Munafo "Model 1"
I am obsessed with large numbers, and as a child I was developing this keen interest just at the time when slide rules were beginning to become obsolete (and were therefore cheap) but calculators were still out of my price range. As a result I developed my own slide rules to overcome the limitations of normal rules.
Model 1
Munafo "Model 1"
I made this at age 14, it is a simplex with 4 nonstandard scales used to calculate the iterated exponential function z=(...((xx)x)...)x = xx(y-1). The instructions are written on the slide between the C+1 and L scales, they read:
Z = X*Y 1. Place 2 on Z1 scale over y on C+1 scale. Below x on Z1 scale read q on C+1 scale. 2. Place 1 on L scale over x on LL scale. Below q on L scale read z on LL scale.
The L scale is linear going from 1 to 11. If the slide and stock are aligned and the value on the L scale is x, then the values on the other three scales can be calculated as follows:
Z1 scale: 102x-10
C+1 scale: 1+2x-K1 where K1 = 1+1/log(2)
LL scale: 22x-K2 where K2=1+log(log(210))/log(2)
The Z1 and LL scales are equivalent, they are just horizontally offset. The LL scale determines the range of values of z that can be calculated and runs from 100.1= 1.26 up to 10100. I'm not sure why I developed such an indirect way to calculate the function, but my goals were to avoid the use of a cursor and to avoid the need to subtract 1 from y. The scales are not particularly accurate but I was mainly interested in just proving that it could work.
Model 3
Munafo "Model 3"
My continued interest in large numbers and nostalgia over my early slide rule attempts led in the early 1990's to this circular rule with a spiral LL scale, produced on a 300-DPI laser printer by a program I wrote myself. My coworkers were quite amused that I had used so much computational power to create an obsolete computing instrument.
All the slide rules I had seen had LL scales that only go up to e10=22026, or a slightly higher limit like 30000 or 100000. I was dissatisfied with this limited range, I wanted to calculate much larger exponents. (The Pickett models 2, 4, and 14 have LL4 scales going up to 1010, but I didn't know this and it still wouldn't have been high enough to satisfy me). This limited range concerned me, and I was also concerned with accuracy. Circular rules offer the potential for greater accuracy by having a greater effective scale length. I realized that the LL scales worked much better if they were joined together into a single continuous scale and on a circular slide rule this scale can be made as a spiral. It is by far the most direct and natural way to implement the LL scale, but apparently it is not obvious. As far as I know, only one commercial slide rule was ever made that uses this idea of a true continuous spiral LL scale — the Fearns, which has an LL scale going from 1.01 to 106, about 3.1 revolutions.
My circular rule has only 3 scales: C, D and a spiral LL scale going from e0.001=1.001 all the way up to e100=2.7×1043. The scale length of the LL scale is about 78 inches, or 2 meters. This completely covers the range of all LL scales on commercial slide rules — most of the better ones have 4 LL scales going from 1.001 to 22000 or from 1.0023 to 1010.
As pictured it is set up to calculate 727=6.6×1022. The number 1 on the C scale has been aligned with 7 on the LL scale, the cursor is aligned with 2.7 and shows the answer. The cursor has no hairline, you're supposed to use the right-hand edge of the cursor as a hairline. Some flaws in the software used to generate the LL scale are evident: 1.00105 and 1.0105 are mislabeled "1.0011" and "1.011" respectively; the number of subdivisions changes at odd places (for example, at 1.48 rather than at 1.45 or at 1.5); some odd numbers were chosen to get labels (like 1400 instead of 1500).
Here is a general discussion of design for convenience and accuracy including much that I have not seen elsewhere.
A brief overview of slide rule history is useful at this point.
Slide rules go back hundreds of years, with new designs keeping almost all elements of a previous popular design to make it easier for people to adjust to the new rule.
The physical form of the slide rule was developed by William Oughtred in 1630 (two logarithmic scales placed side by side to perform multiplication), Seth Partridge (a body or "stock" that holds a slide, making it easier to keep the two pieces together) and John Robertson (the cursor, making it possible to use more than 4 scales and align values from one scale onto another).
