Slide Rules and Table Rappers
Sep. 15th, 2009 12:22 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
pathology_docSlide Rules
These are a lot of fun to play with, but they have one big downside when doing trig as the markings get sort of crowded up near the Sin x = 90 end of the scale. This is a nuisance when you are trying to work out cos x (on a general purpose slide that doesn't cater for it) and its equivalent, sin (90-x), is in an area of the scale where whole degrees have to be interpolated.
Down at the other end of the rule, where x is between 6 and 20-something degrees, things are much easier - fractions of tenths of a degree are easily interpolated and likewise for the tan x scale beneath it. Now of course you can always rearrange sin2x + cos2x = 1, but that involves doing more steps than one can sometimes reliably keep in one's head or on the rule.
Last night it hit me, as it should have long ago - cos x is simply sin x / tan x, isn't it? So if I align the C and D scales, select the relevant sin x, put the cursor on that and then drag tan x under the hairline, the right C index will show me cos x directly as a decimal fraction. Easy!
Next problem - it's all up the 0.9 end of the scale, where interpolation is king and error is rife. And so after a night of sleep on the problem, I had a really bright idea. Divide by three. That brings the cursor down to an area of the D scale where the divisions are clearer, and where the second and third decimal places can then themselves be multiplied by three if the answer isn't easy to reconvert. For example, cos 10.5 degrees...
1) Align C and D scales.
2) Cursor to sin 10.5 degrees; you could read the answer on the D scale, but it's easier just to...
3) Pull 10.5 degrees on the tan scale under the cursor. The right index of the C scale is now at about 985 (i.e. 0.985). Slide the cursor to it, remembering that the exact answer lies at the C scale position, not what your imperfect eye is reading on the D scale.
4) Pull the slide to bring 3 under the cursor (division by 3 on the slide rule); answer is 0.328.
5) Ignore the 0.3; perform 28x3 on the slide rule to reconvert the 0.028 back to its 'original value' ---> 84; i.e. 0.084. Now add the 0.9 (i.e. 0.3 x 3) back in.
Answer is 0.984
The scientific calculator built into Windows returns...
0.98325490756395458455463205643051
I have converted a 0.002 difference to a 0.001 difference, halving the inaccuracy. This is about as good as you can hope to do with a hand held slide rule, and good enough for most day to day applications. In the old days, when there were no computers to cross-check, you'd have gone to a book of tables that some poor bastard had spent his life calculating, or if you had ongoing calculations to do then there were huge slide rules two metres long, fitted with microscopes to read the finest graduations on the scale. A Steampunk techno-geek's wet dream.
I'm sure someone else has come up with this trick before. But I have yet to see it in any of the slide rule manuals I've read online.
The Table Rappers
This is the title of a book my beloved left here when she went back to Canada after visiting last year. It's basically about the séance scene in Victorian Britain. This afternoon, as I was walking to get my lunch, a couple of entirely different interpretations came to mind.
The first was born out of the existence of badly written rape-fics. You know, the sort of dreadfully written Harry Potter dross churned out by barely-literate kids who disdain proof-checking or dictionaries and in which "Draco captured Hermione and rapped her all night long." Well, with that sort of "rapping" translated to tables, I'd be watching out for splinters. You've heard of virtue being its own reward - here's an example where depravity is its own punishment! Although given how dangerous even books can be in the HP universe, forcing sex upon a table in that fanon might be the last thing you'd want to try.
Then of course there's the other sort of table rapping...
Yo. My. Furniture here that you see is the best
'cos it's motherfuckin' sanded smoother than all the rest,
I construct it with love and I varnish with care
And my antique dealer buddies in amazement do stare
'cos the eye cannot see that the stuff is a fake
I can give you Louis Fourteen and your money I'll take
Even Chippendale or Georgian, yeah I know it's a stunt
But my conscience is clear if the look's all you want.
I only wish my carpentry skills were that good. :)
These are a lot of fun to play with, but they have one big downside when doing trig as the markings get sort of crowded up near the Sin x = 90 end of the scale. This is a nuisance when you are trying to work out cos x (on a general purpose slide that doesn't cater for it) and its equivalent, sin (90-x), is in an area of the scale where whole degrees have to be interpolated.
Down at the other end of the rule, where x is between 6 and 20-something degrees, things are much easier - fractions of tenths of a degree are easily interpolated and likewise for the tan x scale beneath it. Now of course you can always rearrange sin2x + cos2x = 1, but that involves doing more steps than one can sometimes reliably keep in one's head or on the rule.
Last night it hit me, as it should have long ago - cos x is simply sin x / tan x, isn't it? So if I align the C and D scales, select the relevant sin x, put the cursor on that and then drag tan x under the hairline, the right C index will show me cos x directly as a decimal fraction. Easy!
Next problem - it's all up the 0.9 end of the scale, where interpolation is king and error is rife. And so after a night of sleep on the problem, I had a really bright idea. Divide by three. That brings the cursor down to an area of the D scale where the divisions are clearer, and where the second and third decimal places can then themselves be multiplied by three if the answer isn't easy to reconvert. For example, cos 10.5 degrees...
1) Align C and D scales.
2) Cursor to sin 10.5 degrees; you could read the answer on the D scale, but it's easier just to...
3) Pull 10.5 degrees on the tan scale under the cursor. The right index of the C scale is now at about 985 (i.e. 0.985). Slide the cursor to it, remembering that the exact answer lies at the C scale position, not what your imperfect eye is reading on the D scale.
4) Pull the slide to bring 3 under the cursor (division by 3 on the slide rule); answer is 0.328.
5) Ignore the 0.3; perform 28x3 on the slide rule to reconvert the 0.028 back to its 'original value' ---> 84; i.e. 0.084. Now add the 0.9 (i.e. 0.3 x 3) back in.
Answer is 0.984
The scientific calculator built into Windows returns...
0.98325490756395458455463205643051
I have converted a 0.002 difference to a 0.001 difference, halving the inaccuracy. This is about as good as you can hope to do with a hand held slide rule, and good enough for most day to day applications. In the old days, when there were no computers to cross-check, you'd have gone to a book of tables that some poor bastard had spent his life calculating, or if you had ongoing calculations to do then there were huge slide rules two metres long, fitted with microscopes to read the finest graduations on the scale. A Steampunk techno-geek's wet dream.
I'm sure someone else has come up with this trick before. But I have yet to see it in any of the slide rule manuals I've read online.
The Table Rappers
This is the title of a book my beloved left here when she went back to Canada after visiting last year. It's basically about the séance scene in Victorian Britain. This afternoon, as I was walking to get my lunch, a couple of entirely different interpretations came to mind.
The first was born out of the existence of badly written rape-fics. You know, the sort of dreadfully written Harry Potter dross churned out by barely-literate kids who disdain proof-checking or dictionaries and in which "Draco captured Hermione and rapped her all night long." Well, with that sort of "rapping" translated to tables, I'd be watching out for splinters. You've heard of virtue being its own reward - here's an example where depravity is its own punishment! Although given how dangerous even books can be in the HP universe, forcing sex upon a table in that fanon might be the last thing you'd want to try.
Then of course there's the other sort of table rapping...
Yo. My. Furniture here that you see is the best
'cos it's motherfuckin' sanded smoother than all the rest,
I construct it with love and I varnish with care
And my antique dealer buddies in amazement do stare
'cos the eye cannot see that the stuff is a fake
I can give you Louis Fourteen and your money I'll take
Even Chippendale or Georgian, yeah I know it's a stunt
But my conscience is clear if the look's all you want.
I only wish my carpentry skills were that good. :)