The Otis King Calculator is basically a
slide rule. This should not be thought to imply that it is a particularly
complicated piece of apparatus suitable only for people with special training;
the slide rule is really a very simple article requiring no special mathematical
knowledge for its operation, yet with it the drudgery can be removed from much
of the calculating work that has to be done in industry and commerce and by
students. The Otis King is even simpler than many slide rules because it
dispenses with special purpose scales. It is also more accurate than the
ordinary slide rule. These two points recommend it to many users.
Because of the cylindrical design of the Otis King it has been possible to
produce a very compact instrument. The scales are 66 inches long and carry many
more graduations than those of an ordinary slide rule. Consequently the user can
read answers comprising several figures directly from the scales instead of
estimating the later figures; the setting of the Calculator is also speedy since
much of the counting of graduations and estimating the value of parts of
graduations is eliminated.
There are two models. MODEL K is suitable for multiplication, division, proportion
and percentages. Besides solving these problems MODEL L gives logarithms, enabling it to be used for working
out powers and roots.
The Calculator consists of three parts, the
chromium-plated holder, the cylinder (which has a knurled top), and the cursor
(engraved with two white indicators). It will be noticed that the knurled top
has a notch cut in it. The purpose of this is to help the user to set quickly to
the "1" on the scale. The "1" is directly in line with the notch.
(Note. The outline drawings below show the correct settings of the
indicators to the scales, and should be carefully followed.)
INSTRUCTIONS
Multiplication and Division
MODEL K. This model has a scale from 1 to 10 mounted on the
holder. The cylinder also has a scale from 1 to 10 on its upper half and the
same scale is repeated on the lower half.
Example: Multiply 2 by
4.
Take the holder in the left hand and open the instrument gently
to its full extent.
Set the bottom indicator to 2 which
will be found about half an inch above the bottom ONE and
slightly to the right of it. (Remember to set to the line and the graduation,
not to the figure itself.)
Moving the cylinder by holding the
knurled top, set the middle ONE to the top indicator. The
middle ONE will be found in line with the notch on the
knurled top and about two inches below it. Do not touch the cursor when
making this movement.
Moving the cursor, slide the top
indicator up to 4, which is about one inch below the notch and a little to the
right of it.
The answer 8 will be read at the bottom
indicator.
Example: Multiply 4 by 4 (on Model K).
Set the
bottom indicator to 4 about one inch above the bottom ONE and slightly to the right of it.
Move the
cylinder to set the middle ONE to the top indicator.
(Remember that the cursor must not be touched during this
movement.)
Moving the cursor, slide the top indicator down
to 4 on the lower half of the cylinder scale.
The answer 16 will
be read at the bottom indicator.
Example: Divide 8 by
2.
Set the bottom indicator to 8 near the top of the scale,
about a quarter of an inch below the top ONE and a little
to the right of it.
Move the cylinder to set 2 about
1 1/2 inches below the notch and a little to the right of
it to the top indicator.
Moving the cursor, slide the top
indicator down to ONE at the middle of the cylinder
scale.
The answer 4 will be read at the bottom
indicator.
Example: Divide 16 by 4.
This is done by the
same method as in the previous division example, except that for the final
movement the cursor is moved upwards to the ONE at the very
top of the scale. (For the second movement the 4 on the lower half of the
scale on Model K can be used, instead of the 4 on the upper half of the
scale; in that case the final movement is to the middle ONE.)
MODEL L. This model
has a scale from 1 to 10 mounted on the holder. The cylinder has a similar scale
on its upper half which is used in conjunction with the holder scale for
multiplication and division. On the lower half of the cylinder there is an
evenly divided scale. This is used in conjunction with the holder scale to find
logarithms.
The instructions given for Model K, for
multiplication and division examples, also apply to Model L except where it
is otherwise indicated. They should be studied first as this may help the user
to locate the position of the numbers on the scales.
Example: Multiply 2
by 4.
Set the bottom indicator to 2.
Move the
cylinder to set the 1 at the beginning of the scale on the upper half of the
cylinder, to the top indicator.
Moving the cursor, slide the top
indicator up to 4.
The answer 8 will be read at the bottom
indicator.
Example: Multiply 4 by 4 (on Model L).
