SECTION ELEVEN

THE DUALISTIC RULE

Duplicated C & D Scales - Special 20" Scales for ordinary calculations - Squares and Square Roots - Cubes and Cube Roots - Log Log Scales.

Duplicated C & D Scales

IN Section 10 we examined the Electrical rule, which includes as part of its scale equipment duplicated C and D scales. We attempted to show that a saving of time can be effected when this type of rule is used. We explained that the c and d scales are identical with the C and D scales, but that the former are positioned so that the p in c is immediately over the 1 and 10 of C and the p in d over the 1 and 10 of D. This arrangement of scales enables the user to multiply or divide by p by merely projecting by means of the cursor index X from D to d or vice versa. We pointed out that a special 2-line Cursor is necessary if full advantage is to be taken of the duplicated scales.

We propose now to study a type of rule which, although closely resembling the Electrical rule in respect of the duplicated scales, is different in some respects.

In this particular rule, shown in Fig. 15, the 1 of c is immediately above Ö10 in C, and similarly, the 1 of d is directly over Ö10 in D. Now Ö10 (which is exactly midway between 1 and 10 of any scale, as will be agreed if a little consideration is given to the logarithms of the numbers) has a value very near to 3·16 and thus is not far removed from p(= 3.14) so at first sight the two rules we are comparing may appear identical in the layout of their duplicated C and D scales.

The slight difference in the relative positions of the "folded" scales means that we cannot with the Dualistic rule multiply or divide by p by simple projection, but as a compensation, the 2-line cursor is not required for the comprehensive use of the Dualistic rule.

(The reader will readily appreciate that an additional broken line can easily be added to the cursor to provide the facility of multiplying and dividing by p, but we do not recommend such an addition. In any event, the point is of no great importance since the other features of the duplicated C and D scales are predominant.)

Apart from work of a specialised nature, probably 90% of the computations effected by slide rule involve the use of the C and D scales only. In the Dualistic rule these scales occupy their usual positions. They are designated by the symbols C₁ and D₁ and lie along the lower edge of the slide, and on the adjacent edge of the stock, respectively, as seen in Fig. 15

The upper margin of the slide, and the edge of the stock adjacent to it, are equipped with modified C and D scales. These are designated by C₂ and D₂ respectively, and are used in conjunction with the C₁ and D₁ scales, as described below.

The extreme margins of the stock and the centre of the slide are provided with a pair of 20" scales. These will be recognised as the principal scales of the 10/20 rule. They may be used separately; they are equivalent to a 20" rule and give the same high degree of accuracy. Scale references P₁, P₂, Q₁ and Q₂. These scales may be used in conjunction with the C₁ and D₁ scales, as will be demonstrated presently.

On the reverse of the slide three scales, LL₁, LL₂ and LL₃, will be found; these are three sections of a continuous log-log scale, extending from 1.01 to 40,000, and are used with the slide inverted in conjunction with the D₁ scale.

C₁ and D₁ Scales

As stated above, these are the C and D scales of the standard type slide rule. The instruction given in Section 4, dealing with the operations of multiplication and division, apply without modification, except that C and D should be read as C₁ and D₁ respectively. In all cases when a calculation involves the use of the C and D scales only, the standard slide rule, or the Dualistic rule, may be used without any discrimination.

C₂ and D₂ Scales

On inspection, it will at once be seen that these two scales are divided in the same manner as the C₁ and D₁ scales, but they are placed differently on the rule. The 1 of the scales C₂ and D₂ is in the middle of the length of the rule. The scales commence at p at the left-hand end of the rule; the readings increase, reaching 10 (or 1) at the mid-point, and then increase, reaching p at the right-hand extremity of the rule.

