Calculation

**pmcgee** · 25th October 2012, 08:41 AM

In case anyone is interested to turn this over in their head ...

It is an old post, but this blogger quoted from "The Village Carpenter by Walter Rose" at The Cobwobbler: The Village Carpenter by Walter Rose -Book Review

- - -
And next time you think you're too tired to do any more work, thing on this little extract:

"Many of our men walked four to five miles daily to work at distant places. Grandfather frequently retold his early feats of strengh and endurence, of walking to and from Swanbourne, for instance ( about 17 miles) , where he did a days work of carpentry, carrying his tools each way. He admitted he was tired on his return."

And they carried their tools! Imagine humping a table saw 4 miles or 17! to a job? I think hand tools would be the order of the day, (they had no such thing as electric tools then anyway) and not that many of them.

Here's an extract on how the usable wood in a fallen tree was calculated. I can't quite wrap my head around it, see what you think:

"For the measurement of trees my father always used a string and a slide rule. As the trunks of trees taper lenghtwise, the middle was taken as the average girth round which the string was passed. I have helped him many times, holding the string carefully with my fingers at the place where it terminated the circumference as measured, afterwards doubling it and then redoubling it twice, with the result that the folds held in my hand were each an eighth of the total circumference. Then he would direct me to drop one eighth part- this an allowence for the bark- and double the remaining seven eighths twice. Each fold of the string was now one-fourth part of the seven eighth of the circumference. He would take the lenght of this with his rule. The measurement thus arrived at represented the "girth", or one side of a squared log, supposing the content of the log to have been square instead of round. On that basis he would then ascertain the cubic content of the log by the use of his slide rule"

What now? Can anyone explain this?

- - -
My answer:

When he says 'doubling' he means doubling over or - really - folding in half. So 7/8ths "doubled" twice gives you 7/32 which is close to the 1/4 (of the original circumference) he states.

Circumference = 2 x pi x r = about 6.3 x r

1/4 of that is about 1.6 x r.

If you imagine a circle with a square drawn inside it, and connect two corners of the square to the centre then you have a right-angle triangle.

Two sides of the triangle are length = r.
So by pythagoras, the side of the square L = the square-root of (r^2 + r^2).

This = (square-root of 2) x r = about 1.4 x r.

So he would over-estimate the size of the square inside the log at the mid-point of the tree ...
it would depend on the taper of the tree how that worked out for you overall.

... Wakeup now!

Paul
- - -
But I realise ... having re-edited this a bit ... it works out much better than that!

The father's estimate is *really* L = 7/32 x 6.3 x r = ... what do you think???

...
..
.

= 1.37 x r. Not too shabby.

I don't know if it is designed in, but that is about a 2% allowance for waste from a perfectly circular log.
If the radius were 2ft=600mm then 2% = 12mm = half an inch.

**Boringgeoff** · 27th October 2012, 11:11 AM

Thanks Paul,
I've been waiting for high brow comments to this thread and am quite surprised that none have forthcome, so some low brow instead. Though not a mathematician I am interested in how people arrive at the same answer via different routes as in the folded string method above.
In my job as a concrete delivery driver I was often asked to calculate the quantity of concrete required for a particular job. This is a simple task if it was a square or rectangle slab with square or rectangular thickenings etc, but if curves or circles were involved I'd be looking for my calculator.
Let's take the case of round holes, say ten verandah post footings of 600mm dia by 600mm deep.
The calculation for this is: r2 x Pi x depth x number of holes = .3 x .3 x 3.141 x .6 x 10 = 1.69 m3. At that time concrete was ordered in increments of .2 m3 so you would round that up to 1.8 m3.
One of our customers told me how to work it out without needing a calculator. I call this Franks method on account of his name was Frank. The same calculation: dia x dia divide by 14 x 11 x depth x number of holes = 6 x 6 \ 14 x 11 x 6 x 10 = 1697 = 1.69 m3.
How this works in your head: 6 x 6 = 36, divide by 14 = a bit over 2.5, x 11 = 27.5, ( call it 28 ), x 6 = 168 x 10 = 1680 = 1.68 m3. Pretty rough but we've still got to round our quantity up to the nearest .2 m3 which gives us 1.8 m3.
The next step in the evolution is what I call Cambos method. Cambo worked his calculation as for a square hole and deducted 20% from the result to achieve the round hole quantity.
Lets check that out: 6 x 6 = 36 x 6 = 216 x 10 = 2160 \ 5 ( 10% x 2 ) = 432 ( call it 430 ) x 4 = 1720 = 1.7 m3. Do our rounding and we get ( Ta daa ) 1.8 m3.
I'm using \ symbol to show division, I'm not sure if this is correct or not. I also noticed that having been out of the industry for over five years the mental maths is not as easy as it once was.

Regards,
Geoff.

**pmcgee** · 27th October 2012, 02:59 PM

11/14 perked my ears up ...

Franks method uses the diameter squared = (2 x radius) squared = 4 x r^2

and then uses 22/7 as the approximation to pi, dividing by 4 to get rid of the one above ... ie 11/14.

Re Cambo's ...

A square of length 2 has area 4. A circle in that square would have diameter=2, so radius=1 ... area = pi =3.14

So the missing part in between is 0.86 out of 4 .. or .. 0.215 out of 1 = 21.5%

I thought it was interesting that when the 1/8th of the rope was discarded it was described as "for the bark etc" when it is not for that reason, and is just part of the algorithm to get the right answer. Humans ... hah!

Cheers,
Paul

**Yawally** · 27th October 2012, 06:46 PM

And I thought a 'slide rule' was when you took your shoes of and slid along the floor to see how far you could go, the peice of string was to measure how long your skid was and then you had a 'pie' to eat after.

**BamBam53** · 27th October 2012, 07:12 PM

And if you don't have a slide rule you can always use log tables.

**dr4g0nfly** · 28th October 2012, 06:57 AM

Originally Posted by BamBam53

And if you don't have a slide rule you can always use log tables.

I've still got and can use both, you had to understand the math to be able to use them.

Calculators, A device for getting the wrong answer to a very high degree of accuracy! Push buttons, write answer, brain not necessary.

**gyropilot** · 28th October 2012, 09:19 PM

I still love and use my Faber Castel 283N.... nothing wrong with slide rules !!
I find the log tables good, but slightly slower...(maybe I'm more used to the slide rule)

Geoff

**joe greiner** · 28th October 2012, 11:29 PM

Slide rule has infinite battery life.

Cheers,
Joe

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