Otis King's Patent Calculator — instructions (Models K and L)
CALCULATOR.
Outline Drawings of Calculator, showing correct
setting of arrows to scales.
ALWAYS SET THE ARROWS
TO THE LINES—NOT TO
THS NUMBERS
ALWAYS SET THE ARROWS
TO THE LINES—NOT Ta
THE NUMBERS
BRITISH MADE THROUGHOUT
CARBIC LIMITED
54, Dundonald Road,
LONDON :: S.W.19.
GENERAL DESCRIPTION.
The Calculator consists of three parts :—
(1) THe HoLper : on which are mounted spiral
(2) THe CyLINDER logarithmic scales.
(3) Tue Cursor (or Indicator), carrying at each end an
engraved arrow, or pointer, which can be set to any
mark, or to which any mark can be set.
DETAILS OF SCALES.
Model K.
ee
Tue Howper carries a logarithmic scale 66 in. long,
from | to 10, graduated as follows :—
By -001 from 1 to 1-12 By :005 from 2 to 4.
OUR yt UZ ge PORE) | agty hee pa (OF
In order that the range of the scale may be unlimited.
decimal points are not marked. Thus the reading 212, for
instance, also serves for 2120, 21200, 21:2, 2:12, etc., etc.
The position of the decimal point in the answer is found by
inspection or by the application of the rules given on pages
7 and 8.
Tue CYLINDER carries two scales in series which are
identical in all respects with the above. The point at which
these two scales meet half-way up the Cylinder is denoted by
the word “ Ong,” and is referred to in the following pages as
the “ Mippie Unity,” or “* Unity.”
Model L.
Tue Houper ScaLe on this Model is similar to the
Holder Scale on Model K.
Tue CYLINDER carries two scales, of which the upper one
is identical with the Holder Scale, and operates with it for
Multiplication, division, &c. There beg no Mippie Unity
On this scale, when calculations of this nature are performed
on the Model L, the Unity (one) at the top or ‘bottom of
the scale, as may be necessary, is used in its place.
The lower scale is graduated from 0 to 1, in 2,000 equal
divisions of -0005 each. This is used in conjunction with the
Holder Scale to give direct readings of the logarithm of any
number, by means of which roots and powers are obtained
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METHOD OF OPERATION.
The following sequence of movements should always
be adhered to in operating the Instrument :—
Movement 1.—Take the Hoxper in the left hand, and open
Instrument gently to full extent. Move
Cursor to set bottom arrow.
Movement 2.—Move CyLinpeR so that number on scale is
set to top arrow (not the arrow to the number).
Do not touch Cursor while moving Cylinder.
Movement’ 3.—Move Cursor to set either arrow as required—
the companion arrow will then point to the
answer (see following examples).
GENERAL RULES.
The following abbreviations are used to render the
descriptions more precise :—
B = Bottom Arrow or Pointer on Cursor.
T = Top arrow or Pointer on Cursor.
Middle Unity on Model K.
Unity = < Available Unity on Model L (see previous
page).
To Muttipty. To Divine.
1. Set B to Multiplicand. 1. Set B to Dividend.
2. Set Unity to T. 2. Set Divisor to T.
3. Move T to Multiplier. 3. Move T to Unity.
4. Read Answer at B.
Problems of Proportion, Practice, Percentage, &c., involve
either one, or a combination of both, of these operations.
EXAMPLES.
The following instructions apply to both bes
K and L.
Multiplication.
Multiply 115 by 23.
Set B to 115. Set Uniry to T. Move T to 23. Read
answer at B: 115 23=2645.
Division.
Divide 833 by 17.
‘Set B to 833. Set 17 to T. Move T to Unity. Read
answer at B : 833—-17=49.
Combined Multiplication and Division.
6x4x9
Solve 775 x2
Set B to 6. Set 7to T. Move T to 4.
6x 4.
(B now indicates answer to —=—).
if
Set 5 to T. Move T to 9.
6x4x9,
Tse
Set 2 to T. Move T to Unity. Read answer at B :—
3-086.
(B now indicates answer to
Proportion.
I. Solve 12:7:: 16: x?
Set B to 12. Set 7 to T. Move B to 16. Read
answer at T. 12:7 :: 16: 9-333.
Il. Solve 18:4::x:53?
Set B to 18. Set 4 to T. Move T to 53. Read
answer at B. 18: 4:: 238-5: 53.