Set
the bottom indicator to 4.
Move the cylinder to set the top
1 just below the notch to the top indicator.
Moving
the cursor, slide the top indicator down to 4.
The answer 16 will
be read at the bottom indicator.
Example: Divide 8 by
2.
Set the bottom indicator to 8.
Move the cylinder
to set 2 to the top indicator.
Moving the cursor, slide the top
indicator down to 1 at the beginning of the scale on the upper half of the
cylinder.
The answer 4 will be read at the bottom indicator.
Logarithms (Model L only)
The lower half of the cylinder on Model L carries an evenly divided scale which
is used for finding the logarithm of any number as follows:
Move
the cursor to set the bottom indicator to 1 at the bottom of the holder
scale.
Move the cylinder to set 000 at the beginning of the evenly
divided scale to the top indicator.
Moving the cursor, slide the
bottom indicator to any number on the holder scale and read its logarithm on the
evenly divided scale at the top indicator.
(Finding antilogarithms
is of course the converse of the above. Having made the preliminary settings of
the bottom indicator to 1, and 000 to the top indicator, the top indicator is
moved to the logarithm and its antilogarithm is found at the bottom
indicator.)
Understanding the Scales
To the user who is unfamiliar
with slide rules, the graduation of the scales of the Calculator may appear a
little confusing, and the following notes of explanation may be
useful.
(a) No final "0"s and no decimal points are printed on the
multiplication and division scales. The user must insert these mentally, for
himself, as he requires them. Thus "57" can be used to represent "57", "570",
"57000", "5.7", ".057", etc.
(b) The value of the small graduations lying between the
larger ones against which numbers are actually marked, i.e. "101", "102", etc.,
can be determined by reference to the marked numbers. Thus between "101" and
"102" the scale is marked with ten graduations and these have the values of
"1011", "1012", "1013", "1014" and so on up to "1019". If it is desired to get a
figure or to read an answer between these small graduations the space must be
divided. Thus "10115" will be set by placing the indicator halfway
between "1011" (which may of course be read "10110", as explained above) and
"1012" ("10120"). With practice the user will find that he can judge other
values such as "10112", "10116", with a fair degree of accuracy.
(c) It will be noticed that the graduations are not of equal value throughout
the scale, and that the lower part of the scale contains many more graduations
than the upper part. Thus the scale from "1" to "2" occupies seven turns of the
spiral, from "2" to "3" occupies only about four turns, from "3" to "4" three
turns, and so on. The result of this is that at the bottom of the scale the user
can set and read numbers comprising more figures than those to which he can set
higher up the scale. Thus the graduations from "101" to "102" are "1011",
"1012", "1013", etc.; from "202" to "204" they represent "2025", "203(0)",
"2035"; and from "62" to "625" they represent "621", "622", "623", "624". It has
been shown in note (b)
how values between, say, "2025" and "2030" can be judged.
As has
been explained in an earlier note, if it is desired to read figures between,
say, "202(0)" and "2025", these must be judged by the user. For example, for
"2021" the indicator mark must be set to a point one-fifth
of the distance between "202" and "2025".
It may be helpful to the
beginner if he draws the portion of the scale on a piece of paper first, and
inserts the actual values that he requires, thus:
(or 200,
.002, etc.)
With a little practice on these lines any
person not previously experienced in using such instruments will find that he
can attain proficiency in using the Otis King.
Further Examples
The user is advised to study the
foregoing instructions first and to attain familiarity by working through the
simple examples several times. When they have been mastered he can proceed to
the following calculations. The instructions apply to both Model K and
Model L, but in the case of Model L it may sometimes be necessary to
use the bottom 1 on the cylinder scale and on other occasions the top
1.
(For the sake of simplicity the bottom white indicator will now
be referred to as B and the top white indicator as T.)
Combined
Multiplication and Division
Solve
6x4x9
7x5x2
(This type of
calculation can usually best be done by alternate multiplication and division of
the individual factors, thus: 6 divide by 7, multiply by 4, divide by 5,
multiply by 9, divide by 2. There is no need to take note of the intermediate
results and except at the end of the calculation it is not necessary to move the
indicator to 1.)
