The maths behind cutting these bevels?

[Phill05]

Extract from
https://www.researchgate.net/institution/Universita-Degli-Studi-Roma-Tre
Jacob and Adam are correct.

 

[J-G]

If you wish.

Addendum: That's a bit like saying Pythagoras invented geometry.

No numbers required, just drawing

View attachment 148106

That is perfectly good for a 'one off' I'll grant you but if you wish to make a number, all identical, at some future date ( ie. you no longer have your compass set) then you must resort to at least arithmetic. -- and No, no-one sugested that Pythagoras 'Invented' geometry, the word is 14c simply meaning 'Earth Measure' - well, in its very basic form.

 

[Jacob]

That is perfectly good for a 'one off' I'll grant you but if you wish to make a number, all identical, at some future date ( ie. you no longer have your compass set) then you must resort to at least arithmetic. --

Wrong again. "you no longer have your compass set" would be the same as having forgotten to make a note of the measurements. Either way you can't reproduce the thing if you don't have one in front of you, or a suitable reference such as a drawing, or dividers set. Or numbers if you want to make a big deal of it and get really confused!

 

[Jacob]

As I said earlier, the size of the basic block MUST be exactly the size you specified in your first post : 55 x 24 x 290 - well that is if you want to use the figures I provided for the distance from the saw to the fence. ANY deviation will need a recalculation......

Wrong again - any deviation would just need "a bit trimming off", unless it's already too short in which case you are completely fooked.

 

[Adam W.]

He didn't and I'm pretty sure no one said that. What he did was to discover some important theorems in the field of geometry and was thus responsible for some major advances in the field of mathematics.

Nice drawing by the way. It nicely shows some mathematical principles. And yes, no numbers required as is the case with the majority of mathematics.

It shows geometric principals, which explain the origin of the Fibonacci sequence before Fibonacci was a twinkle in his daddies eye. It also shows how to draw a fan vault and other complicated stuff.

Maths is still trying to play catchup with geometry.

 

[Sgian Dubh]

Who knew that the simple task (not even that much of a challenge really) of creating specific (preferred, desired?) basic profiles on a lump of wood could result in such a heated discussion? As Jacob said, and I think Adam is reinforcing, creating that profile by starting with a block of wood of the right overall length, width and thickness, plus a few hand held pencil 'gauge' lines, and a bit of hand planing and the job's done: it might take all of, hmm(?) ten minutes, fifteen if you're taking your time. I must admit I can't see a lot of maths involved, except maybe to know that the pencilled in gauge lines should be inset from an edge or end by a known amount followed by using a nice sharp plane, all of which (except the planing bit) I guess counts as maybe simple arithmetic.

Anyway, it's all been very entertaining watching the protagonists batting the topic all over the place. Slainte.

 

[Adam W.]

That is perfectly good for a 'one off' I'll grant you but if you wish to make a number, all identical, at some future date ( ie. you no longer have your compass set) then you must resort to at least arithmetic. -- and No, no-one sugested that Pythagoras 'Invented' geometry, the word is 14c simply meaning 'Earth Measure' - well, in its very basic form.

You don't need to set your compass to anything, just open it and off you go - It's the same result every time.

 

[johna.clements]

I'm not a member of the DUP.

You contend that mathematics did not exist before 1700

No I'm not making it up. The encyclopedia Britannica was published in the age of enlightenment, 1760 or so, which is when amateur gentlemen scientists and mathematicians first started to describe mathematical principals such as the hyperbolic cosine catenary, the funicular curve, Gaussian geometry came later in the 1800's, and the concept of the architect was invented in the 1700's.

Long before that, master craftsmen were using geometry to design and build the great cathedrals without the use of mathematics.

I asked you about Dinosaurs

Dinosaurs were real creatures.

They were only described in about the last two hundred years.

Does that mean that there were no dinosaurs in existence before 1700.

If you think that maths did not exist before it was described then what do you think about dinosaurs.

Dinosaurs are a largely extinct branch of four legged animals. People found dinosaur bones before 1700 but they did not not what they were.

Geometry is an old branch of mathematics. The tree was not defined until recently but the branch is still a branch.

 

[Adam W.]

You contend that mathematics did not exist before 1700


I asked you about Dinosaurs

If you think that maths did not exist before it was described then what do you think about dinosaurs.

Dinosaurs are a largely extinct branch of four legged animals. People found dinosaur bones before 1700 but they did not not what they were.

Geometry is an old branch of mathematics. The tree was not defined until recently but the branch is still a branch.

I did not contend that maths didn't exist before 1700 and I know what dinosaurs are, thank you.

All mathematics does is describe things which already exist.

 

[Phill05]

You contend that mathematics did not exist before 1700


I asked you about Dinosaurs

If you think that maths did not exist before it was described then what do you think about dinosaurs.

Dinosaurs are a largely extinct branch of four legged animals. People found dinosaur bones before 1700 but they did not not what they were.

Geometry is an old branch of mathematics. The tree was not defined until recently but the branch is still a branch.

Can you not read earlier post!!!


