The slide rule

With the slide in the ordinary position and with the indices of the C and D scales in exact agreement, the ratio of the corresponding divisions of these scales is 1. If the slide is moved so that 1 on C agrees with 2 on D, we know that under any number n on C is n × 2 on D, so that if we read numerators on C and denominators on D we have

In other words, the numbers on D bear to the coinciding numbers on C a ratio of 2 to 1. Obviously the same condition will obtain no matter in what position the slide may be placed. The rule for proportion, which is apparent from the foregoing, may be expressed as follows:—

Rule for Proportion.—Set the first term of a proportion on the C scale to the second term on the D scale, and opposite the third term on the C scale read the fourth term on the D scale.

Ex.—Find the 4th term in the proportion of 20 ∶ 27 ∷ 70 ∶ x. Set 20 on C to 27 on D, and opposite 70 on C read 94·5 on D. Thus

It will be evident that this is merely a case of combined multiplication and division of the form, 20 × 70
27 = 94·5. Hence, given any three terms of a proportion, we set the 1st to the 2nd, or the 3rd to the 4th, as the case may be, and opposite the other given term read the term required.^[4]

Thus, in reducing vulgar fractions to decimals, the decimal equivalent of 3
16 is determined by placing 3 on C to 16 on D, when over the index or 1 of D we read 0·1875 on C. In this case the terms are 3 ∶ 16 ∷ x ∶ 1. For the inverse operation—to find a vulgar fraction equivalent to a given decimal—the given decimal fraction on C is set to the index of D, and then opposite any denominator on D is the corresponding numerator of the fraction on C.

If the index of C be placed to agree with 3·1416 on D, it will be clear from what has been said that this ratio exists throughout between the numbers of the two scales. Therefore, against any diameter of a circle on C will be found the corresponding circumference on D. In the same way, by setting 1 on C to the appropriate conversion factor on D, we can convert a series of values in one denomination to their equivalents in another denomination. In this connection the following table of conversion factors will be found of service. If the A and B scales are used instead of the C and D scales, a complete set of conversions will be at once obtained. In this case, however, the left-hand A and B scales should be used for the initial setting, any values read on the right-hand A or B scales being read as of tenfold value. With the C and D scales a portion of the one scale will project beyond the other. To read this portion of the scale, the cursor or runner is brought to whichever index of the C scale falls within the rule, and the slide moved until the other index of the C scale coincides with the cursor, when the remainder of the equivalent values can then be read off. It must be remembered that if the slide is moved in the direction of notation (to the right), the values read thereon have a tenfold greater value; if the slide is moved to the left, the readings thereon are decreased in a tenfold degree. Although preferred by many, in the form given, the case is obviously one of multiplication, and is so treated in the Data Slips at the end of the book.

Inverse Proportion.—If “more” requires “less,” or “less” requires “more,” the case is one of inverse proportion, and although it will be seen that this form of proportion is quite readily dealt with by the preceding method, the working is simplified to some extent by inverting the slide so that the C scale is adjacent to the A scale. By the aid of the cursor, the values on the inverted C (or Ɔ) scale, and on the D scale, can be then read off. These will now constitute a series of inverse ratios. For example, in the proportion

the 4 on the Ɔ scale is brought opposite 3 on D, when under 8 on Ɔ is found 1·5 on D.^[5]

GENERAL HINTS ON THE ELEMENTARY USES OF THE SLIDE RULE.

Before the more complex operations of involution, evolution, etc., are considered, a few general hints on the use of the slide rule for elementary operations may be of service, especially as these will serve to enforce some of the more important points brought out in the preceding sections.

Always use the slide rule in as direct a light as possible.

Study the manner in which the scales are divided. Follow the graduations of the C and D scales from 1 to 10, noting the values given by each successive graduation and how these values change as we follow along to the right. Do the same with the two halves of the A and B scales and note the difference in the value of the subdivisions, due to the shorter scale-lengths.

Practise reading values by setting 1 on C to some value on D and reading under 2, 3, 4, etc., on C, checking the readings by mental arithmetic. To the same end, find squares, square roots, etc., comparing the results with the actual values as given in tables. Practise setting both slide and cursor to values taken at random. Aim at accuracy; speed will come with practice.

When in doubt as to any method of working, verify by making a simple calculation of the same form.

Follow the orthodox methods of working until entirely confident in the use of the instrument, and even then do not readily make a change. If any altered procedure is adopted, first work a simple case and guard carefully against unconsciously lapsing into the usual method during the operation.

Unless the calculation is of a straightforward character, time taken in considering how best to attack it (rearranging the expression if desirable) is generally time well spent.

In setting two values together, set the cursor to one of them on the rule, and bring the other, on the slide, to the cursor line.

