Showing posts with label compasses. Show all posts
Showing posts with label compasses. Show all posts

Tuesday, August 25, 2020

Lesson 6, Rule of Thirds, Part 2 of 2, Serlio

 

The posts in this series  Lessons 1-7  are :

 https://www.jgrarchitect.com/2020/04/lessons.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lessons-2.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-3.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4.html

https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4b-old-first.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lesson-5-addendum.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html
 
https://www.jgrarchitect.com/2020/09/lesson-7-how-to-layout-frame-with-lines.html 
 
 
 
 
 
 
 
 
 
 
 
 

Monday, June 29, 2020

Practical Geometry Lesson 5, Addendum


Why I left out diagram K from Owen Biddle's Plate 1 in his Young Carpenter's Assistant.

Lesson 5 was written for a student today who wants to draw rectangles using practical geometry.
Biddle was writing for the apprentices he worked with in 1805. They needed to know the practical application of geometry for the buildings they worked on - including the curved parts.

This addendum is like one of those long footnotes in an historic report -  a part of the story that's not quite germane to the subject, but ought to be included.


Biddle  identifies each diagram on Plate 1with a letter. There is no diagram for D. However, in his text, between C and E he discusses the mathematical instruments a carpenter should obtain. Perhaps this is D.  I quote him:

- scales of equal parts on the thin ivory or box rule
- a bow pen or compass
- a small piece of gum elastic for rubbing out black lead lines
- a stick of Indian ink
- 2 camel's hair pencils, one large, one small
- a black lead pencil



There is also no J. And there is no text in its place as exists for D. 




Here is K.  

Biddle writes: "Three points (not in a right line) or a small part of a circle being given to find a center which will describe a circle to pass through the points or complete the circle."






                                                     
                                                     Start with a curve a-b .
                                 The curve in Biddle's drawing above is a-b-c.                       










 The curve divided in half:  Swing 2 arcs that are the same length  above above and below the curve: a-c and b-d. Mark where they cross, at f above and below the curve,









Connect  f and f with a line - here dashed. Mark where the line crosses the arc a-b -  I've labeled it g.
This line divides the arc in half. 
If 2 lines were given - here: a-g and g-b , this step would not be necessary. Biddle's diagram  labels his lines a-b and b-c.


Now, the instructions become complex.
Draw it step at a time. And consider that this is only Plate 1 of Biddle's pattern book. He included 43 more Plates for the carpenter's assistant.  

Divide the lines a-g and g-b in half.
This is shown in Biddle's E  and F diagrams. Check Lesson 5. 

Extend the lines which divide  a-g and g-b in half so they intersect at k,
K is the center of the circle which passes the points or completes the circle.

Refer to Biddle's drawing K above for the complete solution, all neatly explained in only one diagram.  



Clearly Biddle thought this information  was essential knowledge for  every carpenter. His next Plates illustrate why. The construction his 'young carpenter's assistant' would be working on involved determining and laying out many curved lines for vaults, arches, windows, stairs and railings.





Plate 2 discusses ellipses: how to draw them using geometry or a trammel, how to find the center and axes of one already drawn.   















Plate 3 is concerned with octagons, arches, groins. the use of trammels, how to divide a line into parts.

I am quite fond of Figure 1, describing " an Octagon within a square." . Simple, quick, even obvious - if you know geometry.

I have seen  painstaking explanations of  how to lay out an octagonal using algebra: quite painful.






Plate 6 reviews raking cornices and "the sweep of a cornice which will bend around a circular wall and stand on a spring."




Plate 31 lays out "the section and elevation of a circular or geometrical stairs". Biddle includes in figure C  "the manner of drawing a bracket for the ends of the circular steps..." and the careful, detailed instructions.




Plates 32-35 - not included here - explain how to layout the newel, the falling moldings, the hand rail for such a stair.






Biddle's Young Carpenter's Assistant, Owen, Biddle, 1805, originally published by Benjamin Johnson, Philadelphia, and Roland and Loudon, New York. Reprint by Dover Publications, Inc. 2006. If you want this book, you can easily order it from them directly. It has an excellent 15 page introduction with bibliography by Bryan Clark Green.


