Fowler's "Universal" Calculator — Instructions, with Supplementary Chapters on Logarithmic Calculation and Circular Calculators
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POOVWE ERS
“UNIVERSAL”
"GALCULATOR
BOORBReSBReBBeBAIMoe JBBSBeErVOB
INSTRUGTIONS
with Supplementary Chapters
on Logarithmic. Calculation,
and Ctreelam Calculators, in
addition to jumerous worked-
ou, cxamples.
by
WILLIAM H. FOWLER,
Whit.Sch., M.tast.C.£. M.1.Mech.£., ete.
FOWLER & Co. "t3e""
Hameson Streer, SALE, MANCHESTER
"Phone? SALa, 1433 "Grams > Scientific, Manchester
SBEEVEBOSEBRBSVoRNs
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Postage ;
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Abroad 2/.
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IOUN NAPIER
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Society of Edinb ; 1 which city a most interesting
tion of personal cs 3 wstruni
facilit
Althon
IGIA, thr
second visit thy
and devices
1 together, id a
the publication of
i by E. M, Hors
ving an mstructive
Ompanied with a
until
that
: Its
profound
nent, amongst
mNntry was
iam College,
nportance
to Fdinburgh
er’ ersonal
one of the
together
me without
ave under
nder nobody
pent a month at Merchiston, returned for a
following summer, with the further intention
1ORN NAFPIAR
of making a third visit the next ye
the untimely death of } er.
8 prevented by
1 sdmiration —§ ff i - ntion
. great that he beg: f onlat ) et tables
which is essentially @ By own use, and to this labour
ves to he
s friend,
devoted some vears of ht ' : ite that d
borne in mind wh
work was as at as at me, The t sb tables were printed
in 1617, and rou : nomer, to an adnuration
second only to
wher
mathe
not
imenta
ality of
lerstood in
Ivanced
efinition of the
enhanced when
his day was
cern conception
exist until long
1 the hol
conception ¢ fractional and
tive values
ithm i ased o1 ¢ risor the
pier's conception of
rom A along
UPPose. a
id point O {
neth 7) at the bh the same ocity
but to move not unifonmiy but wit) velocity, at any pomit
D, proportional to the distance D Y
Lf now F reaches C when © reaches D, then Napier defines
JosePpH Wiitwo} BART
A Pioneer of Mechanical Accurac ind = =Eefi
SIR JOSEPH WHITWORTH, Bart.
A Pioneer of Mechanical Accuracy and Efficiency.
[he increasmg adoption of Napier’s logarithmic syst¢
{ computation s been due in large measure to the ppl
ition to it of graphic princi
for figures, and the f mechanical methods of manipn
lation or Of making comparisons between one ile and
another, coupled with the fact that since man fix counted
vith his fingers, the decim tion has become almost
universal for purposes sckoning its
of convenience for fi
t cc being ad
; obvious lack
to a duo-decima
ted in speci 3 for commercial r 10NS
Engineering has, not inaptly, been descril is the art
néasurement, either of matter, space, force or time, «
ind when extremely accurate measurement is requir
art is cap : gy man ingenuity mid
skill to the utmost,
construct
Rough approximations often do not present much difficulty
but when we get from the domain of rough estimation
m ired accuracy difficulties increase vith amas
rapidity.
Che truth of this is impressed on ry practical ens
who in the pursuit of some ideal of perfection is brought
into close contact with the imperfections of his materials
ind the difficulties of fashioning them to his ends. Up to a
point they yield to his skill and « ‘rience, but as higher
legrees Of perfection are sought, and stores of existing
knowledge are exhausted, he finds that to make progress
he must contribute to them by personal research, and so learn
how much cach generation is indebted to the past, and how
few are the really great pioneers in any field of human
knowledge.
It is this consciousness which impels the writer to pay
a tribute of respect and admiration to Sir Joseph Whitworth
whose high standards of mechanical perfection and workman
ship did so much to raise the reputation of British engineeriny
during the last century.
ples the substitution of scales
MK JOSEPH WHITWORTH, HART,
4 perfect ply sical stramht hue, a plryst al tL surtice,
accurate screw, seem trivial details to the unthinking minc
but they, are the foundations of mechanical perfection and
standards of human measurement of all other thine:
It is easy to express accuracy to any degree in figures as an
abstract idea; it is impossible to do so in terms of matter in
the way all measurements in ? practical world. have to be
frnally translated. One is : y al, the other is an actual
construction and when Sir Joseph Whitworth, the great
Manchester Engineer, showed that dimen ns of the order of
one-hundred-thousandth of an inch id t physically
asured and applied in a work lid. a great deal mor
than talk about it
yineer, Sir Joseph
ind in this respect
is munilicent
scholarships
Apart from | pre-eminen
Whitworth a great public
his memory specially appeals "
founding and endowment of th cheme of
bearing his name, and by the aid of which a number of young
engineers in workshops are annually enabled to enter upon
a university course of scientific and technical education, and
to qualify for the highest posts in the engineering profession.
