Fabula Lignarius

Month: January, 2013


I have been neglecting my blog recently and could give you a whole series of excuses but I won’t. It is not that I lack any subjects to write about it’s rather time that is limited. There are only so many hours in a day.

The last two weeks I have been preparing for the upcoming workshop and now I am finally ready. I wrote a 25 page paper for the participants and made most of the parts for the model they will be building. This morning I found a mistake in the paper, a small one but still significant. I messed up some numbers while calculating the length of a queen post. No worries, no harm is done.

Usually I work at my friends workshop but since it has been quite cold lately I decided to work inside in our spare room. Did you know you don’t need a workshop to do woodworking. Just two square meters of free space is plenty, three would be a luxury. I worked like this before so I am used to it. Back in the days when I lived in an apartment on the fourth flour in Antwerp my living room doubled as a workshop.

on the floor

A wooden floor and a nice solid slab of Sakura (cherry) with a planing stop is all a man could hope for. It would make the perfect shop.

Oh, did you notice the sanding paper in the picture. I know that is really embarrassing I shouldn’t be using that. I needed a small wooden hammer (the one on the pillow) to set my planes and decided to sand away the file marks on the handle. It’s ok you know, sometimes you need to let go of all those preconceptions.

On the Slope

In this post we will look at the differences between a roof
slope defined by a ratio or a roof slope defined by degrees. I have
often questioned why a carpenter would choose one method over the
other. Most carpenters who produce stick frames use angles in
degrees and this works well in combination with a miter saw. Timber
framers however, who use timbers of greater dimensions tend to use
proportions, although some layout methods rely on degrees as well.
I decided to compare both systems in the light of their usefulness
in practice and not only from theoretical point of view. slope3.5 Let’s just consider a
quarter of a circle, 90° in other words a right angle for our case
study. First we will look at a graphical comparison of both
systems. Below I have filled up two similar portions with different
increments. On the right side you see  the increments
representing 10 degrees and on the left side we have increments
that represent proportions. degr. prop. In this example I have
used a run of 30 units since this is what I am used to when working
with a metric Japanese framing square or
sashigane. Don’t worry if you are not familiar
with this particular Japanese square since it could be any framing
square as long as it has the same scale on either arm. The only
thing that’s important to understand is that we are comparing
proportions to degrees. A roof slope would normally not be
described in a ratio to 30 as it is in this drawing, but instead to
10 (or 12 if you were using fractional inches). It is important to
note that if we use proportions/ratio’s the scale doesn’t matter.
You can use any system of measurement: imperial, metric, shaku,
Ell, or
come up with something yourself. A slope of 4/10 is going to be the
same no matter which unit of measure you prefer to use. This is
inherent to the mathematical essence of ratio’s. When we use
degrees a quarter circle is divided in 90° (a right angle). In this
case study we will limit ourselves to an accuracy of 1/2 degree
(for example 46,5°) , and to 0.1 of a rise
unit when we are referring to ratio’s (for example 3.5/10). Greater
accuracy than 1/2 degree is almost impossible to achieve since I
don’t know of any protractor suitable for woodworking that would
allow this. When using ratio’s we limit ourselves to the smallest
increment on our square. Below there is one example of a framing
square with a protractor. At first sight it seems well made,
stainless steel and etched markings. I have never used this one
myself but I know some carpenters who do and I don’t think this
particular model is built with any accuracy in mind. It is
impossible to use the short arm with a greater accuracy than 5mm
and this will not get the job done to my standards. I do not
understand the design of this particular square and therefore it
would not be my weapon of choice. protractor square The Starret protractor, as
shown below, is one I do use for carpentry and lay out purposes.
This is clearly a very accurate tool, originally designed
for machinists and I find it quite useful but I would not consider
it a mandatory tool for your tool chest. For the layout of large
timbers and roof carpentry it’s use is rather limited since it is
not designed for this purpose and it does not substitute a good
framing square. Starret protractor The framing square remains
the essential tool in carpentry layout but also in other woodcrafts as the
