SECTION FIFTEEN

HISTORICAL NOTE

Invention of the Slide Rules - Degree of Accuracy - Common Gauge Points - Marking Special Gauge Points.

NATURAL or hyperbolic logarithms were invented by Napier of Merchiston in 1614, and the system is frequently known by the name of Naperian logarithms. The base of the Naperian system of logs is 27183; this number usually is denoted bye. Common logarithms, namely those to the base 10, are sometimes called Briggsian logarithms; this system is invariably used for ordinary computations.

The first practical application of logs in the form of scales was produced by Professor Gunter in 1620. His instrument consisted of one scale only, and was used in conjunction with a pair of dividers. The slide rule in its modern form was first devised by Wingate in 1626, and the cursor was added by Mannheim in 1851.

Degree of Accuracy

The only criticism we hear advanced against the slide rule is that results obtained with its aid are not always exact. Speaking now of the 10" C and D scales, errors should not much exceed 1 to .2%. Accuracy will depend upon the care taken in manipulating and reading, and upon the accuracy of the instrument itself. All slide rules exhibit small errors in the dividing of the scales if examined critically, but in a good instrument such errors are small, and often difficult to detect. In the course of time, defects develop due to shrinkage or distortion of the rule itself. An old rule frequently displays discrepancies in the lengths of the scales which originally were identical. The effective life of a rule will be considerably lengthened by careful treatment and protection from unnecessary exposure in a hot or moist atmosphere.

The negligence of the shopkeeper who displays for sale slide rules in his window, in the direct rays of the sun, is reprehensible, and indicates ignorance of the merchandise he handles. No slide rule, except the all-metal types which are seldom seen, will retain accuracy and easy movement after prolonged exposure in direct sunlight.

We have attempted to advance the claims of slide rules fitted with duplicate scales, such as the types described in Sections 10 and 11. With this type of rule it is possible to obtain results while using much shorter lengths of scales, and, of course, it follows that the errors, due to discrepancies in the scales, will be smaller.

Interpolated readings are certain to introduce small errors in results, since all we can do in assessing values which do not coincide with an actual graduation of the scales is to estimate their position as though the scales are evenly divided instead of being logarithmic. These errors are smaller perhaps than would be expected. The widest space in the C or D scale of the 10" rule is that lying between 4 and 405. If we set the cursor index exactly in the middle of this space we should no doubt read its position as 4.025. Its true reading should be 40249. We are, however, likely to make larger errors when we estimate other fractions of spaces, since the half-way position is the easiest of all to assess correctly.

All instruments when used are susceptible to errors of varying degrees. If we are asked to name a simple instrument possessing a high degree of accuracy, we immediately think of the engineer's micrometer. In using the instrument, we may, as a result of error in the thread, or zero error, or faulty execution, obtain an error of 0005, i.e. "half a thou". If we are measuring a rod of about ½" diameter, the 0005" error is of the order of 1 in 1000, not far removed from the degree of error we may encounter in a slide rule. If we are measuring the thickness of a sheet of foil of the order of .005", our micro meter error is 10%, something much worse than our slide rule inaccuracies. Again, a work's accountant might criticise the slide rule because it may not give him quite accurately the cost of 4 tons 2 cwt. 3 qrs. of material at £1, 4s. 6d. per ton, forgetting that his weight may be in error to the extent of 1% or more, a larger error than the slide rule will introduce.

When discussing accuracy of slide rule results, points such as those we have mentioned should be remembered.

Gauge Points

In addition to the scale graduations, a few other lines appear in the majority of slide rules. These additional lines, termed gauge points, represent the positions of factors commonly used in calculations.

In nearly all rules the value of p = 3·14159 is marked in the principal scales, p being the constant which enters into calculations relating to circles, spheres, etc. p/4= ·7854, is some times shown by a gauge point, (p/4) d² being the area of a circle of diameter d.

Gauge points, denoted by c and c', appear at 1·13 and 3·57, respectively, in the C scale of many slide rules. The volume of  a cylinder is (p/4) d²1; it may be written in the form (d / Ö( p/4) )² L.

