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Special Scales

We have already seen how most slide rules, in addition to the basic C and D scales used for multiplication, have the A and B scales, useful for problems involving squares and square roots, the K scale for cubes and cube roots, the CI scale for reciprocals, and the L scale on which the common logarithms of numbers on the C or D scale can be read.

The Trigonometric Scales

Still more scales were very common on slide rules. Multiplying numbers by the sine, cosine, or tangent of an angle is something that one doesn't need to be a rocket scientist to have to do on occasion; even being a carpenter will suffice, and, therefore, trigonometric scales were, at least during the 20th Century, a standard feature even on most basic slide rules.

The S scale has marks that are labelled with angles in degrees; the location of each mark corresponds to the position on the C scale where the sine of that angle is found. Similarly, the T scale has marks that are labelled with angles in degrees, and the location of each mark corresponds to the position on the C scale where the tangent of that angle is found.

Thus, the S and T scales can be used for multiplying a number by the sine or tangent of an angle. And, of course, the cosine of 90 degrees minus an angle is the sine of that angle.

The tangent of an angle over 45 degrees is the reciprocal of the tangent of 90 degrees minus that angle. Thus, one uses the T scale to divide instead of multiply for such large angles. The number of degrees for such angles is thus noted in red on most slide rules, in the same way that the numbers on the CI scale usually are.

Usually, the center slide of even a simple slide rule had S and T scales. The simplest way to work with those scales is by flipping the center part over by taking it out of the side of the slide rule first, then flipping it over and putting it back in. But there were windows in the back of the slide rule that permitted using those scales without doing this, although one had to remember how to do that. The L scale was usually between the S and T scales on the back side of the center slide. On these slide rules, the S and T scales tended to be marked in degrees and minutes, and the S scale might work with the A and B scales while the T scale worked with the C and D scales.

The Log-Log Scales

The fancier slide rules also had a set of scales called "log log scales". What these scales let you do was raise numbers to arbitrary powers.

If the K scale, for example, shows the cube of numbers on the D scale by including three scales in the space of one, because adding the logarithm of a number to itself three times produces the logarithm of its cube, then one can cube a number by multiplying its logarithm by three.

And in that case, since a slide rule performs multiplication, a suitable scale should allow exponentiation.

This diagram illustrates a slide rule with a very small-scale log-log scale, matching the K scale. Usually, the log-log scale shown here would instead be broken into three pieces, called LL1, LL2, and LL3, matching the C scale.

The coarse resolution of the diagram above, and its small scale, shows the mark for 1.001 on the LL scale coinciding with the 1 on the K scale, but when the scale is made to correspond with the C scale, on a real 10 inch slide rule, a fourth piece, extending from 1.0001 to 1.001, called the LL0 scale, is worthwhile to provide, even if, for smaller numbers, the D scale will serve.

By shrinking the log-log scale down, I paired a scale with both it and the K scale in the diagram, to make the illustration simpler to understand.

The J scale has been moved so that the 1 on that scale corresponds to the 3 on the K scale. Thus, every number on the J scale is multiplied by three on the K scale.

This displacement means that every number on the MM scale is cubed on the LL scale.

Thus, 1.01 is just past 1.03; 2 is under 8, 3 is under 27, 10 is under 1000.

Notice that the distance between numbers quickly shrinks. Thus, the tick marks between 100 and 1000 represent 150, 200, 250, 300, 350, 400, and 500.

While 10, 100, and 1000 stand in the same relation to each other on the LL scale as 1, 2, and 3 do on the K scale, note that 1.01, 1.02, and 1.03 also stand in approximately the same relationship to each other as 1, 2, and 3 do on the K scale.

This would let the K scale itself stand in for a lower extension of the LL scale, dealing with numbers like 1.001, 1.0001, and 1.00001. So the LL scale is aligned to facilitate that; 1.01 is placed near the index, instead of 10 being placed directly over 100 on the K scale.

