Part 10

The maximum number of figures in the quotient is the difference between the sum of the number of figures in the numerator factors and those of the denominator factors, _plus_ 1 for each factor of the denominator and this results when A has to be set to all the factors of the denominator and all the factors of the numerator except the first brought to B. Each time B is set to a denominator factor or a numerator factor is brought to A, 1 is to be deducted.

Logarithms of numbers are obtained by using the scales _m_ and _n_ and hence powers and roots of any magnitude may be obtained by the procedure already fully explained. The instrument illustrated is made by Messrs. W. F. Stanley & Co., Limited, London.

THE “R.H.S.” CALCULATOR.—In this calculator, designed by Prof. R. H. Smith, the scale-line, which is 50 in. long, is also arranged in a spiral form (Fig. 19), but in this case it is wrapped around the central portion of a tube which is about ¾in. in diameter and 9½in. long. A slotted holder, capable of sliding upon the plain portions of this tube, is provided with four horns, these being formed at the ends of the two wide openings through which the scale is read. An outer ring carrying two horns completes the arrangement.

One of the horns of the holder being placed in agreement with the first factor, and one of the horns of the ring with the second factor, the holder is moved until the third factor falls under the same horn of the ring, when the resulting fourth term will be found under the same (right or left) horn of the holder, at either end of the slot. In multiplication, 100 or 1000 is taken for the second factor in the above proportion, as already explained in connection with Fuller’s rule; indeed, generally, the mode of operation is essentially similar to that followed with the former instrument.

The scale shown on one edge of the opening in the holder, together with the circular scale at the top of the spiral, enables the mantissæ of logarithms of numbers to be obtained, and thus problems involving powers and roots may be dealt with quite readily. This instrument is supplied by Mr. J. H. Steward, London.

THACHER’S CALCULATING INSTRUMENT, shown in Fig. 20, consists of a cylinder 4 in. in diameter and 18 in. long, which can be given both a rotary and a longitudinal movement within an open framework composed of twenty triangular bars. These bars are connected to rings at their ends, which can be rotated in standards fixed to the baseboard. The scale on the cylinder consists of forty sectional lengths, but of each scale-line that part which appears on the right-hand half of the cylinder is repeated on the left-hand half, one line in advance. Hence each half of the cylinder virtually contains two complete scales following round in regular order. On the lower lines of the triangular bars are scales exactly corresponding to those on the cylinder, while upon the upper lines of the bars and not in contact with the slide is a scale of square roots.

By rotating the slide any line on it may be brought opposite any line in frame and by a longitudinal movement any graduation on these lines may be brought into agreement. The whole can be rotated in the supporting standards in order to bring any reading into view. As shown in the illustration, a magnifier is provided, this being conveniently mounted on a bar, along which it can be moved as required.

SECTIONAL LENGTH OR GRIDIRON SLIDE RULES.—The idea of breaking up a long scale into sectional lengths is due to Dr. J. D. Everett, who described such a gridiron type of slide rule in 1866. Hannyngton’s Extended Slide Rule is on the same principle. Both instruments have the lower scale repeated. H. Cherry (1880) appears to have been the first to show that such duplication could be avoided by providing two fixed index points in addition to the natural indices of the scale. These additional indices are shown at 10′ and 100′ in Fig. 21, which represents the lower sheet of Cherry’s Calculator on a reduced scale. The upper member of the calculator consists of a transparent sheet ruled with parallel lines, which coincide with the lines of the lower scale when the indices of both are placed in agreement. To multiply one number by another, one of the indices on the upper sheet is placed to one of the factors, and the position of whichever index falls under the transparent sheet is noted on the latter. Bringing the latter point to the other factor, the result is found under whichever index lies on the card. In other arrangements the inventor used transparent scales, the graduations running in a reverse direction to those of the lower scale. In this case, a factor on the upper scale is set to the other factor on the lower, and the result read at the available index.