eers tor
it may be asked : What has the construction of an accurate
physical straight line, or standard screw. or flat surface to do
with the construction of a logarithmic scale Pais ! That
while the value of a logarithm may be expressed to figures
with any degree of refinement nd used as such withont
difficulty in an arithmetical operation, it is otherwise when
used as an actual physical dimension on a logarithmic scale
for purpose of mechanical measurement and calculation
under varying conditions of temperature, moisture, ete
A logarithmic scale it must be remembered is not a uni
formly divided one. No two adjoining divisions are alike,
ind] when several such scales are required in circular form
with the added feature that they must be capable of strict
concentric rotation in conjunction with three» radial lines
zero, a datum, and a cursor), each having independent
dead-beat movement and capable of being read with little,
Sk JOSEPH WHITWORTH, HAR
or nG, parallax, it will be manifest that the problem of mecan-
“al construction ts not an easy one
of cours unpossible to bring the
accuracy Of a logarithmic scak up to the
even figure table of logarithms, and
rehlnement
Standard
here such accuracy
is require as it is tn certain tronomical
and pl
calculations, the printed tabk ith its tons
of definite values with its longer
always be preferred,
t
arithmetical processes will
The convenience and ; of
chanically divic
for its lack of accuras
tor the calculations ¢
manipulation
wever, atones largely
Fits universal adoption
j \t the same time the potential
errors of construction empt ‘
st ndards of
y of operation
ccount
iasises the necessity of
), Lo insure correspondins
Chis has throughout been the
the d
. aim of Messrs. Fowler in
esign and construction of their Circular Calculators,
and they trust their efforts have not proved unworthy of
their aims :
Separate Copies of this
INSTRUCTION BOOK
may be obtained.
Price 2/6 each, Post Free 2/9.
FOWLER’S
“ UNIVERSAL ”
CALCULATOR.
Fowler's “‘Universal’’ Calculator, like the
Waistcoat pocket instrument consists of a series of concentru
cirewlar scales, logarithmically divided and mounted on a
dial capable of rotation by a thumb nut outside the con
taining case, The scales are equipped with a fixed radial
datum line, and a radial cursor line rotated by a second
thumb nut the rotating scales, cursor line, and operating
mechanism are din an airtight electr plated meta!
containing case, fi with a s3 face so that the scales
are always kept clean, and the instrument 1s
from external injury in a handsome leather wallet which
fits easily into a side pocket Che larg © of the “Univer
enables all the scales to be mounted on one dial, and so of
bein ynchronised and read concurrently It also permit
of the use of larger fivures and easier reading an advantage to
persons of weak ovyesight Another impe¢ ant feature
that the seales are longer and admit of liner graduation and
more accurate reading Chis is specially noticeable in the
‘ Long-Seale ’ which gives a lenght of 194 ins compart
with 10ins, in the ordinary ide-rule and permits of cé
culations being made to three, and sometimes four
ficant figures. The motions of the scales. ai .
effected by gearing adjustment an be m with
ease and nicety. There is none of the objectionable stic
or slackness often so troublesome with the straight slick
owing to changes of climate or temperature, nor is there
iny of the “end switching ’’ often necessary with that
frstrument when working with the full length scale
which induces many use work bitually with the
length. In circular calculators the scales ontinnon
there is no half length.
well-known
preserve TF
LOC RLTHMS.
What they are, What they will do
thin are certain numbers, first disex red_soby
th great 1 1athemi ican : apier, which allow the
irithmetical perations of multiplication and
ordinary
1 by) ‘ npler operations of addition
division to be replace:
and subtraction They l allow difficult calculations of
powers and roots and t mg of trigonometrical ecal-
itions amply effect th great i
with great saving of time ar
mtal labour ”
Definition of a Logparithm,—A ithm (usually written
g ‘ cS i : p between any given
the base, and may
base must be raised
but 10 is Usually
bec se it is the base
in ordinary arithmetical
expression loo y \ So ' ¢-the log of 100
because I would have ; ¢ 2nd power to
equal 100, the base 10 bein; ve understood unicas the
contrary 1s Stated
Considering suc WVO POWCTS 10, the relation between
ton Looo 10,000,
2 4 et
or any number
mder 10, the lo
100
1,000
10,000
te
chmal quality
LOGARITHM
Characteristic and Mantissa of a Logarithm,—Th«
. log is called the characteristic The
called the mantissa Obviously. number
xact multiples of 10 have no-manti
litthe f the preceding shows that for
iy number sis
distinguish numbers
tic is marked
characteristic an be read
hove from numb
character
valitv being indicated by
cterish
often
m when 1 s Ul
itself may : al therefor
The followings help to clear the
the mumber
posit c