The oldest surviving element of the scale names and layout is the set of 4 scales A, B, C, D arranged from top to bottom in that order. This was Mannheim's design around 1855, these scales still appear in the same order and placement on almost all slide rules today. Multiplying with A and B is most convenient because it avoids falling off the end of the scale, a problem discussed below. However, the C and D scales are used more often because they provide greater accuracy. Having all 4 scales makes it possible to take square roots, and allows quickly multiplying something by the square of something else with a single setting, as when computing the area of a circle (a = pi r2: Place 1 on C against r on D; against pi on B read a on A).
Other scales were added later to make other calculations possible, most notably the L scale to take logarithms and antilogarithms (which allows calculating any value of the form xy, yth root of x and logxy) and trig scales (the most common are S, T and ST). Rules in this category include the "modern Mannheim" (A[B.CI.C]D.K, [S.L.T]) and "Rietz" (K.A[B.CI.C]D.L, [S.ST.T]) arrangements.
Modern general-purpose duplex slide rules have lots of extra scales just for convenience. For example, the LL scales allow directly calculating xy in one step, rather than three steps needed with a Rietz (take log of a, multiply by b, take antilog). This is important for financial calculations related to interest, principal, present value, etc.
CI against D Multiplication is the technique of using the CI and D scales to multiply. It avoids the common inconvenience of needing to guess in advance whether the product will be greater than 10. The reason it works is because of the rotational symmetry introduced by having one scale reversed.
C against D Multiplication
Almost all slide rules have C and D scales placed against each other along the lower edge of the slide:
Notice I labeled the right end "10", usually it says "1". There's a reason I used "10" which I'll explain later.
To find z = x × y, place 1 on C against x on D, then against y on C read z on D
Example: multiplying 2 times 3. Place 1 on C against 2 on D; against 3 on C read Z (6) on D.
The inconvenience arises because, when the answer is greater than 10 you have to position the scales a different way: place 10 on C against x on D, then against y on C read z/10 on D.
Example: multiplying 5 times 3. Place 10 on C against 5 on D; against 3 on C read Z/10 (1.5) on D.
Now you see why I label the right end of the scale "10" instead of "1". It helps you remember when to add a decimal point. The rule is: If you set the multiplicand against the 10 index, multiply your answer by a factor of 10.
After a while you can usually figure out in advance which way to do it: if the answer will clearly be bigger than 10, you know to use the second method. However, sometimes (like when multiplying 2.7 by 3.7) it is not so easy to figure out in your head whether the answer will be bigger than 10.
CI against D Multiplication
The inconvenience is avoided entirely if you use the CI scale against the D scale:
Notice the symmetry of this diagram: If you turned it upside down, it would still be the same (except for the numbers and letters). It is because of this symmetry that the rule for multiplying becomes simpler:
To find z = x × y, Place x on CI against y on D (or vice versa). Against 1 on CI (or D) read z on D (or CI); or against 10 on CI (or D) read z/10 on D (or CI).
Example: multiplying 2 times 3. Place 2 on CI against 3 on D (or 3 on CI against 2 on D); against 1 on CI read z (6) on D (or, against 1 on D read z on CI).
Example: multiplying 5 times 3. Place 5 on CI against 3 on D (or 3 on CI against 5 on D); against 10 on CI read z/10 (1.5) on D (or, against 10 on D read z/10 on CI).
Note that in these examples, it doesn't matter which scale is used for x and which for y, the same placement aligns both. Also, the answer can be read on either scale. This elegance and the convenience of not having to use a different setting for answers greater than 10 result from the rotational symmetry of the CI and D configuration: if you turn it over, it's the same (the D scale becomes a CI and the CI becomes a D).
Some slide rules (including the Faber-Castell 2/83N Novo Duplex) have two pairs of double-length scales that each cover half the range from 1 to 10. In the 2/83N they are called W1, W1', W2 and W2'. These are used together to perform multiplication with greater accuracy (because a longer scale makes it easier to align the numbers and read the answer). They are placed on the reverse side of the duplex rule in the positions normally taken by the A B C and D scales on a Mannheim. Here is a representation of the reverse side of the 2/83N:
The index marks r are the square root of 10.
