Slide Rules & Laws of Exponents

Multiplication & Division

Operation Scales Why Example
Multiplication A&B or C&D (X)(Y) = (bx)(by) = bx+y 2 x 4 = 8   (A&B)

14.15 x 200 = 2830 (C&D)

Division A&B or C&D (X)/(Y) = (bx)/(by) = bx-y 800 / 40 = 20   (A&B)

283 / 200 = 1.415 (C&D)
Inversions C&CI (Not shown) X-1 = (bx)-1 = b-x 4-1 = .25

 

Powers & Roots

Operation Scales Why Example
The A scale has two decades for every D decade.  The K scale has three decades for every D decade.  A goes twice as fast and K three times as fast.
Squares / Cubes D&A / D&K XY = (bx)y = bxy 32 = 9  and 33 = 27
Square/Cube Roots D&A/ D&K Y√X = X1/y = (bx)1/y = bx/y √9 = 3 and 3√27 = 3
3/2 Power A&K combine above 93/2 = 27

 

 

Trig. Functions

Operation Scales Why Example
 
Sine/Cosine      
Tangent      
       

 

 

Exponential & Logarithmic Functions

Operation Scales Why Example
 
       
       
       

 

 

  0 1 2 3 4 5 6 7 8 9
10 0.0000 0.0043 0.0086 0.0128 0.0170 0.0212 0.0253 0.0294 0.0334 0.0374
11 0.0414 0.0453 0.0492 0.0531 0.0569 0.0607 0.0645 0.0682 0.0719 0.0755
12 0.0792 0.0828 0.0864 0.0899 0.0934 0.0969 0.1004 0.1038 0.1072 0.1106
13 0.1139 0.1173 0.1206 0.1239 0.1271 0.1303 0.1335 0.1367 0.1399 0.1430
14 0.1461 0.1492 0.1523 0.1553 0.1584 0.1614 0.1644 0.1673 0.1703 0.1732
15 0.1761 0.1790 0.1818 0.1847 0.1875 0.1903 0.1931 0.1959 0.1987 0.2014
16 0.2041 0.2068 0.2095 0.2122 0.2148 0.2175 0.2201 0.2227 0.2253 0.2279
17 0.2304 0.2330 0.2355 0.2380 0.2405 0.2430 0.2455 0.2480 0.2504 0.2529
18 0.2553 0.2577 0.2601 0.2625 0.2648 0.2672 0.2695 0.2718 0.2742 0.2765
19 0.2788 0.2810 0.2833 0.2856 0.2878 0.2900 0.2923 0.2945 0.2967 0.2989
20 0.3010 0.3032 0.3054 0.3075 0.3096 0.3118 0.3139 0.3160 0.3181 0.3201
21 0.3222 0.3243 0.3263 0.3284 0.3304 0.3324 0.3345 0.3365 0.3385 0.3404
22 0.3424 0.3444 0.3464 0.3483 0.3502 0.3522 0.3541 0.3560 0.3579 0.3598
23 0.3617 0.3636 0.3655 0.3674 0.3692 0.3711 0.3729 0.3747 0.3766 0.3784
24 0.3802 0.3820 0.3838 0.3856 0.3874 0.3892 0.3909 0.3927 0.3945 0.3962
25 0.3979 0.3997 0.4014 0.4031 0.4048 0.4065 0.4082 0.4099 0.4116 0.4133
26 0.4150 0.4166 0.4183 0.4200 0.4216 0.4232 0.4249 0.4265 0.4281 0.4298
27 0.4314 0.4330 0.4346 0.4362 0.4378 0.4393 0.4409 0.4425 0.4440 0.4456
28 0.4472 0.4487 0.4502 0.4518 0.4533 0.4548 0.4564 0.4579 0.4594 0.4609
29 0.4624 0.4639 0.4654 0.4669 0.4683 0.4698 0.4713 0.4728 0.4742 0.4757
30 0.4771 0.4786 0.4800 0.4814 0.4829 0.4843 0.4857 0.4871 0.4886 0.4900

 

 

 

http://www.syssrc.com/html/museum/html/sims/javaslide/

http://www.eyrie.org/~dvandom/slide/slideprimer

S and T - Sine and Tangent, these are trigonometric indices.  They run from about 6 degrees to 45 and 90 degrees respectively.  There is no Cosine index because you can just use trigonometry to find the Cosine of one angle from the Sine of another.  There are often red numbers over these scales, representing the Cosine from 6 to 90 degrees and the Tangent from 45 to 90, to help people who don't want to do the trig.


ST - Sine and Tangent are almost the same at very small angles, and this index covers them for angles of about half a degree to about 6 degrees. 
 

CI and DI - The inverse of C and D.  Usually printed in red, to remind the user that these indices go backwards.
 

L and Ln - Logarithm and Natural Logarithm.  The tick-marks are evenly spaced on these scales because everything else is already a logarithm.
 

LL - Called Log-Log indices, these are used for advanced operations.
 


Sine/Tangent: Move the cursor to your angle of interest on the S or T index.  The value of Sine or Tangent will be found on the C and D indices (use whichever index is on the same piece of slide rule as the S and T indices, some rules have the S and T on the slider, others put it on the back of the scale or even on the back of a reversible slider.  Divide the number on the C/D index by ten to get the final answer, since the values for Sine and Tangent (for angles less than 45) fall between zero and one. When using the ST index, divide your result from the C/D index by 100 instead. 

Logarithm: The base-10 logarithm of a number on the C index can be found above or below it on the L index.  Thus, the log10 of 2 is roughly .3.
    
Power of Ten: To raise 10 to a fractional power, find that number on the L index.  Ten to that number will be found on the C index.  Thus, 10^.3 is roughly 2.

Natural Logarithm: The natural logarithm can be found on some slide rules, using the Ln index instead of the L index.  Thus, the natural log (base e) of 2 is between .69 and .7.

Power of e: To raise e (2.71828182845904523536...) to a power between zero and 2.3, line up the cursor on the desired power on the Ln index (which
goes higher than 1 since the marks are closer together).  The answer will be on the C index.  Thus, e^2 is about 7.4.

More Fun With e: If you have the LL indices on your slide rule, one thing you can do with them is find powers of e (and natural logarithms) over
a much wider range.      Put the cursor on a number N on the D index.  The number it lines up with on the LL1 index will be e raised to the power of N/100.  The number it lines up with on the LL2 index will be e raised to the power of N/10 (this is slightly better than using the Ln index, because it's spread out more).  The
number it lines up with on the LL3 index will be e raised to the power N.     So, putting the cursor on 2 on the D index, we get the following results:
     LL1 - e^.02 = 1.0202 (really spread out, good precision)
     LL2 - e^.2 = 1.222
     LL3 - e^2 = 7.4

One bit of warning about the LL3 index...it often uses "M" to mean "thousand."  e^10 is not over 20 million, it's over 20 thousand.