# What Can You Do With A Slide Rule?

There was a time when electronic calculators did not yet exist. This did not stop us from doing complicated things, like going to the moon, figuring out the double helix, or designing the Boeing 747. In those days, when we needed to compute things, we used slide rules which are marvelous and beautiful instruments!

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 were scanned by Clark McCoy of the Oughtred Society. They show the two sides of a particular slide rule in my collection. This is one of the fanciest and 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.

## The Basic Idea

It's clear how to add or subtract two lengths using two ordinary rulers. Slide rules do the same thing, add and subtract lengths, but they don't call them lengths. For example, by calling them logarithms, you can multiply and divide numbers. In fact, I don't know of any slide rule that actually let's you add or subtract numbers. In the heyday of slide rules that was considered a trivial task that you did in your head, or on a piece of paper if you had to.

[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.

## Basic Multiplication

The most basic procedure carried out on a slide rule is the multiplication of two numbers u and v using the C and D scales. These two scales are identical. C is on the slide, and D is on the body. Move the hairline over u on the D scale. Move the slide so that its beginning (marked by 1 on the C scale, and also called the index of the C scale) lines up with the hairline. Move the hairline to the number v on the C scale. Read the result underneath the hairline on the D scale. If the number v projects beyond the end of the slide rule move the end of the slide rule (marked with 10 on the C scale) above u and read the result as before on the D scale underneath the number v on the C scale.

## Why?

Why does this work? The C and D scales show a number x that equals the exponential of the distance of x from the beginning of the C or D scale. So basically you are adding the logarithms of the numbers u and v, and the logarithm of the product equals the sum of the logarithms. This is the fundamental identity underlying all slide rule calculations, and it is worth stating prominently:

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:

1. The real number line is infinite and slide rules have finite length. Hence all scales can only show a part of the real number line. On the C and D scales, any number x is shown as a number between 1 and 10, and it is determined only up to a factor that is an integer power of 10. In other words your slide rule does not usually show the location of the decimal point. You are supposed to understand your problem well enough so you can tell where to put it. The slide rule also does not tell you the sign of your result.
2. Compared to a calculator, a slide rule is severely limited in its accuracy. You can enter and read a number typically to two or three decimal digits only.

## Scales

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.

 Name function 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, B is 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 scales, like and 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 addition of numbers. 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 scales. 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 by subscripts. Ch is the inverse of the hyperbolic cosine. is in the interval . Th is the inverse of the hyperbolic tangent. is in the interval .

Table 1: Common Scales

## One Variable

The power of a slide rule stems from the interplay of the scales and the movements of the slide and the cursor. However, even if your slide was lined up with the scales on the body, but otherwise frozen in place, you could use your slide rule as a lookup table for a large number of formulas. Some of them are listed in Tables 2 and 3. For example, if you wish to compute the expression move the hairline over on the CF or DF scale, and read the result on the W scale.

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

where is the inverse function of . The rows of tables 2 and 3 correspond to , and the columns to .

Note that is not the number under the hairline on the C scale, unless you choose to start on that scale!

 CD CDI CDF CDIF AB W K LL L S T P H

Table 2: One Variable Conversion

 CD CDI CDF CDIF AB W K LL L S T P H

Table 3: More One Variable Conversion

There are some caveats about reading Tables 2 and 3. For example, may have to be in a certain interval, and the tables do not distinguish between different versions of the same scale, e.g., the various LL scales. For the S scale, we only consider the inverse sine function, not the inverse cosine function. So before you use your slide rule as suggested in the tables you'll have to think carefully about what you are doing, which never hurts anyway. The typesetting of some of those formulas is a bit idiosyncratic. They were mostly machine generated, and I did not want to introduce additional errors by excessive manual editing.

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!

## Two Variables

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.

 row entry formula variation result Scale 1 Scale 2 Scale 3 +/- 1 1 1 1 CD CD CD + 2 15 2 1 CD CD CD - 3 2403 1803 1 LL CD LL + 4 139 26 2 CD CDI H + 5 287 83 1 CD AB W - 6 424 168 1 CD LL S -

Table 4: Two Variable Computations

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!

## Three Variables

Suppose we consider a variant of the PLUS procedure where instead of the index we use a number on a fourth scale. Thus we start again by putting the hairline above the number u on scale 1. Then we move the number v on scale 2 underneath the hairline. Next we move the hairline above the number w on scale 3. Finally we read the result on scale 4 underneath the hairline. Scales 1 and 4 are on the body, scales 2 and 3 on the slide. If the scales are D, C, C, D respectively, the answer is uw/v.

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.

## Sophisticated Multiplication and Division

Sophisticated Multiplication sounds like an oxymoron, but it isn't in slide rule lore. We can multiply and divide using the C and D scales, and so in particular we can multiply with and compute reciprocals. Thus there is nothing we can compute with the CI, DI, CF, DF, CIF, and DIF scales that we can't compute with just the C and D scales. The purpose of these additional scales is to make multiplication and division fast and easy by minimizing the number of times and the distances that the slide and cursor have to be moved, particularly when doing repeated division and multiplication. Try it, and you'll see that it is especially convenient if multiplications and divisions alternate. If you have a sequence of multiplications only you can replace some of them with a division by the reciprocal of the relevant factor, using the CI and DI scales. On the *F scales, the number 1 is almost exactly in the middle of those scales, and so by switching to those scales when appropriate one can reduce the distance by which one has to move the slide! If that was their only purpose, the optimal folding factor for the *F scales would have been the square root of 10. It so happens that is close to that square root and works almost as well. In addition however, it makes it possible to multiply or divide by without any slide movement at all. At some stage in the past someone had the quite brilliant idea to approximate the square root of 10 by .