Scales C₂ and D₂ should not be used alone for multiplication and division. A few simple examples will at once show that, although multiplication or division may be effected with their aid, frequently the result is off the scale at the first setting, and cannot be obtained without traversing the slide through its own length. The same dilemma sometimes arises with the C₁ and D₁ scales when multiplying, but the traversing of the slide is rather more easily effected. In any case, there is no advantage gained by using C₂ and D₂ in preference to C₁ and D₁, and since all slide rule users are familiar with the C₁ and D₁ scales, it is advisable to adhere to them. To illustrate this point the reader is asked to compute 6 x 4 using C₂ and D₂ scales. On setting 1 C₂ to 4 D₂ it will be found that the 6 C₂ lies beyond the D₂ scale at the left-hand end. The result may be obtained by traversing the slide. Set the cursor index X to the p near the right-hand end of C₂, and move the slide to bring the p at the left-hand end of C₂ under X. Immediately above 6 C₂ will be found the result, 24, in D₂.

Scales C₁, D₁, C₂ and D₂ used in Conjunction

Computations involving multiplication and/or division are more rapidly effected when using these four scales than when C₁ and D₁ only are employed. The following example is designed to illustrate this feature:

Compute the value of (3·1 x 6·4 x 9·2) / (l·5 x 11·2)

Using C₁ and D₁ only, and performing division and multiplication alternately since this saves time, the following operations are required:
    To 31D₁ set 15C₁.
    Set X to 1C₁. 10C₁ to X. X to 64C₁. 112C₁ to X. X to 1C₁.
   10C₁ to X.
    Result: 109 in D₁ immediately under 92C₁.
Approximation, performed mentally, shows the answer is of the order 10, and the result therefore is 10·9.

Using the four scales:
    To 31D₁ set 15C₁.
    Set X to 64C₂. 112C₂ to X.
Result: 109 in D₂ above 92C₂.

If the rule is used to carry out these two series of operations, it will be found that by using all four scales the number of movements of slide and cursor is greatly reduced and the actual distances through which the slide and cursor are moved in these various operations are very much smaller. Greater accuracy will be attained, because in the course of time all slide rules develop small errors in their scales due to shrinkage or other distortion, and scales which originally were identical differ slightly in length. Critical inspection will almost invariably show that in slide rules which have been in use for some time, the overall lengths of scales on slide and stock differ slightly. With such a rule, imagine multiplication of 12 x 4 is being effected using C₁ and D₁ scales. Set 1C₁ to 12D₁ and the result, 48, appears in D₁ under 4C₁. If the slide scale has, through shrinkage, become, say, slightly shorter than the stock scale, a small error will be seen, the 4C₁ falling just below the 48D₁. Using the Dualistic rule, set 1C₁ to 12D₁, and read the result, 48, in D₂ over 4C₂. With this setting the length of slide scale used is only about 1", and the error will be only about one-sixth of that involved in using the C₁ and D₁ scales, where the length of slide employed is about 6". The same argument applies to any series of operations.

The principle involved in using the four scales of the Dualistic rule is the same as that employed in slide rules generally. Multiplication and division are effected by adding or subtracting logarithms, but with two sets of scales available there are alternative scale readings provided, and the manipulation of the rule is easier and speedier than in the case of the standard slide rule.

In using the Dualistic rule the first factor is selected in the D₁ or D₂ scales. The choice of scales is unrestricted, but it is an advantage to start with that scale in which the first factor lies near the middle of the length of the rule. If the first factor lies between 2 and 6, use the D₁ scale, but if it lies between 6 and 2, start in the D₂ scale. For the factors used subsequently there are alternative scale readings, and the one lying nearest should be used. An example will make this selection of factors clear.

Example: Evaluate 3 x 1·2 x ⁹/4 x 2.5.
    Set 1C₂ to 3D₁ (using X). X to 12C₂. 4C₁ to X. X to 9C₂. 1C₂ to X. X to 25C₁.
Result in D₁ under X is 20·2.

It will at once be noticed that the movements of slide and cursor are small compared with those necessary if the C₁ and D₁ scales are used alone.

