Instructions for the Use of the Fuller Calculator (Models No. 1, No. 2 and Fuller-Bakewell)
STANLEY
TRADE MARK
W.oF. STANLEY, & CO LT?
Head Office, Warehouse and Main Works:
NEW ELTHAM, LONDON, S.E.9
Showrooms:
LONDON : 13, RAILWAY APPROACH, S.E.1
PHONE : HOP 0871-2
GLASGOW: 52, BOTHWELL STREET, C.2
PHONE: CENTRAL 7130
Ref. 30/468/LP/1044/2500
INSTRUCTIONS
FOR THE USE OF
THE
W. F. STANLEY & CO. LTD.
NEW ELTHAM, LONDON, S.E.9
PROFESSOR FULLER’S
CALCULATORS
HAVING A
LOGARITHMIC SCALE OF NUMBERS
Al Feet 8 Inches in Length
GEORGE FULLER, M.Inst.C.E.
FORMERLY PROFESSOR OF ENGINEERING
IN THE QUEEN’S COLLEGE, BELFAST
INSTRUCTIONS
FOR THE USE OF THE
CALCULATOR
STANLE
ESE ATR RD EMAC SI
W.F. ST
ANLEY & Co LTe
MODEL No. |
The Fuller Calculator as used on its support, which is attached
to the end of the box. When not in use the support is kept in a
fitting inside the box.
2
THE
FULLER CALCULATOR
is a logarithmic calculator. Its fundamental principle is precisely
the same as the ordinary Slide Rule, but it differs radically in
mechanical construction.
The principles of logarithmic calculators are too well-known to
those likely to be interested for it to be necessary to enlarge upon
the subject here, especially as it is absolutely unnecessary to have
any knowledge of the subject to use the calculator.
The FULLER CALCULATOR will perform all calculations
involving :
MULTIPLICATION PERCENTAGES and
DIVISION COMBINED MULTIPLICATION
PROPORTION and DIVISION,
giving an.accuracy of I in I0,000.
It costs only a fraction of the cost of an Arithmometer,
and it is far less complicated to use. Its construction is so simple
that there is nothing to get out of order, consequently maintenance
charges are practically nil.
Anyone can calculate. with the Fuller after a brief study of the
following instructions without any mathematical knowledge
whatever.
For Percentage and Proportional Calculations it is the most
efficient calculator of its type in existence.
DESCRIPTION
The Calculator consists principally of a cylinder D about 6
inches high by 3 inches diameter, on which is mounted the spiral
logarithmic calculating scale, which is 500 inches in length.
This revolves and slides on an inner cylinder H', which is held
by ahandle E. The settings are made and the calculations effected
by use of the metal pointers or indexes A & B & F shown in the
illustration,
As the accuracy of a Logarithmic Calculator, other things
being equal, is directly proportional to its length, the vast superiority
of this calculator over all others working on the same principle is
obvious.
The instrument is contained in a mahogany box, which is also
adapted for use as a stand to save the fatigue of holding the
instrument in the hand. See Fig. 1.
Three different models are available. All are similar in
construction but two of them bear additional scales on the inner
cylinder H, a description of which will be found in the following
pages.
MODEL No. |
For calculations involving :
MULTIPLICATION . PERCENTAGES and
DIVISION COMBINED MULTIPLICATION
PROPORTION and DIVISION.
This model has no scale on the inner cylinder H which is occupied
by a table of useful data.
The Spiral Scale is divided as follows :
Each of the primary divisions as far as 650, is divided into
ten parts, and from thence to Iooo into five parts; so that all
numbers of four figures have either a mark upon the scale, or are
midway between two marks. Thus 4786 is shown by a mark ;
also 8432; but 8431 is not shown by a mark, but is midway
between 8430 and 8432. In a large part of the scale the space
between these secondary divisions is large enough to be easily
divided into parts by the eye. Thus many numbers of five figures
are easily shown; for example, 26854. There are the first three
figures at 268, then 5 is at the fifth secondary division, and the
4 must be estimated by the eye as ;%, of the space between 2685
and 2686. As the decimal point is arbitrary the same figures do
not always mean the same amount. Thus to represent 26854,
2685 +4, 268 +54, 26°854, 2:6854,°26854, :026854, etc., the same point
on the scale is used.
To fix the decimal point in the result obtained (though this
may most frequently be determined merely by inspection),
rules will be given founded on the characteristics of the logarithms
of numbers.
The index of the logarithms of numbers
between I000 and ggg9 is 3,
” I00 3, 999°9,, 2,
” To » 99°99» I,
” Is, 9°999 » 0.
” ‘Ty, 19999 » 1,
” ‘Or 4, *09999 », 4,
4 -O0I ,, *00999 ,, 3.
INDEXES OR READERS. (Common to all three Models.)
These are three in number. See figure 1.
(x). & the fixed index.
(2). A the top movable index.
(3). B the lower movable index.
4
The A and B movable indexes actually
consist of two pairs of indexes, namely, one
pair on the left, and one on the right. Those
on the left should be used whenever possible,
as it is easier to read the scale when the
previous graduations are visible. It some-
times happens, however, that when using
the left index a calculation terminates
with the fixed index F disposed immediately
over the bar of the A and B indexes, making
it impossible to read the answer. In-such
cases, which will be rare, the calculation
must be repeated, using the right index.
The bar carrying the movable indexes lies
closely against the cylindrical scale, but the
fixed index stands well away from the scale
to allow the movable bar to pass freely under
it and is pressed down by the thumb of the
left hand when taking a reading.
Either A or B may be used and-usually
it is only possible to use one of them as the
other will be off the scale. Whenever
possible A should be used in preference
to B.
