb-Hepburn House.






When the house was built in 1780, the town of Tinmouth was less than 10 years old. It was the frontier. Paper would have been precious, not generally available for drawing house plans.


The master framer probably used dividers to layout the frame. We can see that he used them to draw the 2'  off set marks on the posts. Look to the bottom right of the post - 2 half circles above a line.

Sheathing was commonly used for diagrams.
I describe one such board found in a barn here:  http://www.jgrarchitect.com/2015/01/a-barn-and-its-daisy-wheel.html

Click the pictures to enlarge them.




25 years later when paper mills had become common, pattern books were popular teaching tools - beginning with basic geometry.


Here is Owen Biddle's Plate I in  Biddle's Young Carpenter's Assistant, 1804:


A and B are illustrations of how to attach paper to a board. C is the T Square.
(E,F,G are diagrams for perpendicular lines and right angles.  J is a 3/4/5 right triangle.K is the circle defined by 3 points not on a straight line.)


Just under the T Square is
H -  the layout of a square using the length of one side.


Biddle describes these engravings as " some of the most useful geometric problems which every carpenter ought to be acquainted with."
He explains that a student should have "a bow-pen or compass". 






 Asher Benjamin's  The American Builder's Companion, 1806, Plate II

has similar diagrams on basic geometry for carpenters.

All figures are explained on the accompanying page.
Fig.  12  is the same diagram as Owen Biddle's  H.

Benjamin writes in his Preface to the Third Edition:
"I have first laid down and explained such problems in Geometry, as are absolutely necessary to the well understanding of the subject."
He begins with

                           Plate I.
                  Practical Geometry.
                       Definitions. 

GEOMETRY, is that Science which treats the descriptions and proportions of magnitudes in general. 











Peter Nicholson's Guide, first published in 1792, in England, begins with geometry. It was updated and reprinted many times in London, New York and Philadelphia.
In his Preface  Asher Benjamin writes that he is "indebted to P. Nicholson's excellent books".

Figure 2 matches Benjamin's Fig.12 and Biddle's H.

This a a print of the actual page, Plate 3 - wear, age spots, and water stains included - in the 10th Edition, 1830.

I have the book in my library - on a long term loan.

.








I  have written this post because of the skepticism I encounter from academics as well as craftsmen.
The use of geometry in construction is often viewed as somehow made up. I suggest doubters read what the master carpenters themselves wrote.


Owen Biddle, Biddle's Young Carpenter's Assistant, originally published 1805, by Benjamin Johnson, Philadephia. Dover (2006) unabridged republication, Dover Publicatons, Inc., Mineola, NY

Asher Benjamin, The American Builder's Companion, first edition published 1806, This print taken from the 6th Edition, 1827; unabridged republication by Dover Publications, Inc., 1969.

Peter Nicholson, The Carpenter's New Guide: Being a Complete Book of Lines for Carpentry and Joinery, Treating Fully on Practical Geometry... 10th edition, John Griggs, Philadelphia, 1830.

Geometry, Taught in 6 Plates

For years I passed over the plates at the front Asher Benjamin's books. At the time, I only wanted to see his buildings, and had no idea why he included plates on geometry and molding profiles.
Now I study them.

The first 5 plates in The American Builder's Companion are instructions on basic geometry because many of his readers were "untaught." Many young men left apprenticeships to seek their fortunes, move west.They still needed to build. Benjamin provided their geometry course.


He begins:

A point is that which has position, but no magnitude nor dimension; neither length, breadth, nor thickness.

Here is Plate 2, the one we might recognize as useful in design. The Figures 3, 4, and 5 describe how to layout perpendicular lines, Figure 12 scribes a square.

By Plate 4 , Figure 3, he is describing "How to find the raking moldings for a pediment" - a semester of academic learning in 6 pages!

It's not just Asher Benjamin who cares about teaching geometry. Peter Nicholson's The Carpenter's New Guide, which ran 13 editions in Britain and the States from 1792 to 1857, spends 126 pages describing what he calls Practical Geometry. He begins with "1. A Point has position but not magnitude." (He's less flowery than Benjamin).

Neither of these pattern-book authors wanted their ideas to only be copied - they wanted their readers to possess the intellectual tools to adapt these designs to their own situations.

Plate 2 above comes from the Dover Publications 1969 reprint of The American Builder's Companion, 6th Edition, published in 1827,