( the number the decimal equival
, : ;
positive quantity, great in © but less than |
ebraic demonstration
6020600
expr antty
sbove
rinus quantity
itrve quantity
expressing it in the second manne
t
1249387 from 1 and adding 1
written | -8750@13
A at - 1249387
the Characterrstic
Mantissa or decimal pa
of course, in both ca
Further, the Mantissa expressed thus is the same | i
075, -75, 5, 75, or any multiple of 10 thereof, i
whether the number be more or less than O
\ table of logs so constructed is greatly simplified
The only part of the logarithm that chang
Characteristic, and its magni le as well as its
determined at }
The M of the logarithm is always sitive, an
+ the same so long as igits composing the number
"
ms is shown in the following table
seen, t only changes that occur are the
I decimal point in the number and the sign of
of the Charact tic
Number Logarithm
750 2. 18
OG613
‘8750613
‘8750615
‘O00 3 -8750613
Che preceding will enable the learner to understand
the follow rules regarding thé characteristic
For numbers ¢ i 1 | the characteristic
less than the digits in the numb ind is positiv Z
For nu ers less than | the characteristic is one morc
than the number of cyphers following the decimal
pot, and 18 ne tive
The mantissa of a logarithm is always positive, and is
the only part of a lo ithm given in tables of logarithms
Confusion in the Ynind of the learner arises sometime
from a failure recognise that the zero position on the
logarithmic scale does not coincide with the zero positior
on the arithmetical scale, of notation, and that when
number falls below 1 the characteristic takes a minus
LOGAVETIMS
» that the mantissa may be al ys positive and
same order
same value for
dependent of the position of the cl cim al j (int
Calculation of Mantissa.—While the characteristic,
arithm can be determined by a m
the determination of the mrantssa
il of aithmetical labour
is a big task
of the methods
the article in
the same. digits in the
or integral
inspection
or decimal part m
and the compilation of «
Space docs not here per
employed Those interester
the /rucyck dia Brit
with the subject
How Logarithms Save
verties Of logarithm
; oA) r 4 : spuci works dealing
saying
Labour, -The labour
t illustrated
multiplication can
subtraction After-
Logarithm
4 : i forming addition
ite “ : ij . 1 r ‘ \ \ tN yn binahons
of circular
calculation
il example
t
urther
movements mental ¢
{t « be shown algebra that if \ and B
factors requiring multiplication, this ye nips se
reduced to addition by the aid of ble of Los arit MD
Since the log of the product uum of th
4 the factors, that is
low (sv
lesired to divide A
reduced to subtraction, that is to
log log A
. |
r | ber A raiser
lesired’ to find the vlue « 1 Nuinbe assert
en power say # : r ee
, ’ t 1woODs '
roots are only powers im decimal! t Tractional or
‘ ~whice os rhe
aperation, by the tid of logarithms, is recuced t my
multiplication, since
lowe At nw low
LOGAKITHM
Taking vow the tr doh cf these pri
not familiar
adding them together we 2 the following results
it may be explained that
A\2 means A square d \d means squ
4% means A cubed \. means cubk ci. Lor'6 a a e090
a“ Lor 3 8020
L-3010
Comparing these logs im the table we find that
\
the Fourth root of A cubed :
| -25! IS 1 -3010—<log20 11-4471 log
Multiplication ha been reduced to dition
} Pp wer
In other ¥
nd it should be added, results accurate
ures could be obtained with just as little
with a more complete table
; mean nth re { the mth power of A to seven § lificant
eva uation of ors ¢ numbers. the Reciprocal mental labour
ler’s Calcul often proves extremely useful
n * mi - 3 yi "« “ nts ¢ Saving time and labour Division.—Let us now take two examples of division
wake ri ' <a numbers 1 to 30 Again very simple, owing to tl small limits of our log
"4 ; 5 table. and the results of which are obvious at sight, viz
I 0000 ; ‘ ‘ ‘ 28.07 =4 06: --]
s0L0 ~ pA 2
$771 : } Subtracting m cach case th the divis
6020 ; : ‘ “on log of the dividend we get
6990 5., 178 1-447)
7782 oe T= 8450
8 a : og 3 6020
is) 9549 los ey j Comparing these results with our table we fined
10 1-000 ‘ 2 30... 1 a 6020—log 4 1 -11389—log 18
: T > be , I ’ 1 or cer } :
Examples of the Use of arithms.—The relationship Uhus the operation of Division is reduced to Subtract
wetween numbers and their logs and s bearing on the
arithmetical operations multiplics division and
Powers and Roots of Numbers.—We will now give
ne or two illustrations showing the application of logs
aluation of powers and roo may demonstrated with to the evaluation of powers, or roots of number
Ti appended short table of 3 from 1 to 30 Ex.
The brevity of the table limits, the demonstrations | t lf A 34 “find
for illustration We know log
i and from table
Therefore log
ery simple examples, but they suffice
Multiplication.—Tuaking first the application of lo
to the operation of multiplication, we have to choose exampl«
to come within the range of the table, and the results in each
bvious Referring to table we find 1°4313 ts log 27.