Whether the final result appears in D₁ or D₂ depends upon which scales, C₁ or C₂, were used for the intermediate factors; the determination presents no difficulty. Very often, especially in short computations, the order of the result is already known, and the slide rule is used to obtain an accurate figure. In such cases it is only necessary to glance at the results lying in D₁ and D₂ under X. These values differ in the ratio of Ö10 to 1, i.e. about 3·16 to 1. In such a case the appropriate reading will be obvious.

In longer computations, when the result is not obvious, a rough approximation should be made to determine the position of the decimal point. This approximation will also disclose in which of the two scales D₁ or D₂ the result lies.

The following method for determining in which scale the result lies may be preferred, and the reader is advised to spend a few minutes making himself familiar with it, since it applies also to the 20" scales which will be dealt with later. The method may, at first reading, sound complicated. It is, in fact, very easy of application and has earlier been explained in Section 9, but we think some repetition here may be desirable. We have in mind also the fact that an experienced slide rule user may be reading this section without having perused the earlier notes.

Every multiplication, or division, or combined multiplication and division, involves using two factors in the C scale. In a multiplication the 1 (or 10) C is set to some value in D, and the result found in D opposite the multiplying factor in C. In division the divisor in C is first set, and the result read opposite the 1 (or 10) C, and in multiplication/division, the divisor in C is set and the quotient obtained opposite another factor in C. In applying the method - which we believe to be original -it is only necessary to observe whether the two factors are both in the same C scale or whether one is in C₁ and the other in C₂. If the two factors are selected in different sections of the C scale, the result is obtained by crossing from D₁ to D₂, or vice versa; if both factors lie in the same part of the C scale, the result will be found in that part of the D scale in which the number being multiplied or divided appeared.

A simple example may assist. Suppose multiplication of 8 x 3 is desired. There are six different ways of obtaining the result, 24, they are:

the cursor index X being used in setting where necessary.

In the first and third settings of the slide the two factors 1 and 3 lie in the same section of the C scale, namely, both in C₂ in the first, and both in C₁ in the third method. In both the result lies in the section of the D scale in which the factor 8 was chosen. In the other four methods the factors 1 and 3 lie in opposite sections of the C scale, and the result is always in the opposite section of the D scale from that in which the first factor 8 was selected.

In the simple example just cited it is easy to determine in which part of the D scale the result will be found, but in a longer one it is advisable to record the various operations as now suggested. When the first slide setting is made, note which section of the D scale is used and jot down D₁ or D₂ as the case may be. If in the next operation the two factors used are in the same section of the C scale, take no further notice of them, but if they are in different sections of the C scale write a stroke thus, /, following the D₁ or D₂. Proceed in this way, making a stroke each time scale C₁ and C₂ are both used in any one setting of the slide, the second stroke cancelling the first by changing it into a x, so the record starting with, say D₁, would next become D₁/, and then D₁ x. At the end of the computation the record will finish either with D₁, or D₁/, or D₁ x. If the last symbol is a stroke, the final result will lie in the D₂ scale; in other cases it will lie in D₁.

Example: Evaluate (2·8 x 93 x 107 x 46) / (18 x 52 x 29).

Under X read 472 in D₁, and 1495 in D₂.

The symbols when written down in line result in D₁ x ; the indication is that the result is in D₁. Approximation gives 46 and the result is 47·2.

The 20" Scales

The scales lying along the top and bottom edges of the face of the stock designated by the symbols P₁ and P₂ respectively, together form a 20" logarithmic scale, and in combination with a similar pair of scales placed in the middle of the slide and designated by Q₁ and Q₂ form the equivalent of a 20" slide rule.

When a higher degree of accuracy than can be derived from the 10" C and D scales is desired, the P and Q scales should be used. Inspection of the illustration will show the additional dividing which has been made possible by the use of these long scales.

Multiplication and division are effected by using the P and Q scales and the cursor index X. The method given earlier for determining whether the final result should be read in D₁ or D₂ may be adopted when there is any doubt as to whether the result appears in P₁ or P₂. This method has already been dealt with fully and need not be repeated. Two examples are now given to illustrate the use of the 10/20 scales.