This is done by placing a number on the log-log scale as follows: first, its natural logarithm is taken (just as the sine is taken first for the S scale), and then the common logarithm of that is taken to fit in with the other scales for multiplication. 1.01 to the 100th power is very nearly equal to e, and 1.001 to the 1000th power is even closer, and so on. So this is where we meet natural logarithms on a slide rule.

This diagram, placing the K scale directly against the LL scale, illustrates this more clearly, showing the two cases, the first where the correspondence shows with an alignment based on natural logarithms, and is an approximation approached in the limit, and the second where the correspondence is exact, and shows with an alignment based on common logarithms.

Incidentally, while most slide rules with log-log scales aligned them based on natural logarithms, one maker (Pickett) did use the common logarithm alignment. Also, some very old slide rules did have a log-log scale which was matched to the K scale for simplicity. Others, including the very first slide rule to have a log-log scale as illustrated in Roget's original paper, aligned them with the A and B scales. On some such slide rules, the normal log-log scale was called the E scale, and the reciprocal log-log scale was called the F scale.

Many, but not all, slide rules with log-log scales also had reciprocal log-log scales (LL/3, LL/2, LL/1, LL/0 or LL03, LL02, LL01 and LL00) with numbers on them that were the reciprocals of those on the corresponding log-log scale, to allow numbers between 0 and 1 to be raised to powers as well without taking an extra step to calculate their reciprocals.

The original versions of the Keufel & Esser 4080, 4081, and 4082 slide rules made prior to 1947 used the C and D scales with the normal log-log scales, but saved space by using the A and B scales with the reciprocal log-log scales; this arrangement was the subject of a 1939 patent (U. S. patent 2,170,144). On those slide rules, the normal scales were the LL1, LL2, and LL3 scales; the LL00 scale contained the reciprocals of the numbers on the LL2 and LL3 scales, and so combined what would normally be called the LL02 and LL03 scales, and the LL0 scale ranged from 0.999 to .905, and had the reciprocals both of the numbers on the LL1 scale and what is normally called an LL0 scale, so it combined the LL00 and LL01 scales.

The consistent use of the C and D scales with both the normal and reciprocal log-log scales was credited, on a page in Pickett's 1954 catalog, where the up-to-date nature of the rules on their slide rules was noted, to a 1912 patent by Franklin S. Beckett for a circular slide rule, U.S. Patent 1,042,755. However, Seinen Yokota's slide rule from 1907 also had full-length log-log and reciprocal log-log scales which operated in conjunction with the C and D scales, and so it was mistaken to credit Beckett with that invention.

Most slide rules with both log-log and reciprocal log-log scales put the LL0 and LL/0 scales on the other side of the slide rule from the rest of the log-log scales; this was because all the ordinary scales were on the front, but there wasn't quite enough room for both log-log scales in full on the back. One of the few slide rules that avoided this was the Deci-Lon slide rule from Keufel and Esser, which is today one of the slide rules most sought after by collectors. It placed the four log-log scales on one side, and the four reciprocal log-log scales on the other.

The Faber-Castell Novo-Duplex, however, did even better, having all four segments of both log-log scales on one side. The Aristo 0969 StudioLog and the Aristo 0972 HyperLog were two other slide rules which had all four segments of both the normal and reverse log-log scales on one side and so did the Pickett Model 2, 3, and 4, N2, N3, N4, and 803, and the Flying Fish 1003.

The Faber-Castell Novo-Duplex (also sold as the Novo-Biplex in some countries) were also unique in having two sets of double-length scales for square roots which slid against each other, allowing higher accuracy in multiplication.

Incidentally, the log-log scale on the slide rule was first invented in 1815 by Dr. Peter M. Roget, a doctor of medicine, who is also the originator of Roget's Thesaurus. Although his invention was published in a scientific journal, it was largely forgotten, and the log-log scale was independently re-invented much later and on more than one occasion.