Set B to 6. Set 7 to T. Move T to 4.
(B now indicates answer to
6x4
7
)
Set 5 to T. Move T to 9.
(B now indicates answer to
6x4x9
7x5
).
Set 2 to T. Move T to 1. Read answer at B:
3.086.
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.
II.
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 proportion, viz. 214,
to T. 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 9.)
Percentages
I.
What is 5%
(a) of 162 ?
(b) off 162 ?
(c) on 162?
Set B to 162 (capital amount or quantity). Set 1 to T.
The instrument is now set to solve percentage problems involving
% OF, % OFF and
% ON 162.
(a)
Move T to 5 (rate %). Read answer at B: 5% of 162 =
8.1.
(b)
Move T to 95 (100rate %). 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.
II.
What % of 3735 is 4.54 ?
Set B to 3735. Set 1 to T. Move B to 4.54.
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 1 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 throughout a series of problems, the instrument may be set to the
constant terms, and the answers found by subsequent movements 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 3/32, 7/32, 15/32, 29/23.
Set B to 32. Set 1 to T (32 and 1 being the constants in this series).
Move B in succession to 3, 7, 15, 29 and read the corresponding answers at
T, viz. .09375, .21875,
.46875, .9062.
Money Calculations
In Sterling
calculations 1/2 new pence is set as 005. Thus the setting
for 22 1/2 np is 225. For
£3.01 1/2 np the setting is 3015; do
not forget the '0' in such amounts.*
I.
If 54 articles cost £39.225,
what is the price of 15? 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 £10.895.
II.
Find interest on £675 at 6 1/2% p.a. for
29 days.
(£675 x
6.5
100
x
29
365
)
Set B to 675. Set 1 to T. Move T to 6.5.
Set 365 to T. Move T to 29. Read answer at B:
£3.486.
In some calculations it may be found preferable to invert the
setting of the instrument and to work to the "1"s 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 UPPER CYLINDER SCALE. When this scale is 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 close cylinder in.
Move T to bottom 1. Set top 1 to T.
To open cylinder out.
Move T to top 1. Set bottom 1 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.
THE LOWER CYLINDER SCALE. Where involved
expressions occur above or below the line, the Otis King Calculators offer
valuable advantages over the ordinary slide rule, which, even if engraved with
log-log scales, cannot solve the following, whereas Model L will give all
powers and roots, fractional or otherwise, of all numbers without limit, and
solve any expression, however extended. The following expression is given as an
example:
1.0083.1x363 x 4000
6 x5260000 x
421.82
= .2495.
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 Calculator. The intermediate stage in dealing
with the above problem is to simplify it into the following:
1.025 x 3.98
x 4000
6 x 12.11 x
900.1
The process for effecting this is as follows:
To LOGARIZE (i.e. find the logarithm representing a
number).
Set B to bottom 1 of holder scale. Set
".000" of lower cylinder scale to T. Move B to number
(antilogarithm), and read mantissa at T.
To DELOGARIZE (i.e. to find the number represented by a
logarithm).
Set B to bottom 1 of holder scale. Set
".000" of lower cylinder scale to T. Move T to mantissa. Read
antilogarithm (number) at B.
To ascertain any Power or Root of any number
POWERS
Multiply the logarithm of the number by the
index of the power and take the antilogarithm of the product.
Example: What is
1.0083.1
? Log. of 1.008 =
0.0035.
0.0035x3.1 =
0.01085. Antilog. of 0.01085 =
1.025. Therefore
1.0083.1 =
1.025.
ROOTS
Divide the logarithm of the number
by the index of the root and take the antilogarithm of the
quotient.
Example: What is 363 ?
Log. of 63 = 1.7993. 1.7993χ3 =
0.5998. Antilog. of 0.5998 =
3.98. Therefore 363 =
3.98.
Compound Interest
Find the amount that £250 will become
in 14 years at 5 1/2% compound interest.
(a)
Set B to 1. Set 000 to T. Move B to 1055
(100+5 1/2%). Read log. .0232 at
T.
(b)
Set B to 232. Set 1 (beginning of upper cylinder scale)
to T. Move T to 14. Read 325 at B.