PROELL’S POCKET CALCULATOR is an application of the last-named principle. It comprises a lower card arranged as Fig. 21, with an upper sheet of transparent celluloid on which is a similar scale running in the reverse direction. For continued multiplication and division, a needle (supplied with the instrument) is used as a substitute for a cursor, to fix the position of the intermediate results. A series of index points on the lower card enable square and cube roots to be extracted very readily. This calculator is supplied by Messrs. John J. Griffin & Sons, Ltd., London.

CIRCULAR CALCULATORS.

Although the 10 in. slide rule is probably the most serviceable form of calculating instrument for general purposes, many prefer the more portable circular calculator, of which many varieties have been introduced during recent years. The advantages of this type are: It is more compact and conveniently carried in the waistcoat pocket. The scales are continuous, so that no traversing of the slide from 1 to 10 is required. The dial can be set quickly to any value; there is no trouble with tight or ill-fitting slides. The disadvantages of most forms are: Many problems involve more operations than a straight rule. The results being read under fingers or pointers, an error due to parallax is introduced, so that the results generally are not so accurate as with a straight rule. The inner scales are short, and therefore are read with less accuracy. Special scale circles are needed for cubes and cube roots. The slide cannot be reversed or inverted.

THE BOUCHER CALCULATOR.—This circular calculator resembles a stem-winding watch, being about 2 in. in diameter and ⁹⁄₁₆in. in thickness. The instrument has two dials, the back one being fixed, while the front one, Fig. 22 (showing the form made by Messrs. W. F. Stanley, London), turns upon the large centre arbor shown. This movement is effected by turning the milled head of the stem-winder. The small centre axis, which is turned by rotating the milled head at the side of the case, carries two fine needle pointers, one moving over each dial, and so fixed on the axis that one pointer always lies evenly over the other. A fine index or pointer fixed to the case in line with the axis of the winding stem, extends over the four scales of the movable dial as shown. Of these scales, the second from the outer is the ordinary logarithmic scale, which in this instrument corresponds to a straight scale of about 4¾in. in length. The two inner circles give the square roots of the numbers on the primary logarithmic scale, the smaller circle containing the square roots of values between 1 and 3·162 (= √(10)), while the other section corresponds to values between 3·162 and 10. The outer circle is a scale of logarithms of sines of angles, the corresponding sines of which can be read off on the ordinary scale.

On the fixed or back dial there are also four scales, these being arranged as in Fig. 23. The outer of these is a scale of equal parts, while the three inner scales are separate sections of a scale giving the cube roots of the numbers taken on the ordinary logarithmic scale and referred thereto by means of the pointers. In dividing this cube-root scale into sections, the same method is adopted as in the case of the square-root scale. Thus, the smallest circle contains the cube roots of numbers between 1 and 10, and is therefore graduated from 1 to 2·154; the second circle contains the cube roots of numbers between 10 and 100, being graduated from 2·154 to 4·657; while the third section, in which are found the cube roots of numbers between 100 and 1000, carries the graduations from 4·657 to 10.

What has been said in an earlier section regarding the notation of the slide rule may in general be taken to apply to the scales of the Boucher calculator. The manner of using the instrument is, however, not quite so evident, although from what follows it will be seen that the operative principle—that of variously combining lengths of a logarithmic scale—is essentially similar. In this case, however, it is seen that in place of the straight scale-lengths shown in Fig. 4, we require to add or subtract arc-lengths of the circular scales, while, further, it is evident that in the absence of a fixed scale (corresponding to the stock of the slide rule) these operations cannot be directly performed as in the ordinary form of instrument. However, by the aid of the fixed index and the movable pointer, we can effect the desired combination of the scale-lengths in the following manner. Assuming it is desired to multiply 2 by 3, the dial is turned in a backward direction until 2 on the ordinary scale lies under the fixed index, after which the movable pointer is set to 1 on the scale. As now set, it is clear that the arc-length 1–2 is spaced off between the fixed index and the movable pointer, and it now only remains to add to this definite arc-length a further length of 1–3. To do this we turn the dial still further backward until the arc 1–3 has passed under the movable pointer, when the result, 6, is read under the fixed index. A little consideration will show that any other scale length may be added to that included between the fixed and movable pointers, or, in other words, any number on the scale may be multiplied by 2 by bringing the number to the movable pointer and reading the result under the fixed index. The rule for multiplication is now evident.

