Instructions — Precision Slide Rule No. 2/83N, 62/83N with enlarged scale range
Faber Castell 2/83N (and 62/83N) instructions in English
INSTRUCTIONS
Precision Slide Rule
No. 2/83N, 62/83N
for Mechanical and Electrical Engineers,
with enlarged
Students of Technical Universities,
scale range
Engineering Colleges, Polytechnics
The CASTELL 2/83 N with enlarged scale range
Many satisfied users of the 2/83 N slide rule, in a sprit of constructive criticism, have suggested that the arrangement of the scales on the front and back should be improved. Our Development
Department has taken advantage of the experience gathered during these last few years and utilised the latest knowledge acquired. Apart from numerous minor improvements, the scales A, B, DI
(on the front) and LL00, CI, D and LL0 (on the back) have been added.
Contents
The Scales of the pocket slide rule 62/83 N with graduation length 5" (12.5 cm) The Small.Angle Scale ST and the Mark ρ on Scales C and D
The Scales of the Slide Rule 2/83 N
Calculations with Complex Numbers
Calculations with the Root Scales W1, W1', W2, W2'
How to read the Scales with graduation length of 10 inches
How to read the Scales with graduation length of 20 inches
Calculations with the Mantissa Scale L
The Exponential Scales LL1, LL2 and LL3 for positive exponents; LL01, LL02 and LL03 for negative exponents
Multiplication
Multiplication with Scales C and D of back of slide rule
The Natural Logarithms
Division
Powers of e
Combined Multiplication and Division
Roots of e
Formation of Tables
Powers of any numbers desired
Calculations with Scales CI and DI
Roots of any numbers desired
Calculation with the Square Scales A and B
The Logarithms to "Base 10"
Squares and Square Roots
Logarithms to any desired base
Cubes and Cube Roots
Meaning of the Scale Marks
Calculations with the Scales CF, DF and CIF
The Cursor
Calculations with the Pythagorean Scale P
The Care of the Slide Rule
Calculations with the Trigonometrical Scales S, T1 and T2
The scales of the pocket slide rule 62/83 N with graduation length 5" (12.5 cm)
From graduation figure 1 to graduation figure 1.2.
(Section of scale reading 1 to 2)
From graduation figure 2 to graduation figure 3.
(Section of scale reading 2 to 5)
From graduation figure 5 to graduation figure 7.
(Section of scale reading 5 to 10)
Each interval is equivalent to 2 sub sections. An accurate
Each interval is equivalent to 5 sub sections. This provides an Each interval is equivalent to 10 sub.sections. This provides an
reading can be taken of the values corresponding to 3 places.
accurate reading of the values corresponding to 3 places, if the accurate reading of the values corresponding to 2 places, which
The odd numbers are obtained by halving the distance between last figure is a 5.
more over are identified by graduation marks.
two graduation marks.
Other intermediate values must be estimated. Example: To set to 318, first find 3.1.7.5 by halving the distance between 315 and 320, and then move the cursor line slightly to the right.
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (1 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
The Scales of the Slide Rule 2/83 N
All scales are marked with reference to the basic scales C and D. At the right.hand end of the slide rule they bear the mathematical formula based on the numbering of the basic scales.
The cursor, entirely surrounding the slide rule, enables each stage of the calculation to be connected with the next, throughout all the scales on the front and back of the slide rule.
Scales on the front of the slide rule:
T1 tan 0.1 x (cot)
1st tangent scale
T2 tan x (cot)
2nd tangent scale
on upper body
cube scale
K x3
fixed square scale
A x2
fixed x.folded scale
DF πx
movable x.folded scale
CF πx
movable square scale
B x2
on slide
reciprocal x.folded scale CIF 1 :πx
reciprocal basic scale
CI 1 : x
movable basic scale
C x
fixed basic scale
D x
fixed reciprocal basic scale DI 1 : x
sine scale
S sin 0·1 x (cos) on lower body
arc scale for small angles ST arc 0.01 x
Pythagorean scale
P √(1 . (0.1 x)2)
Scales on the back of the slide rule:
LL03 e.x
exponential scales for negative exponents
LL02 e.0·1 x
LL01 e.0.01x
on upper body
LL00 e.0·001
2nd movable root scale
reciprocal basic scale
mantissa scale
movable basic scale
1st movable root scale
(√10)x
W2' √(10x)
CI 1 : x
on slide
L
½ log x
C
x
W1' √x
1st fixed root scale
W1
2nd fixed root scale
W2
√x
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (2 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
fixed basic scale
and exponential scale
D
x
LL0 e0.001x
on lower body
LL1 e0·01x
additional exponential scales for positive exponents LL2 e0·1x
LL3 ex
How to read the scales with graduation length of 10 inches, e.g. C, D, CF, DF, CIF
The following should be noted:
The slide rule does not show the actual place of decimals to which a number belongs. For example, the 6 shown on the slide rule may equally well denote 6, 0.6, 60, 600, 6000 or 0.006, and so
forth.(An exception is provided by the exponential scales) The position of the decimal point is ascertained afterwards, by a rough calculation with round figures. In most practical calculations it is known
in advance, so that no further rules for determining the decimal point are required. It is the basic scales C and D that give the clearest idea of the way in which the scales are sub.divided. Once
familiar with the graduation of these two scales, we shall be able to understand the others likewise.