The value of ( p/4)  is 1·13 approximately. If the gauge point c is set to the value of the diameter of a cylinder on D, the volume of the cylinder may be read on A coincident with the length, 1, on B. For some values of d and 1, the result will be off the scale when c is set to diameter. In such cases if c'  is used, the result will be obtainable.

The gauge point M is seen in scales A and B in some makes of rule. Its virtual value is 1/p = ·3183. To find in one setting of the slide the area of the curved surfaces of a cylinder, we set M to diameter in A, and read over the length in B the area of curved surface in A.

Other gauge points may be found, their inclusion or omission being dependent upon the decision of the manufacturers or designers of the rule. We mention the following, which are the commonest:

r' at 3438, and r" at 206255, in scale C, give the numbers of minutes and seconds in a radian respectively. These gauge points may be used for finding the values of trigonometrical functions of small angles. For any small angle, say, less than 2°, the sin and the tan may be taken as identical. If we set the r' mark to the graduation in scale D, representing one-tenth of the number of minutes in the angle, the sin or tan may be read in D under the 1C or l0C.

Example: Find the sin or tan of 22'.
    Set r' to 2·2D. Read sin or tan in D under l0C = ·0064.

If the angle is expressed in seconds, the r" is used in a similar manner.

A third gauge point rg occasionally may be seen between 6·3 and 6·4 on scale C; this is used in the same way when the angle is expressed in the centesimal system.

If we remember that the sin 1° = tan 10 = ·0174, we shall have no difficulty in inserting the decimal points in results obtained when using these gauge points.

A gauge point is sometimes placed between division 114 and 116 in scales A and B; this is called the gunner's mark, and is used in certain calculations relating to artillery.

The value of g = 32·2' (per sec.)², the gravitational acceleration imposed on freely falling bodies near the earth's surface, is occasionally indicated by a gauge point. g is used frequently by engineers in problems concerning dynamics. A gauge point at 746 - the number of watts equivalent to one horse-power - is sometimes to be found.

The inclusion of many gauge points in a slide rule is to be deprecated. The only one we think deserves its place is p and possibly p/4.

If any number enters frequently into our calculations, it is fairly easy to add a gauge point to register its position. The mark should be scribed with a razor blade broken so as to provide a sharp corner, and a square should be used to ensure the line lies at right angles to the length of the rule. It is exasperating to add a gauge mark and then find its position is not quite correct, and we have found a safe method to adopt is first to paste a small piece of paper on the scale, and very lightly pencil the mark on the paper. The position of the mark should be very carefully checked, and, if necessary, corrected. The mark can now be cut through the paper into the scale, care being exercised to avoid cutting too deeply, the paper removed, and a trace of printer's ink rubbed into the cut impression, after which the scale may be polished. If neatly executed, a fine black line will result. Lines registering gauge points should stand slightly off the scales with which they are associated, in order to avoid confusion with the divisions of the scales.

The signs Quot./+1 and Prod./-1 which appear at the left and right-hand ends respectively of certain makes of rules, are of little consequence, and we would prefer not to mention them. They are designed to assist in ascertaining the numbers of digits in a product or quotient. We have, in Section 4, given rules for determination of the position of decimal points, based on the position of slide relative to the stock. We have shown that if when multiplying the slide is set so that it protrudes to the right of the stock, the number of digits in the product is one less than the sum of digits in the two factors. Another way of expressing the same rule is to say: If when multiplying, the result lies to the right of the first factor the digits in the product are one less than those of the two factors. The sign Prod./-1at the right-hand end of the stock is a reminder of the rule when expressed in this manner. The sign Quot./-1 similarly reminds us that the quotient will contain one more digit than the difference between the digits of the dividend and divisor if the result appears to the left of the dividend. When the result in a multiplication lies to the left of the first factor, the number of digits of the product is equal to the sum of the numbers of digits in the two factors, and in division the number of digits in the quotient is equal to the difference between the digits of dividend and divisor when the result is found on the right of the dividend.

© Hodder Stoughton, reproduced with permission.