6 ; 20 jx 4=:28 Therefore we say 35
bYUSY
find 6989
miven Aare uniple
fractional inches
uy
" tiny
present no chit ulty
yould only be
table and to tak e ths ofthat valuc
vould be the log
ould be the answer required
The above examples suffices
metical labour can be reduced 1
mmcceeding chapters it will b
Logarithmic Seales
t
tums in
rv, and the
but
por
umber correspar
rreatly
logarithms
by the
LOGARITHMIG SGALES.
Che Mechanical Operation of Logarithms.
Straight Logarithmic Scales.—The earliest form of
jowarith mic e for calculating purposes > the
sliclo-rile 4 il } ) t f two strat
the logarithms
1 ' § > &§, 9, 10, and with the int
these numbers ubdivided im
Straig
similarly grad sont
a logarithmic manner
pace will permit Tine
slide past aud o r, and are
lent sliding Ca ght angles
irrangeead
mean
the logarithmic value
med mechanically, Thi £ repre
itions of a table o gar sut as such printed tables
available «givin ies of logarithms t
subtractions
decimals r numbers from 1 to LOO,000
t Ho contrivance of this |} unless
vive calculated results with the same
those obtainable by ordinal irithmetical meth
he aid of a printed table
Nevertheless, mechanically operated scales e results
accurate enough for all wdinary caleulatio anc the
iormous advantage of being able to do thi rapidly sirnple
inanipulation of a cale, with little or no ‘ntal labour, ha
made the logarithm ( tire-savit
1 nsable instrument calculating
to perform
lortur y the mathematical
sables < mparatively short
reading though the Iength of the
factor when accuracy is essential
It has been shown that the
sarithm
wcourals
Sci 8 tar
iracterishe o1 tegral prrt
{ the Jogarithm of any number can be
rime numbers 1, 2, 3, 4, 5, 6 be I
mntergers or any multiple of 1) thereof
to the mantis
read at sight, and
7 10 may represent simpli
It iS only in respect
rithm that the valuc
ifferentiated from that of a printed table,
r decimal part of a lo
ia &
rit
A LOGARITHMIC
circilar Ones on account of theu : oon
the uifficulties of « structing a jocurate
pocket iistrument af s kind at ~asonabh
and) until Messr wier & Co. turned their
biect they had nevér been isfactorily overcom
by suitable construction an adequate length of s
proy ided. tu allow this to be largely overcome
Defects of the Straight Slide-Rule.—The straight
ide-rule has many rent detects
Cireular Logarithmic Sc
op rated exactly n the same ay as stra
scales, that is vith one ile rotat
yncentric OF an two ing equipp
It is not capable fiording readings of much practical
value unless it is about 10 in. (254mm eT
is inconvenient for the pocket,
ide is liabie to get dirty and stick
and make setting and idling troublesome
Changes of climate or temperature produce errors
entric circular scale
The res n operation often cannot be read unless the
lide is drawn out completely and reversed or “* end switched operation of ack m and subtraction being
tl the aid of < <a adial Datum-line ™m conjunc
Chis causes confusion and waste of time, ancl kk ; many 1 of ,
rotating racial Cursor-lint
ers to work habituall ; half-leneth , whict
~iuces the accuracy of the instrument, because k rithmic Each method has intace hut
sraduation is not uniform, and’‘the intervals between thx is the best, a is aclopted in tl
prime numbers of the basic scé (No. 1) diminish rapidly Calculators rmits of hed | fa ‘Long ule
sone progresses fron 10. This is shown by the following ‘xtending + d several less liability
figures which represetit percentage of the total length of conh setting factors sults \
f the dle Occupied by respective intervals ulation Datum
raight
Per cent ot ; m i j Om} therefore
cent ¢ ’ Barter
total lent! 7 nvent : es i 10)
Intervals ' : 3! 10)
coincide
No “end switching : ‘ ; i auing,
- | limut-
fhe distance bet : Wi ¢ séen, is only
venth « it bet 0 ’ cannot be subdi : :
intensified and ther » half-len
to the sa i ¢ enience of this
ations of subdivision and accurate
by : c subdivision of the intervals must be of a
decimal character and therefore by 2. 5 or 10 On tt ontrary, the length of
by extending it round the circum
It. is “¢ from these fact \ kk ithmic scales
n spiral fashion, © iture that |
for multiplication and division require ngth to allow
ble graduation and of setting and reading, Sacrifig:
of lengtl means sacrifice of accuracy and clarity. How greatly this contributes
compared th the dinary ‘(Oin. straight slide-rule,
be judged from the follov parti
of in Fowlers Long-scale Caler
to accurate calculation,
These defects of the straight logarithmic slide-rule ar
recognised, and many efborts have been made to prodn mas
TH
Fowler’s Circular Long-Scales.