III. Divide 8975 in the proportions 83 : 79 : 33: 19.
Set B to 8975. Set sum of required proportions,
viz., 214, to T. Without altering the setting of the
Instrument, move T in succession to 83, 79, 33, 19,
and read the corresponding proportions at B, viz.,
3481, 3313, 1384 and 797. (On Model L this
calculation necessitates ‘“‘ closing in” the Cylinder.
See page 6.)
x Percentages.
I. What is 5% (a) of 162?
(b) off 162?
(c) on 162?
Set B to 162 (capital amount or quantity). Set
Uniry to T.. The Instrument is now set to solve
percentage problems involving % Or, % Orr and
% On 162.
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(a) Move T to 5 (rate %). Read answer at B :—
597 of 1620.1.
(b) Move T to 95 (100-rate %). Read answer
at B: 5% off 162=153-9.
(c) Move T to 105 (100+-rate %). Read answer
at B: 5% on 162=170°1.
Als II. What % of 3735 is 4-54?
Set B to 3735. Set Uniry to T. Move B to 4:54
4 Read answer at T : °12155%.
III]. What is the percentage of profit on cost where
goods purchased for £5,760 are sold for £9,420 ?
Set B to 5760 (capital). Set Unity to T. Move
B to 9420 (selling price). Read answer at T :—
163:5. Percentage of profit=63:5% (163-5 - 100).
Constant Factors.
I. In cases where one pair of factors is repeated through-
out a series of problems, the Instrument may be set to the
constant terms, and the answers found by subsequent move-
ments of the Cursor only.
In Percentage Example I, for instance, the Instrument
being set to the constant terms 162: 100%, any percentage
of, off or on 162 will be shown at B when T is moved to the
relative figure, e.g., Move T to 45. Read answer at B:—
45% of 162=72-9. Move T to 126. Read answer at B :—
26% on 162=204-1, and so on.
II. Decimalise 4, %, #, 3%.
Set B to 32. Set Unity to T (32 and | being the
constants in this series). Move B in succession to
3, 7, 15, 29 and read corresponding answers at T,
viz., (09375, -21875, -46875, -9062.
Sterling Calculations.
The Sterling items must be reckoned as decimals of
pounds, shillings or pence as best suits the problem.
I. If 54 articles cost £39 4s. 6d., what is the price of 15 >
(a) Set B to (£) 39-225. Set 54 to T. (The cost
of any number of articles at this price can now
be obtained by moving T to the number
required.) Move T to 15. Read answer at B
15 articles cost £10895, i.e., £10 17s. 11d.
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Alternatively :-—
(b) Set B to 784-5 (shillings). Set 54 to T. Move
T to 15. Read answer at B:—15 articles cost
217-9 shillings, i.e., £10 17s. 11d.
II. Find Interest on £675 at 64% = a. for 29 days.
6:5
Oey? aa Xap
Set B to 675.. Set Unity to T. Move T to 6:5.
Set 365 to T. Move T to 29. Read answer at
B: £3-486, ie. £3 9s. 82d. (If preferred, multiply
further by 240, and read answer at B: 836:75 pence
ie. £3 9s. 83d.)
In some calculations it may be found preferable to invert
the setting of the Instrument and to work to the Unities on
the Holder Scale instead of to those on the Cylinder Scale.
In this case, the answer is of course read at the Pointer
opposite to the one indicated in the foregoing examples.
Model L.
THE _Upprr CYLINDER ScaLeE.—When this Scale 1s used
in conjunction with the Holder Scale to perform the types of
Calculations described in the preceding pages, it will be noted
that upon occasion the Cylinder becomes closed in or opened
out too far for the Pointer on the Cursor to be moved to the
required figure. In this case proceed as follows, without
altering the setting of the Instrument :—
To Crose Cytinper In. Move T to bottom UNity.
Set top Unity to T.
To Open CyLinper Out.—Move T to top Unity.
Set bottom Unity to T.
The Pointer can then be set to the required figure and the
calculation completed. This operation may be performed
during any calculation and does not affect the process or
answer in any way.
