Part 32

FARMAR’S PROFIT-CALCULATING RULE.—The application of the slide rule to commercial calculations has been often attempted, but the degree of accuracy required necessitates the use of a long scale, and generally this results in a cumbersome instrument. In Farmar’s Profit-calculating Rule the money scale is arranged in ten sections, these being mounted in parallel form on a roller which takes the place of the upper scale of an ordinary rule. The roller, which is ¾in. in diameter, is carried in brackets secured to each end of the stock, so that by rotating the roller any section of the money scale can be brought into reading with the scale on the upper edge of the slide and with which the roller is in contact. This scale gives percentages, and enables calculations to be made showing profit on turnover, profit on cost, and discount. The lower scale on the slide, and that on the stock adjacent to it, are similar to the A and B scales of an ordinary rule. The instrument is supplied by Messrs. J. Casartelli & Son, Manchester.

CONSTRUCTIONAL IMPROVEMENTS IN SLIDE RULES.

The attention of instrument makers is now being given to the devising of means for ensuring the smooth and even working of the slide in the stock of the rule. In some cases very good results are obtained by slitting the back of the stock to give more elasticity.

In the rules made by Messrs. John Davis & Son, a metal strip, slightly curved in cross section as shown at A (Fig. 30), runs for the full length of the stock to which it is fastened at intervals. Near each end of the rule, openings about 1 in. long are made in the metal backing through which the scales on the back of the slide can be read. To prevent warping under varying climatic conditions both the stock of the rule and the slide are of composite construction. The base of the stock is of mahogany, while the grooved sides, firmly secured to the base, are of boxwood. Similarly the centre portion of the slide is of mahogany and the tongued sides of boxwood. Celluloid also enters into the construction, a strip of this material being laid along the bottom of the groove in the stock. A fine groove runs along the centre of this strip in order to give elasticity and to allow the sides of the stock to be pressed together slightly to adjust the fitting of the slide. As a further means of adjustment the makers fit metal clips at each end of the rule, so that by tightening two small screws the stock can be closed on the slide when necessary.

In the rule made by the Keuffel and Esser Company of New York, one strip is made adjustable (Fig. 32).

THE ACCURACY OF SLIDE RULE RESULTS.

The degree of accuracy obtainable with the slide rule depends primarily upon the length of the scale employed, but the accuracy of the graduations, the eyesight of the operator, and, in particular, his ability to estimate interpolated values, are all factors which affect the result. Using the lower scales and working carefully the error should not greatly exceed 0·15 per cent. with short calculations. With successive settings, the discrepancy need not necessarily be greater, as the errors may be neutralised; but with rapid working the percentage error may be doubled. However, much depends upon the graduation of the scales. Rules in which one or more of the indices have been thickened to conceal some slight inaccuracy should be avoided. The line on the cursor should be sharp and fine and both slide and cursor should move smoothly or good work cannot be done. Occasionally a little vaseline or clean tallow should be applied to the edges of the slide and cursor.

That the percentage error is constant throughout the scale is seen by setting 1 on C to 1·01 on D, when under 2 is 2·02; under 3, 3·03; under 5, 5·05, etc., the several readings showing a uniform error of 1 per cent.

A method of obtaining a closer reading of a first setting or of a result on D has been suggested to the author by Mr. M. Ainslie, B.Sc. If any graduation, as 4 on C, is set to 3 on D, it is seen that 4 main divisions on C (40–44) are equal in scale length to 3 main divisions on D (30–33). Hence, very approximately, 1 division on C is equal to 0·75 of a division on D, this ratio being shown, of course, on D under 10 on C. Suppose √(4·3) to be required. Setting the cursor to 4·3 on A, it is seen that the root is something more than 2·06. Move the slide until a main division is found on C, which exactly corresponds to the interval between 2 and the cursor line, on D. The division 27–28 just fits, giving a reading under 10 on C, of 74. Hence the root is read as 2·074. For the higher parts of the scale, the subdivisions, 1–1·1, etc., are used in place of main divisions. The method is probably more interesting than useful, since in most operations the inaccuracies introduced in making settings will impose a limit on the reliable figures of the result.

For the majority of engineering calculations, the slide rule will give an accuracy consistent with the accuracy of the data usually available. For some purposes, however, logarithmic section paper (the use of which the author has advocated for the last twenty years) will be found especially useful, more particularly in calculations involving exponential formulæ.

APPENDIX.

NEW SLIDE RULES—FIFTH ROOTS, ETC.—THE SOLUTION OF ALGEBRAIC EQUATIONS—GAUGE POINTS AND SIGNS ON SLIDE RULES—TABLES AND DATA—SLIDE RULE DATA SLIPS.

THE PICKWORTH SLIDE RULE.—In this rule, made by Mr. A. W. Faber, the novel feature is the provision of a scale of cubes (F) in the stock or body of the rule. From Fig. 33 it will be seen that the scale is fixed on the bevelled side of a slotted recess in the back of the rule. The slide carries an index mark, which is seen through the slot and can be set to any graduation of the scale; in its normal position it agrees with 1 on the scale. The C scale on the face of the rule is divided into three equal parts by two special division lines, marked II. and III., which, together with the initial graduation 1 of the scale, serve for setting or reading off values on the D scale. Similar division lines are marked on the D scale.