_Rule for Multiplication._—_Set one factor to the fixed index and bring the pointer to 1 on the scale; set the other factor to the pointer and read the result under the fixed index._

With the explanation just given, the process of division needs little explanation. It is clear that to divide 6 by 3, an arc-length 1–3 is to be taken from a length 1–6. To this end we set 6 to the index (corresponding in effect to passing a length 1–6 to the left of that reference point) and set the pointer to the divisor 3. As now set, the arc 1–6 is included between 1 on the scale and the index, while the arc 1–3 is included between 1 on the scale and the pointer. Obviously if the dial is now turned forward until 1 on the scale agrees with the pointer, an arc 1–3 will have been deducted from the larger arc 1–6, and the remainder, representing the result of this operation, will be read under the index as 2.

_Rule for Division._—_Set the dividend to the fixed index, and the pointer to the divisor; turn the dial until 1 on the scale agrees with the pointer, and read the result under the fixed index._

The foregoing method being an inversion of the rule for multiplication, is easily remembered and is generally advised. Another plan is, however, preferable when a series of divisions are to be effected with a constant divisor—_i.e._, when _b_ in (_a_)/(_b_) = _x_ is constant. In this case 1 on the scale is set to the index and the pointer set to _b_; then if any value of a is brought to the pointer, the quotient _x_ will be found under the index.

_Combined Multiplication and Division_, as (_a_ × _b_ × _c_)/(_m_ × _n_) = _x_, can be readily performed, while cases of continued multiplication evidently come under the same category, since _a_ × _b_ × _c_ = (_a_ × _b_ × _c_)/(1 × 1) = _x_. Such cases as _a_/(_m_ × _n_ × _r_) = _x_ are regarded as (_a_ × 1 × 1 × 1)/(_m_ × _n_ × _r_) = _x_; while (_a_ × _b_ × _c_)/(_m_) = _x_ is similarly modified, taking the form (_a_ × _b_ × _c_)/(_m_ × 1) = _x_. In all cases the expression must be arranged so that there is _one more factor in the numerator_ than _in the denominator_, _1’s being introduced as often as required_. The simple operations of multiplication and division involve a similar disposition of factors, since from the rules given it is evident that _m_ × _n_ is actually regarded as (_m_ × _n_)/(1), while (_m_)/(_n_) becomes in effect (_m_ × 1)/(_n_). It is important to note the general applicability of this arrangement-rule, as it will be found of great assistance in solving more complicated expressions.

As with the ordinary form of slide rule, the factors in such an expression as (_a_ × _b_ × _c_)/(_m_ × _n_) = _x_ are taken in the order:—1st factor of numerator; 1st factor of denominator; 2nd factor of numerator; 2nd factor of denominator, and so on; the 1st factor as _a_ being set to the index, and the result _x_ being finally read at the same point of reference.

EX.—(39 × 14·2 × 6·3)/(1·37 × 19) = 134.

Commence by setting 39 to the index, and the pointer to 1·37; bring 14·2 to the pointer; pointer to 19; 6·3 to the pointer, and read the result 134 at the index.

It should be noted that after the first factor is set to the fixed index, the _pointer_ is set to each of the _dividing_ factors as they enter into the calculation, while the _dial_ is moved for each of the _multiplying_ factors. Thus the dial is first moved (setting the first factor to the index), then the pointer, then the dial, and so on.

_Number of Digits in the Result._—If rules are preferred to the plan of roughly estimating the result, the general rules given on pages 21 and 25 should be employed for simple cases of multiplication and division. For combined multiplication and division, modify the expression, if necessary, by introducing 1’s, as already explained, and subtract the sum of the denominator digits from the sum of numerator digits. Then proceed by the author’s rule, as follows:—

_Always turn dial to the_ LEFT; _i.e._, _against the hands of a watch_.