All scales marked in red run in the opposite direction (reciprocally) from right to left, the exceptions being the extended supplementary graduations which are provided to enable a calculation to
be continued in the case of border line values just below 1 (beginning of graduation) or just above 10 (end of graduation).
Let us now have a look at the basic scales C and D on the front of the slide rule, the reading . and setting . exercises being carried out by means of the long cursor line or the index.1 (beginning of
scale) or index.l0 (end of scale), as the case may be.
A section of the graduation.range from I to 2
(Scales C and D)
From guide.number 1 to guide.number
1·1
10 sub.divisions of 10 intervals each (=
1/100 or 0·01 per graduation mark)
Here an accurate reading can be immediately taken to 3 places (e.g. 1.0.1). By halving the space between two graduation marks, 4 figures can be accurately set (e.g. 1.0.1.5). In all cases the final
number must then be a 5.
A section of the graduation.range from 2 to 4
(Scales C and D)
From guide.number 3 to guide.number 4
10 sub.divisions of 5 intervals each (= 1/50
or 0·02 per graduation mark)
Here an accurate reading can be immediately taken to 3 places (e.g. 3.8.2). The last number is then always even (2, 4, 6, 8). If the intermediate spaces are halved, this provides the uneven
numbers (1, 3, 5, 7, 9) as well (e.g. 3.8.3).
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (3 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
From guide.number 8 to 9 and
guide.number 10
10 sub.divisions of 2 intervals each (= 1/20
or 0·05 per graduation mark)
A section of the
graduation.range from 4 to 10 (Scales C and D)
Here an accurate reading can be taken to 3 places, when the last number is a 5 (9.0.5). By halving the intermediate spaces it is even possible to take an accurate reading to 4 places. Here again the
last number is always a 5 (9.0.7.5).
How to read the scales with graduation length of 20 inches: W1 W1' W2 W2'
These scales are provided along the edges and adjacent edges of the slide, on the back of the slide rule, and run from 1 to 3.3 at the bottom and from 3 to 10 at the top. Their use results in doubled
accuracy. They are sub.divided differently, however, from the scales on the 10 inch graduations.
Graduation range from 1 to 2.
This section is first of all divided into ten sub.divisions, marked 1·1, 1·2, 1·3, 1·4... 1.9. Each of these, in turn, is divided into ten further sub.divisions, but the latter are not numbered, owing to
lack of space. Finally, a small stroke is also provided to show the exact centre between these graduation marks. Readings can be taken as follows: 1.1.2.5, 1.3.1.5, 1.4.4.5, 1.5.2.5, 1.7.1.5...
1.9.7.5.
Graduation.range from 2 to 5.
Hero again, this section is first of all divided into tenths, but they are not numbered, except for the graduation marks corresponding to the values 2, 2.5, 3, 3·5. 4, 4.5 and 5. The user must
recognise the remaining tenths for himself, i.e. the values 2·1, 2·2, 2.3 ... to 4·7. 4.8, 4·9.
Further tenths are entered In between these tenths but without "marked centres". We thus hove the following values, starting with 2 and without using the decimal point: 2.0.0, 2.0.1, 2.0.2. 2.0.3,
2.0.4, 2.0.5, 2.0.6, etc., up to 4.9.7, 4.9.8, 4.9.9, 5.0.0.
Graduation.range from 5 to 10.
In this range, the tenths first of all appear, as before, but in between these it is only the fifths that are provided.
Starting with 5, we thus have the following graduation marks: 5.0.0, 5.0.2, 5.0.4, 5.0.6, 5.0.8, 5.1.0, 5.1.2, etc. up to 9.9.6, 9.9.8, 1.0.0.
The reading range of these scales exceeds by far these possibilities. Further intermediate values, however, must be estimated.