In using the rule for cubes or cube roots the slide is drawn to the right, this movement never exceeding one-third of the length of the D scale. With this limited movement, and with a single setting of the slide, the values of ∛̅a, ∛(a × 10), and ∛(a × 100)) (a being less than 10 and not less than 1) are given simultaneously and without any uncertainty as to the scales to use or the values to be read off.

To Find the Cube of a Number.—The marks II. and III. on D divide that scale into three equal sections. If the number to be cubed is in the first section, I. on C is set to it; if in the second section, II. on C is set to it; if in the third section, III. on C is set to it. Then, under the index mark on the back of the slide will be found the significant figures of the cube on the scale F. If I. on C was used for the setting, the cube contains 1 digit; if II. was used, 2 digits; if III. was used, 3 digits. If the first figure of the number to be cubed is not in the units place, the decimal point is moved through n places so as to bring the first significant figure into the units place, the cube found as above, and the decimal point moved in the reverse direction through 3n places.

To Find the Cube Root of a Number.—The index mark is set to the significant figures of the number on scale F, and the cube root is read on D under I., II. or III. on C, according as the number has 1, 2 or 3 digits preceding the decimal point. Numbers which have 1, 2 or 3 figures preceding the decimal point are dealt with directly. Numbers of any other form are brought to one of the above forms by moving the decimal point 3 places (or such multiple of 3 places as may be required), the root found and its decimal point moved 1 place for each 3-place movement, but in the reverse direction.

THE “ELECTRO” SLIDE RULE.—In this special rule for electrical calculations, made by Mr. A. Nestler, the upper scales run from 0·1 to 1000, and are marked “Amp.” and “sq. mm.” respectively. The lower scale on the slide running from 1 to 10,000 is marked M (metres), while the lower scale on the rule (0·1 to 100) is marked “Volt.” The latter scale is so displaced that 10 on M agrees with 0·173 on the Volt scale. The four factors involved are the current strength (in Amp.); the area of a conductor (in sq. mm.); the length of the conductor (in metres); and the permissible loss of potential (in volts). Having given any three of these, the fourth can be found very readily. On the back of the slide are a scale of squares, a scale of cubes and a single scale corresponding to the D scale of an ordinary rule. Hence, by reversing the slide, it is possible to obtain the 2nd, 3rd and 4th powers and roots of numbers. In another form of the rule, the scale of metres is replaced by one of yards, while instead of the area of the conductor in sq. mm., the corresponding “gauge” sizes of wires are given.

THE “POLYPHASE” SLIDE RULE.—This instrument, made by the Keuffel & Esser Company, New York, has, in addition to the usual scales, a scale of cubes on the vertical edge of the stock of the rule, while in the centre of the slide there is a reversed C scale; i.e., a scale exactly similar to an ordinary C scale but with the graduations running from right to left. The rule is specially useful for the solution of problems containing combinations of three factors and problems involving squares, square roots, cubes, cube roots and many of the higher powers and roots. It is specially adapted for electrical and hydraulic work.

THE LOG-LOG DUPLEX SLIDE RULE.—The same makers have introduced a log-log duplex slide rule, in which the log-log scale is in three sections, placed one above the other, these occupying the position usually taken up by the A scale. These scales are used in the manner already described (page 86), but some advantage is obtained by the manner in which the complete log-log scale is divided, the limits being e^{¹⁄₁₀₀} to e^⅒ (on Scale L.L. 1); e^⅒ to e (on Scale L.L. 2); and e to e^{10} (on Scale L.L. 3), e being the base of natural or hyperbolic logarithms (2·71828). In this way a total log-log range of from 1·01 to 22,000 is provided, meeting all practical requirements. These log-log scales are read in conjunction with a C scale placed at the upper edge of the slide. A similar C scale, but reversed in direction, is placed at the lower edge of the slide, this having red figures to distinguish it readily. The adjacent scale on the body of the rule is an ordinary D scale, and under this is an equally-divided scale giving the common logarithms of values on D. In the centre of the slide is a scale of tangents.

It will be understood that a “duplex” rule consists of two side strips securely clamped together at the two ends, forming the body of the rule, the slide moving between them; hence both front and back faces of the rule and slide are available, graduations on the one side being referred to those on the other by the cursor which extends around the whole. In this instrument, the scales on the back face are the ordinary scales of the standard rule with the addition of a scale of sines which is placed in the centre of the slide. It will be evident that this instrument is capable of dealing with a very wide range of problems involving exponential and trigonometrical formulæ.

SMALL SLIDE RULES WITH MAGNIFYING CURSORS.—Several makers now supply 5 in. rules having the full graduations of a 10 in. rule, and fitted with a magnifying cursor (Fig. 34). This forms a compact instrument for the pocket, but owing to the closeness of the graduations it is not usually possible to make a setting of the slide without using the cursor. This, of course, involves more movements than with the ordinary instrument. It is also very necessary to use the magnifying cursor in a direct light, if accurate readings are to be obtained. If these slight inconveniences are to be tolerated, the principle could be extended, a 10 in. rule being marked as fully as a 20 in., and fitted with a magnifying cursor. The author has endeavoured, but without success, to induce makers to introduce such a rule.