_Note dial movements only; ignore those of the pointer._

_Each time 1 on dial agrees with or passes fixed index_, ADD _1 to the above difference of digits_.

_Each time 1 on dial agrees with or passes pointer_, DEDUCT _1 from the above difference of digits_.

Treat continued multiplication in the same way, counting the 1’s used as denominator digits as one less than the number of multiplied factors.

EX.—(8·6 × 0·73 × 1·02)/(3·5 × 0·23) = 7·95 [7·95473+].

Set 8·6 to index and pointer to 3·5. Bring 0·73 to pointer (noting that 1 on the scale passes the index) and set pointer to 0·23. Set 1·02 to pointer (noting that 1 on the scale passes the pointer) and read under index 7·95. There are 1 + 0 + 1 = 2 numerator digits and 1 + 0 = 1 denominator digit; while 1 is to be added and 1 deducted as per rule. But as the latter cancel, the digits in the result will be 2 − 1 = 1.

When moving the dial to the left will cause 1 on the dial to pass _both_ index and pointer (thus cancelling), the dial may be turned back to make the setting.

It will be understood that when 1 is the _first_ numerator, and 1 on the dial is therefore set to the index, no digit addition will be made for this, as the actual operation of calculating has not been commenced.

In the Stanley-Boucher calculator (Fig. 23) a small centre scale is added, on which a finger indicates automatically the number of digits to be added or deducted; the method of calculating, however, differs from the foregoing. To avoid turning back to 0 at the commencement of each calculation, a circle is ground on the glass face, so that a pencil mark can be made thereon to show the position of the finger when commencing a calculation.

_To Find the Square of a Number._— Set the number, on one or other of the square root scales, to the index, and read the required square on the ordinary scale.

_To Find the Square Root of a Number._—Set the number to the index, and if there is an _odd_ number of digits in the number, read the root on the inner circle; if an even number, on the second circle.

_To Find the Cube of a Number._—Set 1 on the ordinary scale to the index, and the pointer (on the back dial) to the number on one of the three cube-root scales. Then under the pointer read the cube on the ordinary scale.

_To Find the Cube Root of a Number._—Set 1 to index, and pointer to number. Then read the cube root under the pointer on one of the three inner circles on the back dial. If the number has

1, 4, 7, 10 or −2, −5, etc., digits, use the inner circle. 2, 5, 8, 11 or −1, −4, etc., „ „ second circle. 3, 6, 9, 12 or −0, −3, etc., „ „ third circle.

_For Powers or Roots of Higher Denomination._—Set 1 to index, the pointer to the number on the ordinary scale, and read on the outer circle on the back dial the mantissa of the logarithm. Add the characteristic (see p. 46), multiply by the power or divide by the root, and set the pointer to the mantissa of the result on this outer circle. Under the pointer on the ordinary scale read the number, obtaining the number of figures from the characteristic.

_To Find the Sines of Angles._—Set 1 to index, pointer to the angle on the outer circle, and read under the pointer the _natural sine_ on the ordinary scale; also under the pointer on the outer circle of the back dial read the _logarithmic sine_.

THE HALDEN CALCULEX.—After the introduction of the Boucher calculator in 1876, circular instruments, such as the Charpentier calculator, were introduced, in which a disc turned within a fixed ring, so that scales on the faces of both could be set together and ratios established as on the slide rule. Cultriss’s Calculating Disc is another instrument on the same principle. The Halden Calculex, of which half-size illustrations are given in Figs. 24 and 25, represents a considerable improvement upon these early instruments. It consists of an outer metal ring carrying a fixed-scale ring, within which is a dial. On each side of this dial are flat milled heads, so that by holding these between the thumb and forefinger the dial can be set quickly and conveniently. The protecting glass discs, which are not fixed in the metal ring but are arranged to turn therein, carry fine cursor lines, and as these are on the side next to the scales a very close setting can be made quite free from the effects of parallax. This construction not only avoids the use of mechanism, with its risk of derangement, but reduces the bulk of the instrument very considerably, the thickness being about ¼in.

