Part 9

With the slide set as above, the 8th, 9th, etc., powers of 1·1 cannot be read off; but it is seen that, according to (2) in the foregoing, the missing portion of the E scale is that part of the upper scale (2 to about 2·6) which is outside the rule to the left. Hence placing 1·1 to 10 on D, the 8th, 9th, etc., powers of 1·1 will be read off _on the upper part_ of the E scale. In general, then,

If _x_ on the _lower_ line is set to 1 on D, then _x^n_ is read directly on that line and _x_^{10_n_} on the upper line.

If _x_ on the _upper_ line is set to 1 on D, then _x^n_ is read directly on that line and _x_^{_ⁿ⁄₁₀_} on the lower line.

If _x_ on the _lower_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read directly on that line and _x^n_ on the upper line.

If _x_ on the _upper_ line is set to 10 on D, then _x_^{_ⁿ⁄₁₀_} is read directly on that line and _x_^{_ⁿ⁄₁₀₀_} on the lower line.

These rules are conveniently exhibited in the accompanying diagram (Fig. 14). They are equally applicable to both the E and -E scales of the 10 in. rule, and include practically all the instruction required for determining the _n_th power or the _n_th root of a number. They do not apply directly to the 20 in. rule, however, for here the relation of the lower and upper scales will be _x^n_ and _x_^{100_n_}.

EX.—Find 1·167^{2·56}.

Set 1·167 on E to 1 on D, and over 2·56 on D read 1·485 on E.

EX.—Find 4·6^{1·61}.

Set 4·6 on upper E scale to 1 on D, and over 1·61 on D read 11·7 (11·67) on E.

EX.—Find 1·4^{0·27} and 1·4^{2·7}.

Set 1·4 on E to 10 on D, and over 2·7 on D read 1·095 = 1·4^{0·27} on lower E scale and 2·48 = 1·4^{2·7} on upper E scale.

EX.—Find 46^{0·0184} and 46^{0·184}.

Set 46 on upper E scale to 10 on D, and over 1·84 on D read 1·073 on lower E scale and 2·022 (2·0228) on upper E scale.

EX.—Find 0·074^{1·15}.

Using the -E scale, set 0·074 to 1 on D, and over 1·15 on D read 0·05 on -E.

The method of determining the root of a number will be obvious from the preceding examples.

EX.—Find ^{1.4}√(17) and ^{14}√(17).

Set 17 on E to 1·4 on D, and over 1 on D read 7·56 on upper E scale and 1·224 on lower E scale.

EX.—Find ^{0·031}√(0·914).

Set 0·914 on -E to 3·1 on D, and over 10 on D read 0·055 on upper -E scale.

When the exponent _n_ is fractional, it is often possible to obtain the result directly with one setting of the slide. Thus to determine 1·135^{¹⁷⁄₁₆} by the first method we find ¹⁷⁄₁₆ = 1·0625, and placing 1·135 on E to 1 on D, read 1·144 on E over 1·0625 on D. By the direct method we place 1·135 on the E scale on 1·6 on D, and over 1·7 on D read 1·144 on E. It will be seen that since the scale D is assumed to run from 1 to 10 we are unable to read 16 and 17 on this scale; but it is obvious that the _ratios_ (1·7)/(1·6) and (17)/(16) are identical, and it is with the ratio only that we are, in effect, concerned.

Since an expression of the form _x_^{-_n_} = (1)/(_x^n_) or ((1)/(_x_))^{_n_}, the required value may be obtained by first determining the reciprocal of _x_ and proceeding as before. By using both the direct and reciprocal log.-log. scales (E and -E) in conjunction however, the required value can be read directly from the rule, and the preliminary calculation entirely avoided. In the Davis form of rule, the result can be read on the -E scale, used in conjunction with the D scale of the rule, _x_ on E being set to the index mark in the aperture in the back of the rule.

EX.—Find the value of 1·195^{−1·65}.

Set 1·195 on E to the index in the left aperture in the back of the rule, and over 1·65 on D read 0·745 on the -E scale.

