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

Edward Shaw - uses the tools

This picture of a 1854 construction site had me hoping, even if it was idealized.

The architect - wearing the stove pipe hat - holds dividers as he measures something on the drawing for the observant and expectant carpenters. In the foreground on the grass is a carpenter square, a hammer, and a large compass.

Maybe Edward Shaw's pattern book, The Modern Architect, published in 1854, would mention geometry! Maybe I'd find mention of proportions in a paragraph about something else!

Well, he does say that a main floor window's height should not be more than double its width. Room length, breadth and height and height are mentioned in relationship to each other. But then he states that 10 ft is the desired height... There is great advice for the carpenter and homeowner about foundations, lath and plaster, and 'warming'. Fun, but not what I hoped for.

Shaw's life (1783-1859) spans the change from custom to repetitive parts in construction. The picture shows a building being balloon framed with 2x's , not posts and beams. The drawing in the illustration is being measured and scaled up by dividers, an ancient tool, not a modern architect's scale with regular increments. Almost anyone can draw circles with a compass. In the time Shaw practiced master carpenters and architects knew how to use compasses for design, layout and framing of rectangular buildings.

The book includes extensive explanation of how to lay out columns, scrolls for hand rails, and molding details that would require a hand held compass. The large compass shown would have been for stepping off foundations and wall locations based on the drawing made by the small compass. Or it is possible that the 'compass' is  perhaps a level, folded up.

The picture is the cover of the Dover Publications reprint of Shaw's book. Inside is a reprint of the etching in black and white. It is too dark to reproduce well. For a look at the original print try: http://www.historicnewengland.org/preservation/your-older-or-historic-home/articles/pdf149.pdf . It is part of a good article on a mid-19th century Maine builder in the SPNEA journal, 1967. SPNEA (Society for Preservation of New England Antiquities) is now Historic New England.