The posts in this series  Lessons 1-7  are :

 https://www.jgrarchitect.com/2020/04/lessons.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lessons-2.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-3.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4.html

https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4b-old-first.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lesson-5-addendum.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html
 
https://www.jgrarchitect.com/2020/09/lesson-7-how-to-layout-frame-with-lines.html

Tuesday, June 16, 2020

Practical Geometry Lessons, Lesson 5: Rectangles


Today these skills are not required knowledge for builders. We have steel carpenter squares that have true 90* corners, as well as levels and  lasers. 


The carpenter squares shown here are some of the earliest made in the States. They were made in Shaftsbury and N. Bennington, VT, 1825-60. Some are on display at the Bennington Museum; all are available for study.

The 1503 woodcut at the end of this post includes a square being used for a layout. That square might not have matched the square of another builder.

Practical geometry taught how to 'prove' that an angle was 'true'. Carpenters today still make their work 'true'.  










This "Geometric Problem'  and its solution was particularly important when carpenter squares were not necessarily true: the square corner was not always accurate, not dependably 90*.

This is the end of Lesson 5.
The carpenter's assistant who masters these problems is now ready to assist in layout and framing. Maybe he (no recorded 'shes' that I know of) will go on to learn design. 



For more ways to draw a square  see Drawing a Square, Parts 1 and 2.
Part 1: https://www.jgrarchitect.com/2019/12/practical-geometry-drawing-square-with.html
Part 2: https://www.jgrarchitect.com/2020/01/practical-geometry-drawing-square-with.html



 
After-thoughts and questions:

How did carpenters carry a 10' rod?  Did the rod fold, or come in pieces that could be connected with pegs,  perhaps leather sleeves?
I have seen one in a medieval print, part of which is shown here.

The builder on the left uses a square for layout. Below him is an axe, a saw, and a level with a triangular hole and a plumb bob.
The builder in the center holds a 10' rod.   The landscape to his right is the site where he will layout a building, or perhaps land.
 A 'rod' when used in land surveying is 16.5 feet long. It is also called a 'perch' or 'pole'. How did they carry that awkward length? 'Links' and 'chains' are also used in surveying, so perhaps a rod could be a length of chain.

 Woodcut by Gregor Riesch, Margarita Philosophical, published in Baden,1503.

 

The posts in this series  Lessons 1-7  are :

 https://www.jgrarchitect.com/2020/04/lessons.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lessons-2.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-3.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4.html

https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4b-old-first.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lesson-5-addendum.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html
 
https://www.jgrarchitect.com/2020/09/lesson-7-how-to-layout-frame-with-lines.html

 

Sunday, May 31, 2020

Owen Biddle's "The Young Carpenter's Assistant"


 Owen Biddle's pattern book The Young Carpenter's Assistant was published  in Philadelphia in 1805.
I am rereading it, looking especially at his use of Practical Geometry.



His first drawings, A and B, on Plate 1 are a plan and section of a drafting board. His instructions include how to construct the frame and the panel, and how to locate the small wood buttons which will keep the dampened paper in place while it dries by the fireplace to become "smooth as a drum head" .*
Drawing C is a T Square whose construction is also described.
He lists the tools a student would need; including a small compass, often called a 'bow pen'.

The rest of Plate 1, Plates 2, 3, with Biddle's descriptions explain "the most useful geometric problems every Carpenter ought to be acquainted with."*

I read between the lines: What was common knowledge that did not need explanation?

- 'Carpenter's Assistants' might have been men who needed to learn more about construction but had not had regular apprenticeships. Many apprentices ran away before completing their training; the most famous one is Benjamin Franklin.
 -  The carpenter's assistant had little experience with paper for architectural drawings. He used a convenient board, a compass, a straight edge, and an awl when he needs to work out a design. These designs are often found on boards during renovation.
-  Biddle specified the compass to be small because there are many different sized compasses. The ones shown here are in a cabinet maker's shop. The biggest is about 16" tall
-  Geometry was a practical tool in construction.

Biddle had set up a school to teach 'architecture' to carpenters.  Assistants might not have been trained to draw in their apprenticeships. Those who designed might not trained under a master builder.  Biddle himself was a contractor becoming an architect.