Fig, 2.—Showing Position
of Indexes A and B when
correctly adjusted
TO ADJUST THE INDEXES
Before attempting to calculate it is as well to see that the
Indexes A and B are in correct adjustment.
Referring to the illustration, it will be seen that they are fixed
exactly the length of the spiral scale apart.
When the index A is set to the beginning of the scale, the index
B should coincide with the last division on the scale. Should it
not so coincide owing to the bar being out of adjustment, viz., not
parallel to the axis of the cylinder, it can be adjusted by means
of the screws fixing the bar to the inner cylinder at the top. See
gure 2.
P is a pivoting screw. T is the tightening screw and C is not
really a screw at all, but a Cam. If 7 is released and C turned, ,
the bar will be seen to move from side to side, with respect to the
Axis of the instrument. When it is in correct alignment, tighten
T and the rule is ready for use.
INSTRUCTIONS FOR USING
THE FULLER CALCULATOR
MODEL No. 1
The foregoing details of construction show that in operating
the Calculator there can only be two different movements, viz.,
the moving of the Scale or the moving of the Indexes A and B. The
former is a multiplying movement and the latter a dividing
movement.
Therefore taking any factor of any calculation, if it is a
Numerator it must be brought to the Index by moving the Scale,
but if it is a Denominator it must be set by moving the Index
A or B to the Scale.
Obviously, the same form of movement cannot be made twice
in succession, that is, if the last movement was multiplying (moving
the Scale), the next must be a dividing movement (moving the
Index) to complete the sequence and give a result.
When no factor exists, the sequence of movement is completed
by taking 1 as the factor. For instance, in simple or continuous
multiplication the dividing movement is carried out using x as the
factor, and moving the Index accordingly.
The Sequence of Movement is therefore the same whether
for Multiplication, Division, or both combined.
The only other points to remember in this connection are that
the first and last movements must always be multiplying
(moving the Scale), and the Fixed Index F is used on these
occasions only. That is, a multiplying factor is first of all set
to the Fixed Index and no further attention is paid to this Index
until the answer is read under it.
Note Carefully.
When the Indexes A and B are to be moved, the term set is used
When the Cylinder is to be moved, the term bring is used.
EXAMPLE OF MULTIPLICATION
173, X 24
———— = 4152.
I
Factor 173 is multiplying, therefore bring 173 to the Fixed
Index F. The next movement must be dividing and the denom-
inator factor is I understood, therefore set the movable Index
A or B tor on the Scale.
The next movement must be multiplying, therefore bring 24.
(240) to the movable Index A or B. The answer, 4152, is now
under the Fixed Index F.
173 X 24 X 12
= 49824.
IXxXI
Having obtained the above answer, suppose we find it necessary
to multiply further, say by 12 to bring feet to inches.
Simply continue the sequence of movements. The last move-
ment was multiplying, therefore divide by 1 by setting the Index
A or B to r on the Scale, and then multiply by bringing 12 to
the movable Index, A or B. The answer, 49824, is now under the
Fixed Index F. It should be noticed that the accuracy of the
last figure 4, can be checked at once mentally.
EXAMPLE OF DIVISION
286 X I XI
= II-g16. — —
24 _ 24 IL
Being the multiplying factor 286 to the Fixed Index F.
Set the Index A or B to the dividing factor 24. To complete
the sequence of movements, multiply by 1 understood by bringing
i on the Scale to the Index A or B. The answer, 11.916, is under
F Index.
To divide further by, say, II set the Index A or B to II on the
Scale, complete the operation by multiplying by 1 understood,
bringing I on the Scale to the Index. The answer, 1.0833, is
under F Index.
286 xX I
== 1:0833.
COMBINED MULTIPLICATION AND DIVISION
25 xX 22 xX 16
oo = E9704.
II X 29 X I4
Bring 25 to F. Divide by setting A to 11. Multiply by
bringing 22 to A or B. Divide-by setting A or B to 29. Multiply
by bringing 16 to A or B. Divide by setting A or B to 14.
Complete sequence by bringing 1 to A or B. The answer 1+9704
(correct to four places) is under F.
It will be observed that these are operations of merely adding
and subtracting lengths on the Scale, adding for multiplication and
subtracting for division.
7
The following Tables cover all’ types of multi-
plication and division and set out the sequence
of operations very clearly.
When the indexes are to be moved the term Set is used.
When the cylinder is to be moved the term Bring is used.
MULTIPLICATION
Bring (a) to F
Bring (a) to F Set A to 100
et A to 100 Bring (8) to A or B
(a xb) Bring (8) to A or B (@ x bx) Set A to Ioo
Product read at F Bring (c) to A or B
Product read at’ F,
Bring (a) to F
Set A to Loo
Bring (6) to A or B It will be seen that a
et A to 100 similar sequence of opera-
(axb xexd) Bring (c) to A or B tions applies to finding the
Set A to I00 product of any number of
Bring (Z@) to Aor B factors.
Product read at F
DIVISION
Bring (a) to F Bring (a) to F
“a Set A or B to (m) axb } Set A or B to (m)
m | Pring r00 to A wi Bring (0) to A or B
Quotient read at F Quotient read at F
Bring (a) to F Bring (a) to F
axb Xe Set A or B to (m) a Set A or B to (m)
O ] Bring (0)todorB Bring r00 to A
m Set A to Ioo mxXn Set A or B to (x)
Bring (c) to A or B Bring I00 to A
Quotient read at F Quotient read at F
Bring (a) A F bn) Bring (a) to F
b et A or B to (m Set A or B to (m)
axe Bring (2) fo on B axb x6 Bring (0) to A or B
et A or B to (n a Set A or B to (n
mxn Bring 100 to A mxn Bring (c) to A ” B
Quotient read at F Quotient read at F
It will be seen that a similar sequence of operations applies
to the division of the product of any number of factors by the
product of any number of other factors.