* Long-scal nstrument
liameter, the lLong-ac
my 2¢
tleneth of S04
strum
ength
tr
ent
of 50 an
ument 31 in.
ith of nearly
+1
conveTuenths
tdi curs are Operat
ind raputy, even mn a
ually well in any clima
truction secures a de
VIOUSI' 1
FOWLER
SRSAL *' CALCULATOR,
Description of Scales,
‘UNI
fhe instrument contains six separate circular scales
irranged concentrically on a sin dial, so that they all
rotate together,
The scales are rotated by the knurled nut at the top
Che scales have a common ro line, which rotates with
the scales, and also a common “ cursor ™ line which is rotated
by the nut at the side. See illustrations at front of book.
Commencing with the largest circle, and proceedi
inwards to the smallest, these ales are as follows :
Scales No, 1 (The ** Short-Scale '' for Multiplication
or Division).—This (the Primary-Seale of the instrument)
consists of a single circle, graduated clockwise, and logarith-
nically to represent the nine primary spaces between the
numbers | to 10, or any mult or division of 10 thereof,
The scale, being a circle, the beginning and the end of it
coincide, If this point be designated “ Zero the scale may
be regarded
as beginning at 0 and ending at 1
as beginning at 1 and ending at 10
as beginning at 10 and ending at 100, et
arners are often a little puzzled by this
Che point to gr s that 1¢ ri s lowarithmic, both
regards the ipal div subdivisians,
mally conin y 10, 5 or
space pert
The user may < n to the prime nun
pleases, provided it be a ninitiple of 10, and that the sub
bers any
given proportionate values
space between first pal numbers
rked 10 and a \ ‘ 5 30 cent
the whole circumference ubdivi ints 100, each
quivalent to -Ol of tt value attaching the primary
umber, i.e. the first subdivision may re nt bel JO
101 or 1010, etc,, according as wnity is taken to represent 1
i0, 100, 1000, et very tenth division is figured for
onvenience of reference
(
14 “ UNIVER CULATOR DESCRIPTION
In the second primary space there are 50 subdivisions
each equivalent to -02 of the value attaching to the primar
number, so the first subdivision may represent 2-02, 20-2
202, et
Che m subdivisions in this space are figured 22, 24
etc,, for convenience of reference, and the odd ones (23, 25
etc.) distinguished with a longer line.
In each of the four spaces 3-4, 5, 5—6, 6-7, there are 20
subdivisions, each ‘O5 of the value assigned
sraduations following
Che half of these four
to the prime numbers,
3 may read -305, 3-05
spaces aré marked 35,
In each of the prime space S-9, 9-10, there are 10
subdivisions, h equivales of the prime number, so
that graduations following 7 ht read -71, 7-1, 71, or 710
etc
Owing diminishing distances between the prim
numbers it is impossible to subdivide the le uniformly,
but every effort has been made to produce a clear well
ninated scale by avoid excessively close division
lines, or by overcrowding the reference figure
Changes in magnitude of subdivision have been
as few as | msistent with their decimal charact
The values of the subdivisions in the primary
follows ;
Primary Spaces lto2 2to3
Subdivisionequals § 0-01 0-02
too : ’
[experience shows that spaces which can t tit with the
Cursor-line or the Datum-line are preferable for accurate
setting and reading to finer subdivisions that produce a
pallisading effect difficult to read.
It has been pointed out that the prime divisions and
numbers indicating them along the scale may represent
any multiple of 10 thereof—that js to say, 3 may represedt
. 30, 300, etc. but the subdivisions must have
proportionate values,
UNIVERSAL CALCULATOK DESCRIPTION,
lf, for example, the number $0 on the ale represented
300, the reacir following would 305, 310, 315
but if the same prime number were taken to repr
the readings immediately following would be
3+15, 3-2, ete
Learners ould further bear in mind, when setting
reading with the Cursor-line or Datum-line that the scale
] rithmic throughout, and that 30-5 on the scale is rather
more than hall-way between 30 and 31. Attention to this
point makes for accuracy wh setting, or reading, positions
on the scales,
Seale No. 2 (The Recipro
like Scale No, 1. except that it
(a point to be remembered when setting or re:
he scale g s by aid of Cursor-line or Datum-line th:
reciprocal f any reading on Scale No. 1
used with any other scale provided ‘the value, whose
is desired, is read from Scale No. 1.
often. permits the work of calculation to be
and 24.)
No. 3 (The Logarithm Scale). —This
of a single circle graduated clockwise, Further,
: graduations are uniform, not logarithmic, and numbered
from 0 to 1-00 at intervals of 10 divisions. thus
05, “10, 15, 2 ‘90, 95, 1-00
This gives 200 graduations in the circumference, and as
each of these graduat 3 can be split with the Cursor-line
values can be read accurately to four figures at least
Scale No, 3 is used with Seale No. 1 as follows
If the logarithm of a number is required, the Cursor-line
is set over the number on Scale ». 1, and the mantissa
of the log of the number is read on Scale No. 3 under the
Cursor. The characteristic of the logarithm is determined
as explained in the chapter on logaritlims.