Tue Lower Cy.inper Scate.—Where involved expres-
sions occur above or below the line, the Otis King’s Calculators
offer valuable advantages over the ordinary slide rule, which,
even if engraved with log-log scales, cannot solve the follow-
ing, whereas Model L. will give all powers and roots, fractional
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or otherwise, of all numbers without limit, and solve any
expression, however extended. The following expression is
given as an example :—
10088? x */63 x 4000
6 x5/260000 x 421-8
All involved expressions must be replaced by their
numerical value before the problem can be dealt with, and
this prior process is, of course, common to both the slide
rule and the Otis King’s Calculator. The intermediate stage
in dealing with the above problem is to simplify it into the
following :—
= ‘2495,
1-025 x 3-98 4000
6x 12:11 900-1
The process for effecting this is as follows :—
To Locarize—(i.e., find the logarithm representing a number).
Set B to bottom Unity of Holder Scale. Set ‘‘ -000”
of lower Cylinder Scale to T. Move B to number (anti-
logarithm), and read mantissa at T.
To DerLocarize—(i.e., to find the number represented by
a logarithm).
Set B to bottom Unity of Holder Scale.
of lower Cylinder Scale to T. Move T to mantissa.
antilogarithm (number) at B.
Set “ -000”
Read
To ascertain any Power or Root of any number.
Powers. Roots.
Multiply the logarithm of
the number by the index of
the power and take the anti-
logarithm of the product.
Example :—What is |-008%1
Log. of 1-008=0-0035.
0-0035 x 3-1=0-01085.
Antilog. of 0:01085=1-025.
Therefore |-008*-4=1-025.
Divide the logarithm of
the number by the index of
the root and take the anti-
logarithm of the quotient.
Example :—What is °\/63.
Log. of 63=1-7993.
1-7993 ~3=0-5998.
Antilog. of 0:5998=3-98.
Therefore *\/63=3-98.
TO DETERMINE POSITION OF DECIMAL POINT.
The following rules apply to both Models K and L :—
A number having n figures to the left of the decimal point shall be
designated as having +n places.
A decimal number having n cyphers to the
right of the decimal point, between the decimal point and any number other
than 0, shall be designated as having —n places.
To Determine Position of Decimal Point (continued)
Thus the numbers—
5430000, 674, 81:2, 7:82, 0-45, 0:0421, 0-00675;
have +7, +3, +2, +1, +0, —I, —2 places
Multiplication.
To find the number of places (p) in the product (P =X x Y).
Let X have m places, and Y have n places.
Rute [, p=m+n or pem+nel,
(a) When the result is below the original setting p* m+n.
Example : 3x4 (m=1; n=1).
Set B to 3. Set Unity to T. Move T to 4 (This is below setting.)
“p=m+n=2. Answer =12.
(b) When the result is above the setting, the product has m--n-1 places
Example: 3 x3 (m=1; n=1).
Set B to 3. Set Uniry to T. Move T to 3. (This i is abcns sete)
p=m+n-l=1. Answer =9.
Division.
To find the number of places (q) in the quotient (Q -%).
Rue I. q=mn or q=m-n-+l as follows :—
(c) When the result is above the setting, the quotient has m-n places.
Example: 34 (mel; nl).
Set B to 3. Set 4 to T. Move T to Unrry. (This is above setting.)
qemn=0, Answer = 0°75,
(d) When the result is below the setting, the quotient has m-n+ 1
places.
Example : 5+ 4 (m=1; n=1).
Set B to 5. Set 4 to t Move T to Unity. Read answer at B. (This
is below setting.) ..qg=I-1+1=+1. Answer=1-25.
Calculations involving Multiplication and Division.
Rute III. Two methods inay be used in working out, complex problems
involving both multiplication and division. They are :—
(1) Taking numerator and denominator alternately.
(2) Taking all the numerators'first and thé dividing consecutively by the
denominators.
Of these two methods, only the latter can be used if the position of the
decimal point is required. If the other is used, the decimal point must be
found by inspection.
First multiply consecutively the series of factors in the numerator and ther
divide consecutively by the factors of the denominator.
Take the algebraic sum of the places in the factors of the denominator
from the algebraic sum of the places in the factors of the numerator, and to
this result add the algebraic sum of the results obtained from the application
of Rules I and II to the several steps of the problem.
Ries 432 x32-4 x0-0217 x 0-98
ample * 9-00000621 «412000 x 0-175 x4-71
No. of places in factors of numerator =3 +2+(-1)+0= +4.
=141-14...
+s 9» «9 ~«99~—oy~—p GEnominator = -5+6+0+1 =+2.
; _ Difference = +2,
Results of various steps in calculation=—I+1+1 ..=+1.
Number of places in answer. = +3,
Answer = 141-14. .
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