8
TO FIX THE DECIMAL POINT
In fixing the decimal point, we make use of the Characteristic
of the logarithm of the number. This Characteristic is simply
the number of figures before the decimal point, minus one: thus, the
Characteristic of 294:386 is 2, because there are three figures
before the decimal point.
EXAMPLES
The Characteristic of 4360 is 3
1 4°36 iso
» *436 is —I
” ”
” ”»
Similarly, if there is a nought immediately following the decimal
point, the Characteristic will be —z; if two noughts, —3; and so
on, thus :—
The Characteristic of 0436 is —2
” 00436 is —3
” ”
In Multiplication, the Characteristic of the Product is the sum
of the Characteristics of the factors, plus 1 for every time a factor
is brought to the lower movable index B, instead of the upper
movable index A.
48:42 X +06434 = 3°115. In this case, 6434 is brought to B,
so that the Characteristic of the product = 1 —2 + I = 0
‘, there is one figure before the decimal point in the product
_ which is therefore 3.115.
13:28 X 142°7 = 1895. In this case, neither factor is brought
to B, therefore the Characteristic of the product is I -+ 2 = + 3,
so that there are four figures before the decimal point.
14 X 12 X 3 X 2 X ‘277 = 279'2. In this case, 2 is brought
to B, so that the Characteristic of the product is I + I + 0-+0
—1 + 1 = + 2. There are therefore three figures before the
decimal point.
In Division, the Characteristic of the Quotient is the alge-
braical difference between ‘the sum of the Characteristics of the
9
factors of the numerator and the sum of the Characteristics of the
factors of the denominator. To this difference 1 is added every -
time a factor of the numerator is brought to B, and ris subtracted
every time B is set to a factor of the denominator.
4°75 X 3°5 X 2°75
= 285:0
“1604
In this case, 2:75 is brought to B, so that the Characteristic
of the quotient is (o + o + 0) — (—1) + 1=+2.
There are, therefore, three figures before the decimal point.
2I:75 X 15:25 X 8+333 X 238 X 2240
== 3173.
268°75 xX 1728
In this case 15°25, 8-333 and 238 of the numerator are brought
to B, and B is set to 268-75 and 1728 of the denominator.
The Characteristic of the quotient is, therefore (I + 1 +0 +
2+ 3)—(2 + 3) +3—2= + 3 and there are four figures before
the decimal point.
to
LOGARITHMS, POWERS AND ROOTS
To obtain powers not higher than the seventh, the quickest
way is by direct multiplication.
For higher powers and roots. Place the upper movable
index (A) to the number, and read the scales (V and M), These
added together give the mavtissa of the logarithm of the number.
To this the characteristic has to be added. The characteristic of the
logarithm of a number greater than unity is ove less than the number
of figures in the integral part of that number. Thus the character-
istic of 5432 is 3, of 543:+2 is 2, of 54°32 is I, and of 5 +432 is o.
Multiply or divide the resulting number by the power or root,
as shown above. Then place the cylinder so that it reads on the
scales (NV and M) the decimal part of the quotient. The power
or root is then at the index (A). In the result the number of figures.
before the decimal point is ove more than the number in the integral
part of the above quotient.
The scale (N) is read from the top divided spiral line and (M/)
from the vertical edge of the scale (NV).
Examples. 51, on placing (A) to 500, scale (N) reads -68
and scale (M) :o1897, which gives the logarithm of 5 = -69897,
the characteristic being 0. Then -69897 xX 13 = 9:08661. Now
placing the cylinder so that it reads -0866r on scales (NV and M) the
index (A) reads 12207, and the required power is 1220700000,
having Io figures, as the integral part of the above quotient is 9.
a
a/ rat, on placing (A) to 741, scale (N) reads +86 and scale (M)
‘00982, which gives the logarithm of 741=2'86982, the
characteristic being 2. Then 286982 + 5 = +57396. Now placing
the cylinder so that it reads -57396 on scales (N and M) the index
(A) reads 37495, and the required root is 3:7495, having one figure
before the decimal point, as the integral part of the above quotient
is 0.
II”
ROOTS OF DECIMAL FRACTIONS
Write them as vulgar fractions, and multiply numerator and
denominator by ten or a power of ten, so that the denominator
may have a complete root. Then take the required root of the
numerator by the method given above, and of the denominator
by inspection.
Thus
Jt = Vt = af = v 40
7 Io , Io® To
— — 3
Jo = R/ 4 = AS A V 40.
Io? 10? Io
— _ a
5 / +5386 _. A/ 586 A / 3809- __ 58600
oo roe W080
TT ——_-— 3 o———
a /0008s — 2/2 / B50 V5
a Toe Io To?
(-0434) a)! = (aie) Gaon
The facility of obtaining and working with logarithms of
numbers gives the rule a great additional value.
NOTE.—The Scales N and M have been replaced in
Model 2 by a very long open scale on the inner cylinder.
This model is specially recommended for calculations
involving the extended use of logs.
TABLES
The tables printed on pages 27-32 have been made and selected
as those considered most useful. Owing to out want of a decimal
system, it has been deemed most important to have a series of
tables which give for our measures of weight, length, time, etc.,
the equivalent decimal fraction of the larger for successive numbers
of the smaller unit. This enables results to be obtained without
the necessity of reduction. Thus to find the area of a rectangle
whose sides are 24’ 64” and 43'5%”. The table gives by inspection
+5208 and +4583 opposite 64” and 54” respectively, so that the
area is obtained by multiplying 24-521 by 43°458. The result,
as shown by the calculator, is1065-°6. I=fthe parts of a square foot
are required in twelfths, the table shows that +6 of a foot is
equivalent to 74 twelfths, and the result reads 1065—7}.