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

Let's assume that is positive, and the roots are real. If is negative we ignore that fact and worry about the signs of the solutions later. As an exercise you may want to figure out what happens when the roots of the quadratic equation are complex. If the solutions are and we have

So we want to find two numbers and that add to and multiply to . We move the hairline over on the D scale, and place the beginning or end of the slide under the hairline (choosing whichever causes the smaller projection of the slide beyond the body). Now the product of any pair of numbers on the D and CI scales (or on the DF and CIF scales) is equal to . Your slide rule now contains a table of pairs of numbers that all have the same product. All that's left to do is to move the hairline until we find a pair of numbers on the D and CI scales (or DF and CIF scales) that add to . Computing the sums mentally as we move the hairline is a pleasant exercise that requires no external help. Once we have the pair of numbers we can figure out the sign of the roots from the signs of and .

## Cursor Marks

Many slide rules have special purpose marks on the cursors, in addition to the hairline. As an illustration, here is a list of calculations that can be accomplished with the cursor marks on the Faber Castell Slide Rule pictured above.
• Put the hairline over the diameter of a circle on scale D or C and read the area of the circle under a special mark on scale A or B.
• The same marks can be used with the CI scale to compute the volumes of cylinders.
• Two marks facilitate instant conversion of kW and hp (PS) on the scales C and D.
• To multiply a number with 3.6 (and an integer power of 10) put the hairline over it on the C or D scale, and read the result under a special mark on the CF or DF scale. Of course, reversing that procedure let's you divide by 3.6.

## Specific Slide Rules

Table 5 lists scales on some specific slide rules. Numbers indicate the number of scales present. For example, 8 LL scales usually means 8 distinct scales, 2 C scales usually means there is a C scale on each side of the slide rule. A green entry means the scale is on the slide, a black it is on the body. Name is the name of the slide rule. Sides lists how many sides are used (one or both, or one and a half in the case where the slide is reversible but there are no scales on the back of the body). Scales lists the total number of scales. The table is sorted by decreasing total number of scales. Marks lists how many marks are on the cursor, including the hairline, and the remaining columns indicate the specific scales as listed above. The last column gives reference numbers corresponding to notes that follow the table.

 Name Sides Scales Marks C D CI DI CF DF CIF A B R,W K LL L S T ST P H Sh Ch Th Notes Pickett N4 2 33 2 2 2 2 1 1 1 1 2 8 1 1 2 1 2 1 9 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 Pickett N803 2 28 2 2 2 2 1 1 1 1 1 1 2 1 8 1 1 1 1 6,7 Aristo MultiLog 0970 2 24 6 2 2 1 1 1 1 1 1 1 1 8 1 1 1 1 Aristo Studio 2 23 6 2 2 1 1 1 1 1 1 1 6 1 1 2 1 1 Post Versalog 2 23 2 2 2 1 1 1 1 2 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 Pickett N600 2 22 2 2 2 1 1 1 1 1 1 1 6 1 1 1 1 8 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

Table 5: Specific Slide Rules

Notes:

1. The LL0 scale is merged with one of the D scales.
2. The W scales enable multiplication and division with increased accuracy, effectively providing a rule with a length of two feet.
3. The slide is reversible. The scales one the back side are S, and 3 LL scales.
4. One additional scale, labeled BI, shows .
5. Has rulers and instructions on the back of the scale.
6. LL scales are merged with their reciprocals, e.g., LL1 with LL01.
7. Has a special DFM scale on the body. It is folded at M which is the base 10 logarithm of e.
8. Also has an Ln (natural logarithm) scale.
9. Has the following additional scales:
• 3 cube root scales on the body
• A pair of scales folded about the base 10 logarithm of e.
• A natural logarithm scale on the slide.
10. One edge of the slide rule protrudes and can be used as a ruler.
11. This rule is available as part of a "programmed self instruction kit". Also has instructions on the back of the rule.
12. Is made of clear plastic to be used on an overhead projector, for teaching purposes. Comes with a 48 page instruction manual. Comes in a box labeled "All Metal Slide Rule".
13. Fine example of a special purpose slide rule. Made for electronic calculations specifically in the Cleveland Institute of Electronics. Back Side has tables and formulas helpful in electronics. There are two special purpose scales. One shows , the other shows . One L scale has base 10, the other base e.

## Home Work

I'm employed by an institution of higher learning, and I feel compelled to assign some home work problems:
• Think about ways of making slide rules more accurate.
• Why is it that there seem to be no slide rules that have L scales on both the slide and the body, thus enabling addition and subtraction?
• Think carefully about the base of the logarithm used on a slide rule.
• Slide rules work because the logarithm of the product is the sum of the logarithms. Could you use functions other than logarithms? It's an excellent exercise to work out the answer, but if you can't or won't, but you must know, click here!
• How do you solve a quadratic equation with complex roots, using your slide rule?
• Build your own slide rule.