Example: Evaluate (13·65 x 23·4) / 39·6 .
    Set X to 1365P₁. 396Q₂ to X. X to 234Q₁.
Result is 807 in P₂.

The value in P₁ under X is 255 and it is obvious that this result is incorrect. 396 appeared in Q₂ and 234 in Q₁, therefore the result must be in P₂, since the first factor, 1365, is in P₁. It is quite unnecessary to write down the symbols, but if, for illustration only, we do so, they will be P₁ /. The stroke at the end indicates the final result is in the opposite scale to that in which the first factor, 1365, was found: 13 into 39 is 3, and 3 into 24 gives 8 as an approximate result. Now, with two values under X, 807 and 255, there is no difficulty in selecting the correct one and at the same time inserting the decimal point. Result: 8·07.

A longer example is now given:

Evaluate (4·4 x 69·2 x 24·6 x 1·246 x 36) / (15·1 x 82.2 x 18·6 x 28·1).

The symbols written in line should appear P₂ x x, showing the result is in P₂. It is 519.

Approximation. - 4·4 into l5 is slightly over 3, which divides into 69 about 20: 20 into 82, say, 4, 4 into 24 gives 6, 6 into 18 is 3, and 3 into 36 gives 12: 12 times 1·2 is 14 approximately and we are left with 14/28 = ·5 as the approximate result. The actual result is, therefore, ·519; it lies in P₂, as indicated by the symbols.

The procedure explained in Section 4 for determination of position of decimal point, may be used if desired, but we strongly recommend the approximation method as being easier and safer. In a long computation there is a risk that a factor may be inadvertently omitted in the slide rule manipulation. The approximation if carefully made will disclose the error - another sound reason for making it.

The reader is now advised to practise the use of this new rule by working through a few simple examples, the results of which may easily be checked. It is confidently predicted that when familiarity with the scales is attained the rule will make an appeal as being superior to the standard type. The difficulties - if there are  - have now been dealt with and the remaining instruction deals with simple points.

Squares and Square Roots

The relative positions of the 10" and 20" scales give a ready means of evaluating squares and square roots. The square of any number is obtained by projecting by means of X, the number from either Q₁ or Q₂ into C₁. For example, 1·6, when projected from Q₁ into C₁, gives 2·56. When projecting from Q₂ to C₁, the squares are 10 times the actual values of the numbers engraved along the C₁ scale. 5 in Q₂ lies immediately above 2·5 in C₁, and this value must be read as 25. Readers now familiar with the A, B, C and D scales of a standard rule will notice the similarity in procedure. They will also notice the higher degree of accuracy possible with the longer scales.

Square roots are obtained by the reverse process of projecting from C₁ into Q₁ or Q₂. Square roots of numbers from 1 to 10 are obtained by projection from C₁ into Q₁, and square roots of numbers from 10 to 100 by projection from C₁ into Q₁. When a number whose square root is desired lies outside the range 1 to 100, the procedure outlined in Section 5 should be used, reading Q₁ for the left-hand half of scale A, and Q₂ for the right-hand half.

Scales P₁, P₂ and D₁ may be used for squares and square roots if preferred; the procedure will be obvious from the instructions given above.

Cube and Cube Roots

Cubes are easily obtained by setting the 1 (or 10) of C₁ to the number in D₁ the cube lies in D₁ immediately below the number in Q₁ or Q₂. To cube 2·2 set 10C₁ to 2·2D₁ set X to 2·2Q₁ and read in D₁ under X the result, 10·65, the decimal point being inserted by inspection.

Cube roots are evaluated by setting X to the number in D₁ and then moving the slide until the value in D₁ coincident with 1 (or 10) C₁ is the same as the number in Q₁ or Q₂ under X. Suppose the cube root of 2 is required. Set X over 2 in D₁ now move the slide about an inch to the right of its mid- position, and then carefully adjust it until the value in D₁ coincident with 1 C₁ is the same as the value in Q₁ under the cursor index X; this value will be found to be 1 ·26, which is the cube root of 2 (1.25992).