12
DIRECTIONS FOR
PERFORMING CALCULATIONS
INVOLVING PERCENTAGES AND RATIO
For rapidity combined with accuracy, the Fuller Calculator
is probably the most efficient instrument in existence for calcu-
lating Percentage Costs and all Proportional Values.
When either of the movable indexes is at one number and the
fixed index at another, and the cylinder is turned into any
other position, though the numbers at the indexes will be different
oheir ratio will remain constant.
Example.—To convert francs and centimes into sterling money,
supposing exchange 25f. 25c. for £1. The ratio between centiines
and pence is 2525 to 240. Place the cylinder so that the fixed
index is at 2525, and make one of the movable indexes point to
240. Then on moving the cylinder to read off different numbers of
centimes at the fixed index, the corresponding value in pence will
be read at the movable index.
Wages Table.—To find the wages for different times at 35s.
per week of 57 hours. Place the cylinder so that the fixed index
is at 57, and make one of the movable indexes point to 420, the
number of pence in 35s. Then on moving the cylinder to read off
different numbers of hours at the fixed index, the corresponding
wages in pence will be read at the movable index.
To determine Percentages.—Set the fixed index F to the
total number or quantity and the movable indexes to the I00
and 1000 marks which are at the top and bottom of the scale. Then
bring each of the component numbers in turn to the fixed index F,
when the percentage will be shown by whichever of the movable
indexes is upon the scale.
13
Example.—What percentage of 840 are the following numbers?
336 231 73°5 and = 47-25
» 40% 27°5% 8+75% 5 625%
Bring 840 to the fixed index and set the movable indexes to the
ends of the scale, that is, the 100 and 1,000 marks respectively ;
now shift the scale to bring 336 to the fixed index. The movable
index then shows the percentage to be 40. Then bring the follow-
ing numbers in turn to the fixed index, when the percentage will
be simultaneously found at the movable index.
To Add or Subtract a Percentage.—Bring 100 to the fixed
pointer and set the movable index to roo plus or minus the required
percentage. The percentage ratio is now set and any amount
brought to the fixed index will reveal the corresponding amount
under the movable index A or B.
Example.—Add 2}% to {40; £120; {60. Bring 100 to the
fixed index F and set movable index A to ro2} or 102+5. Bring
£40; {£120; and £60 in succession to the fixed index F and the
respective answers will be found under the movable index 4A,
namely {41 ; £123; and £61°5.
To subtract 2$% the procedure is exactly the same, but the
movable index B would be set to 100 —2} or 97-5,
14
INSURANCE BROKERAGE CALCULATIONS BY THE
FULLER CALCULATOR
56h gy Tue are 10%, 15%, 25%) $%, 44% and 45% of
Ss. 3d. .
Bring roo to the fixed index to represent 100% and set the
movable index to £586: 9125, the decimal equivalent of £586 18s. 3d.;
then bring each of the percentages to the fixed index; when which-
ever of the movable indexes is upon the scale will show the answer
as follows :—
£ os. d.
10% — 388:69125 or 58 13 Io
15% — 88-035 or 88 0 9
25% — 14673 or 146 14 7
$% — 2°9345 or 218 8
44% — 26411 or 26 8 3
45% — 264+11 or 264 2 2
£586:9090 = ss £586 18 3
Example 1.—{£60,000 @ 5/6%—£165. Bring 600 (for 60,000)
to the fixed index and set the upper movable index A to 20; then
bring 5:5 (for 5/6) to the lower movable index B when the index
shows the answer to be 165.
Example 2.—£5,000 @ 7/5%—£18 ros. tod. Bring 500 (for
5,000) to index F and set the index A to 20; then bring 7.417
shillings to the lower index, when the index F reads £18541.
Example 3.—{12,000 @ 10/6%—£63.. Bring 129 (for 12,000)
to index F and set the index B to 20; then. bring 10°5 (10/6) to:
index A, when index F shows the answer as £63.
Example 4.—£400 @ 10/6%-—£2 2s. od. When: dealing with
small amounts it is sometimes more convenient to read the answer
in shillings instead of in pounds and decimals, so bring 400. to
index F, as usual, but place the index A at Ioo (x) instead of at
the division 20. Then bring 10-5 (shillings) to the index A, when
the index F gives the answer as 42/-.
15
FULLER CALCULATOR © MODEL No. 2
MODEL No. 2
This is a Fuller Calculator with two extra Scales on
NRE i the inner Cylinder in place of the Table of Data.
Lote se ele : (1) A Scale of Logarithms to four decimal places.
oe eel . (2) A Scale of Sines from 5° 45’ up to 88°.
——— | ) INSTRUCTIONS FOR USING THE
| } LOGARITHM SCALE
eo 3
rere . — A logarithm consists of two portions; a whole number portion,
-_ a sh or characteristic, and a decimal fraction or mantissa.
rere a
Sarena wrarerare For numbers less than unity the characteristic is minus, for
ONCE example :
S een | The log. of 0:4821 = 1-6831, or — 1 + +6831.
' , St:
Bibi oe This may also be expressed as a quantity which is all negative
PROT PE LN P thus ; — +3169. :
Quantities in this form are much more easily handled when
CILPPPPCL LOY 1 calculating with a slide rule, than quantities which are partly
alti positive and partly negative. This fact has been made use of
AT in graduating the logarithm scale of the Fuller Calculator.
