# B Common Slide Rule Scales

Created in about 1850, the Mannheim slide rule layout had four scales which became referenced as A, B, C, and D. The A and B scales (on the stock and on the slide of the rule, respectively) indicate the square of a number found on the C or D scale (slide or stock, respectively). From the popularity of the Mannheim design, these labels for these primary scales continued to be used on most slide rules over the next century of their development. As new scales were added to future models, other simple labels were provided, many of which became standards. The table below lists some of the most common labels of scales that can be found on many popular slide rules.

Scale Use Operation
C main scale, on slide number $$x$$, where $$1 \le x\le 10$$
D main scale, on stock number $$x$$, where $$1 \le x\le 10$$
A square, on stock $$x^2$$
B square, on slide $$x^2$$
L log scale $$\log x$$
K cube scale $$x^3$$
S inverse sine – angle whose sine is given on C/D scale $$\sin^{-1}(x/10)$$
T inverse tangent – angle whose tangent is given on C/D scale $$\tan^{-1}(x/10)$$
ST small angle inverse sine/tangent $$\sin^{-1}(x/100)$$ $$\approx\tan^{-1}(x/100)$$
SRT same as ST, also converts radians to degrees $$\sin^{-1}(x/100)$$ $$\approx\tan^{-1}(x/100)$$ $$\approx x/100$$
T1 same as T $$\tan^{-1}(x/10)$$
T2 inverse tangent – for larger values of $$x$$ $$\tan^{-1}(x)$$
CI reciprocal of C scale; on slide $$1/x$$
DI reciprocal of D scale; on stock $$1/x$$
CF C scale, folded, usually at $$\pi$$; on slide $$\pi x$$
DF D scale, folded, usually at $$\pi$$; on stock $$\pi x$$
CF/m C scale, folded, at $$2.3$$; on slide $$x/\log e$$
DF/m D scale, folded, at $$2.3$$; on stock $$x/\log e$$
CIF folded reciprocal C scale; on slide $$1/(\pi x)$$
DIF folded reciprocal D scale; on stock $$1/(\pi x)$$
LL3, LL3+, Ln3 log-log scale; exponential function $$e^{ x}$$
LL2, LL2+, Ln2 log-log scale; exponential function $$e^{ x/10}$$
LL1, LL1+, Ln1 log-log scale; exponential function $$e^{ x/100}$$
LL0 log-log scale; exponential function $$e^{ x/1000}$$
LL00 log-log scale; exponential function $$e^{-x/1000}$$
LL01, LL1-, Ln-1 log-log scale; exponential function $$e^{-x/100}$$
LL02, LL2-, Ln-2 log-log scale; exponential function $$e^{-x/10}$$
LL03, LL3-, Ln-3 log-log scale; exponential function $$e^{-x}$$
N1,N01, etc. log-log scale; base-10 $$10^{x}$$, $$10^{-x}$$, etc.
Ln natural logarithm $$\ln x$$
P Pythagorean scale $$\sqrt{1-(x/10)^2}$$
Sq1, R1, W1 square root (1-10) $$\sqrt{x}$$
Sq2, R2, W2 square root (10-100) $$\sqrt{10x}$$
SH, Sh, Sh1 hyperbolic sine $$(e^{x/10}-e^{-x/10})/2$$
Sh2 hyperbolic sine, extended $$(e^{x}-e^{-x})/2$$
TH, Th hyperbolic tangent $$(e^{x/10}-e^{-x/10})/(e^{x/10}+e^{-x/10})$$
SI, TI; TI1, TI2, etc. reciprocal of S, T, etc. $$\sin^{-1}{(1/x)}$$ , etc.

The following special scales are discussed in the vignette British Thornton Special Scales and can be found on several models of this Maker’s slide rules.

Special British Thornton Scales
Sd inverse differential sine – angle [degrees] for which Sd is given on C scale $${\rm Sd}^{-1}(10x)$$
Td inverse differential tangent – angle [degrees] for which Td is given on C scale $${\rm Td}^{-1}(10x)$$
ISd inverse differential inverse sine – sine for which ISD is given for inverse angle on C scale [1/degrees] $${\rm ISd}^{-1}(1/10x)$$
ITd inverse differential inverse tangent – tangent for which ITd is given for inverse angle on C scale [1/degrees] $${\rm ITd}^{-1}(1/10x)$$
Ps Pythagorean scale, sine $$\sqrt{1-(x/10)^2}$$
Pt Pythagorean scale, tangent $$\sqrt{x^2-1}$$

For the above table, the functions used are defined by

• $${\rm Sd}(x) = x/\sin x$$
• $${\rm Td}(x) = x/\tan x$$
• $${\rm ISd}(x) = x/\sin^{-1} x$$
• $${\rm ITd}(x) = x/\tan^{-1} x$$

Special scales found on the Hemmi Model 153 are discussed in the vignette Hemmi 153 Scales. Many of the labels used are the same as those above, however they have very different meanings. A synopsis of these scales, also found on a few other slide rules as well, is provided below.

Special Hemmi Scales (e.g., Model 153)
L standard linear scale $$0\le y \le 1$$
$$\theta$$ angular measure (deg.) $$\sin^{-1}(\sqrt{y})~~$$ (deg.)
$$R_\theta$$ angular measure (rad.) $$\sin^{-1}(\sqrt{y})~~$$ (rad.)
$$G_\theta$$ Gudermannian function, $${\rm gd}(u) \equiv \sin^{-1}[\tanh(u)]$$ $$\tanh^{-1}(\sqrt{y})$$
$$P$$ sine of angular measure $$\sqrt{y}$$
$$Q$$ sine of angular measure $$\sqrt{y}$$
$$Q'$$ extended sine scale $$\sqrt{1+y}$$
$$T$$ tangent of angular measure $$\tan[\sin^{-1}(\sqrt{y})]$$