It may be noted in passing that the log.-log. scale affords a simple means for determining the logarithm or anti-logarithm of a number to any base. For this purpose it is necessary to set the base of the given system on E to 1 on D, when _under_ any number on E will be found its logarithm on D. Thus, for common logs., we set the base 10 on E to 1 on D, and under 100 we find 2, the required log. Similarly we read log. 20 = 1·301; log. 55 = 1·74; log. 550 = 2·74, etc. Reading reversely, over 1·38 on D we find its antilog. 24 on E; also antilog. 1·58 = 38; antilog. 1·19 = 15·5, etc.

For logs. of numbers under 10 we set the base 10 to 10 on D; hence the readings on D will be read as one-tenth their apparent value. Thus log. 3 = 0·477; log. 5·25 = 0·72; antilog. 0·415 = 2·6; antilog. 0·525 = 3.·35, etc.

The logs. of the numbers on the lower half of the E scale will also be found on the D scale; but a consideration of Fig. 14 will show that this will be read as _one-tenth_ its face value if the base is set to 1 on D, and as _one-hundredth_ if the base is set to 10.

For natural, hyperbolic, or Napierian logarithms, the base is 2·718. A special line marked ε or _e_ serves to locate the exact position of this value on the E scale, and placing this to 1 on D we read log._{_e_} 4·35 = 1·47; log._{_e_} 7·4 = 2·0; antilog._{_e_} × 2·89 = 18, etc. The other parts of the scale are read as already described for common logs. Calculations involving powers of _e_ are frequently met with, and these are facilitated by using the special graduation line referred to, as will be readily understood.

If it is required to determine the power or root of a number which does not appear on either of the log.-log. scales, we may break up the number into factors. Usually it is convenient to make one of the factors a power of 10.

EX.—3950^{1·97} = 3·95^{1·97} × 10^{3 × 1·97} = 3·95^{1·97} × 10^{5·91}.

Then 3·95^{1·97} = 15, and 10^{5·91} (or antilog.) 5·91 = 812,000. Hence, 15 × 812,000 = 12,180,000 is the result sought.

Numbers which are to be found in the higher part of the log.-log. scale may often be factorised in this way, and greater accuracy obtained than by direct reading.

The form of log.-log. rule which has been mainly dealt with in the foregoing gives a scale of comparatively long range, and the only objection to the arrangement adopted is the use of a separate slide.

_The Jackson-Davis Double Slide Rule._—In this instrument a pair of aluminium clips enable the log.-log. slide to be temporarily attached to the lower edge of the ordinary rule, and used, by means of a special cursor, in conjunction with the C scale of the ordinary slide. In this way both the log.-log. and ordinary scales are available without the trouble of replacing one slide by the other. Since the scale of exponents is now on the slide, the value of _x^n_ will be obtained by setting 1 on C to _x_ on E and reading the result on E under _n_ on C.

By using a pair of log.-log. slides, one in the rule and one clamped to the edge by the clips, we have an arrangement which is very useful in deducing empirical formulæ of the type _y_ = _x^n_.

_The Yokota Slide Rule._—In this instrument the log.-log. scales are placed on the face of the rule, each set comprising three lines. These, for numbers greater than 1, are found above the A scale while the three reciprocal log.-log. lines are below the D scale. Both sets are used in conjunction with the C scale on the slide. Other features of this rule are:—The ordinary scales are 10 in. long instead of 25 cm. as hitherto usual; hence the logarithms of numbers can be read on the ordinary scale of inches on the edge of the rule. There is a scale of cubes in the centre of the slide and on the back of the slide there is a scale of secants in addition to the sine and tangent scales.

_The Faber Log.-log. Rule._—In this instrument shown in Fig. 15, the two log.-log. scales are placed on the face of the rule. One section, extending from 1·1 to 2·9, is placed above the A scale, and the other section, extending from 2·9 to 100,000, is placed below the D scale. These scales are used in conjunction with the C scale of the slide in the manner previously described. The width of the rule is increased slightly, but the arrangement is more convenient than that formerly employed, wherein the log.-log. scales were placed on the bevelled edge of the rule and read by a tongue projecting from the cursor.

Another novel feature of this rule is the provision of two special scales at the bottom of the groove, to which a bevelled metal index or marker on the left end of the slide can be set. The upper of these scales is for determining the efficiency of dynamos and electric motors; the lower for determining the loss of potential in an electric circuit.