In multiplying factors, as 57 × 0·1256, take the fractional value first. It is easier to set 1 on C to 1256 on D and read under 57 on C, than to reverse the procedure. When both values are eye-estimated, set the cursor to the second factor on C and read the result on D, under the cursor line.

In continuous operations avoid moving the slide further than necessary, by taking the factors in that order which will keep the scale readings as close together as possible.

SQUARES AND SQUARE ROOTS.

We have seen that the relation which the upper scales bear to the lower set is such that over any number on D is its square on A, and, conversely, under any number on A is its square root on D, the same remarks applying to the C and B scales on the slide. Taking the values engraved on the rule, we have on D, numbers lying between 1 and 10, and on A the corresponding squares extending from 1 to 100. Hence the squares of numbers between 1 and 10, or the roots of numbers between 1 and 100, can be read off on the rule by the aid of the cursor. All other cases are brought within these ranges of values by factorising with powers of 10, as before explained.

The more practical rule is the following:—

To Find the Square of a Number, set the cursor to the number on D and read the required square on A under the cursor. The rule for

The Number of Digits in a Square is easily deducible from the rule for multiplication. If the square is read on the left scale of A, it will contain twice the number of digits in the original number less 1; if it is read on the right scale of A, it will contain twice the number of digits in the original number.

Ex.—Find the square of 114.

Placing the cursor to 114 on D, it is seen that the coinciding number on A is 13. As the result is read off on the left scale of A, the number of digits will be (3 × 2) − 1 = 5, and the answer is read as 13,000. The true result is 12,996.

Ex.—Find the square of 0·0093.

The cursor being placed to 93 on D, the number on A is found to be 865. The result is read on the right scale of A, so the number of digits = −2 × 2 = −4, and the answer is read as 0·0000865 [0·00008649].

Square Root.—The foregoing rules suggest the method of procedure in the inverse operation of extracting the square root of a given number, which will be found on the D scale opposite the number on the A scale. It is necessary to observe, however, that if the number consists of an odd number of digits, it is to be taken on the left-hand portion of the A scale, and the number of digits in the root = N + 1
2, N being the number of digits in the original number. When there is an even number of digits in the number, it is to be taken on the right-hand portion of the A scale, and the root contains one-half the number of digits in the original number.

Ex.—Find the square root of 36,500.

As there is an odd number of digits, placing the cursor to 365 on the L.H. A scale gives 191 on D. By the rule there are N + 1
2 = 5 + 1
2 = 3 digits in the required root, which is therefore read as 191 [191·05].

Ex.—Find √0·0098.

Placing the cursor to 98 on the right-hand scale of A (since −2 is an even number of digits), it is seen that the coinciding number on D is 99. As the number of digits in the number is −2, the number of digits in the root will be −2
2 = −1. It will therefore be read as 0·099 [0·09899+].

Ex.—Find √0·098.

The number of digits is −1, so under 98 on the left scale of A, we find 313 on D. By the rule the number in the root will be −1 +1
2 = 0, and the root is therefore read as 0·313 [0·313049+].

Ex.—Find √0·149.

As the number of digits (0) is even, the cursor is set to 149 on the right-hand scale of A, giving 386 on D. By the rule, the number of digits in the root will be 0
2 = 0, and the root will be read as 0·386 [0·38605+].

Another method of extracting the square root, by which more accurate readings may generally be obtained, is by using the C and D scales only, with the slide inverted. If there is an odd number of digits in the number, the right index, or if an even number of digits the left index, of the inverted scale Ɔ is placed so as to coincide with the number on D of which the root is sought. Then with the cursor, the number is found on D which coincides with the same number on Ɔ, which number is the root sought.

Ex.—Find √22·2.

Placing the left index of Ɔ to 222 on D, the two equal coinciding numbers on Ɔ and D are found to be 4·71.

Note that under the cursor line we have the original number, 22·2, on A, and from this the number of digits in the root is determined as before.

The plan of finding the square of a number by ordinary multiplication is often very convenient. The inverse process of finding a square root by trial division is not to be recommended.

To obtain a close value of a root or to verify one found in the usual way, the author has, on occasion, adopted the following plan:—Set 1 (or 10) on B to the number on the A scale (L.H. or R.H. as the case may require), and bring the cursor to the number on D. If the root found is correct, the readings on C under the cursor and on D under the index of C, will be in exact agreement.

If 1 on B is placed to a number n on the L.H. A scale, the student will note that while root n is read on D under 1 on C, the root of 10 n is read on D under 10 on B. Hence, if preferred, the number can be taken always on the first scale of A and the root read under 1 or 10 on B, according to whether there is an odd or even number of digits in the number. Obviously the second root is the first multiplied by √10.

In raising a number to the third power, a combination of the preceding method and ordinary multiplication is employed.