% ‘$79 ' i ‘
CRC EO The scale has been figured to read both ways, from right to
eo : left and from left to right. One set of readings (right to left)
baresyere td a is marked + and deals with numbers of unity or more. The
| err eee | other reading is marked — and deals with numbers of less than
* = Ic AS . | ' To find the logarithm of a number ::.
we il om If any number on the main scale be brought to the fixed index
Pe a fe——far F, the logarithm of that number automatically appears on the
a a | a ' inner cylinder under the index L, at the top of the movable cylinder.
rt = co If the number dealt with is greater than unity, the plus reading
na mg “fest is taken, but if it is less than unity, the minus reading is the correct
Two-thirds full size LOG. 2 = +3010. i — One.
Fig. 3 . _ 17.
16
EXAMPLES
Find the log. of 4.4480. Bring 4448 to F and under L read:
+ +6481, or — +3519. As the number dealt with is greater than
unity, obviously the plus reading is correct.
To find the log. of -2590. Being less than unity, the log.
will be minus. Bring 2590 to F, and under L read: — +5868.
Suppose the log. of a still smaller number is required, say
“02590, obviously the reading will be the same, prefixed by the
characteristic “1,” ¢.e., — 1°5867.
To find the antilog. of any number, the procedure is, of
course, the reverse of the foregoing.
To find the value of (24-2)?3. Bring 24:2 to F, and under
the index L, read: +3837, the mantissa of the log.
The characteristic is 1, and the complete log. is 1:3837. Mul-
tiply this by 2°3 by usual method, and the result will be 3+1827;
set the mantissa -1827 to the Index L, and under the index F,
read : 15233, the antilog.
The answer is therefore -- 15233.
To find the value of (-3642) **. Set +3642 to F and the log.
= — +4387. (Being less than unity, the negative value is taken.)
Multiply this by 4:2 by usual method, and the result will be
— 1°8425.
Bring — +8425 to L, and read 1437 at #, which makes the
answer ‘01437.
18
THE SINE SCALE
This scale occupies the lower half of the inner cylinder. Like
the other scales it is a spiral, having a total length of approxi-
mately 32 ft. resulting in a very open reading.
Each division on the scale from 5° 45° to 48° represents one
minute, but from 48° onwards each division represents 5 minutes.
This scale is recommended to Engineers and Surveyors ~ for
solving any expressions involving the use of Sines. or Cosines.
Calculations in latitude and departure can be solved in a fraction
of the time spent in working with tables, and triangles can be solved
with great rapidity and accuracy.
INSTRUCTIONS FOR USE
If any angle on the Sine Scale is brought to the Index S Fig. 3,
the Sine of the angle will be found on the movable cylinder against
the fixed index F.
‘, Bringing any angle on the Sine Scale to the Index S is equiva:
lent to setting F to, the actual value of the Sine of the angle
concerned.
Solution of Triangles.
From the general formula :—
B a b Cc
sn A sinB sinC
c 4 6 sin A ce sin A
a= — = ay
sin B sin C
asin B csin B
A C b= =
b sin A sin C
a sin C 6 sin €
c= = a
sin A sin B
hence: Given two angles and one side or two sides and_ the
angle opposite one of them we can solve the triangle by using
one of the above formule.
19
Example I. Let A = 75°
C = 24°
b = 126 yards.
Then B = 180° — (75° + 24°) = 81°
To find a.
6 sin A 126 x sin 75°
as - =>
sin B sin 81°
Thus the calculation is performed as in ordinary combined
multiplication and division, except that the index S is used for
setting the sine values. ,
Move the cylinder until its index S marks 81° on the scale
of sines: set the movable index to 126: move the cylinder until
its index S marks 75° on the scale of sines : read a (12323) on the
movable index. .
te, & = 123°23
To find c.
b sin C
C= =
_ sin B sin 81°
Move the cylinder until its index S marks 81° on the scale of
sines: set the movable index to 126: move the cylinder until
its index S marks 24° on the scale of sines: read ¢ (51888) on the
movable index.
126 x sin 24°
1.6, ¢ == 51:+888 yards.
Where the sine of an angle greater than go° is involved, we ©
can make use of the following :—
sm A = + sin (180°— A).
Example IT.
Let A = 42°
C = 4I°
6 = 120 yards
“ B= 97°
To find c, b sin C 120 X sin 41°
C¢= ————_———_ SO TO
sin B sin 97°
sin 97° = sin (180° — 97°) = sin 83°.
Move the cylinder until its index S marks 83° on the scale
of sines: set the movable index to 120: move the cylinder until
its index S marks 41° on the scale of sines: read c (79318) on the
movable index.
1.€, € = 79°318 yards.
20
To tind a.
6 sin A
120 X sin 42°
a> = — prea
120 X sin 42°
sin B sin 97° sin 83°
Move thé cylinder until its index S marks 83° on the scale
of sines: set the movable index to 120: move the cylinder until its
index S marks 42° on the scale of sines: read a (8091) on the
movable index.
1é, Qa 80-91 yards.
Example III. Two sides and one angle given.
Let a@ = 71°3 yards.
: = I09‘0 yards.
Bo = 54° 15!
To find A.
Since a b
sn A sin B
.. osin A =asin B
; asin B
sin A = =
b 10g
Move the cylinder until its index S marks 54° 15’ on the scale
of sines: set the movable index: to 109: move the cylinder to ,
bring 71:3 to the movable index: read A (32° 3’ 40") against
the index S on the scale of sines.
71°3 X sin 54° 15/
To find C. -
C = 180° — (A + B) = 180° — (32° 3! 4o” + 54° 15’ 0”)
= 93° 41" 20”,
To find c.
bsin C
— =
sin B sin 54° 15’ 0”
(Note: sin 93° 41’ 20” = sin 86° 18’ 40”.)