_The Perry Log.-log. Rule._—In this rule, introduced by Messrs. A. G. Thornton, Limited, Manchester, the log.-log. scales are arranged as in Fig. 16, the E scale, running from 1·1 to 10,000, being placed above the A scale of the rule, and the -E or E^{−1} scale running from 0·93 to 0·0001, below the D scale of the rule. These scales are read in conjunction with the B scales on the slide by the aid of the cursor.

The following tabular statement embodies all the instructions required for using this form of log.-log. slide rule:—

When _x_ is greater than 1.

_x^n_ Set 1 on B to _x_ on E; over _n_ on B read _x^n_ on E _x_^{-_n_} Set 1 on B to _x_ on E; under _n_ on B read _x_^{-_n_} on E^{−1} _x_^{_ⁱ⁄ₙ_} Set _n_ on B to _x_ on E; over 1 on B read _x_^{_ⁱ⁄ₙ_} on E _x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E; under 1 on B read _x_^{_⁻ⁱ⁄ₙ_} on E^{−1}

When _x_ is less than 1.

_x^n_ Set 1 on B to _x_ on E^{−1}; under _n_ on B read _x^n_ on E^{−1} _x_^{-_n_} Set 1 on B to _x_ on E^{−1}; over _n_ on B read _x_^{-_n_} on E _x_^{_ⁱ⁄ₙ_} Set _n_ on B to _x_ on E^{−1}; under 1 on B read _x_^{_ⁱ⁄ₙ_} on E^{−1} _x_^{_⁻ⁱ⁄ₙ_} Set _n_ on B to _x_ on E^{−1}; over 1 on B read _x_^{_⁻ⁱ⁄ₙ_} on E

If 10 on B is used in place of 1 on B, read _x_^{_ⁿ⁄₁₀_} in place of _x^n_ on E, and _x_^{-_ⁿ⁄₁₀_} in place of _x_^{-_n_} on E^{−1}. If 100 on B is used, these readings are to be taken as _x_^{_ⁿ⁄₁₀₀_} and _x_^{-_ⁿ⁄₁₀₀_} respectively.

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 _n_th 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^2_n_. 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.

FULLER’S CALCULATING RULE.—This instrument, which is shown in Fig. 18, consists of a cylinder _d_ capable of being moved up and down and around the cylindrical stock _f_, which is held by the handle. The logarithmic scale-line is arranged in the form of a helix upon the surface of the cylinder _d_, and as it is equivalent to a straight scale of 500 inches, or 41 ft. 8 in., it is possible to obtain four, and frequently five, figures in a result.

Upon reference to the figure it will be seen that three indices are employed. Of these, that lettered _b_ is fixed to the handle; while two others, _c_ and _a_ (whose distance apart is equal to the axial length of the complete helix), are fixed to the innermost cylinder _g_. This latter cylinder slides telescopically in the stock _f_, enabling the indices to be placed in any required position relatively to _d_. Two other scales are provided, one (_m_) at the upper end of the cylinder _d_, and the other (_n_) on the movable index.

In using the instrument a given number on _d_ is set to the fixed index _b_, and either _a_ or _c_ is brought to another number on the scale. This establishes a ratio, and if the cylinder is now moved so as to bring any number to _b_, the fourth term of the proportion will be found under _a_ or _c_. Of course, in multiplication, one factor is brought to _b_, and _a_ or _c_ brought to 100. The other factor is then brought to _a_ or _c_, and the result read off under _b_. Problems involving continuous multiplication, or combined multiplication and division, are very readily dealt with. Thus, calling the fixed index F, the upper movable index A, and the lower movable index B, we have for _a_ × _b_ × _c_:—Bring _a_ to F; A to 100; _b_ to A or B; A to 100; _c_ to A or B and read the product at F.

The maximum number of figures in a product is the sum of the number of figures in the factors and this results when all the factors except the first have to be brought to B. Each time a factor is brought to A, 1 is to be deducted from that sum.

For division, as _a_/(_m_ × _n_), bring _a_ to F; A or B to _m_; 100 to A; A or B to _a_; 100 to A and read the quotient at F.