Multiplication
a·b
This is chiefly carried out with the main scales C and D of the front of the slide rule.
Example: 2.45 x 3 = 7.35
Set the 1 of the slide of the rule (C 1) above the 2·45 on the lower scale of the slide rule (D
245), move the cursor line (Important: Whenever the cursor line is mentioned in this and the following
chapters, this will always refer to the long, central hairline on the front and the back of the cursor.) Into
position above the 3 of the lower slide scale (C 3) and read the product . 7·35 . underneath the
cursor line on the lower scale of the slide rule (D 735).
Multiplication with Scales A and B: see below
Example: 2.04 x 3.18 = 6.487. Place C 1 above D 2.04, place the cursor line above D 3.18, and read the result . 6.49 . on D, likewise underneath the cursor line.
Example: 11.45 x 4.22 = 48·35. Place C 1 above D 11.45, place the cursor line above C 4.22, and read the result . 48·35 . on D, likewise underneath the cursor line.
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (4 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
In calculations on the lower scales C and D it sometimes happens that with the setting C 1 above the first factor on scale D the slide projects too far to the right, so that it is no longer possible to
set the second factor on C.
Transposing of the slide
Example: 7.5 x 4.8 = 36
in this case the slide is pushed to the left until the beginning C 1 is replaced by the end of the
slide C 10 (marked 1) above the first factor on scale D.
This operation is termed transposing the slide. It can be avoided if, in case of need, C 10 (end of slide) is immediately placed above the first factor. An experienced user knows at once which
method of setting to adopt, i.e. "beginning of slide, C 1, above first factor" or "end of slide, C 10 (marked 1), above first factor".
Examples for practice:
Setting: "Beginning of slide C 1 above first factor": 1.82 x 3.9 = 7.1; 0.246 x 0.37 = 0.091; 213 x 0·258 = 54·95
Setting: "End of slide C 10 above first factor": 4.63 X 3.17 = 14.68; 0.694 X 0.484 = 0.336
This operation of transposing of the slide is not necessary, in practical use, with the "x.displaced scales" CF, DF, in con junction with scales C and D (see below).
Multiplication with Scales C and D of back of slide rule
a·b
Multiplications can also be carried out with the scales C and D on the back of the slide rule. It is necessary, however, to use the main cursor line for all settings and reading.
Example: 3.63 x 1.41 = 5.12. Use the cursor line to place C1 above D 3.63, then move cursor line to C 1.41 and read below on D the result, 5.12.
The C and D scales of the back of the slide rule are especially convenient for compound calculations, in conjunction with the root scales (see below) and the exponential scales (see below).
Division
a/b
Use the cursor line to place the numerator on D and the denominator on C opposite each other; the result can then be found underneath the beginning of the slide of the rule, C 1, or under the end
of the slide, C 10. Division with A and B: see below.
Example: 9.85 ÷ 2.5 = 3.94
First bring the cursor line into position above the numerator, 9.85, on the lower scale D of the
body of the slide rule, then place the denominator 2.5 (on the C graduation) underneath the
cursor line. The numerator and the denominator are now level with each other, and underneath
the beginning of the slide, C 1, the result 3·94 can be found on scale 0.
Examples for practice: 970 ÷ 26·8 = 36.2; 285 ÷ 3.14 = 90.7; 7500 ÷ 835 = 8.98; 0.685 ÷ 0.454 = 1·509; 68 ÷ 258 = 0.264
Combined Multiplication and Division
(a · b · c) / (d · e · f)
These calculations are mainly carried out with the scales C and D on the front of the slide rule.
Example: (13.8 x 24.5 x 3.75)/(17.6 x 29.6 x 4.96) = 0.491
Always start with the division and then let multiplication and division alternate. No readings need be taken of the intermediate results. We thus first use the cursor line to place D 1.3.8 and C 1.7.6
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (5 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
level with each other (division). No reading is taken of the result, approximately 0.8 underneath C 10 on D, and the multiplication by 24.5 is carried out immediately, by placing the cursor line on
C 2.4.5. The result (about 1.9 on D) is then divided by 29.6, by leaving the cursor line in its present position and bringing C 2.9.6 into position underneath it. The next operation is the
multiplication of the result (0.65 underneath C 10 on D) by 3.7.5, the final operation being the division by 4.96, In the same manner. The result 0.491 can then be found underneath C 10 on D.