On the front face, Fig. 24, the fixed ring carries an outer evenly-divided scale, giving logarithms, and an ordinary scale, 1–10, which works in conjunction with a similar scale on the edge of the dial. The two inner circles give the square roots of values on the main scales as in the Boucher calculator. On the back face, Fig. 25, the ring bears an outer scale, giving sines of angles from 6° to 90° and an ordinary scale, 1–10, as on the front face. The scales on the dial are all reversed in direction (running from right to left), the outer one consisting of an ordinary (but inverse) scale, 1–10, while the three inner circles give the cube roots of values on this inverse scale. As the fine cursor lines extend over all the scales, a variety of calculations can be effected very readily and accurately.

SPERRY’S POCKET CALCULATOR, made by the Keuffel and Esser Company, New York (Fig. 26), has two rotating dials, each with its own pointer and fixed index. The S dial has an outer scale of equal parts, an ordinary logarithmic scale, and a square-root scale. The L dial has a single logarithmic scale arranged spirally, in three sections, giving a scale length of 12½in. The pointers are turned by the small milled head, which is concentric with the milled thumb-nut by which the two dials are rotated. The gearing is such that both the L dial and its pointer rotate three times as fast as the S dial and pointer. All the usual calculations can be made with the spiral scale, as with the Boucher calculator, and the result read off on one or other of the three scale-sections. Frequently the point at which to read the result is obvious, but otherwise a reference to the single scale on the S dial will show on which of the three spirals the result is to be found.

_The K and E Calculator_, also made by the Keuffel and Esser Company, is shown in Figs. 27 and 28. It has two dials, of which only one revolves. This, as shown in Fig. 27, has an ordinary logarithmic scale and a scale of squares. There is an index line engraved on the glass of the instrument. The fixed dial has a scale of tangents, a scale of equal parts and a scale of sines, the latter being on a two-turn spiral. The pointers, which move together, are turned by a milled nut and the movable dial by a thumb-nut, as in Sperry’s Calculator, Fig. 26.

SLIDE RULES FOR SPECIAL CALCULATIONS.

ENGINE POWER COMPUTER.—A typical example of special slide rules is shown in Fig. 29, which represents, on a scale of about half full size, the author’s Power Computer for Steam, Gas, and Oil Engines. This, as will be seen, consists of a stock, on the lower portion of which is a scale of cylinder diameters, while the upper portion carries a scale of horse-powers. In the groove between these scales are two slides, also carrying scales, and capable of sliding in edge contact with the stock and with each other.

This instrument gives directly the brake horse-power of any steam, gas, or oil engine; the indicated horse-power, the dimensions of an engine to develop a given power, and the mechanical efficiency of an engine. The calculation of piston speed, velocity ratios of pulleys and gear wheels, the circumferential speed of pulleys, and the velocity of belts and ropes driven thereby, are among the other principal purposes for which the computer may be employed.

THE SMITH-DAVIS PIECEWORK BALANCE CALCULATOR has two scales, 11 feet long, having a range from 1d. to £20, and marked so that they can be used either for money or time calculations. The scales are placed on the rims of two similar wheels and so arranged that the divided edges come together. The wheels are mounted on a spindle carried at each end in the bearings of a supporting stand. The wheels are pressed together by a spring, and move as one.

To set the scales one to the other, a treadle gear is arranged to take the pressure of the spring so that when the fixed wheel is held by the left hand the free wheel can be rotated by the right hand in either direction. When the amount of the balance has been set to the combined weekly wage the treadle is released locking the two wheels together, when the whole can be turned and the amounts respectively due to each man read off opposite his weekly wage. The Smith-Davis Premium Calculator is on the same principle but the scales are about 4 feet 6 inches long and the wheels spring-controlled. Both instruments are supplied by Messrs. John Davis & Son, Ltd., Derby.