A few early slide rules that had only the LL2 and LL3 scales called them M and N, and some others called them LU and LL.

Also, as the early practise on slide rules was just to allocate the letters A, B, C and D to the four main scales on the rule, whatever their type, the letters E and F were sometimes used for log-log scales, and sometimes for many other possible types of scale. The arrangement of scales was typically:


and on one slide rule, the E scale might be an L scale, and the F scale might be a K scale, on another, the E scale might be the LL3 scale and the F scale might be the LL2 scale; on yet another, the E scale might combine the functions of the LL2 and LL3 scale, and operate with the A and B scales, while the F scale might similarly combine the functions of the LL02 and LL03 scales.

Slide Rule Scales: A Listing

Also, some slide rules had other special scales. A few of the more common of those, as well as the scales we have already discussed, are listed below:

Slide rules with the scales for calculating reactance and resonant frequency often also had smaller versions of those scales on the back (repeated, and labelled with explicit magnitudes) to assist in locating the decimal point in such calculations.

One particular slide rule had a very large assortment of special scales. That was the Pickett N16-ES slide rule, designed by one Chan Street of Street Laboratory and Industries of El Segundo, California. This was a special-purpose slide rule designed for electronics calculations.

Even some of its ordinary scales were special; the S and T scales were actually SI and TI scales, giving the reciprocals of sines and tangents. This was done with the intent of facilitating polar to rectangular conversions. A Keufel and Esser slide rule, and some Hemmi slide rules, also had reversed sine and tangent scales, which were labelled as inverse scales in some cases.

The scales on the rule were organized into several groups.

The Hemmi 269 slide rule was a slide rule which had stadia scales (that is, an H and V scale, simply labelled SIN COS and COS^2) which worked with the C and D scales, rather than the K scale. In addition, it had some extra scales:

The Aristo Geodat had a two-part 1 - cos(theta) scale that operated with the C and D scales, in addition to a two-part stadia scale set that worked with the C and D scales like that of the Hemmi 269 and a scale used to enter the cotangent of half the angle in a calculation, for small angles running from one degree to one-tenth of a degree; this scale wrapped around, starting and ending in the middle of the rule. The same model number, 0958, was applied to versions of this rule using angles in degrees or grads (400 to the circle).

The Hemmi 266 slide rule, in addition to having scales for working out a*b by adding log(a) to log(b), thus finding 10^(log(a)+log(b)), had scales for finding sqrt(a^2+b^2) and 1/(1/a+1/b); the former were called the P and Q scales, the latter the T1 and T2 scales.

Thus, the P scale is very definitely not the same as the one found on some European slide rules; instead of numbers being bunched together on the right, here they are bunched together on the left, but more fundamentally, these scales perform addition rather than multiplication, and so logarithms aren't involved in their construction.

On this slide rule, unlike the Hemmi 153, to be discussed next, the P and Q scales were short, not running the full length of the slide rule, so that they could share space with the T1 and T2 scales.

The Hemmi 153 slide rule had a unique complement of scales. In addition to a basic set of normal logarithmic scales, it had P and Q scales for finding sqrt(a^2+b^2) which were longer than those of the Hemmi 266, but in addition, it had a set of scales that worked in conjunction with those scales to give the values of both trignometric and hyperbolic functions.

It has a scale labelled with the Greek letter theta, giving angles in degrees. Corresponding positions on the R sub theta scale give its equivalent in radians, on the P and Q scales its sine, and on the T scale its tangent. As well, the G sub theta scale gave the Gudermannian of the value on the R sub theta scale, so that starting with a value on the G sub theta scale, its hyperbolic sine was in the corresponding position on the T scale, and its hyperbolic tangent in the corresponding position on the P and Q scales.

The P and Q scales on the Hemmi 153 were marked from 0 to 10, with the Q' scale going from 10 to 1.414.