Biddle has  includes 2 drawings of the same construction. Biddle first explains the design, Plate 15. Then he details how to make the construction drawing become an inviting illustration for a potential client, Plate 16.

Here is Plate 15. In his description Biddle says he gives 'the lines of a pitch pediment frontispiece'.  He writes about 'Lines' as Serlio did: the Lines for layout and design. He adds that the  "Column is made 10 diameters in height." The use of diameter of the column as a measurement is discussed by Palladio.


Biddle says for the best appearance "the door should be as wide as  half its height"*. The scale on the right side of the drawing is labeled '10 diameters'. Did the 'door' consisted of just the part that's hinged? No, Biddle's dimensions include the door and its surround. Here is the door with a height of 10 units divided into 2 red squares, each  5 units square.

As I followed Biddle's notes and dimensions on the drawing I learned how he thought about construction and design: his unit is the door and its frame: the 'door'  itself was not the measure; the door frame between the columns is. The column's height includes its base and its capital, but not the lower base  which is noted as 8" high. This allows the base to be adjusted to fit the vagaries of the location.





Of course the fanlight is a semicircle. The frame is half the height of the squares of the door. 
The key stone of the semicircular fanlight was the center of the square, the height of the pediment 3/4 of the square.

The roof pitch falls exactly along the Lines which cross the square and locate its 3/4 mark.

The notes on the right side about the height of the little columns in the architrave and the  return of the eaves reference the span of the door which is derived from the height of the columns.




Biddle notes at the bottom of the actual door its width: 9 parts.  the parts lay out the panels. The panels themselves are 3 parts wide. Their length and spacing is also determined by the 9 parts.
The fan light's rays are also set by the 9 units. . The tic marks for the 9 units extended to the arc of the fan light layout the fins. The rhythm for the main fins is 1,2,3,2,1.  In the center of the 'petals' of the fan lights are more tic marks - the centers of the arcs that create the tracery.

I tried to draw this clearly. What resulted was a jumble of lines. I'll try words:
Biddle began with the diameter of the column. He used 10 diameters to create 2 squares for door and 3/4 of the square for a fanlight and architrave. His 5 unit width is divided into 10 units, 9 of which are the physical door. The Lines inform all his design.
Biddle uses 10 parts because he supposes
"that the door is for a town house with a narrow front..."  The parts will be in proportion to each other, but take up less space.


The  square shown here has already been divided into 4 equal parts. The Lines divide the 4 rectangles into 5 rectangles. The bottom and top lines defining the square are thus divided into 5 equal parts.



Plate 16

Here is the same entrance shaded. He writes " ...the student should make it his business to understand the effects of light and shade..."  He explains how to think about shading in 2 paragraphs, where "the shade should be strong", where the part should be "bright".

Plates 17 and 18  are similar: one drawing setting up the design itself, the next explaining how to shade the design.  









* Plates 1, 2, 3, 15, and 16, and their descriptions
Owen Biddle, The Young Carpenter's Assistant; or, A System of Architecture Adapted to the Style of Building in the United States, 1805, Benjamin Johnson, Philadelphia, and Ronalds & London, New York. Reprint 2006, Dover Publications, Inc.

2 good biographies of  Biddle: 1) the introduction to the Dover Edition of Biddle book by Bryan Clark Green. 2) an article by Micheal J. Lewis in  American Architects and Their Books to 1848, edited by Hafertepe and O'Gorman, 2001, U.Mass Press.







Sunday, April 5, 2020

PRACTICAL GEOMETRY LESSONS - Introduction and Lesson 1










 

The posts in this series  Lessons 1-7  are :

 https://www.jgrarchitect.com/2020/04/lessons.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lessons-2.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-3.html

 https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4.html

https://www.jgrarchitect.com/2020/04/practical-geometry-lesson-4b-old-first.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lessons-lesson-5.html

https://www.jgrarchitect.com/2020/06/practical-geometry-lesson-5-addendum.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-1_21.html

https://www.jgrarchitect.com/2020/08/lesson-6-rule-of-thirds-part-2-serlio.html
 
https://www.jgrarchitect.com/2020/09/lesson-7-how-to-layout-frame-with-lines.html
 
 
 


Wednesday, January 8, 2020

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.