Examples for practice: (38.9 x 1.374 x 16.3)/(141.2 x 2.14) = 2.883; (1.89 x 7.68 x 8.76)/(0.723 x 4.76) =36·95
Formation of Tables
a/b:c/d:e/f
For forming a table, first set the slide rule the required "equivalence"; mutual conversions of measurements, weights and other units can then be carried out. Once we establish that 1 inch = 25.4
mm, for example, C 1 Is placed over the corresponding value; for 75 lbs = 34 kgs., the two values are placed opposite each other on C and D.
Example; To convert yards into metres. 82 yards = 75 metres.
Place C 75 above D 82. This produces a table, and reading may be taken as follows:
42 yards are 38.4 metres; 2.8 yards are 2·56m; 640yards are 585 m; 16m are 17.5 yards; 128 m
are 140 yards, and so forth.
Examples for practice:
1 metre of material costs DM 45. Place C 1 above D
1 inch = 25.4 mm ("Equivalence": 26" . 66 cm). Place C 1 (left.hand 1 of C) above D
Exchange rate: US$ 1.. = DM 4.
45 and take the following readings with the cursor
2.5.4 (2.5.4 on D) and take the following readings by the aid of the cursor line:
Place C 10 above D 4.0.0 and take the following
line:
17 inches = 43.2 cm
readings with the aid of the cursor line:
3.20 metres of material costs DM 144.
38 inches = 96.6 cm
US$ 2.61 = DM 10.44 US$ 4.73 = DM 18.92
2.40 metres of material costs DM 108.
If, in the formation of tables, It is no longer possible to set and read off individual values, because the slide of the rule projects too far, we again transpose the slide, i.e. "hold the setting" by
placing the cursor line over C 1. The slide is then pushed along until C 10 replaces C 1.
Calculations with Scales CI and DI
The movable scale CI
This is sub.divided from 1 to 10, so that its graduation system corresponds to that of the scales C and D, but it takes the opposite direction and is therefore coloured red. Its use enables various
types of calculation to be performed. (This manipulation can be arrived by using scales CF and DF, as explained below).
1) If, for a given number a, the reciprocal 1 ÷ a is required, this is set on C or CI and the reciprocal read off, above it, on CI or on C. The reading can be taken without adjusting the slide, with
the use of the cursor alone. Experienced slide.rule users can also operate without zero setting, if they only refer to scales C and CI (and, above all, do not allow themselves to be confused by
1/a
the scale D appearing below).
Example: 1 ÷ 8 = 0·125, 1 ÷ 2 = 0.5; 1 ÷ 4 = 0.25; 1 ÷ 3 = 0.333.
(2) Multiplication likewise can be carried out with the scales D and CI. (Division by reciprocal = multiplication). This method is popular with many slide rule users.
axb
Example: 0.66 x 20.25 = 13.37.
Proceed as with division, i.e. first place the cursor line above 0.66 on D, then place the 20.25 on the CI underneath the cursor line; the product, 13.37, on D, can now be read off under C 1.
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (6 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
3) The following shows how simply products with a number of factors can be
calculated:
Example: 0.66 x 20.25 x 2.38 = 31.8.
axbxc
Multiply the first two factors as above; with C 1 above 13.37 (intermediate results) we immediately have the first setting required for multiplication by the next factor (by the method learnt first at
the top of p. 8). Consequently, we now bring the cursor line into position above C 2.38. The result, underneath it, on D, is 31.8. This could be followed immediately by a further multiplication, by
bringing the next factor, on CI, into position underneath the cursor line and reading off the result, under C 1 (or C 10, as the case may be), on D. We thus multiply alternately with the aid of D and
CI (see above, for method) and with the aid of C and D (first method - see above).
(4) Combined Multiplication and Division.
This likewise can be advantageously performed with the scale CI. Example: 36.4 / (3.2 x 4.6) = 2.473
First the division, i.e. cursor line above D 3-6-4, then C 3.2 underneath the cursor line. (Intermediate result: 11.37, underneath C 1). With CI above D 11.37, we already have
the first setting for the subsequent multiplication by 1/64 which is performed by the aid of the scale CI (1/c). The cursor line is thus now brought into position above CI 4.6,
the result - 2.472 - likewise being found beneath cursor line.
a/bxc
Example for practice: 44 / (4.85 x 3.66) = 2.48; 4.774 x /(0.63 x 1.24) = 6.11; 23.1 / (2.73 x 17.9) = 0.473
The trigonometric and exponential calculations provide further fields of use for the CI scale.
The fixed scale DI
This is subdivided like Scale D but runs in the opposite direction and is therefore coloured red.
1) In conjunction with D, DI gives the reciprocal. It is set and read with the cursor line. This offers the advantage that the two scales are provided permanently opposite each other, forming a table.