Iog X sin 93° 41’ 20”
Move the cylinder until its index S marks 54°15’ 0” on the scale
of sines ; set the movable index to 109: move the cylinder until
its.index S marks 86° 18’ 40” on the scale of sines : read ¢ (13402)
on the movable index.
4.005 C = 134-02 yards.
2i
THE
FULLER-BAKEWELL
CALCULATOR
FOR ENGINEERS AND SURVEYORS
The replacing of the table of constants on the fixed cylinder H
(fig. I, page 2) of the FULLER CatcuLator by two logarithmic
scales, one of cosines squared and the other of sines multiplied by
cosines, is due to the suggestion of Mr. W. N. BAKEWELL, M.I.C.E.,
and it will be seen from the following that this alteration gives
very great power to the instrument for the calculations required
when surveying and levelling with the Tacheometer.
The formula for the horizontal distance between the Tache-
ometer station and the reading staff, when the latter is held vertical,
is: @d = 100 S! Cosa + K Cos. a. ‘
where d@ = the horizontal distance,
S1 = stadia intercept
@ = angle of the telescope,
kK = the constant of the instrument
(o when the telescope is anallatic)
For the difference of level between the centre of the telescope
of the tacheometer and the point where its axis cuts the vertical
staff: v = AB Sin. a Cos. a + K Sin. a.
where v = vertical height,
AB= Inclined distance to staff,
a = angle of the telescope.
To explain the use of the Calculator a model of a Field Book
for a Tacheometrical survey is given on page 24.
DESCRIPTION OF THE SCALES
The two Scales on the inner cylinder H, Fig. 1, are separated
by a Zero mark, which is just over an inch up from the handle.
(1) The Scales of Cosines Squared occupies about an inch
of the cylinder below the Zero, and is subdivided to ten minutes
up to ro deg., and from thence up to 35 deg. (which is the full
extent of the Scale) it is subdivided to five minutes.
22
(2) The Scale of Sines multiplied by Cosines occupies
most of the cylinder above the Zero. It starts at 0° 35’, and
extends to 45°, subdivided as follows :—
From 0° 35’ up to 26°, to single minutes,
From 26° up to 39°, to 5 minutes.
and the remainder of the scale to 10 minutes.
TO READ THE SCALES
The readings on Scales r and 2 are taken by means of the two
indexes L and S (fig. 3, page 16). L is used for taking readings
on the Scale of Sines multiplied by Cosines, from 5° 46’ to 45°,
and S is used for readings on the remainder of the Scale and also
for all readings on the Cosines Squared Scale.
23
LER
| 6g
Q 5
E | & ;
8 | 7) me)
m4 [| 8 5
7 | 8 a DIREC
» i .
® & ge m 2 Z, plicity itself and i ator for Tacheometri
4 ; E re # S a 2 + S < s as follows :— etrical Calculations
» e Of an od vr ira) o 4 (xt) Brin To find d an
> mW °Q + Hn 0 § the lower dv.
8 2 ea a NS X Koo4 this also brin index S (Fig. 3) to Z
Fe S| e: " N 4 3 (2) Set A or Bt gs the fixed index F ero. Incidentall
4 a g 2 fora) iS (3) Bri to the standia i to Zero. y;
3 oo Hs J SN mg the index S to re he ecePe St
5 rons! Sa) on the S . e verti
fl BI 8 the Horizontal distance Squared (beloe the Zero) and seed
[os | 8 AS 5 cylinder. istance at A or B, which Zero) and read
q eel a x a (4) Bring the index L ever is on the :
v é = aS S on the scale of Sin to the vertical angle of
be Aa © ase i a H Zero and read the es multiplied by Cosi the telescope
So: a || aS is on the Scal vertical height at sines, above the
fe) °. L ane g Note—The decimal point at A or B, which ever
a 2 ae : : s the in is arbitra ;
a 3 ay °® 8 ag J the followin, ak ie fails to give as with a slide rule.
y © oMo4 a ao q ° l g table gives the si v for angles of
eal " qr 0'% 2 ‘ © fon no g ~ e sin. cos. for ang] of less than 35"
rm an Ky , Not a I . gles from 1’ to 34’
BS | a8o¢ B R ¢ @ + a r | 00029 || 23 of
al ; rm BA gg | me Sane ° a “00058 t "00378 25
Oo 2 S) 8X8 > 3 g m a4 st 3 00087 i "00407 26 ‘00727
© 2 - ae) a a + 3 Aw 4 OOI2I * 00436 2 "00756
q By LES | 2 + 0 Se 8 ors i "00465 28 "00785
A a | aba3|s mS eo gg 4 < jor, ak | eeist 00814
OS ; D 0% 9 y 8 HS gg Z “00204 "00524 "00842
Ss 7 Hw > Ya) : ‘B. 8 . 19 . 3° ’
we qi 7 02 © +r 0 @ 8 00233 20 00553 31 00872
S AS & an oo” eo © Be 9 -00262 00582 "00902
2 ro q re Sad os v0 00291 3e -oo611 3 00930
A as " 2 oR i - "00320 oa ‘00640 m1 “00960
. SB: omene) So 2 0034 3 “00669 ‘00990
gs BF S SAS «0 arc 9 2
By, : ene) mo ws @ BD 4 -00698
q mek oa INHO H ae o° YSes ga When, theref 9
. 4 TO a , refo :
E . 7 = 4 aa NAHH ai | the number opposite t is below 38’, St has to b
a o # s 6 > wD x rel 4 eld book given th, ia the observed angie oO be multiplied by
Ss a 5 8 o ay © 1 eo height v is found e last vertical ang] gle. Thus in the mod
5 B > ° g + _ H coe manner by the rand multiplying ror 6 observed is 15’, and in
5 BS + | 7 re ? an . When the T alculator. y "00436 in the ordina "
2 | a3 ar + ot as it will b acheometer used ry
| ey +H RS + o K e seen that t has not an ;
a) ms |° ° RF os FF 8 . ag K cos. @ has to be © ed etd Of an anabatic telescope
° 9 se most usual ; when, h ce, as found ab
HS fe) + gs _ al, K m , however above
a f q » of 8 K =1' 6" ay be taken f , @ is below 21°, which
WHOd 40 “ON § 8 gg Piles Fo. oS A would bh or K cos, a’; » which
; the hei ave to be ad ; and su
OH a i a be ad eight v, whe added to each ppose
quot a ” = added to S! , n the angle a i value of d.