Extract from
https://www.researchgate.net/institution/Universita-Degli-Studi-Roma-Tre
Jacob and Adam are correct.

Phill

 

[johna.clements]

I did not contend that maths didn't exist before 1700 and I know what dinosaurs are, thank you.

All mathematics does is describe things which already exist.

And master craftsmen were using a branch of mathematics which had not been defined just like the dinosaur bones.

 

[johna.clements]

Yes but all maths is only useful if numbers are ultimately applied. Mathematical concepts are developed from past observations of the real world in order to predict what will (or 'might' in statistical maths) happen in the future. Such concepts provide a framework but need numbers to turn them into reality.
My blood was made to boil recently by that young lady on the box, Prof(even) Hannah Fry asserting that EVERYTHING depends on maths. No, it's the other way round, maths depends on everything, or at least, everything we understand.
Brian

Maths is everywhere.

 

[Adam W.]

And master craftsmen were using a branch of mathematics which had not been defined just like the dinosaur bones.

Geometry, they were using practical geometry, which existed before Euclid and Pythagoras came along and described it using mathematics.

I realise that it may be a difficult concept to grasp..

 

[Jacob]

Can you not read earlier post!!!


Extract from
https://www.researchgate.net/institution/Universita-Degli-Studi-Roma-Tre
Jacob and Adam are correct.

Phill

Yes interesting.
One greatly overlooked instrument is the divider (or pair thereof). It made precision engineering possible in a non mathematical age and uses a process not unlike Calculus to obtain precision.
Still extremely useful if e.g. you want to mark up a bit of wood with precisely equal intervals, for dovetails etc.
Dividers probably gave us the duodecimal system as 12 is the lowest common multiple of 1,2,3,4 and very convenient for making things, whereas including 5 takes you up to 60 as LCM.
Decimal system suited people who counted with their fingers (accountants etc) though toes were included in some places - we still have the "score" in English.

 

[Yojevol]

Yes interesting.
One greatly overlooked instrument is the divider (or pair thereof). It made precision engineering possible in a non mathematical age and uses a process not unlike Calculus to obtain precision.
Still extremely useful if e.g. you want to mark up a bit of wood with precisely equal intervals, for dovetails etc.
Dividers probably gave us the duodecimal system as 12 is the lowest common multiple of 1,2,3,4 and very convenient for making things, whereas including 5 takes you up to 60 as LCM.
Decimal system suited people who counted with their fingers (accountants etc) though toes were included in some places - we still have the "score" in English.

They're wrongly named really. They should be 'multipliers', used to lay out multiples of a base dimension. It's difficult to divide a line without using trial and error although it is possible with a bit of geometry added in to the process.
Brian

 

[Jacob]

They're wrongly named really. They should be 'multipliers', used to lay out multiples of a base dimension. It's difficult to divide a line without using trial and error

It's easy. Yes trial and error. That's the whole point. This is what seems to have been generally forgotten.
You step out your divisions as near as you can judge and look at the error.
Say 3 divisions - you then divide the error by 3 as near as you can judge and adjust the dividers to fit.
Step it out again and if necessary divide the much smaller error again.

or though it is possible with a bit of geometry added in to the process.
Brian

not necessary

 

[johna.clements]

Geometry, they were using practical geometry, which existed before Euclid and Pythagoras came along and described it using mathematics.

I realise that it may be a difficult concept to grasp..

Yes Euclid and Pythagoras defined parts of mathematics in the branch called geometry. More is being defined as time progresses. I would not be surprised in new branches are added in the future.

This is the same way dinosaurs where defined but did exist before then. We will discover more groups of extinct species as we continue to dig around on this planet. But what we have not yet discovered does exist, we just haven't seen it yet.

 

[johna.clements]

They're wrongly named really. They should be 'multipliers', used to lay out multiples of a base dimension. It's difficult to divide a line without using trial and error although it is possible with a bit of geometry added in to the process.
Brian

Draw a line at an angle to the line to the line you want you want to divide.

Mark that line with units. Say three.

Draw a from the end of the first line to the last unit mark.

Set up a square on that line with a straight edge on the other face,

Slide the square down the straight edge until it reaches the next unit mark.

Mark where the square crosses the first line.

If there are three unit marks then you have marked a third. If you had four then you have marked a forth.

No need for trial and error.

 

[Jacob]

Draw a line at an angle to the line to the line you want you want to divide.

Mark that line with units. Say three.

Draw a from the end of the first line to the last unit mark.

Set up a square on that line with a straight edge on the other face,

Slide the square down the straight edge until it reaches the next unit mark.

Mark where the square crosses the first line.

If there are three unit marks then you have marked a third. If you had four then you have marked a forth.

No need for trial and error.

Can be done with dividers if you do the step out process along the line first but make sure the error is +, then make this line your second line, to which you have the dividers already set very precisely to 1/3.
So it's trial and error by a different route.

 

[Adam W.]

View attachment 148113

Extract from
https://www.researchgate.net/institution/Universita-Degli-Studi-Roma-Tre
Jacob and Adam are correct.

What's the article that this comes from, as the link only directs to the institution?