2) As with CI (see above) multiplication can also be carried out by the aid of DI. (Result appears above D l or D 10).
Example: 6.43 X 2.96 = 19.03. Cursor line above D l 6-4-3, second factor 2-9-6 on C under cursor line, read off result 19.03 above D 1.
3) It is above all with the function 1/(a x b) that the Scale DI is of advantage.
Example: 1/ (0.284 x 0.12). First carry out the multiplication on C and D, i.e. C 1 above D 2-8-4, cursor line above C 1.2; the intermediate result (0.03408) appears underneath, on D; the result,
29.34, Is likewise found on DI, underneath the cursor line on D.
The use of DI in trigonometrical calculations should also be noted.
Calculation with the Square Scales A and B
Multiplication and division can also be carried out on Scales A and B. For this purpose, use is made of the long cursor line.
Example: 2.5 X 3 = 7·5
Place the long cursor line above A 2.5, move B 1 into position under the cursor line, move the
latter into position above B 3, and find above it, on A, the result 7.5.
For division, the numerator on A and the denominator on B are first of all placed opposite each other by the aid of the cursor line (first of all above the numerator on A), after which the result can
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (7 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
be road off above B 1 or B 10. For this purpose, however, the cursor line must again be used.
a2
Squares and Square Roots
The square of any number is obtained by setting the cursor line over this number on scale D, the result is shown above it on scale A, under the cursor line. We can, however, also use the first and
last graduation marks at either end of the scales.
Examples:
1.662 = 2.75
2.492 = 6·2
3.222 = 10.24
When setting and taking readings one should get accustomed to work only with groups of figures and
determine the decimal value thereafter by a rough estimate. The figures should be rounded off liberally, as √a
this rough estimate is merely made to avoid getting tenfolds or tenths of the actual cursor.
Further examples: 1.3452 = 1.81; 4·572 = 20.9; 0·7652 = 0.585; 67.32 = 4530.
Square Roots are extracted by reversing this procedure. If, therefore, the cursor line is placed on a figure on A, then the square root can be read off on D, underneath it. Extracting square roots is,
however, not as simple as finding the square, as may be seen from the following example: Extract √3.65. Place the cursor line on A 3.65 and find underneath on D the square root 1·91. As we
have learned that the position of the decimal point is not shown on the slide rule, we could also set the cursor line over A 36.5 (in the second half of scale A between 10 and 100). Underneath we
would find the root 6·04, which is obviously the wrong result.
Before extracting the square root, we must decide If the number must be set on the right or the left half of the scale. At the left are the numbers from I to 10, and at the right those from 10 to 100.
If the number lies outside the scale range 1 to 100, it should be factorised by hundreds to bring significant figures within these limits.
Examples: √1936. We factorise √1936 = √(100x19.36) = 10 x √19.36 = 10 x 4.4 = 44
√0.00378 = √(37.8/10 000) = √37.8/100 = 6.15/100 0.0615
√145.8 = √(100 x 145.8) = 10 x √1.548 = 10 x 1.207 = 12.07
√507 000 = √ (10 000 x 50.7) = 100 x √50.7 = 100 x 7.12 = 712
Square and square root can also be determined by the aid of the root scales, with greater accuracy (see below).
Cubes and Cube Roots
(Cube scale K' see Immediately below)
a3
The cube scale K consists of three equally long units 1-10, 10-100, 100-1000, each one third of the length of the C, D and CI scales. To find the cube of a number, the cursor line is set to the base
number on scale C and the cube of that number is read off on K.
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (8 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
Examples:
1·673 = 4·66
2·343 = 12·8
5·773 = 192
When extracting a cube root, we use the opposite procedure. We set on K and read off on D.
3√
a
Examples: 3√4.66 = 1.67; 3√29.5 = 3.09; 3√192 = 5.77
When the base number is below 1 or above 1000, we must factorise it by powers of 10 to bring it within the interval of 1-1000 (similar to the method applied for square roots, see above).
If the cube scale K is employed with the Square Scale A, powers with the exponents 2/3 and 3/2 can be obtained with the aid of the cursor line.
Example: 7.53/2 = 20.5 Set the cursor line over 7.5 on A and read underneath on K the answer 20.5.
Example: 1322/3 = 25.9 Set the cursor line over 132 on K and read underneath on K the answer 2.69.
a3/2
a2/3
Calculations with the Scales CF, DF and CIF
(1) Formation of Tables
a/b:c/d:e/f
Since with the " folded" scales CF and DF, the value 1 is more or less in the centre, they provide an advantageous means of continuing the calculations, in the formation of tables, thus avoiding
the necessity for transposing of the slide when calculating on C and D.