To Find the Cube of a Number.—Set the L.H. or R.H. index of C to the number on D, and opposite the number ON THE LEFT-HAND scale of B read the cube on the L.H. or R.H. scale of A.

By this rule four scales are brought into requisition. Of these, the D scale and the L.H. B scale are always employed, and are to be read as of equal denomination. The values assigned to the L.H. and R.H. scales of A will be apparent from the following considerations.

Commencing with the indices of C and D coinciding, and moving the slide to the right, it will be seen that, working in accordance with the above rule, the cubes of numbers from 1 to 2·154 (= ∛10) will be found on the first or L.H. scale of A. Moving the slide still farther to the right, we obtain on the R.H. A scale cubes of numbers from 2·154 to 4·641 (or ∛10 to ∛100). Had we a third repetition of the L.H. A scale, the L.H. index of C could be still further traversed to the right, and the cubes of numbers from 4·641 to 10 read off on this prolongation of A. But the same end can be attained by making use of the R.H. index of C, when, traversing the slide to the right as before, the cubes of numbers from 4·641 to 10 on D can be read off on the L.H. A scale over the corresponding numbers on the L.H. B scale. Hence, using the L.H. index of C, the readings on the L.H. A scale may be regarded comparatively as units, those on the R.H. A scale as tens; while for the hundreds we again make use of the L.H. A scale in conjunction with the right-hand index of C.

By keeping these points in view, the number of digits in the cube (N) of a given number (n) are readily deduced. Thus, if the units scale is used, N = 3n − 2; if the tens scale, N = 3n − 1; while if the hundreds scale be used, N = 3n. Placed in the form of rules:—

N = 3n − 2 when the product is read on the L.H. scale of A with the slide to the right (units scale).

N = 3n − 1 when the product is read on the R.H. scale of A; slide to the right (tens scale).

N = 3n when the product is read on the L.H. scale of A with the slide to the left (hundreds scale).

With decimals the same rule applies, but, as before, the number of digits must be read as −1, −2, etc., when one, two, etc., cyphers follow immediately after the decimal point.

Ex.—Find the value of 1·4³.

Placing the L.H. index of C to 1·4 on D, the reading on A opposite 1·4 on the L.H. scale of B is found to be about 2·745 [2·744].

Ex.—Find the value of 26·4³.

Placing the L.H. index of C to 26·4 on D, the reading on A opposite 26·4 on the L.H. scale of B is found to be about 18,400 [18,399·744].

Ex.—Find the value of 7·3³.

In this case it becomes necessary to use the R.H. index of C, which is set to 7·3 on D, when opposite 7·3 on the L.H. scale of B is read 389 [389·017] on A.

Ex.—Find the value of 0·073³.

From the setting as before it is seen that the number of digits in the number must be multiplied by 3. Hence, as there is −1 digit in 0·073, there will be −3 in the cube, which is therefore read 0·000389.

The last two examples serve to illustrate the principle of factorising with powers of 10. Thus

Cube Root (Direct Method).—One method of extracting the cube root of a number is by an inversion of the foregoing operation. Using the same scales, the slide is moved either to the right or left until under the given number on A is found a number on the L.H. B scale, identical with the number simultaneously found on D under the right or left index of C. This number is the required cube root.

From what has already been said regarding the combined use of these scales in cubing, it will be evident that in extracting the cube root of a number, it is necessary, in order to decide which scales are to be used, to know the number of figures to be dealt with. We therefore (as in the arithmetical method of extraction) point off the given number into sections of three figures each, commencing at the decimal point, and proceeding to the left for numbers greater than unity, and to the right for numbers less than unity. Then if the first section of figures on the left consists of—

1 figure, the number will evidently require to be taken on what we have called the “units” scale—i.e., on the L.H. scale of A, using the L.H. index of C.

If of 2 figures, the number will be taken on the “tens” scale—i.e., on the R.H. scale of A, using the L.H. index of C.

If of 3 figures, the number will be taken on the “hundreds” scale—i.e., on the L.H. scale of A, using the R.H. index of C.

To determine the number of digits in cube roots it is only necessary to note that when the number is pointed off into sections as directed, there will be one figure in the root for every section into which the number is so divided, whether the first section consists of 1, 2, or 3 digits.

Of numbers wholly decimal, the cube roots will be decimal, and for every group of three 0s immediately following the decimal point, one 0 will follow the decimal point in the root. If necessary, 0s must be added so as to make up complete multiples of 3 figures before proceeding to extract the root. Thus 0·8 is to be regarded as 0·800, and 0·00008 as 0·000080 in extracting cube roots.

Ex.—Find ∛14,000.

Pointing the number off in the manner described, it is seen that there are two figures in the first section—viz., 14. Setting the cursor to 14 on the R.H. scale of A, the slide is moved to the right until it is seen that 241 on the L.H. scale of B falls under the cursor, when 241 on D is under the L.H. index of C. Pointing 14,000 off into sections we have 14 000—that is, two sections. Therefore, there are two digits in the root, which in consequence will be read 24·1 [24·1014+].