Seale No. 4 for Cubes and Cube Roots, also *‘ Long-Scale
Multiplication and Division."’
Chis extends round three circles and is graduated clockwise.
16 GALCULATOR DESCRIIVTION
Seale No z eubes of readings on Scale No. 4
Scale No, 4 gives cabe-roots 61 i on Seale No, 1
For locating positions of « ubes and cube-roots the fe lowing
tables, constructed mentally, are conven nt :—
For Numbers between The Cube is betwee
1 and 10 P Wee eS Sox | anc i)
10 and 20 ; Seve 1,000 and 8.000
20 and 30 : Saverseciias 8,000 and 27,000
30 and 40 27,000 and 64,000
ete
et
The converse is of course true, viz :-
The Cube-root is betw
Bade crblssivatentmdhe od brated
For Numbers between
Tand 1,000 , 16
1,000 and 8,000 ... aa Fc canter 10 and 20
Ja) ee A whee 20 and 30
64,000... ...c0ee: eaaies 80 and 40
ete
nly necessary to re member
are 8, 27 and 64, then to udd three
8,000 and
27,000 and
‘ etc
To construct abo
that the cubes of 2
eyphers
When multiplying and dividing with Scale No 4. don't
forget that Factors must be set on that Scale, and an
songht for on that scale Don’t mix Scale No.1 and $
No, 4,
Squares and Square-Roots, “Squ and = square-roots
can be obt byt of a two-circl ule and in Fo
Ve Pox i le" id “Magnum Lo
is provided which
ha single movement
considerations
i ior i ircle scale, and the
prevented
fanumt the number
The Square Root of a number can be found by uStng
Sca‘e } ] single ¢ je) in conjunchon with Scale NO. 2;
as follo
number (whose square root 1s requires on
UNIVERSAI CALCULATOR PESCRIPTION,. 17
Scale No. 1 und the Datum, and set the Cursor over the
Unity line of the dial
Rotate the dial until the reading of the Datum on
No. | is the same as the reading of the Cursor on Scale
(3) The reading so found is the Square Root required,
Che above operations amount to finding the f-logarithm
of the numb the logarithm of the square root, since
Scales No, J] and No aduated in opposite directions
and rotation the dial divides thé allotted value between
Datum and Cu , the whole rithm) into two equal
parts when the readings are qi
This method may be em yoes of Fowlers’
Calculators fitted with a Reciprocal Scak
Che magnitude of the squares and square roots can be
approximately locate dd mentally by following tables :
For Numbers between Che Square is between
Tand 10 1 and (7 agra
10 and 20 100 and 400
20 and 30 ‘ 10 and 900
30 and 40 .. 900 and 1,600
10 and 50 1,600 and 2,500
50 and 60 a 9500 and 3,600
ott otc
Che converse of the above is equally tru
rs between
100
100 and 100 10 and
400 anc 900 , ‘ 20 and
900 and 1,6 : 30 and
600 and 2,50 ; 10 and 50
2,500 and 3,6 50 and 60
etc OF
Square root id’ between
Note.—For numbers less than unity
The Sauare becomes Less than the Numbes
The Square-root becomes Grenter than the Number,
Is * UNIVERSAL " CALCULATOR DESCKIPTION
All powers and roots can of course be got by rid of logs as
explained in ¢ hapter on logs
Nort It will occasionally be found that the square
root given by above method is an imaginary root, but there
is no difficulty in detecting this, and the 1 correct
value should then be sought
Scale No. 5 (Sines of Angles),—This is a scale of angles
graduated clockwise round the inner, and then continued
round the outer, circumference of a common circle The
scale ranges from 35 minutes to 90 deg
Angles are read on Scale No. 5
Natural Sines read on Scale No, |
Logarithmic Sines are read on Scale No, 3.
Che range of angles on the circles art follows
Inner Cire from 35 mins. to 5 deg. 45 mins’
Juter Circ from 5 deg. 45 mins. to 90 deg
Notr.—The scale of angles is not divided ‘ decimally
like all the other scales, but into degrees and minutes, as
are always expressed in terms of degrees, minutes
and seconds, though seconds are seldom used
360 deg. a circle, 60 mins I deg., GO secs 1 min
fhe luations of the vie are as follows:
From 35 mins. to 1 deg. the intervals 1 min
Ideg. to 3 2 mins
to 8 5 mins
10 mins
20 mins
1 dey
2 ces
Scale No. (Tangents of Angles).—This.is a scale
ited clockwise round the inner circumierence
ofa sing circle
The scale ranges fron
Angles are read on Scal
Natural Tangents are read on Scale
‘ ONIVERS/ CALCULATOR DESCRIPTION,
Logarithmic Tangents are read on Scale No,
The graduations are as follows :—
From 5 degs, 45-mins 8 degs, Intervals = 5 mins.
to 12 ‘ : = LO
to 30 : is 20
to 45, 30
”
1uge Points.—In making practical calculations ther
are certain multiplying factors which often occur, and to
save time and trouble in setting them, it convenient to
have them rked on Scale No. | and Seale No, 4.