Example: 75 lbs = 34 kgs. Place C 3-4 above D 7-5; this provides the conversion from English lbs. into kilogrammes.
Beyond 50 kgs. however (C 5), no further readings can be taken. Here we switch over to the upper scales CF and DF, and the slide rule can be set to the necessary values with the aid of the cursor
line.
If the "equivalence" In question (e.g. 75 lbs = 34 kgs.) is not known, but the formula 1 lb = 0.454 kgs., for example,
is known, then CF 1 is placed underneath DF 4-5-4, and this likewise provides the conversion from lbs Into kgs.
(2) Multiplication
axb
If; in multiplication on C and D, the second factor proves "unsettable", or if the method of transposing of the slide has to be adopted, this can be avoided by setting CF to the second factor and
taking a reading of the result on DF.
Example: 2.91 x 4 = 11.64. Place C 1 above D 2-9-1, and place the cursor line above CF 4. Above it, on DF, a reading may be taken of the result (11.64).
Examples for practice: 18.4 x 7.4 = 136.1; 42.25 x 3.7 = 156.3; 1.937 x 6 = 11.62.
(3) Multiplication and Division by π
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (9 of 20) [03/03/2001 13:28:27]
ax π
a/ π
Faber Castell 2/83N (and 62/83N) instructions in English
The change-over from the C and D scales to the CF or DF scale can be carried out direct with the cursor line and results in multiplication by π.
Example: 1·184π = 3·72. With the slide in the zero position (C 1 above D 1 and C 10 above D 10) the cursor line is brought into position above D 1-1-8-4, and the result - 3.72 - is read on DF,
likewise beneath the cursor line.
The converse operation results in division by π.
Examples: 18.65/π = 5·94. Place the cursor line above DF 1-8-6-5 and read the result - 5·94 - on D.
Examples for practice:
Area of an ellipse: F=a x b x π ; F=5·25 x 2·22 x π = 36.6.
Place C 10 above D 5-2-5, bring the cursor line into position above C 2-2-2, and read the result - 36.6 - on DF, without
any need to take a reading of the intermediate result - 11.65 - on D.
Length of a circular arc: s = α r π / 180 ; s =26.2 x 352 x π / 180 = 161
We start with the division, i.e. use the cursor line to bring C 1-8 and D 2-6-2 opposite each other. No reading need be taken of the intermediate result 0.1455 (under C 1). The multiplication by
352 is carried out by placing the cursor line under 3-5-2. (Intermediate result: 51·2, on D). The multiplication by is again carried out by switching over to the top part and by finding the result 161
underneath the cursor line on DF.
(4) The CIF Scale operates in conjunction with CF and DF just as CI does with C and D scales.
Examples for multiplication by a number of factors:
2.23 x 16·7 x 1.175 x 24·2 = 1059.
Solution: CI-2·23 placed above D-16·7 by the aid of the cursor line; the latter is placed above CF-1·175. CIF 24.3 under the cursor line. Read result, 1059, on DF, above CF 1.
0·53 x 0·73 x 39.1 x 0·732 = 11.07.
Solution: CI-0.53 placed above D-0·73 with the aid of the cursor line; the latter is placed above CF-39·1. CIF 0.732 underneath cursor line; Read the result 11.07, on DF, above CF 1.
Calculations with the Pythagorean Scale P
This scale represents the function y = √(1 - (0.1x)2); it operates in conjunction with D (= x), of which the values must be read from 0.1 to 1. The graduation runs in the opposite direction and is
therefore coloured red.
If D is set to the value x, then the corresponding value y = √(1 - x2) can be read on P; conversely, If D is set to y. the value x = √(1 - y2) can be read on P.
Example: y =√(1 - 0.82) = 0·6; x = √(1 - 0.62) = 0.8
If, therefore, we set D to x = 0.8, we find the value y = 0·6 on P, and vice versa. Example: sin α
= 0·134; cos α = 0.991 If D is set to the sine, the cosine is obtained on P, and vice versa.
Example:
Calculate the active current and the reactive current of a circuit absorbing 35 A with a cos ϕ of 0.8.
Ip = I x cos ϕ = 35 x 0·8 = 28 (A);IQ =I x sin ϕ = 35 x 0·6 = 21 (A).
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (10 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
Place C 35 above D 10 and find at D 8 (for a cos ϕ of 0.8), on C, the active current 28; underneath D 8 the value 0.6 (i.e. sin ϕ = 0·6) is at the same time found on P. If the cursor is now moved
into position onto D 6, the wattless current 21 is found above it on C.
