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Obsolete technology geekery
So in the effort to get me posting about anything, and prevent my journal from idling for another month, I'm going to talk about a toy I bought myself on eBay a few weeks ago.
It's a slide rule.
It's a moderately nice one, though the sight on it is slightly askew, such that if I don't touch it, the line isn't quite vertical. It's only a 10 inch model; the really accurate 20-inch and longer ones aren't selling on eBay, or are too pricey.
It has scales L, C, D, A, B, K, Ci, CF, DF, CiF, S, T, ST, LL1, LL2, LL3, LLO, and LLOO.
Just think - there was a time when every engineering student would know what all of those scales meant. I've included a discussion of what those scales mean, and what it means I can compute with this, below the cut.
Okay, so I should really include a close-up of what my particular slide rule looks like, but despite the obsolescence of the technology google has plenty of slide rule pictures to give you the general gist if you've never seen one before. I assume that you have at least seen a slide rule used to multiply before, or can get a handle on how that would work based on the wikipedia article and this java sliderule applet.
Anyway, as to what those scales mean, the "L" scale is linear, and goes from 0 at one end of the slide rule to 1 at the other end. Then, in terms of the other scales, when the rule is zeroed:
D = 10^L
C = D
A = D^2
B = C^2
K = C^3
Ci = 1/C
CF = C * π
DF = D * π
CiF = Ci * π
S = arcsin(D/10)
T = arctan(D/10)
ST = arcsin(D/100) or arctan(D/100) (The two functions are sufficiently close for arguments from 0.01 to 0.1)
LL1 = e ^ (D/100)
LL2 = LL1^10 = e ^ (D/10)
LL3 = LL2^10 = e ^ D
LLO = e ^ (-A/1000)
LLOO = LLOO^(0.1) = e ^ (-A/10)
The center slide contains the scales C, Ci, CiF, and CF on one side and B, K, C, and Ci on the other. All other scales are on the fixed portions of the rule. The way I've written the relationships above, the C=D relationship can be messed with by sliding the center piece, but all other relationships have to hold.
Note that this means that C goes from 1 to 10, A from 1 to 100, LL3 from e to just over 20,000 (e^10), and LLOO from 0.902 to about 0.00005 (e^-10).
Now, what can be done with this? Obviously, we can now look up natural and base 10 logs and antilogs, trig functions, cube and square roots, and multiplication by π without moving the center piece. Sliding the C and D scales (or A and B) past each other lets us multiply, since by doing that we're essentially adding values on the L. Combine that with log lookup, and it would seem like we're just a stone's throw away from exponentiation. In fact, that can be done directly on the slide rule.
Say you want to do 2.5^3.5 (I can usually read three digits of accuracy off this slide rule, but let's keep it simple). First, locate 2.5 on the LL scales; it's near the right end of LL2. Then, slide the right end of C so as to line up with 2.5 on LL2. Now, with the scale shifted I've changed the relationship between C and D to:
C = D / (ln 2.5)
So:
LL2 = e^D = e ^ ((ln 2.5)*C/10)
That means that if I now find 3.5 on the C scale, I can look up the answer on LL3, since
LL3 = LL2^10 = e ^ ((ln 2.5)*C)
And I find that the answer is approximately 24.6. Xemacs's calculator says 24.7052942196, so I was wrong with that last digit. As I said, the sight's off a bit.
By combining the LL scales with Ci or CiF, I can also take nth roots (since taking the root is just exponentiation to 1/n). I have yet to discover any non-obvious uses of the trig scales, though clearly I can use them to use the result of a trig function in a multiplication or division without needing the intermediate result.
Update: Specifically, the make/model of my slide rule is a "LOG LOG DECIMAL TRIG ACU-MATH No. 130", and the sight is just enough off from vertical that I can notice it.
It's a slide rule.
It's a moderately nice one, though the sight on it is slightly askew, such that if I don't touch it, the line isn't quite vertical. It's only a 10 inch model; the really accurate 20-inch and longer ones aren't selling on eBay, or are too pricey.
It has scales L, C, D, A, B, K, Ci, CF, DF, CiF, S, T, ST, LL1, LL2, LL3, LLO, and LLOO.
Just think - there was a time when every engineering student would know what all of those scales meant. I've included a discussion of what those scales mean, and what it means I can compute with this, below the cut.
Okay, so I should really include a close-up of what my particular slide rule looks like, but despite the obsolescence of the technology google has plenty of slide rule pictures to give you the general gist if you've never seen one before. I assume that you have at least seen a slide rule used to multiply before, or can get a handle on how that would work based on the wikipedia article and this java sliderule applet.
Anyway, as to what those scales mean, the "L" scale is linear, and goes from 0 at one end of the slide rule to 1 at the other end. Then, in terms of the other scales, when the rule is zeroed:
D = 10^L
C = D
A = D^2
B = C^2
K = C^3
Ci = 1/C
CF = C * π
DF = D * π
CiF = Ci * π
S = arcsin(D/10)
T = arctan(D/10)
ST = arcsin(D/100) or arctan(D/100) (The two functions are sufficiently close for arguments from 0.01 to 0.1)
LL1 = e ^ (D/100)
LL2 = LL1^10 = e ^ (D/10)
LL3 = LL2^10 = e ^ D
LLO = e ^ (-A/1000)
LLOO = LLOO^(0.1) = e ^ (-A/10)
The center slide contains the scales C, Ci, CiF, and CF on one side and B, K, C, and Ci on the other. All other scales are on the fixed portions of the rule. The way I've written the relationships above, the C=D relationship can be messed with by sliding the center piece, but all other relationships have to hold.
Note that this means that C goes from 1 to 10, A from 1 to 100, LL3 from e to just over 20,000 (e^10), and LLOO from 0.902 to about 0.00005 (e^-10).
Now, what can be done with this? Obviously, we can now look up natural and base 10 logs and antilogs, trig functions, cube and square roots, and multiplication by π without moving the center piece. Sliding the C and D scales (or A and B) past each other lets us multiply, since by doing that we're essentially adding values on the L. Combine that with log lookup, and it would seem like we're just a stone's throw away from exponentiation. In fact, that can be done directly on the slide rule.
Say you want to do 2.5^3.5 (I can usually read three digits of accuracy off this slide rule, but let's keep it simple). First, locate 2.5 on the LL scales; it's near the right end of LL2. Then, slide the right end of C so as to line up with 2.5 on LL2. Now, with the scale shifted I've changed the relationship between C and D to:
C = D / (ln 2.5)
So:
LL2 = e^D = e ^ ((ln 2.5)*C/10)
That means that if I now find 3.5 on the C scale, I can look up the answer on LL3, since
LL3 = LL2^10 = e ^ ((ln 2.5)*C)
And I find that the answer is approximately 24.6. Xemacs's calculator says 24.7052942196, so I was wrong with that last digit. As I said, the sight's off a bit.
By combining the LL scales with Ci or CiF, I can also take nth roots (since taking the root is just exponentiation to 1/n). I have yet to discover any non-obvious uses of the trig scales, though clearly I can use them to use the result of a trig function in a multiplication or division without needing the intermediate result.
Update: Specifically, the make/model of my slide rule is a "LOG LOG DECIMAL TRIG ACU-MATH No. 130", and the sight is just enough off from vertical that I can notice it.