picture below illustrates. It depicts John Browne ,1660-1670, the instrument maker. Although he is not a carpenter the fact that he is holding a framing square and the platonic solids seen in the background illustrates that he was also a mathematician who studied Euclid. (Thanks to Sim Ayers for pointing out the correct information about this picture)John Brown, The
Description and Use of the Carpenter's Rule The next picture shows the
typical tables etched in a modern western style framing square.
Note that this square has decimal inches instead of fractional
inches. Maybe it’s because I grew up in a part of the world where
measure is expressed decimally that I feel uncomfortable using
fractions. I used fractional inches for quite some years when
working in the US and never became a great fan of it. Just a
personal point of view, read not of any
In the end a carpenter should be able to
adapt to any given situation and this includes a different measure
system. chappell framing
square These are two sashigane
one with a metric scale and the other with a shaku/sun scale. You
can read more about them here.
omote The sashigane has two
sides one with a regular scale and the other with the square root
scale to lay out hip rafters for example. urame Let’s leave the numbered
pieces of steel behind and get back to our comparison of ratio’s
and degrees. We noticed that the degree increments are equally
spaced but the proportional increments are not. The linear distance
measured between the proportional lines on the circumference of our
circle becomes smaller. Here is another version of the drawing that
shows the proportions in relation to a square. proportion to
sashigane There are actually many
more lines on the drawing, one for every mm on the short arm and
one for every 1/2 degree, but to keep the drawing clear they are
not shown here. Drawing this was fun, I couldn’t figure out an
efficient way to draw the lines of the proportional slopes and
ended up drawing many lines manually which was not only boring but
quite time consuming as well.  I need to get better at this
since repetitive tasks should be very straight forward with CAD. If
you are interested in the process, which I am afraid you are not,
it goes like this. And no I don’t want to be annoying but I do feel
the compulsive need to share the procedure with you. Click
‘enter’ to repeat the previous command, hover the cursor close to
the previous line to select it’s ‘end point’ and start the new line
from there, type ‘1000’ for the length of the line, click
‘tab’ to select the dialog box for the angle of the line and click
at the reference point which was already established. This
procedure takes me about 4 seconds – that is if I do it correctly
and of course that’s not always the case and then I lost another
six seconds in between the lines. On top of that I could have
continued endlessly, in the true sense of the word, forever and
ever and ever. This brings us to the next observation. There is a
difference between both systems when we look at the amount of
divisions of our right angle. Employing an accuracy of 1/2
degree means we have 180 different angles to choose from when using
degrees as our unit of measure to define a roof slope. If we
employ proportions, in theory we  should have an endless
amount of angles to choose from. The rise in
our proportional method could have any possible length, even then
we are still able to define an angle smaller then 90° with it. The
angular value between 4 and 5 degrees is the same as the angular
value between 28 and 29 degrees, that is 1°. This is different when
we use the proportional method along the arms of a framing square.
The angular value between 8.3/10 and 8.6/10 is about 1° but the
angular value between 28.3/10 and 30/10 is also 1°! 8.3:10-8.6:10 28.3:10-30:10 In one occasion 1° is
represented by 3 increments on the scale of the framing square and
in the other occasion 1° is represented by 17 increments. Very
important to keep in mind during the design process. When a
designer uses degrees for his roof slopes when drawing his plans,
he might have a tendency to come up with angles like 37,15°. But if
the carpenter uses ratio’s when laying out his roof, things can
become unnecessarily complicated. You could convert all the angles
to ratio’s but this is just another step where mistakes can creep
in and you end up with strange numbers which are unpleasant to work
with. I believe that  a design should be made using the same
system that will be employed in it’s execution just for the sake of
simplicity. To convert a roof slope to an angle we divide the
numerator by the denominator which gives us the tangent value of
our roof slope. We then use the inverse tangent function (tan⁻¹ or
arctangent) to get the angle expressed in degrees.


3.5/10 = 0,35 tan⁻¹=


37,15° tan = 0,756667783 x 10 =

In the next post we will continue to look at the
differences and reflect them to their practical value. The question
remains whether the method we choose will suit our needs on the
construction site?