Only a limited number of factors or 1uge points "can
be given in this way. A great many must inevitably be
omitted and sought for in the Book of Instructions relating
to the “ Unive L" Calculator It is hoped however that
the selection shown below will prove generally convenient
Pom Multipher,
1 -4142
7321
Dia. of Circle of Area | : 1284
4
ecs in Arc equal to radiu 57 -298
lo convert com, logs to
hyperbolic lo 8025!
Metres to yards...... 0936
Yards to metres...... 9144
Inches to centimetres “54
Centimetres to inclies 3937
Kilogrammes to Ibs : ney 2046
sodbbdece : 1536
E.H.P. are . a 746
FOWLER'S
** UNIVERSAL ” CALCU
Operation of Scales.
Multiplication Scale No. 1 alone.—
lo find product of ries of factors
t dial till A comes under datum
Set cursor to unity
Set dial till B comes ander cursor
Read product A x B under datum)
Continuing the operations for further
Set cursor to unity
Set dial till C comes under cursor
d product A inder datum)
Continuin
Set cursor to units
Set dial till D comes under cursor
(Read product A . D ander datum
Repeat as many times as there are factors
The movements of the scales and cursor described above
have been equivalent practically to the adding together o!
the logs of A, B, C, and D n movements altogether
three for the first pair of factors and two for each factor
ifterwards
The number of movements can be reduced by using Seale
No. | and No. 2 in conjunction, as will be shortly shown but
whatever the method adopted, the final effect amounts to
adding together the logs, of the multipliers, and subtractiny
the logs of the divisors
ivision Scale No. 1 alone,—Consider the ‘fraction
with one numerator and one denominator-
Set dial till A comes under datum
Set cursor to B.
Set unity to cursor
Read answer under datum
UNIVERS " CALCULATOR OPERATION 21
A. B
Next consicler the fraction ar with two
factors in the numerator and one in the denominator
Set dial till A comes under datum.
Set cursor to ¢
Set dial till B is under cursor
Read answer under datum
Now consider a faction with several factors ti thr
Numerator us well in the denominator
it makes no difference to the answer whether all the
top factors » multiplied together, and then divided by all
the bottom factors multiplied ‘together; or, whether the
top and bottom factors are wo d in pairs as single fac tions
and finally united in 4 group
Obviously, also the insertion of the factor 1, in the numer
ator, or the denominator, can make no difference. Learners,
however, will find it easier, at first, to work sut hac ompound
fraction by taking the factors alternately from numerator
snd denominator, and to@nable them to do this in a routine
manner the factor | should be inserted in the fraction as often
as may be necessary to make the numerator ¢ ontain one mor
factor than the denominator :
{ny compound fraction can then be worked as follows
: Aa Cc 1xBxcC
Fraction 71 Wor 1 a
Set factor A under datum
Set cursor to M.
Set factor B to cursory
Sct cursor to |
Set factor C to cursor
Read answer under datum
Fracrion. ———. Work
Mi N
Set factor A under datum
Set cursor to M
Set factor B to cursor
Set cursor to N,
Set factor 1 to cursor
Read answer under datum
UNIVERSAI CALCULATOR OPERATION
rachon — work as
Mx NXP
Set factor A under datum
Set cursor to M
Set factor B to cursor
Set cursor to N
Set factor 1 to cursor
Set cursor to P
Set factor 1 to cursor,
Read answer under datum
In all thesé examples the routine is alike.
The factors : n alternately from the numerator and
the denominator beg i with the numerator
The Dial is always turned for Multipliers.
The Cursor is always turned for Divisors.
The Datum used only for First Factor and Answer,
Rapid Action with Fowler’s Calculator,—If the reader
has followed the previous routi methods of manipulation
he will be able to understand some shorter ones, and d vise
further ones for himself, according to circumstances; recog
nising that they all depend on the same principles, viz., the
the addition of logarithms of factors for purpose of multi-
plication, and their bstraction for purpose of division
few examples are here given for illustration.
As there are several ways of working a problem with
arithmetic, so there are several ways of working with a
Fowler's Calculator, and when the user becomes familiar
with its manipulation he will discover how movements
ean be curtailed and time saved by using the Reciprocal
Seale (No. 2) in conjunction with the Primary Scale (No 1)
I 1 Multiplying three factors
Find value of A x B x €
Set A on Scale 1 under datum
Set cursor to B on Scale 2
Set C on scale | under cursor
Read answer on Scale 1 under datum (3 movements.)