Ex.—Find ∛0·162.

As the divisional section consists of three figures, we use the “hundreds” scale. Setting the cursor to 0·162 on the L.H. A scale, and using the R.H. index of C, we move the slide to the left until under the cursor 0·545 is found on the L.H. B scale, while the R.H. index of C points to 0·545 on D, which is therefore the cube root of 0·162.

Ex.—Find ∛0·0002.

To make even multiples of 3 figures requires the addition of 00; we have then 200, the cube root of which is found to be about 5·85. Then, since the first divisional group consists of 0s, one 0 will follow the decimal point, giving ∛0·0002 = 0·0585 [0·05848].

Cube Root (Inverted Slide Method).—Another method of extracting the cube root involves the use of the inverted slide. Several methods are used, but the following is to be preferred:—Set the L.H. or R.H. index of the slide to the number on A, and the number on ᗺ (i.e., B inverted), which coincides with the same number on D, is the required root.

Setting the slide as directed, and using first the L.H. index of the slide and then the R.H. index, it is always possible to find three pairs of coincident values. To determine which of the three is the required result is best shown by an example.

Ex.—Find ∛5, ∛50, and ∛500.

Setting the R.H. index of the slide to 5 on A, it is seen that 1·71 on D coincides with 1·71 on ᗺ. Then setting the L.H. index to 5 on A, further coincidences are found at 3·68 and at 7·93, the three values thus found being the required roots. Note that the first root was found on that portion of the D scale lying under 1 to 5 on A; the second root on that portion lying under 5 to 50 on A; and the third root on that portion of D lying under 50 to 100 on A. In this connection, therefore, scale A may always be considered to be divided into three sections—viz., 1 to n, n to 10n, and 10n to 100. For all numbers consisting of 1, 1 + 3, 1 + 6, 1 + 9—i.e., of 1, 4, 7, 10, or −2, −5, etc., figures—the coincidence under the first section is the one required. If the number has 2, 5, 8, or −1, −4, −7, etc., figures, the coincidence under the second section is correct, while if the number has 3, 6, 9, or 0, −3, etc., figures, the coincidence under the last section is that required. The number of digits in the root is determined by marking off the number into sections, as already explained.

Cube Root (Pickworth’s Method).—One of the principal objections to the two methods described is the difficulty of recollecting which scales are to be employed and with which index of the slide they are to be used. With the direct method another objection is that the readings to be compared are often some distance apart, the maximum distance intervening being two-thirds of the length of the rule. To carry the eye from one to another is troublesome and time-taking. With the inverted scale method the reading of a scale reversed in direction and with the figures inverted is also objectionable.

With the author’s method these objections are entirely obviated. The same scales and index are always used, and are read in their normal position. The three roots of n, 10n and 100n (n being less than 10 and not less than 1) are given with one setting and appear in their natural sequence, no traversing of the slide being needed. The readings to be compared are always close together, the maximum distance between them being one-sixth of the length of the rule. The setting is always made in the earlier part of the scales where closer readings can be obtained, and finally, if desired, the result may be readily verified on the lower scales by successive multiplication.

For this method two gauge points are required on C. To conveniently locate these, set 53 on C to 246 on D; join 1 on D to 1 on A with a straight-edge and with a needle point draw a short fine line on C. Set 246 on C to 53 on D, and repeat the process at the other end of the rule. The gauge points thus obtained (dividing C into three equal parts) will be at 2·154 and 4·641, and should be marked ∛10 and ∛100 respectively.^[6]

Ex.—Find ∛2·86, ∛28·6 and ∛286.

Set cursor to 2·86 on A and drawing the slide to the right find 1·42 under 1 on C, when 1·42 on B is under the cursor. Then reading under 1, ∛10 and ∛100, we have

∛2·86 = 1·42; ∛28·6 = 3·06 and ∛286 = 6·59.

It will be seen that factorising with powers of 10, we multiply the initial root by ∛10 and ∛100. Obviously the three roots will always be found on D, in their natural order and at intervals of one-third the length of the rule. The number of digits in the roots of numbers which do not lie between 1 and 1000, is found as before explained.

In any method of extracting cube roots in which the slide has to be adjusted to give equal readings on B and D, the author has found it of advantage to adopt the following plan:—The cursor being set to, say, 4·8 on A, bring a near main division line on B, as 1·7, to the cursor; then 1 on C is at 1·68 on D. The difference in the readings is two small divisions on D, and moving the slide forward by one-third the space representing this difference, we obtain 1·687 as the root required. With a little practice it is possible to obtain more accurate results by this method than by comparing the reading on D with that on the less finely-graded B scale.