To multiply z = x × y: Place 1 or 10 on W2'/W1' against x on W2/W1; against y on W1'/W2' read z on W1/W2 (or read z/10 if 10 was placed against x). If that places y off-scale then instead place r on W1'/W2' against x on W1/W2; opposite y on W1'/W2' read z on W2/W1 (or read z/10 if y is greater than r). Always follow the rule that if x is set against 1 or 10, the answer is read on the scale directly adjacent to y, and if x is set against r, the answer is read from y to the opposite W scale using the cursor.
Here are 4 examples using a setting of the left index mark to x:
Example 1: multiplying 1.5 times 2. Place 1 on W2' against 1.5 on W2; against 2 on W2' read z (3) on W2.
Example 2: multiplying 1.5 times 4. Place 1 on W2' against 1.5 on W2; against 4 on W1' read z (6) on W1.
In these two we read the answer with the cursor ("opposite" rather than "against" the multiplier)
Example 3: multiplying 5 times 1.5. Place r on W1' against 5 on W1; opposite 1.5 on W2' read z (7.5) on W1.
Example 4: multiplying 5 times 4. Place r on W1' against 5 on W1; opposite 4 on W1' read z/10 (2) on W2.
Half the time you need to set a right index mark to x. Here are 4 more examples:
(Note: in these two examples the illustration fails to show correct alignment with the 8.)
Example 5: multiplying 8 times 1.5. Place 10 on W1' against 8 on W1; against 1.5 on W2' read z/10 (1.2) on W2.
Example 6: multiplying 8 times 5. Place 10 on W1' against 8 on W1; against 5 on W1' read z/10 (4) on W1.
And another two cases using the cursor to read the answer:
Example 7: multiplying 2 times 6. Place r on W2' against 2 on W2; opposite 6 on W1' read z/10 (1.2) on W2.
Example 8: multiplying 2 times 2. Place r on W2' against 2 on W2; opposite 2 on W2' read z (4) on W1.
Reversed Split Square-Root Scales
As you can see, using the W1-W2 scales has two inconveniences — the need to guess which index mark to start with (a problem familiar in ordinary C-against-D multiplication) and (50% of the time) the need to use the cursor to read the answer on the opposite scale from y.
The first of these inconveniences can be eliminated by reversing the placement and orientation of the two scales on the slide (or, just by removing the slide and putting it back in upside down):
Notice I have also simplified the scale names, because now there is a new symmetry and the instructions would be the same if all W1 and W2 were replaced with W1' and W2'.
With this scale layout the instructions for multiplication become: If x and y are both on W2, align x on one W2 with y on the other W2 using the cursor. Against either r read z on W1.
If x and y are both on W1, align x on one W1 with y on the other W1 using the cursor. Against either r read z/10 on W2.
If x is on W2 and y on W1 or vice-versa, place x against y. Against 1 read z, or against 10 read z/10.
Here are several examples:
Example 1: multiplying 1.5 times 2: both are on W2. Align 1.5 on W2 with 2 on W2 using the cursor; read z (3) on W1 against r on W2.
Example 2: Multiplying 5 times 6: both are on W1. Align 5 on W1 with 6 on W1 using the cursor; read z/10 (3) on W2 against r on W1.
Example 3: multiplying 1.5 times 4. Place 1.5 on W2 against 4 on W1; read z (6) on W1 against 1 on W2.
Example 4: multiplying 2 times 9. Place 2 on W2 against 9 on W1; read z/10 (1.8) on W2 against 10 on W1.
Example 5: multiplying 7 times 8. Align 7 on W1 with 8 on W1 using the cursor; read z/10 (5.6) on W1 against r on W2.
Notice in all these examples that both permutations of x and y occur simultaneously with the same slide position. In other words, multiplying x times y uses the same position as multiplying y times x. Also, the answer can always be read in two places: on the slide against an index mark on the stock, or on the stock against an index mark on the slide. As with CI-against-D method, these things result from the fact that there is greater symmetry in the new arrangement.
Acknowledgments
Bob Finch corrected my instructions for the original W1/W2/W1'/W2'
method. Thanks also to the sliderule-trade mailing list for helping me
figure out how to get the one rule I always wanted (a 5-inch log-log
rule! Yeahy!) and how much I should pay.
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