UNIVER CALCULATOR OFLRATION
Ex 2,—Multiplying five factors,
Find value of A x B Cc D
Set A on Scale 1 under datum
Set cursor to B on Scale 2
t C on Scale 1 under cursor
Set cursor to D on Scez y
Set E on Scale l under cursor
Read answer on Scale 1 under datum. (5 movements,}
Ex. 3.—Operate similarly for any odd number of factors
it is interesting to compare the above example with the
9 movements necessary when using Scale No. 1 alone (ses
20), or by comparing it with the movements of an
ordinary Straight Slide-rule with its intermittent “ end-
switching.”” This is only one ef many illustrations that
could be given
4.—To Multiply any Even Number of Factors,
juct of four factors A, B, C, D, or any other
1 number, is required. It can be obtained quickly with
Seales No. ] and No. 2 by adding a factor | to make the
even number of factors into an odd number, thus:
A Bx Cy D 1 Then proceed exactly as in Examph
2 above
Set A on Seale | under datum
Set cursor to B on Scale 2.
Set C on Scale | under cursor
Set cursor to D on Scale 2
Set 1 under cursor.
Read answer on Scale | under datum (5 movements
Rapid Division with Scales No. 1 and No, 2 with Even
Number of Factors in the denominator,
EXAMPLE 1 Find value of A
5B
Set A on Seale | under datum
Set Cursor to B on Scale 1.
Set C on Scale 2 under cursor
Read answer on Scale | under datum. (8 movements
UNIVERSAL! cALcul
Examp_e 2—Find valut
OPERATION
Here the artifice may be adopted of msertin
factor 1, into the denominator
number of factors, thus
Then proceed as follows
A on Scale | under datum
Set cursor to B on Scale 1
Set C on Scale 2 under cursor
Set cursor to D on Scale 1
Set 1 under cursor.
Read answer on Scale 1 under datum
to
* at contain
(5 move
mm
nts
2 divisions p%
FOWLER'S
“ UNIVERSAL ** CALGULATOR,
Practical Worked-out Examples,
Multiplication of Two Factors on Short Scale No. 1,
fultiply 12-5 by 5 -62
?-S on Scale 1 under datum.
s the 8th graduation st the 12,
Set Cursor to |.
Set dial till 5-62 on Scale 1 comes under Cursor.
his is between the 55 and the 60; the act point being
ke the 56, and two-fifths of the
xt divisio1 ;
just under Sci 1 under Datum
this as 71 ‘9.
Kx. 2.—Multiply 0347 by 2-8 (on the Short Scale No, 1)
Set 347 on Scale 1 under Datum
This lies be cn 30 and 35;
the 9th div
f the next d
Set the Cursor to
Set dial till 28 comes under Cursor
Read answer (just over 97) on Scale | under Datum,
By vis inspection it will be seen that the answer must
be in. the hbourhood of -09
There - we write our answer a
-()f
By act multiplication, the correct answer is -09716,
, upproximation by the instrument
the exact point being
» the 345 and two-fifths
viven by the Calculator
showing how cle
Multiplication of Two Factors on the Long Scale (No.4).
under the Cursor (this is the first
on the Long Seale)
9 under Datum.
$7 on the L ile under Datum,
This is the 7th sduation past the 34
UNIVERSAI CALCULATOR EXAMPLES
t Carsor to |
Set dial till 28 o 1¢ Long Scale comes under Datum.
Read answer 9 n the onter of the three circles forming
the Long Scale, the Datum
This is really , as shown in Ex. 2, and is a very
close approximation to the true value -09716
From the above it will be seen that, where greater accuracy
is desired, it is often advantageous to use the Long Scale
Multiplication of 3 Factors on the Short or Long Scale,
The method is precisely the same whichever Scale
used, so it will be described only for the rt Scale,
Find product of -0347 x2 63-5
Set 347 on f 1 under Datum
Set Cursor
Set dial till on Scale | comes under Cursor
All the above settings as shown in Ex. 2
Set Cur to |
Set di: ll 635 on Scale 1 comes under Cursor
Chis is the 7th division the 60.
Read answer, 6-17, Scale 1 under Datum
The position of the decimal point in the answer is judged
by inspection.
“By actual multiplication the correct answer is 6-16966,
showing 4 close approximation by the use of the instrument
Multiplication of 4 or more factors on the Short or Long
Scales,
Pind product of -0347
Set 347 under Datum
Set Cursor to |
Set dial till 28 comes under Cursor.
Read product 347 = 2-8 under Datam
Next set Cursor iin to |
Set cial till € comes under Cursor.
Read product -0347 x 2-8 « 63-5 under Datum,
Apain set Cursor to 1,
t dial till 49 comes undef Cursor.
Read product. 0347 3-5 & 4-9 under Datum.
——————————
UNIVERSAL “' CALCULATORN BNAMPLUES
rhis, if using the Short Seale, comes just short of midv
between the 30 and the first division past the 30; and
should estimate the answer as 30-23 (midway being 30 +25)
The process after setting the first factor under the Datum
is a succession of settings of Cursor and of factors on the
scale, and of finally reading the product under the Datum
The whole operation begins at the Datum and ends there
it consists In sum of turning the several factors past a fixed
point, and reading the total of angular movements at the end
It matters not whether angular move nents of the dial
and C