The numbers on the C and D scale, from 1 to 10, had common logarithms going from 0 to 1, and a number on one of those scales was at the same horizontal position as that of its logarithm on the L scale.

In the case of the P and Q scales, one has to first divide the number on the scale by ten, and then its square would be found on the L scale; 0 is above 0, 10 is above 1, and 5 is above 0.25; the square of 1/2 (0.5) is 1/4 (0.25).

The P scale is also labelled sin, and the Q scale is also labeled cos.

There was also a Q' scale, which continued the Q scale to larger numbers.

The scale labelled theta runs from 0 to 90, with 45 exactly in the middle, above .5 on the L scale; it is symmetrical, and the markes on the scale become very close together at the edges, and are furthest apart in the center.

30 degrees on the theta scale is above 5 on the P scale. And, indeed, the sine of 30 degrees is 0.5. So the theta scale is organized so that the tick mark for a given angle from 0 to 90 degrees is above the square of the sine of that angle on the L scale. Thus, the sine of 45 degrees is 1/sqrt(2), so 45 degrees on the theta scale is above 1/2 on the L scale.

Knowing how the theta scale is arranged, then the R theta and G theta scales are easy enough to derive: R theta shows the angles on theta in radians instead of degrees, and below the Gudermannian of a number on the R theta scale is that number on the G theta scale.

Considering that the number on the P scale divided by 10, going from 0 to 1, is the sine of the angle as shown in degrees on the theta scale or in radians on the R theta scale (and the L scale shows the sine squared, as noted), that the T scale, going from 0 to infinity, shows the tangent of that same angle is easy enough to accept.

Finding sqrt(a^2 + b^2) using the P and Q scales is a way to find the hypoteneuse of a right triangle given its other two sides. Only if the hypoteneuse is 1 could one look upon a and b as corresponding to sine and cosine. So if we take 0 on the Q scale, move it to where an angle is on the theta scale, the cosine (times 10) of that angle appears under 10 on the P scale, which is how its label is justified.

And now it becomes possible to see how the G theta scale is useful.

Find a number on the G theta scale, and in the corresponding position on the R theta scale is the angle that is its Gudermannian.

In the corresponding position on the P scale is that angle's sine. So look up a number on the G theta scale, on the P scale will be the sine of the Gudermannian of that number, which is its hyperbolic tangent.

Similarly, on the T scale will be the tangent of the Gudermannian of that number, which is its hyperbolic sine.

And if one moves the left index, zero, on the Q scale over a number on the G theta scale, then under the right index of the P scale will be (multiplied by ten, of course) the cosine of the Gudermannian of that number. The secant of the Gudermannian of a number is its hyperbolic cosine, so one way to find that is by taking the result we obtain, and then using the C and CI scales to find its reciprocal.

A second method is possible: just as the sum of the squares of the sine and cosine of any angle is always one, in the case of hyperbolic functions, the square of the hyperbolic cosine minus the square of the hyperbolic sine of the same argument is always one.

The hyperbolic sine, as noted above, can be found on the T scale.

Then look for the hyperbolic sine on the Q scale, and read the hyperbolic cosine on the Q' scale below it - since the numbers on the Q' scale are the ones found a full scale length ahead of those on the Q scale, they're the ones derived from q' = sqrt(1 + q^2).

As I found an image of the Hemmi 153 slide rule that I could use, from an old drafting catalogue, I have provided it here:

however, much better images are available elsewhere on the Web, if you would like a better look at this historic slide rule.

The Flying Fish 1018 slide rule, a plastic slide rule made in Shanghai, had both P and Q scales accompanied by theta, R theta, G theta, and T scales like on the Hemmi 153, and I and J scales similar to the T1 and T2 scales of the Hemmi 266, except they also ran the full length of the rule, with the J scale extended by a J' scale.