Example: apparent power 530 kVA, effective power 428 kW. To find reactive power and cos ϕ:
Place C 530 above D 10, bring cursor into position above C 428, find the value (0.807) for the cos ϕ underneath it, on D; trace this value on P, by means of the cursor, and find the required
reactive power - 313 kW - above it, on C.
Roots of numbers only just below 1 and 100 can be found by means of this scale with an increased degree of accuracy:
Example:√0.925 = √(1− 0.075 = √(1-(0.2739)2) = 0.9618. We form the equation: 1 - z = 1 - 0.925 = 0.075.
Set the cursor on D 75 and find 0.2739 on W1 the cursor is now placed on D 2739 and the reading 0.9618 Is taken below it, on P.
Calculations with the Trigonometrical Scales S, T1 and T2
The trigonometric scales T1, T2 and S are sub-divided decimally; in conjunction with the basic scales C and D they indicate angular functions or, if read in the reverse direction, the angles.
When the T1, T2 and S scales are used in conjunction with the D, P and CI scales, as trigonometrical tables, the following must be noted:
The S scale, when the black figures are read, provides, in conjunction with the D scale (black), a table of sines; the same applies when the red figures are read, in conjunction with the P scale
(red). In the case of small angles the first procedure is the more accurate, while in the case of wide angles preference is given to the second method.
The S scale, when the red figures are read, provides, in conjunction with D (black), a table of cosines; the same applies when the black figures are read, In conjunction with P (red). In the case of
wide angles the former process Is the more accurate, while with small angles preference is given to the latter method.
The two T scales, when the black figures are read, provide, in conjunction with the D scale (black), a table of tangents; the same applies when the red figures are read, in conjunction with CI
(red).
The two T scales, when the red figures are read, provide, in conjunction with the D Scale (black), a table of cotangents; the same applies when the black figures are read, in conjunction with CI
(red).
sin 13° = 0.225 / S 13° (black) -D 0.225 (black)
sin 76° = 0.97 / S 76° (red) - P 0.97 (red)
cos 11° = 0·982 / S 11° (black) - P0·982 (red)
cos 78° = 0.208 / S 78° (red) - D 0.208 (black) These settings are carried out with the slide rule set at zero, by the aid of the cursor line.
tan 32° = 0.625 / T1 32° (black) - D 0.625 (black)
tan 57° = 1·54 / T257° (black) - D1.54 (black)
or T1 18° (black) - CI 3.08 (red)
cot 18° = 3.08 / T218° (red) - D3·08 (black)
75°
(red)
D
0.268
(black)
or
T2 75° (black) - CI 0.268 (red)
T
cot 75° = 0.268 / 1
To proceed from the sine of angle to its cosine (or vice versa) no reading need be taken of the angle. These pairs of values are to be found one under the other on D and P. The reading of the angle
can also be avoided when proceeding from the tangent to the cotangent, as these pairs of values are to be found one under the other on C and CI. An intermediate reading of the angle need only be
taken when proceeding from the sine or the cosine to the tangent or cotangent. Since, when reading the functions, these can be obtained either on D or on CI, multiplications and divisions can in
many cases follow immediately. It is only when the reading is taken on P that the value must be transferred to the main scales.
Further examples of the use of the trigonometrical and Pythagorean scales In a right-angled triangle.
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (11 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
(1) Example: Given: a = 2; b = 3. Find: c and α ; Formula: a x (1/b) = tan α ; a x (1/c) = sin α;
C 1 above D 2, cursor on CI 3; result for a (33.7) found on tan scale.
Cursor on 33.7 of the sine scale; value for c (3.6) found on CI.
(2) Example: Given a = 8; b = 20. Find c and α
C 10 above D 8, cursor on CI 20; find (21.8°) on tan scale.
Place cursor on 21.8 on sine scale; value for c (21.55) found on CI.
(3) Example: Given: α = 20; b = 8. Find c and α
C 1 above D 20, cursor on CI 8; value for α (68.2°) found on tan scale (T2).
Cursor to 68.2 on sine scale; value for c (21·55) found on CI.
(4) Example: Given: c = 5; α = 36.87°; Find: a and b. Formula: a = c x sin α; b = c x cos α
C 5 above D 10, cursor on 36.870 on sine scale; value for a (3) found on C.
The value (0.8) for cos a is found on the P scale at the same time, and the cursor moved to D 8.
The value for b, i.e. 4, can now be read on the C scale.