yo 33197] e ** * pa eeitipiyine & by ci multiplied Sy ain a ose ax’, K is to
x =1'5 ft., anda DY Sih & COS @ instead of o vi a; the error
uoTye}g JO ‘ON | a az° is only +037 ft. nly by sin. a when
|
24,
TABLES AND FORMULAE
FOR USE WITH
FULLER’S CALCULATING RULES
. Round | Square Plate
Cubic Rod Bar \
Ins. rft, long, | x ft.xx7 |? ft. xr ft,
1” diam. xT’, xe
Ibs. Ibs, Ibs, tos,
Brass, cast, . 2. . 1 « 8 +298 2:81 3°58 43:0
» wire coe - oe +308 2°91 3°70 444
Bronze. . . 2. 1 « te ee "303 2°86 3°64 43°97
Copper, sheet . . . . . «© «| 1318 2°99 3°81 45°75
” hammered, see | 1322 3:03 3°86 463
Iron, cast oo . 1257 2°42 3:08 370
» Wrought , oo. ’ +278 2:62 3°33 40:0
Lead . . ww ew ee ‘ "412 3:88 4:04 59°
Stel . 2. 2. 2. ww . | 1283 2:67 3°40 40°8
Zinc ny . "252 238 3'03 36:3
Cubi enacity Mod, Mod.
Foot, | Sqins, | issticity | Rupture
: Ibs, Ibs. Ibs, Ibs,
Cast iron eee ee | 44g 16,500 17,000,000 —
Wrought iron 8 6 « | 480 65,000 29,000,000 —
Steel bars se ee | 490 145,000 35,000,000 —
» plates oe —_ 80,000 — —
Elm . oo... we 34 14,000 1,000,000 7,500
Fir, Red Pine. . . . . 37 13,000 1,600,000 8,000
.» Spruce se ee 37 12,000 1,600,000 11,000
» Larch . . .. . 33 9,500 1,100,000 7,500
» Yellow Pine. . . . 29 — _ 7,000
Oak, English . . 53 15,000 1,500,000 12,000
» American . 54 10,000 2,000,000 10,000
Teak . 48 15,000 2,400,000 15,000
27
Decimats or A Decree or Hour. | BirMINGHAM Wire GAUGE. Decimais oF A Foor. DEcIMALS oF A Cwr.
Min. Deg. Min, Deg. Min. Deg. No. Ins, No. Ins, = ; ; ow ; ve a} ewt,
1 ory | ar ]-35" ar |-6833 | or | 300] ar | -o32 | orogr || 6" | «sro4xl] "| Gog] S10 | «3303 | S10 | -eeg6
2 | 70333 22 | -3067 “|? ; ‘284 22 | 7028 4 | :02083 + | °52083 2 | 0179 TI | +3482 20 | +6786
3 | 05 23 | +3833 1 43 1 °7167 3 | +259 23 | -025 q| 03125 al «53125 3 | 0268 12 | 3871 ar | +6875
4 | 0666 |] 24 | +4 44 | °7333 4 | 1238 || 24 |} -o22 Raney Pp ree ah Sae7 | ig fecter | oe | segee
5 0833 25 |°4167 1 45 [+75 5 220 25 02 5 | 04208 g| -25208 3 | -oqn6 14/378 23 | 7084
6 26 | +4333 46 | +7667 6 203 26 018 a” | -0G25 3° | «3625 2 | 0336 1 | 3839 24/9143
§ 1167 38 ae aR | P38 Nee 38 oe | 107292 $l +57292 7 | (0625 16 | +3929 25 | °7232
8 | +1333 28 | +4667 48 |°8 8 ‘165 28 O14 : - 8 08333 || 7° 38333 8 | conry 17 | «4or8 26 | +4322
9 tts 29 | +4833 49 | -8167 9 +148 29 ‘O13 1} -09374 4] 30374 9 | -0803 18 | «4107 24'f941t
10 | +1667 30 45 50 | +8333 10 +134 30 ‘O12 4? | 192378 vl tore 10 | 0805 r9| -4196|13 01-45
Ir | +1833 |} 3r | -s167 | 51 | +85 Ir ! +120 31 | -o1 t eee t eng tr | cooks | a2 | tee © | e289
12 {+2 32) | °5333 52 | 8667 12 "109 32 +009 3 4 | a ; R
3 4 | +125 $ 625 12 | -1o07r 21 | '4375 2 1.*7679
13 | 2167 33 4°55 53 | 8833 13 “095 33 “00 &i 13541 $) +6354 13 | ‘1161 22 | +4464 3 | 7768
ra | 77383 ff 34 | 507 | S49 re | SoS 34 | (307 g | 14583) £ | -64583| 14] +125 | 23] 4554) 4 | +7857
16 | .26 35 [1°95] 55 | 9te7 | 18 | 022 38 | oes "B| $15625 “$! +65625|| 15] +1330] 24 | +4643 51° 7946
16 | +2667 || 36 | +6 56 | +9333} 16 | +065 36 | 004 2° 16666 || 8° ‘66666 16 | -1429}| 25 | +4732 6 | +8036
17 | (2833 37, | “6167 37 | '98 4 1058 : 4 | °17707 $| +67707 17} 1518 26 | +4822 7 \*8125
18 13 38 | 6333 38 | 9667 8 ote i +1875 } 6875 18 | +1607 27 457 8) 8214
19 | +3167 39 «| 65 59 | 9833 19 ‘O42 ar ‘19791 21 +6979 Ig] ‘1696/12 o 9 | 8303
