There are many pages about slide rules on the web, and you can still buy brand new slide rules (40 years old but never used, and still in their factory supplied box) in various places. The purpose of this particular and quite idiosyncratic slide rule page is to describe common scales used on slide rules, and the kind of mathematical expressions that could be evaluated with those scales.
A subsidiary of this page describes a software package that let's you enter an expression and will tell you how that expression can be evaluated with a slide rule. If you are looking for that software go here.
The two images on this page show the two sides of a particular slide rule in my collection. This may be one of the fanciest and perhaps most beautiful slide rules ever made, a Faber Castell Novo Biplex 2/83 N. It's made of plastic, and has 30 scales and 11 cursor marks. The rule is about 13.5 inches long and 2.25 inches wide. You can click on the pictures and see an enlarged image, but that doesn't come close to holding the real thing in your hands. It feels heavy and solid. The slide and cursor move with silky smoothness and yet they stay in place wherever you let go of them. The lettering is crisp and detailed, and pristine! No space is wasted, but the information is not crowded either. Every scale has a purpose.
German made slide rules of that time (the late 1960s) usually come with an accessory plastic ruler. This particular slide rule has a ruler (not shown) that lists common formulas and physical data on one side. Those may be useful for slide rule calculations. However, the other side of that ruler has a detailed list and explanation of common notations in set theory! This is about as useless for slide rule calculations as a list of large mammals. Apparently this slide rule was made when the "new math" was at its zenith and Faber Castell wanted its share of the action.
[Jeff Weiner brought to my attention that actually there are some slide rules that can add and subtract, specifically the Pickett Microline 115 and the Pickett 901 rules.]
A slide rule consists of three parts: the body, the slide, and the cursor. The body and the slide are marked with scales. The cursor has a hairline that facilitates accurate positioning of the cursor at a specific point on some scale. There may be other marks on the cursor that are used for specific and special purposes.
It's convenient to think of the logarithm as the common (base 10) logarithm, and the length of the slide rule as one unit, but you can also think of log meaning the natural logarithm, and the length of the slide rule being log(10) units.
The multiplication of two numbers exhibits two important properties of slide rule calculations:
All other scales on a slide rule are referenced to the C and D scales. Following is a list of scales commonly found on slide rules. For each scale we list the name (like C), the function underlying it (like ), and some explanations or comments.
|C, D|| The basic scales. C is on the slide,
D on the body.
|CI, DI|| CI is on the slide,
DI on the body.
|CF, DF|| CF is on the slide,
DF on the body.
|CIF, DIF|| CIF is on the slide,
DIF on the body.
|A, B|| A is on the body,
on the slide.
|R, W|| May come with subscripts to
distinguish and , and have a prime
attached to distinguish location on the body or slide.
These scales are labeled R
(Root) or W
(Wurzel). The radical symbol may also be used.
|K||This scale usually occurs by itself, rather than as a member of a pair.|
|LL, E||or|| This is one of
the scales that show the decimal point. Usually there are several
where is in the interval
|L|| The only scale on a slide rule that
has a constant increment. Usually on the slide. If there was one such
scale on the slide and one on the body they could be used for the
|S||,|| Lists the
angle for which of .
On slide rules, all angles are measured in degrees, and reside in the interval
. The scale usually lists both
and , using the identity
|T||,|| Similar to
the S scale. is in the interval ,
is in and . There may be a similar scale of in the interval
in which case subscripts may be used to distinguish the
|ST|| showing the angle (in
degrees) in the unit circle for an arc of length where is
in the interval . For such small arcs, within the
accuracy of a slide rule, the angle (measured in radians), the sine,
and the tangent are all equal.
|P|| for in the interval
. The Pythagorean Scale.
|H|| for in the interval
. There may be another scale for in
and the two scales may be distinguished by subscripts.
|Sh|| is the inverse of
the hyperbolic sine. is in the interval If a scale
is present for in the scales may be distinguished
|Ch|| is the
inverse of the hyperbolic cosine. is in the interval .
the inverse of the hyperbolic tangent. is in the interval .
More generally, if you choose a number on a scale corresponding to the function (as listed in Table 1), and you read the corresponding number on a scale corresponding to the function , then
Note that is not the number under the hairline on the C scale, unless you choose to start on that scale!
As the tables clearly indicate, if you move the hairline over any number on any scale at all, and read the number on the same scale right under the hairline, you'll get that very same number back!
Of course the number of possibilities is vastly increased by allowing the slide to move. We consider two procedures, PLUS and MINUS, involving scales 1, 2, and 3. Scales 1 and 3 are on the body, scale 2 is on the slide.
