The creation of accurate log scales - without computers

Bald Eagle

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I am frequently impressed by the amount of calculation that can be done using a simple slide rule. This also led to me to marvel at the fact that people were making accurate log scales - without computers - a LONG time ago. I'm really curious how they went about laying those scales out by hand.

Logarithms and slide rules have existed in some form since 1617, and slide rules dominated as the tool for rapid calculation up until the mid to late 1970's.

But there is little to no information on how a physical, drawn logarithmic scale was produced - and reproduced - accurately and in large enough quantity for a mass market. It is of course easy to do with computers - there are plenty of graphical programs and pdf's of logarithmic scales out there, but I simply cannot find any information about how it was done _before_ computers. Is there a "ruler and compass" equivalent method for laying out a logarithmic scale?

It seems that not only have the slide rules disappeared, along with people who know how to use them, but also all of the records for their fundamental construction.

Any help in locating reliable sources of detailed information would be greatly appreciated!
 

Chince

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Found a few things in my limited time - dont think i dug up any specific info about how the devices were built. But found some references and a resource.

Below book is the "History of Mathematics Vol2 by D.E. Smith" p205 is the one here and mentions some of the earlier primitive methods, doesnt really say much about precision though.


r27Kd07WUj.png

At the bottom,
"On this entire topic see F.Cajori, A history of the Logarithmic Slide Rule, New York, 1909; hereafter referred to as Cajori, Slide Rule."


-All that I got for now
 
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JWW427

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Don't forget the simple but elegant Abacus.
The slide rule is probably far older than 1654. We probably only have fragments of the advanced math from Prediluvian times when Atlantis ruled. Ive heard that we are given truncated math and higher physics so that regular folks won't catch on to the secrets of the universe that only the PTB know.

If I had to guess from all the mysteries and coverups we've uncovered so far, it would be that the slide rule has always been with us, but someone just wrote that it was invented in 1654. Or they re-discovered it from some cave or secret archive.

AB.jpeg

As for ancient Egypt, I thought we were taught that the Arabs invented zero?

e-1.jpeg


The 360 degree circle, the foot and its 12 inches, and the "dozen" as a unit, are but a few examples of the vestiges of Sumerian Mathematics, still evident in our daily lives.

 
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Silent Bob

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It's a bit like the question about old architecture and statues etc that we can't make today. We've been massively dumbed down and de-skilled to the point that we find it hard to believe we were ever capable of such things, especially with the fake historical narrative. They could work out the logs using multiplication and subtraction, just with many stages - very slow and tedious work. I came across a great book once called 'The Quadrivium' which explains how they created a lot of the shapes and made measurements, very clever stuff.


According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD
1606181139573.png

Top Left: Arithmetic. Top Right: Geometry. Bottom Left: Music. Bottom Right: Astronomy


I taught maths for a while and loved looking into ancient maths, even before I discovered stolen history! I think there is a lot to discover there. For instance, when I taught Pythagoras I also talked about his mystery school a bit. The really interesting point from his theory really sent my head spinning when I think about it. In the early days his theory caused huge outrage, I'll try and explain why - the upshot is that it suggests we live in a fake or manufactured reality......

So if you know this theory it is that the square of the two shorter sides of a right angled triangle added together will equal the square of the longest side. The classic example is the 3, 4, 5 triangle, 3 sq = 9 , 4 sq = 16, 5 sq = 25 and we can see that 9 + 16 = 25 - no problem here. The trouble is when you have a triangle with the two shorter sides of say 1cm each. To get the long side we add 1 sq + 1 sq = 2, so the long side is the square root of 2. This caused the outrage, as the square root of 2 is an irrational number, i.e. it goes on forever. It was felt that in nature everything could be expressed as a ratio (fraction) but the square root of 2 cannot. The way I picture it is if you measure out two sides of a triangle at 1m, then try and measure the long side you can never get an exact measurement. Imagine you have a fractal ruler that you can zoom in on - mm, then micro, nano, pico, femto etc - you could literally zoom in forever, and the edge of the triangle would never match up with the any of the measurement marks, even though you could theoretically keep zooming in forever. This is why they were upset, they didn't think reality worked like that and hence this theory was heresy! It feels like a computer glitch to me, endless loop!

Also Velikovski told us that our calendar used to have 360 days exactly at one time. Hence 360 degrees in a circle. 12 months of 30 days exactly, which is how long it actually took the moon to orbit back then. We see 3, 6 and 9 all over here! Something happened to distort all of this, those extra 5 days at the end of the year after christmas always feel like they don't belong to the rest of the year, or is that just me? So we've gone from perfect numbers to distorted ones, 365 and a quarter days in a year and 29 and a bit days in a month - yuck!
 

Forrest

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A-Ha.
I kept searching, and came across this:
Post automatically merged:

Also:
Nice find! There are any number of arithmetic-approximation ways to get to a rough log scale, he is showing one. It could also be done by guess and by gosh, just make some marks on a stick about where you think the numbers should be, within reason. The key to refining the accuracy is to get to his Figure P.8, even if that means violating any number of local, state, and federal statutes, where two such crudely-made scales are then offset and compared to each other, quite like a slide rule. The markings on one, or the other, or both, are then adjusted to match each other more closely. Then slide them to a different amount of offset and repeat. Not much math required. It's akin to a binary search, by bracketing and reducing the error at each step.

