# Slide Rules for special purposes

### Theory and Construction

Although most slide rules have gauge points to aid specific calculations. For example pi, 3.1415... is usually marked. A special slide-rule is one that has whole scales inscribed in such a way that calculations relating to specific problems can be resolved quickly, simply and accurately. They have the advantage that a problem can be resolved almost as quickly and with little more trouble than it takes to write the numbers down. In some circumstances this is important, for example when a flight heading has to be corrected to account for prevailing winds. A specific scale should be more accurate than using a general scale. For example for a calculation involving Barometric pressure at, or around sea level numbers in the range 710-770 (mm of mercury) will be used. This range is compressed into less that 15mm on a general purpose rule. A slide rule is also less bulky than a chart or graph.

Of course a special rule can be used for one, or a very limited number of different problems. They are complicated and difficult to produce, especially when the scale markings are calculated by hand. This is a serious flaw and limits the range of applications in which special rules were used. Finally, suitable blank rules were difficult to obtain.

Despite these problems, many special rules were constructed and used. The following section describes briefly the theory and examines a particular case study of an oil tonnage calculator.

#### The Principle of Special Rules

Special rules operate by adding lengths proportional to the values of the functions represented on their scales. The rule shown in the above figure solves the equation

(*) Q + R - S =0

where lengths Q=f_1(q), R=f_2(r) and S=f_3(s). q, r and s are variables marked on the scales from some origin 0.

For example, in the standard slide-rule, if Q is taken as the Scale A the squared scale, R is the C scale, the proportional scale on the slider and S is D, the proportional scale on the stock we solve s=r sqrt(q).

In order to prevent confusion over which scales should correspond it is easiest to inscribe Q on the upper part of the stock and R contiguous with Q on the slider. The result is then read using an index line on the slider against S which is the lowest scale. This corresponds to the process of division using three proportional scales on a slide rule and is depicted in the following figure.

#### Construction steps

To construct a special scale we should perform the following steps.

1. Arrange the equation in a proper form for preparing a slide rule. In the case there are a number of forms, choose the one which will be easiest to operate
2. Arrange the terms in the equation so that the quantity usually solved for comes at the end and the signs of the terms alternate as nearly as possible. This will ensure as far as possible that it will be unusual to run out of scale and that as many of the scales as possible will increase to the right.
3. Choose the limits of the variables and select a constant of proportionality for scales so the longest scale will not exceed the length of the rule.
4. Plot the scales and fix them to the rule. If there is no index, the position of one of the scales will have to be located by solving an example.
5. The finished rule should have all the scales properly marked and [1] recommends the description of the method of operation should appear on the rule.

[Figure - usual layout of scales]

By using two or more sliders and scales it is possible to introduce more terms in equation (*). A full worked example to solve the equation

H=240 s [ (T_s - T_R)/145 ]^1.3

for H, given s, T_s and T_R is given in [1]. The solution uses six scales, two sliders and a fixed central ``middle support rule''.

### The Magnameta Oil Tonnage Calculator

Note: The images are large - between 400k and 500k.

#### The purpose of this rule

The Magnameta Oil Tonnage Calculator, British Patent GB919063, is a slide rule specifically constructed to deal with the problems relating to the loading of oil tankers. As the instruction book says,

It is intended primarily for use in handling loading problems, when the ability to estimate the weight per tank with speed and accuracy is of prime importance.

It continues,

The rule, which in spite of its size is neither heavy nor cumbersome, carried under the arm, is the answer to this problem. The official figure must under these circumstances be worked out later.

#### A description of the instrument

The rule is of standard design with a single slider and cursor. The instrument has five scales denoted A-E. The primary scales A\&E, marked on the stock, are 783mm in length. The other scales are marked on the slider and measure 381mm.

A is the tank capacity, calibrated from 15000 to 50000 cubic feet capacity. A and C are International Specific Gravity values from 0.6 to 1.08. Apart from running in opposite directions the scales are identical and are in reality logarithmic scales running from 33.292 to 59.972 with markings giving the specific gravities. D is marked in American Petroleum Industry (A.P.I.) specific gravity values. E is calibrated in tons for values 450-1500 tons per tank.

The cursor has three lines, marked I, II and III from right to left. The lines II and III correspond to multiplication by 1.015 and 1.12 respectively to allow calculations in long, metric or short tons respectively. The cursor is also fitted with a clamping device which holds it in one place relative to the slider. Loading problems always involve oil of a fixed specific gravity and being able to accurately fix the cursor has obvious advantages. Lastly, the rule has six index points on the slider.

#### Use of the instrument

To perform a calculation the cursor is fixed at a specific gravity and then the slider, with fixed cursor, is moved to a corresponding volume on the top scale A. The weight is read directly off scale E below using the appropriate index point. This is exactly as described above. Different index points are used to give weights in long, metric and short tons.

The rule can be used for tanks calibrated in water tons and cubic meters (metric water tons) by using lines II and II on the cursor and A.P.I specific gravities can be converted to International specific gravity units by allowing the cursor to run freely and reading across scales C and D.

#### It's construction and use

Patent GB919063 was granted to the inventor, Guy William Farrier Sangwin, the complete specification having been published Fed 20, 1963. The calculations for the scale and the printing of the plates took three years to complete. The prototype rule, with scales printed onto plastic and laminated onto a wooden base survives, although the left most 150mm of scales A and E have been badly damaged by water (The rule was rescued from a garage). This particular rule was never produced commercially because of the introduction of cheep electronic calculators in the late 1960's.

#### The instructions

The instruction book that accompanies the rule has been scanned and made available. Please remember, these images are Copyright © 2002 Chris Sangwin. If you wish to make use of them please contact me. I would simply like to know who has found this information useful and interesting.

### Fluid Discharge Calculator

This double slide rule was made by W. F. STANLEY & Co. Ltd., LONDON, and also marked Ernest H. Essex Ass: M. Inst: C.E. The slide rule was used to calculate the flow and discharge of fluids through pipes and culverts of different size and material. Kutter's Formula is marked on the back of the the rule for calculation of ovoid culverts and circular pipes. Length 10 1/4 ins. Width 2 7/8 ins.

### References

[1] Arnold Hoelscher
Graphic Aids In Engineering Computation (1952)

[Slide rule page] [Chris Sangwin]