Slide Rules - Home

Browser note:

In the table below, the From and To entries give the range of numbers found on the scale, exclusive of any scale extensions.

The Definition maps a value v in that range to the horizontal position of that value on the scale, from 0 at the left end to 1 at the right end.

The Relation links the scale to the basic C,D scales. Assuming a closed slide, moving the hairline to x on D yields the given value on the described scale.

(The inverse relation -- the C,D value corresponding to a given value on the scale -- is 10 to the power of the definition.)

Label From v To v Definition Relation
2pi 1/(2pi) ~ .159 10/(2pi) ~ 1.59 logv - log(1/(2pi)) x / (2pi)
theta 0 90 v/90 90 logx
theta, Stheta 0 90 (sin°v asin°(p/10) = asin° sqrt logx
A, B 1 100 logv / 2 x²
C, D 1 10 logv x
CF, DF sqrt 10 ~ 3.16 10 sqrt 10 ~ 31.6 logv - log sqrt 10 ( sqrt 10)x
CF, DF pi ~ 3.14 10pi ~ 31.4 logv - logpi pix
CF/M, DF/M ln 10 ~ 2.30 10 ln 10 ~ 23.0 logv - log ln 10 (ln 10)x
CFM, DFM loge ~ .434 10 loge ~ 4.34 logv - log loge (loge)x
CI, DI 1 10 1 - logv 10/x
CIF 1/pi ~ .318 10/pi ~ 3.18 log(10/v) - logpi 1/(pix)
db 0 20 v 20 logx
Gtheta 0 infinity (sin atan sinhv asinh tanr = asinh tan asin sqrt logx
H 25/pi² ~ 2.53 2500/pi² ~ 253 1 - ((logv - log(25/pi²)) / 2) 1/(4pi²x²)
K 1 1000 logv / 3
L 0 1 v logx
LI, L 0 1 1 - v 1 - logx
LL0, LL0+ exp(.001) ~ 1.001 exp(.01) ~ 1.010 3 + log lnv exp(x/1000)
LL00, LL0-, LL/0 exp(-.01) ~ .990 exp(-.001) ~ .999 3 + log ln 1/v 1/exp(x/1000)
LL01, LL1-, LL/1 exp(-.1) ~ .905 exp(-.01) ~ .990 2 + log ln 1/v 1/exp(x/100)
LL02, LL2-, LL/2 exp(-1) ~ .368 exp(-.1) ~ .990 1 + log ln 1/v 1/exp(x/10)
LL03, LL3-, LL/3 exp(-10) ~ .00005 exp(-1) ~ .368 log ln 1/v 1/expx
LL1, LL1+ exp(.01) ~ 1.010 exp(.1) ~ 1.105 2 + log lnv exp(x/100)
LL1' exp(.001) ~ 1.001 exp(.01) ~ 1.010 log((v-1) / lnv) NA
LL2, LL2+ exp(.1) ~ 1.105 e ~ 2.72 1 + log lnv exp(x/10)
LL3, LL3+ e ~ 2.72 exp(10) ~ 22026 log lnv exp(x)
Ln 0 ln(10) ~ 2.30 (log e)v lnx
P 0 sqrt .99 ~ .995 1 + log sqrt (1 - v²) sqrt (1 - x²)
P', Q' 10 sqrt 200 ~ 14.1 (v²/100) - 1 10 sqrt log(10x)
P, Q 0 10 v²/100 p := 10 sqrt logx
R1, Sq1, sqrt 1 sqrt 10 ~ 3.16 2 logv sqrt x
R2, Sq2, sqrt sqrt 10 ~ 3.16 10 (2 logv) - 1 sqrt (10x)
Rtheta, Sr 0 pi/2 ~ 1.57 (sinv r := asin(p/10) = asin sqrt logx
S 18/pi ~ 5.73 90 log(10 sin°v) asin°(x/10)
S 1.8/pi ~ .573 90 log(10 sqrt sin°v) asin°(x²/100)
Sh1, Sh2 asinh(.1) ~ .1 asinh(1) ~ .881 1 + log sinhv asinh(x/10)
Sh2, Sh1 asinh(1) ~ .881 asinh(10) ~ 3.0 log sinhv asinhx
SI 18/pi ~ 5.73 90 1 - log(10 sin°v) asin°(1/x)
SRT, ST 1.8/pi ~ .573 18/pi ~ 5.73 logv - log(1.8/pi) (1.8/pi)x
T 0 infinity (sin atanv tanr = tan asin sqrt logx
T, T1 18/pi ~ 5.73 45 log(10 tan°v) atan°(x/10)
T2 45 90-18/pi ~ 84.3 log tan°x atan°x
Th atanh(1/10) ~ 0.1 infinity log(10 tanhv) atanh(x/10)
TI 18/pi ~ 5.73 45 1 - log(10 tan°v) atan°(1/x)
TI2 45 90-18/pi ~ 84.3 1 - log tan°v atan°(10/x)
Tr1 atan(.1) ~ .1 pi/4 ~ .785 log(10 tanv) atan(x/10)
Tr2 pi/4 ~ .758 atan(10) log tanx atanx
x 0 pi/2 ~ 1.57 2v/pi (pi/2) logx

CF, DF

Range: sqrt 10 ~ 3.16 to 10 sqrt 10 ~ 31.6
Definition:logv - log sqrt 10
Relation:( sqrt 10)x
Description: A few slide rules (the Hemmi 250 is the only one I know of off-hand) have CF/DF scales folded at the square root of 10, on the grounds that an off-scale setting is impossible rather than merely unlikely. I think a short scale extension is a better alternative.

CF/M, DF/M

Range:ln 10 ~ 2.30 to 10 ln 10 ~ 23.0
Definition:logv - log ln 10
Relation:(ln 10)x
Description: Folded at ln(10), for easy conversion between common and natural logarithms.

CFM, DFM

Range:loge ~ .434 to 10 loge ~ 4.34
Definition:logv - log loge
Relation:(loge)x
Description: Folded at log(e), for easy conversion between common and natural logarithms.

P, Q

Range:0 to 10
Definition:v²/100
Relation:p := 10 sqrt logx
Description:

The matched P and Q scales are one of the very few instances of a slide rule calculation that isn't essentially ether a table look-up or an addition of logarithms.

The position of a number is proportional to its square. So, while adding lengths x and y on C/D scales yields exp(ln(x)+ln(y)) = xy, adding lengths x and y on P/Q scales yields sqrt (x²+y²). Handy for converting rectangular to polar form.

Not to be confused with the P scale common on European slide rules, these scales are found on only a handful of rules. I know of the Hemmi 153 (and its Post-labelled counterpart), the slightly simpler Post 1459, the 20-inch Hemmi 154, the Lafayette F-686, and the SIC 1570 (the latter two both made by Relay).

S

Range:1.8/pi ~ .573 to 90
Definition:log(10 sqrt sin°v)
Relation:asin°(x²/100)
Description: Some slide rules, mostly basic simplex ones, have an S scale that reads sin(x) on A/B rather than C/D, thereby obtaining the range of an ST/S pair at the cost of accuracy.