The magnifying cursor, supplied by Messrs. A. G. Thornton, Limited, has a lens which fills the entire cursor. It has a powerful magnifying effect, and the change from the natural to the magnified reading is less abrupt than with the semicircular lens.

THE CHEMIST’S SLIDE RULE.—A slide rule, specially adapted for chemical calculations, has been introduced recently by Mr. A. Nestler. In this instrument the C and D scales are as usually arranged; but, in place of the A and B scales, there are a number of gauge points or marks denoting the atomic and molecular weights of the most important elements and combinations. The scales on the back of the slide are similarly arranged, so that by reversing the slide the operations can be extended very considerably. The rule finds its chief use in the calculation of analyses. Thus, to find the percentage of chlorine if s grammes of a substance have been used and the precipitate of Ag.Cl. weighs a grammes, we have the equation, x = (Cl.)/(Ag.Cl.) × (a)/(s). Hence, the mark Ag.Cl. on the upper scale of the slide is set to the mark Cl. on the upper scale of the rule, when under a on the C scale is found the quantity of chlorine on D. By setting the cursor to this value and bringing s on C to the cursor, the percentage required can be read on C over 10 on D.

The rule is also adapted to the solution of various other chemical and electro-chemical calculations.

THE STELFOX SLIDE RULE.—This rule, shown in Fig. 35, has a stock 5 in. long, fitted with a 10 in. slide jointed in the middle of its length by means of long dowels. By separating the parts the compactness of a 5 in. rule is obtained. The upper scales on the rule and slide resemble the usual A and B scales. The D scale on the lower part of the stock is in two sections, the second portion being placed below the first, as shown in the illustration. The centre scale on the slide corresponds to the usual C scale, while on the lower edge of the slide is a similar scale, but with the index (1) in the middle of its length. The arrangement avoids the necessity of resetting the slide, as is sometimes necessary with the ordinary rule, and in general it combines the accuracy of a 10 in. rule with the compactness of a 5 in. rule; but a more frequent use of the cursor is necessary. This rule is made by Messrs. John Davis & Son, Limited, Derby.

ELECTRICAL SLIDE RULE.—Another rule by the same makers, specially useful for electrical engineers, has the usual scales on the working edges of the rule and slide, while in the middle of the slide is placed a scale of cubes. A log-log scale in two sections is provided; the power portion, running from 1·07 to 2, is found on the lower part of the stock, and the upper portion, running from 2 to 10^3, on the upper part of the stock. The uppermost scale on the stock is in two parts, of which that to the left, running from 20 to 100 and marked “Dynamo,” gives the efficiencies of dynamos; that on the right, running from 20 to 100 and marked “Motor,” gives the efficiencies of electric motors. The lowest scale on the stock, marked “Volt,” gives the loss of potential in copper conductors. The ordinary upper scale on the stock is marked L (length of lead) at the left, and KW (kilowatts) at the right; the ordinary upper scale on the slide is marked A (ampères) and mm^2 (sectional area) at the left, and HP (horse-power) at the right. Additional lines on the cursor enable the electrical calculations to be made either in British or metric units.

THE PICOLET CIRCULAR SLIDE RULE.—A simple form of circular calculator, made by Mr. L. E. Picolet of Philadelphia, is shown in Fig. 36. It consists of a base disc of stout celluloid on which turns a smaller disc of thin celluloid. A cursor formed of transparent celluloid is folded over the discs, and is attached so that the friction between the cursor and the inner disc enables the latter to be turned by moving the former. By holding both discs the cursor can be adjusted as required. The adjacent scales run in opposite directions, so that multiplication and division are performed as with the inverted slide in an ordinary rule. The outer scale, which is two-thirds the length of the main scale, enables cube roots to be found. Square roots are readily determined and continuous multiplication and division conveniently effected. Modified forms of this neatly made little instrument are also available.

OTHER RECENT SLIDE RULES.—Among other special types of slide rule, mention should be made of the Jakin 10 in. rule for surveyors, made by Messrs. John Davis & Son, Limited, Derby. By the provision of a series of short subsidiary scales, the multiplication of a sine or tangent of an angle by a number can be obtained to an accuracy of 1 in 10,000. The Davis-Lee-Bottomley slide rule, by the same makers, has special scales provided for circle spacing. The division of a circle into a number of equal parts, often required in spacing rivets, bolts, etc., and in setting out the teeth of gearwheels, is readily effected by the aid of this instrument. The Cuntz slide rule is a very comprehensive instrument, having a stock about 2¼ in. wide, with the slide near the lower edge. Above the slide are eleven scales, referable to the main scales by the cursor. These scales enable squares and square roots, cubes and cube roots, and areas and circumferences of circles to be obtained by direct reading. A much more compact instrument could be obtained by removing one-half the scales to the back of the rule and using a double cursor.