(c)
Set B to 1. Set 000 to T. Move T to
.325. Set 1 (beginning of upper cylinder scale) to
T. Move T to 25. Read 528.5 at B. (Answer:
£528 10s.)
Approximation Method for finding Square Roots and Cube Roots
without the use of Logarithms
The following method may be
used for finding approximate square roots and cube roots on
Model K:
Example: To find the cube root of 9.
Estimate it as, say, 2. Work out 2x2x2 on the
Calculator, giving 8 at bottom indicator. Keeping cylinder in same
position, move bottom indicator to 9. Read 225 at top indicator. Note
difference between estimate and 225 = 25; divide by 3 = 83, and add to original
estimate = 2083.
Example: To find the square root of 87.
Estimate it as, say, 9. Work out 9x9 on the Calculator,
giving 81 at bottom indicator. Keeping cylinder in same position, move
bottom indicator to 87. Read 9665 at top indicator. Note difference
between estimate and 9665 = 665; divide by 2 = 333, and add to original estimate
= 9333.
Finding the Decimal Point
In common with all slide
rules, the Otis King gives answers which do not show the position of the decimal
point. The simplest way of deciding where the decimal point comes is by
inspection and for this method it may sometimes be helpful to make a mental
calculation with approximate figures. Thus 11.03 multiplied by
20.45 gives 2257 on the Calculator. It is roughly 10
multiplied by 20, which equals 200, so the decimal is placed after the third
figure 225.7.
Where the calculations are too
involved for the above method to be used, the decimal point can be determined by
the following rules, which apply both to Model K and
Model 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.
Calculations involving Multiplication and Division
RULE III.
Two methods may 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 then 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 then 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.
Results of various steps in calculation = -1 + 1
+ 1
= +1
Number of places in answer
= +3
Answer = 141.14.
Notes on determining position of Decimal Point
Note
1.
Thus:
5430000.00
(7 figures are to left of
decimal point)
has +7 places
81.2
(2 figures to left of decimal
point)
has +2 places
2.
0.45
(no figures to left of decimal
point, and no noughts between decimal point and first figure other than 0,
to right of decimal point)
has 0 places
0.0421
(one nought between decimal
point and first figure to right of decimal point)
has 1 places
3.
Left of decimal point
Right of decimal point
No. of places
5430000
.
+7
674
.
+3
81
.
2
+2
7
.
82
+1
0
.
45
0
0
.
0421
1
0
.
00675
2
4.
MULTIPLICATION. RULE I
When in the third movement of a multiplication calculation
the cursor is moved downwards the number of places in the product
(i.e. figures to the left of the decimal point) is equal to the sum of the
number of places in the two terms of the calculation. If the cursor is
moved upwards the number of places in the product is one less than
the sum of the number of places in the two terms of the calculation.
5.
DIVISION. RULE
II
When in the third movement of a division calculation the
cursor is moved upwards the number of places in the quotient is
equal to the number of places in the dividend minus the number of places
in the divisor. If the cursor is moved downwards the number of
places in the quotient is one more than the number of places in the
dividend minus the number of places in the divisor.
In summary form, the rules for finding the position of the decimal
point are:
When the cursor moves up, in multiplication, the result
has m+n1 places.
When the cursor moves up, in
division, the result has mn places.
When the cursor
moves down, in multiplication, the result has m+n
places.
When the cursor moves down, in division, the result has
mn+1 places.
The rules are expressed by the following diagram which it is suggested the
user should cut out and fix to the cursor of his calculator with adhesive
transparent tape.
X χ
m+n1 mn
X χ
m+n mn+1
CARBIC LIMITED 54, Dundonald Road, London,
S.W.19 Printed in England
Notes:
Only approximately (logarithmic scales !), but the
nonlinear correction would be way less than the width of a mark scale, so
linear interpolation can be considered to be exact for all practical purposes.
In the earlier (pre-decimalisation) version of the
manual, this sections reads: "The Sterling items must be reckoned as decimals
of pounds, shillings or pence as best suits the problem.", as in the even
earlier 8-page
instructions. (back)
This Otis King Manual has been HTML'ized by Andries
de Man from a copy provided by Dick Lyon,
courtesy of Ray Hems. Thanks to Dick Lyon for proofreading and discussing this
page.
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