Part 29

In rules with no -E scale the value of x^{-n} is obtained by the usual rules for reciprocals. We may either determine x^n and find its reciprocal or, first find the reciprocal of x and raise it to the nth power. The first method should be followed when the number x is found on the E scale.

EX.—3·45^{−1·82} = 0·105.

Set 1 on C to 3·45 on E, and under 1·82 on C read 9·51 on C. Then set 1 on B to 9·5 on A, and under index of A read 0·105 on B.

When x is less than 1 the second method is more suitable.

EX.—0·23^{−1·77} = ((1)/(0·23))^{1·77} = 4·35^{1·77} = 13·5

Set 1 on B to 0·23 on A, and under index of A read (1)/(0·23) = 4·35 on B.

Set 1 on C to 4·35 on E, and under 1·77 on C read 13·5 on E.

As with the Davis rule, the exponent scale C will be read as ⅒th its face value if its R.H. index (10) is used in place of 1.

SPECIAL TYPES OF SLIDE RULES.

In addition, to the new forms of log.-log. slide rules previously described, several other arrangements have been recently introduced, notably a series by Mr. A. Nestler, of Lahr (London: A. Fastlinger, Snow Hill). These comprise the “Rietz,” the “Precision,” the “Universal,” and the “Fix” slide rules.

THE RIETZ RULE.—In this rule the usual scales A, B, C, and D, are provided, while at the upper edge is a scale, which, being three times the range of the D scale, enables cubes and cube roots to be directly evaluated and also n^{³⁄₂} and n^⅔.

A scale at the lower edge of the rule gives the mantissa of the logarithms of the numbers on D.

THE PRECISION SLIDE RULE.—In this rule the scales are so arranged that the accuracy of a 20 in. rule is obtainable in a length of 10 in. This is effected by dividing a 20 in. (50 cm.) scale length into two parts and placing these on the working edges of the rule and slide. On the upper and lower margins of the face of the rule are the two parts of what corresponds to the A scale in the ordinary rule; while in the centre of the slide is the scale of logarithms which, used in conjunction with the 50 cm. scales on the slide, is virtually twice the length of that ordinarily obtainable in a 10 in. rule. The same remark applies to the trigonometrical scales on the under face of the slide. Both the sine and tangent scales are in two adjacent lengths, while on the edge of the stock of the rule, below the cursor groove, is a scale of sines of small angles from 1° 49′ to 5° 44′. This is referred to the 50 cm. scales by an index projection on the cursor.

If C and C′ are the two parts of the scale on the slide and D and D′ the corresponding scales on the rule, it is clear that in multiplying two factors 1 on C can only be set directly to the upper scale D; while 10 on C′ can only be set directly to the lower scale D′. Hence if the first factor is greater than about 3·2, the cursor must be used to bring 1 on C to the first factor on D′. Similarly, in division, numerators and denominators which occur on C and D′ or on C′ and D cannot be placed in direct coincidence but must be set by the aid of the cursor.

Any uncertainty in reading the result can be avoided by observing the following rule: If in setting the index (1 or 10) in multiplication, or in setting the numerator to the denominator in division, it is necessary to cross the slide, then it will also be necessary to cross the slide to read the product or quotient.

THE UNIVERSAL SLIDE RULE.—In this instrument the stock carries two similar scales running from 1 to 10, to which the slide can be set. Above the upper one is the logarithm scale and under the lower one the scale of squares 1 to 100. On the edge of the stock of the rule, under the cursor groove, is a scale running from 1 to 1000. An index projecting from the cursor enables this scale to be used with the scales on the face of the rule, giving cubes, cube roots, etc.

On the slide, the lower scale is an ordinary scale, 1 to 10. The centre scale is the first part of a scale giving the values of sin n cos n, this scale being continued along the upper edge of the slide (marked “sin-cos”) up to the graduation 50. On the remainder of this line is a scale running from right to left (0 to 50) and giving the value of cos^2n. In surveying, these scales greatly facilitate the calculations for the horizontal distance between the observer’s station and any point, and the difference in height of these two points.

On the back of the slide are scales for the sines and tangents of angles. The values of the sines and tangents of angles from 34′ to 5° 44′ differ little from one another, and the one centre scale suffices for both functions of these small angles.

THE FIX SLIDE RULE.—This is a standard rule in all respects, except that the A scale is displaced by a distance (π)/(4) so that over 1 on D is found 0·7854 on A. This enables calculations relating to the area and cubic contents of cylinders to be determined very readily.