It may assist to observe that:
    If the number lies within the range 1-10 its cube root will be found in Q₁ and coincident with 1 C₁.
    If the number lies within the range 10-31 (p³) its cube root will be found in Q₁ and coincident with 10C₁.
    If the number lies within the range 31-100 its cube root will be found in Q₂ and coincident with 1 C₁.
    If the number lies within the range 100-1000 its cube root will be found in Q₂ and coincident with 10C₁.

Numbers from 1 to 1000 have cube roots from 1 to 10. If the number whose cube root is required is not within the range 1-1000 it should first be altered by moving the decimal point three, or multiples of three, places to right or left to bring the number within that range. The cube root should then be found as detailed above, and finally the decimal point of the result should be moved back one place for each step of three places made in the original number.

Example: Find the cube root of 116,300.
    Moving the decimal point three places to the left alters the figure to 116·3, which lies within the 1 to 1000 range. The cube root of 116·3 is 4·88. The decimal point must now be moved one place to the right, giving the actual result as 48·8.

In evaluating cube roots it is a good plan to find the nearest integral result mentally.

Example: Find the cube root of ·682. First move the decimal point three places to the right, so that the number becomes 682. The cube of 5 is 125, which is well below 682. Try the cube of 7; 7 x 7 = 49 (say 50); 7 x 50 = 350, still too small; try 9; 9 x 9 = 81; and 9 x 80 = 720. The required cube root is less than 9. Set X over 682D₁ and the slide so that 10C₁ is over 9D₁. Now move the slide slowly to the left until the reading in D₁ below 10C is the same as that in Q₂ under X. These identical values are 8·8. The cube root of 682 is 8·8, and of ·682, ·88.

Squares, square roots, cube and cube roots, maybe evaluated easily with the aid of the log-log scales, often with a higher degree of accuracy than can be attained with the P and Q scales.

Log-log Scales

When the log-log scale is used the slide should be inverted so that the surface which generally is underneath is brought uppermost, or, if the log-log scale is fitted to a separate slide, the slides should be interchanged. The log-log scale provides a means of effecting unusual computations. Its most useful property is the ease with which powers and roots may be evaluated, even when the power root is a mixed number.

Suppose the value of 8·4^1.79 is required. Set 8·4LL₃ to 1D₁, then immediately above 1·79D₁ will be found the result, 45·1, in LL₃. If the index of the power is negative, e.g. 8·4^-1.79, the value of 8·4^1·79 should first be evaluated and the reciprocal of this be found, using the C and D scales;

i.e. 8·4^-1·79 = 1 /45·1 = ·0222.

To evaluate ^4.15Ö1·31; to 415D₁ set 1·31LL₂, and then use X to project 10D₁ into LL₁ and read the result 1·067.

Results outside the range of the log-log scale may be obtained by the methods suggested in Section 6.

Logarithms to any base may be obtained by setting the base in LL  to l of D₁, or 1 of D₂ and projecting the number whose log is required from the log-log scale into D₁ or D₂. The log so obtained will be complete, comprising characteristic and mantissa. Common logarithms are found by setting the l0LL₃ to 1D₁. It will be seen that immediately below 100LL₃ stands 2D₁, 2 being the log of 100. Below 1000 stands 3 and below 10,000 stands 4. To obtain the logs of numbers in LL₁ and LL₂ the cursor index must be used to project into D₁. If the number whose log is required lies towards the left-hand end of the log-log scale, l0LL₃ should be set to 10D₁, or 1D₂.

Natural logarithms are obtained by setting the value 2·7183 near the left-hand end of the LL₃ scale to the 1 of D₁. This setting will enable all the natural logs of numbers within the range of the log-log scale to be read without moving the slide; the numbers in LL₁ and LL₂ being projected into D₁ by using the cursor index X.

It is useful to remember that the 10th powers of numbers in LL₁ lie immediately below in LL₂, and the 10th power of numbers in LL₂ lie immediately below in LL₃.

© Hodder Stoughton, reproduced with permission.

Return to
   Dualistic rule in Collection