20 _| °3333 40 | 6067 70 ‘035 } ‘20832 4 | +70832 20 | +1786 I 3080 10 | +8393
$) 21874 Ri ‘71874 21 | +1875 2 ‘5179 ir “Ba8e
MULTIPLIERS FOR CONVERTING, ; aa 2% | -22916 +) 72918 22 | -1964 3 "52 t Been
Notre.—The converse of these is obtained by dividing by the number instead of multiplying. } 23958 $i +7395 23 | '2054 4 $357 3 By3
to hyperbolic lo | 2*3026 3° 25 9° “75 24 | +2143 '
Feet to links %P : a I*5151 4 | +26041 $j} + 76041 25,| +2232 6 15536, 15 “8839
Square feet to square ‘links a 2°2957 £ | +27083 4 | +77083 26 | +2322 4 | +5625 16 | ° 9) 9
Acres to square yards Pr 2.7.0) 2) +28125 Ql +78125 27 | +24r1 8) 5714 17] ‘got
Tons to pound. Cd 2-7) } | -29166 4 79166 || 10 | -25 9 | +5803 18 | +9107
Lbs. per sq. in, to Ibs. persq. foot se . ct rad $| -30208 $j) +80208 1 | +2589 10 | +5893 19 | *9196
Lbs. avoir, to grains, . Sr 7000 a’ | 3i2s 4 8128 2 | +2079 rr | +3982 20 | +9286
Cubic feet to gallons. Toho hh rts 2355 “a1 o292 Z| 82292 3|+2768|| 121-6071] 2 | +9375
Rood masonry 2 ft. thick to cub, yds." ee ee ee 24 $| +3229 $ 8 23 | go2 32 | -oab4
Rod brickwork 1’ 14” yy oe ee ww 11 °333 / 4° "33333 }) 10° | *83333 4) °2857 13 (625 | 23 | «985
Metres tofeet . 6. 6 8 8 ee ee ee ee 32809 : 4] 34374 a 84374 5 | 2946 14 | 625 | 23 95 4
Inches to millimetres. 25 ‘4 : 4 | -35416 i 85416 6 | +3036 15 | 6339 2 9643
Square metres to square 5 10°764 | 36458 a! -86458 7 | +3125 16 | +6429;| 25 | +9732
Square inches tosquare millimetres . . 2. 6. 6 7 ew ew 645 °14 4 | 375 3 | +875 8| 3214 17| 6518] 26 | «9822
Cubic metres to cubic feet er SS 35'°317 g| 38541 §| +8854 9 | +3303 18 | +6607 27 | «grt
Cubic inches to cubic millimetres see eee ewe} 16386 ae 4 a2 8028 .
Grammes to grains . . . . « « sone 15 °432 £ | °39583 +) 9593
Kilogrammes to lbs. 6. ee ee 72046 $ | 40625 5 190625 Dectwaus or a La.
Tons totonneaux, . 6. 6 6 6 ee ee ee 10160 5: 41666 | rr gi
Gallons to litres tofgotis’ 2 of 8 tt qe 4] +42707 fe £92707 | ; : a
Kil mmetres to foot Ibs. soe ee . 9375 on Ibs oz. s oz. ¢
Kilogram, on square millimetre to Ibs, on square inch» 1. 1422 t tayox wi 94701 | 0156 % | +3125 || 10h | +6562
Miles tokilometres . . 0. 06 1 1 ew ee we el 16093 1° 45833 ¢ | +95833 % | 10312 54| +3437 || rr | 68475
Hectares to acres. i 2°4711 *y 488 2. -96875 ||" #| +0468 6 | +375 11g | +7187
Miles per hour to feet per second . . ‘ ‘ : 1 +467 4 7 i ‘g7916|| I 0625 || 64 | +4062 ) 12 | +75
Knots to feet per second Pr rr a 1'688 £ | +4791 , 979 gi| 1} | -0937 7 | +4375 || 124 | +78x2
Cubic feet of water tolbs. 2 0. 1 6 ee ew ee en 62°425 $] 48958 Z| +9895 ie 09 7) 488? | a3" | brs
¢ tol Bee Pb pb pit ap o's eee | 24 | srso2|) 8 | +5 134 | +8437
t C ine er 4 J
one atmesphere to o Ibs, pe sq. foot . 2 1 1 ew ew ww ef 216 3 +1875 84] +5312 T4 875
” ” ” kilogs. "per sq.metre . . « 6 «© « « «| 10333 34 | 2187 9 | +5625 || 144 | «9062
” ” », Millimetre of mercury) . . . . « « « 760 e322 fect per second per second, 4 125 of | +5937 |) 15 9375
" ” * fest of watr 7. 33°9 TEL. = 550 ft. Ibs, per second, 4% | '2812]| to | +625 || 15% | +9687
+? Lad ”
28 r
29
Decimats of A Pounp, D, oF Yzar, D. oF AN AcRE. NATURAL Sinus,
: Deg.|| 0” 10 -| 20° | 30% 0 o
a 4. £ s. a £ D. Y. ry Acre. 3 ‘ ° ass 5 6 7a
+ | -002 1.2} | +0604 I 0027 1 ‘00625 46 |i 7x