Now, there is the possibility of operator error, such that the series of corrections to the markings doesn't converge monotonically to an exact* solution. Maybe a mark is made in the wrong direction here and there. It might be faster and more reliable to make up THREE such nearly-identical scales, then compare them to each other in the same sequence used for the Three-Plate Method. Call the scales A, B, and C. Compare and adjust A and B. Then set B aside and compare A to C and adjust. Then set A aside compare C to B, adjust and continue in this rotation.

*'Exactly' is relative to the length of the sticks, the linewidth of the marks, etc. Longer sticks and thinner marks make it more accurate.
 
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Silent Bob

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A-Ha.
I kept searching, and came across this:
Post automatically merged:

Also:
I found the 2nd link much easier to understand than the first one. If we were making slide rules today then I would be the one working out all the values and could design the scale but I would definately need a skilled craftsman like the guy from the first link to make the physical rule itself!
 

6079SmithW

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It's a bit like the question about old architecture and statues etc that we can't make today. We've been massively dumbed down and de-skilled to the point that we find it hard to believe we were ever capable of such things, especially with the fake historical narrative. They could work out the logs using multiplication and subtraction, just with many stages - very slow and tedious work. I came across a great book once called 'The Quadrivium' which explains how they created a lot of the shapes and made measurements, very clever stuff.


According to Plato, a core of mathematical knowledge – later known as the Quadrivium – was essential for an understanding of the Universe. The curriculum was outlined in Plato’s Republic. The name Quadrivium means four ways, but this term was not used until the time of Boethius in the 6th century AD
View attachment 3184
Top Left: Arithmetic. Top Right: Geometry. Bottom Left: Music. Bottom Right: Astronomy


I taught maths for a while and loved looking into ancient maths, even before I discovered stolen history! I think there is a lot to discover there. For instance, when I taught Pythagoras I also talked about his mystery school a bit. The really interesting point from his theory really sent my head spinning when I think about it. In the early days his theory caused huge outrage, I'll try and explain why - the upshot is that it suggests we live in a fake or manufactured reality......

So if you know this theory it is that the square of the two shorter sides of a right angled triangle added together will equal the square of the longest side. The classic example is the 3, 4, 5 triangle, 3 sq = 9 , 4 sq = 16, 5 sq = 25 and we can see that 9 + 16 = 25 - no problem here. The trouble is when you have a triangle with the two shorter sides of say 1cm each. To get the long side we add 1 sq + 1 sq = 2, so the long side is the square root of 2. This caused the outrage, as the square root of 2 is an irrational number, i.e. it goes on forever. It was felt that in nature everything could be expressed as a ratio (fraction) but the square root of 2 cannot. The way I picture it is if you measure out two sides of a triangle at 1m, then try and measure the long side you can never get an exact measurement. Imagine you have a fractal ruler that you can zoom in on - mm, then micro, nano, pico, femto etc - you could literally zoom in forever, and the edge of the triangle would never match up with the any of the measurement marks, even though you could theoretically keep zooming in forever. This is why they were upset, they didn't think reality worked like that and hence this theory was heresy! It feels like a computer glitch to me, endless loop!

Also Velikovski told us that our calendar used to have 360 days exactly at one time. Hence 360 degrees in a circle. 12 months of 30 days exactly, which is how long it actually took the moon to orbit back then. We see 3, 6 and 9 all over here! Something happened to distort all of this, those extra 5 days at the end of the year after christmas always feel like they don't belong to the rest of the year, or is that just me? So we've gone from perfect numbers to distorted ones, 365 and a quarter days in a year and 29 and a bit days in a month - yuck!
Bob, please make a thread with your understanding of this non mainstream mathematics, but for laymen like me

Thanks!

Smith
 

Silent Bob

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[/QUOTE]
Bob, please make a thread with your understanding of this non mainstream mathematics, but for laymen like me

Thanks!

Smith
[/QUOTE]

I was thinking of starting a thread like that, I'll start with the fibonacci sequence - there is a very interesting secret hidden in that one that blew me away the first time I stumbled across it!
 

Bald Eagle

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I found the 2nd link much easier to understand than the first one. If we were making slide rules today then I would be the one working out all the values and could design the scale but I would definately need a skilled craftsman like the guy from the first link to make the physical rule itself!
Well, numerically, obviously the 2nd one is pretty easy - that would just be a manual equivalent of graphing it out in software and printing it.
But of course measuring out distances on a scale out to 2 or 3 decimal places is easier said than done.

The 1st article just describes the gadget that makes positioning the scriber to that accuracy possible, as well as the peripheral stuff to guide the scriber and assisting in making the rapid scribing of a lot of different lines.
As I understand it, it's a dial that turns the feed-screw, with a worm gear attached to THAT in order to rotate it in small increments VERY precisely.

So it's a lot less "magical" geometry than I thought it might be, and just applying precision machining to achieve great accuracy.

Not sure if any still exist, but it would be interesting to see more details of the K&E dividing engine - size, weight, price... :D
 

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