PLUS: Select u on scale 1 (on the body), align it with the index of scale 2 (on the slide), move the hairline to v on scale 2, and read the result on scale 3 (on the body), underneath the hairline. For example if the scales involved are D, C, and D, the result would be the product, uv.
MINUS: Select u on scale 1, align it with v on scale 2 on the slide, move the hairline to the index of scale 2, and read the result on scale 3 on the body, underneath the hairline. For example, if the scales involved are again D, C, and D, the result is the quotient, .
What happens if we use other scales? Assuming a (very hypothetical) slide rule that has all the scales listed above both on the body and on the slide, these two procedures let you evaluate 3,540 different expressions in 4,394 different ways. Six examples are given in Table 4. Click here to see a similarly organized pdf file (of several hundred pages) showing all the possibilities.
In general, if is the function corresponding to scale 1 (again, as listed in Table 1), the function corresponding to scale 2, and the function corresponding to scale 3, then the result that you read on scale 3 is
where the base of the logarithm is the length of the slide rule and exp is the inverse function of log. The symbol indicates whether to use the plus or the minus procedure.
The first three rows of Table 4 show the most common operations on a slide rule: product, quotient, and power.
The last three rows show less common formulas that can be evaluated. Thus, according to the fourth row, to compute follow the PLUS procedure with scales 1, 2, and 3 being D, CI, and H, respectively. The first number in that row, 139, indicates the entry in the pdf table, 26 means it is the 26th distinct formula in the table, and 2 means it's the second way to evaluate this particular formula. These numbers are not important for the example, but they illustrate the organization of the pdf table. Caveats apply even more so than to the one variable Table 2 and 3 above. The variables have to be in certain ranges, and you may have to be judicious about which variant of the relevant scale you use to read your result.
Of course, slide rule manuals do not list thousands of formulas. They describe basic principles and then people can figure out how to use slide rules to best advantage for their particular applications. There are more pedestrian ways to compute but if you have to evaluate such expressions many times you'll find the shortcut eventually. Once you have it you can impress your friends and coworkers!
The last example in Table 4 requires an LL scale on the slide. When I went to high school our work horse slide rule was the Aristo Scholar 903. One version of it has a body and cursor with one side, but a slide with two sides. The back of the slide shows several LL scales. So prior to doing this calculation you need to turn the slide around. This gives you a very strange slide rule without a C scale. For years I have wondered for what kind of application one would want to turn the slide on the Aristo Scholar, and after writing this web page I know!
With the 13 scales assumed here, there are 24,314 distinct such expressions, filling 2,143 printed pages that you can view or download here. The four columns following the mathematical expression give the scales 1, 2, 3, and 4 being used.
As discussed above, one thing slide rules can do that calculators can't is create tables. Here is an intriguing application of that idea that I found in the Post Versalog Slide Rule Instructions, Frederick Post Company, 1963. That readable little book describes very many applications of slide rules.
Suppose we want to find the roots of the equation
|Aristo Hyperlog 0972||2||31||6||2||2||1||1||1||1||1||1||1||1||8||1||1||1||1||1||2||2||1||1|
|Faber Castell Novo-Biplex 2/83 N||2||30||11||2||2||2||1||1||1||1||1||1||2,2||1||8||1||1||2||1||1||1,2|
|Aristo MultiLog 0970||2||24||6||2||2||1||1||1||1||1||1||1||1||8||1||1||1||1|
|Faber Castell 52/82||2||22||7||2||2||1||1||1||1||1||1||1||3||1||1, 1||2||1||1||4|
|Aristo Scholar 0903 LL||1.5||15||4||1||1||1||1||1||1||3||1||1, 1||1||1||3,4|
|Faber Castell 111/54||1.5||14||5||1||1||1||1||1||1||3||1||1||1||1||3,4,10|
|Faber Castell 57/89||1.5||14||5||1||1||1||1||1||1||2||1||1,1||2||1||10|
|Pickett Electronic N-515||1||11||1||1||1||1||1||1||2||1||1||13|
|Aristo Scholar 0903||1||10||4||1||1||1||1||1||1||1||1||1||1|
|Pickett Trig Projection Rule||1||9||1||1||1||1||1||1||1||1||1||1||12|
|Pickett Microline 120||1||9||1||1||1||1||1||1||1||1||1||1||11|
|Faber Castell Mentor 52/80||1||7||5||1||1||1||1||1||1||1||5|
|Pickett Microline 160||1||7||1||1||1||1||1||1||1|