THE BEGHIN SLIDE RULE.—We have seen that a disadvantage attending the use of the ordinary C and D scales, is that it is occasionally necessary to traverse the slide through its own length in order to change the indices or to bring other parts of the slide into a readable position with regard to the stock. To obviate this disadvantage, Tserepachinsky devised an ingenious arrangement which has since been used in various rules, notably in the Beghin slide rule made by Messrs. Tavernier-Gravêt of Paris. In this rule the C and D scales are used as in the standard rule, but in place of the A and B scales, we have another pair of C and D scales, displaced by one-half the length of the rule. The lower pair of scales may therefore be regarded as running from 10^{n} to 10^{n + 1}, and the upper pair as running from √(10) × 10^{n} to √(10) × 10^{n + 1}. With this arrangement, without moving the slide more than half its length, to the left or right, it is always possible to compare all values between 1 and 10 on the two scales. This is a great advantage especially in continuous working.

Another commendable feature of the Beghin rule is the presence of a reversed C scale in the centre of the slide, thus enabling such calculations as a × b × c to be made with one setting of the slide. On the back of the slide are three scales, the lowest of which, used with the D scale, is a scale of squares (corresponding to the ordinary B scale), while on the upper edge is a scale of sines from 5° 44′ to 90°, and in the centre, a scale of tangents from 5° 43′ to 45°. On the square edge of the stock, under the cursor groove, is the logarithm scale, while on the same edge, above the cursor groove, are a series of gauge points. All these values are referred to the face scales by index marks on the cursor.

THE ANDERSON SLIDE RULE.—The principle of dividing a long scale into sections as in the Precision rule, has been extended in the Anderson slide rule made by Messrs. Casella & Co., London, and shown in Fig. 17. In this the slide carries a scale in four sections, used in conjunction with an exactly similar set of scale-lines in the upper part of the stock. On the lower part of the stock is a scale in eight sections giving the square roots of the upper values. In order to set the index of the slide to values in the stock, two indices of transparent celluloid are fixed to the slide extending over the face of the rule as shown in the illustration. As each scale section is 30 cm. in length, the upper lines correspond to a single scale of nearly 4 ft., and the lower set to one of nearly 8 ft. in length, giving a correspondingly large increase in the number of subdivisions of these scales, and consequently much greater accuracy.

In order to decide upon which line a result is to be found, sets of “line numbers” are marked at each end of the rule and slide and also on the metal frame of the cursor. In multiplication, the line number of the product is the sum of the line numbers of the factors if the left index is used, or 1 more than this sum if the right index is used. The illustration shows the multiplication of 2 by 4. The left index is set to 2 (line number, 1), and the cursor set to 4 on the slide (line number, 2); hence, as the left index is used, the result is found on line No. 3. Similar rules are readily established for division. The column of line numbers headed 0 is used for units, that headed 4 for tens, and so on; one column is given for tenths, headed −4. The square root scale bears similar line numbers, so that the square root of any value on the upper scales is found on the correspondingly figured line below.

THE MULTIPLEX SLIDE RULE differs from the ordinary form of rule in the arrangement of the B scale. The right-hand section of this scale runs from left to right as ordinarily arranged, but the left-hand section runs in the reverse direction, and so furnishes a reciprocal scale. At the bottom of the groove, under the slide, there is a scale running from 1 to 1000, which is used in conjunction with the D scale, readings being referred thereto by a metal index on the end of the slide. By this means cubes, cube roots, etc., can be read off directly. Messrs. Eugene Dietzgen & Co., New York, are the makers.

THE “LONG” SLIDE RULE has one scale in two sections along the upper and lower parts of the stock, as in the “Precision” rule. The scale on the slide is similarly divided, but the graduations run in the reverse direction, corresponding to an inverted slide. Hence the rules for multiplication and division are the reverse of those usually followed (page 30). On the back of the slide is a single scale 1–10, and a scale 1–1000, giving cubes of this single scale. By using the first in conjunction with the scales on the stock, squares may be read, while in conjunction with the cube scale, various expressions involving squares, cubes and their roots may be evaluated.

HALL’S NAUTICAL SLIDE RULE consists of two slides fitting in grooves in the stock, and provided with eight scales, two on each slide, and one on each edge of each groove. While fulfilling the purposes of an ordinary slide rule, it is of especial service to the practical navigator in connection with such problems as the “reduction of an ex-meridian sight” and the “correction of chronometer sights for error in latitude.” The rule, which has many other applications of a similar character, is made by Mr. J. H. Steward, Strand, London.

LONG-SCALE SLIDE RULES

It has been shown that the degree of accuracy attainable in slide-rule calculations depends upon the length of scale employed. Considerations of general convenience, however, render simple straight-scale rules of more than 20 in. in length inadmissible, so that inventors of long-scale slide rules, in order to obtain a high degree of precision, combined with convenience in operation, have been compelled to modify the arrangement of scales usually employed. The principal methods adopted may be classed under three varieties: (1) The use of a long scale in sectional lengths, as in Hannyngton’s Extended Slide Rule and Thacher’s Calculating Instrument; (2) the employment of a long scale laid in spiral form upon a disc, as in Fearnley’s Universal Calculator and Schuerman’s Calculating Instrument; and (3) the adoption of a long scale wound helically upon a cylinder, of which Fuller’s and the “R.H.S.” Calculating Rules are examples.