The Relay 158 slide rule did not have the G theta scale, but it did have both full-length P and Q scales, as well as full-length T1 and T2 scales, labelled I and I' on that rule. However, while the same letter, Q, was used as a label for the scale on the slide, it ran in the opposite direction to the P scale (labelled P1 here, with the two scales as P1 and P2 rather than P and P', as with the Q and Q' on the Hemmi 153).

Also interesting, and also made by Hemmi, were the Lafayette Radio 99-7102 Vectorlog or F-686 slide rules. Their complement of scales was:

Sh1 Sh2 DF A|B DF Th CI C/LL1'|D LL3 LL2 db
Tr1 Tr2 P' P|Q ST Sr Sth C|D LL03 LL02 LL01

The db scale was uniform and linear, like an L scale, but going from 0 to 20.

The P and Q scales were the same as those on the Hemmi 153.

The C/LL1' scale was a normal C scale, but extended slightly to the right with an LL1' portion going from 1 to 1.1' which has been described as intended to enhance the accuracy of calculations with the log-log scales.

Since these slide rules already had conventional hyperbolic function scales, quite understandably a G theta scale like that of the Hemmi 153 would have been felt to have been unnecessary, and conventional hyperbolic scales would have been percieved as far more useful and convenient.

The G theta scale is the subject of a 1937 U.S. patent, number 2,079,464, with Hisashi Okura as inventor; the other scales of this family date from 1929, with Sadatoshi Betsumiya and Jisuke Miyazaki as inventors. Jisuke Miyazaki is also the author of the book "Development History of the Slide Rule".

While the P and Q scales of this type are mostly found on Japanese-made slide rules, since this included slide rules made in Japan for American brands, they were popular enough in the United States that these scales, as the V1 and V2 scales, were included on the Dempster RotaRule, a slide rule made in the United States. This rare circular slide rule has many interesting aspects that make it attractive to collectors; it is also extremely rare.

One important feature was a clutch mechanism that locked the cursor to the outer ring of the slide rule until the user pressed on the cursor to unlock it.

After postwar price controls caused it to cease production, another version was sold as the Boykin Model 510 RotaRule. Later, the Pickett 110-ES circular slide rule, made in Japan for Pickett, also included these scales; it is considered to have been similar to the RotaRule, but did not appear to me to have includes all the scales it did, although it also seems to have included the extra-long scales for high-precision multiplication as well.

At first, it seems as though it would not be possible to have a scale like the G theta scale on a slide rule without scales like the P and Q scales. The various Gudermannian identites make the inverse Gudermannian equal to an inverse hyperbolic function of a trig function, and the Gudermannian itself equal to an inverse trig function of a hyperbolic function. Either way, it is the argument to a trig or hyperbolic function that is on the 'outside', not the value of the function. On the Hemmi 153, one looks up an angle on the theta or R theta scale, and finds the value of the sine or tangent on the P or T scale; on a conventional slide rule, one looks up an angle on either the S or T scale, and the sine or tangent are both found on the C scale.

However, let's not be quite so fast to dismiss the possibility.

Some slide rules, such as those with the Darmstadt arrangement, have a different kind of P scale, the Pythagorean scale. If we think of the S scale as the "theta" scale, and the C scale as the "sine" scale, the P scale becomes the "cosine" scale. So if the equivalent of a G theta scale were added following that interpretation of the scales, it could be used to find the hyperbolic tangent and the hyperbolic secant at least, and then further scales could be added to continue the pattern.

Slide rules for business use sometimes had a linear scale which was used for calculating the number of days between two dates. A similar scale could be used to make a perpetual calendar: the days of the week might be symbolized by a pattern of tick marks like this:

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with short ticks representing the weekend, and the longest ticks representing Monday, Wednesday, and Friday so that the alternating pattern allows recognizing any day of the week quickly. The scale of days of the week could be positioned against the scale showing the days of the year by bringing, further along the scale, a mark for the century opposite a mark for the year of the century.

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