(5) Example: Given: c = 21.54; b = 20. Find: a and α
C 2154 above D 10, cursor to be placed at C 2 (for b = 20) and find the value (21.80) for a on the cos scale, but take the
reading 0·372 at the same time from the P scale. Push the slide along to the left by one scale-length. Move cursor to D
0.372 and find the value for α, i.e. 8, on C.
For the scalene triangle we have the formula a/ sin α = b / sin β = c / sin
Example: Given: a = 38.3; α = 52°; β = 59°; γ = 69°;
Find: b and c.
Place C 383 above S 52°; above S 59° and S 69° we now find the results (41.7 and 45.4) on C.
The Small-Angle Scale ST
For the functions of small angles from 0.55 to 6° the lower part of the body of the slide rule
bears the ST Scale ( arc 0.01 x) with the formula sin α ≈ tan α ≈ arc α
The ST scale operates in conjunction with scale D (or C).
All further calculations in this column are carried out solely with the aid of the cursor line.
and
the Mark ρ on Scales C and D
The functions of small angles can also be determined with the mark ρ = π/180 = 0.01745 in
accordance with the formula:
arc α = 0.01745 x α = ρ x α
In series calculations: set C 1 above ρ on D; underneath the angle-value on C, read the result on
D.
Examples for practice:
sin 2.5° ≈ tan 2.5° ≈ arc 2.5° ≈ 0.0436; sin 0.4° ≈ tan 0.4° ≈ arc 0.4° ≈ 0.00698; sin 0.0052° ≈ tan 0.0052° ≈ arc 0.0052° ≈ 0.0000908
Set the angle-values on the arc-scale ST. Read the functions from scale C (with zero setting) or Example: sin 3° ≈ tan 3° ≈ arc 3° = 0·0524. Place beginning of slide, C 1, above D 3; the result
from scale D (with the aid of the cursor line).
0.0524 can now be read on D, underneath ρ on C.
For calculation of the cosine and cotangent functions of angles over 84.5°
Example: cos 88° = sin 2° ≈ arc 2° 0 x 2 = 0·0349
Example: cos 88° = sin 2° ≈ arc 2° = 0·0349
cot 86·5° = tan ·5° ≈ arc 3·5° ≈ ρ x 3.5 = 0.0612
cot 86·5° = tan 3·5° ≈ arc 3.5° = 0.0612
Place the cursor line above the angle-value on the scale ST, and read the result from C (with
This is a simple multiplication; we thus place the beginning of the slide, C 1, above ρ on D, then
zero setting) or on D, underneath the cursor line.
the cursor line above the second factor on C, and read the result underneath it, on D.
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (12 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
For conversion of arcs into angular degrees:
Examples for practice::arc 6.28 = 360°; arc 1.11 = 63.5°; arc 0.04 = 2.29°; arc 0.007 = 0.402°; arc 0.64 = 36.7°; arc 0.32 = 18.35°
C 1 or C 10 above the ρ mark on D, then place cursor line above the arc-measurement on D. The
Set the arc on the C or D Scale. The degrees can be read from the arc-scale ST (with the aid of
angular degrees can be read above it, on C. (Increased accuracy is achieved when using the ρ marks on
the cursor line).
scales W1 and W1'.)
Calculations with Complex Numbers
Two complex values Ξ = 7.5 e i π/8 and ς = 3.4 e i π/10 are to be added. They are first of all concerted, in accordance with the Euler Equation
R x e iϕ = R (cos ϕ + i sin ϕ ) to the form (a + i b).
For calculations on the slide rule it is best to express these magnitudes vectorially: as follows:
= 7.5/22.5° and ς = 3·4/18° we can now calculate as follows:
(1) Place C 75 above D 10, move cursor to S 22·5° and read the value of b1 (2.87) on C. At the same time find the value of cos ϕ (0·924) on the P scale. Move cursor to D 924 and find the value of
a1 (6.93) on C.
(a1 + i b1) = 6.93 + i 2.87.
(2) Place C 34 above D 1, move cursor to S 18°, and find the value for b2 (1.05) on C. At the same time, find the value for cos ϕ (0.951) on P. Move cursor to D 951 (red extended graduation) and
find the value for a2 (3.24) on C.
(a2 + i b2) = 3·24 + i 1.05
a1 + a2 = 6.93 + 3.24 = 10.17 and i (b1 + b2) = i(2.87 + 1.05) = i 3·92.
The result is thus: = (10.17 + i 3.92)
If the result is to appear in vectorial form, we calculate thus:
Place C 10 above D 392, move cursor to CI 1017, and find the value of ϕ (21.07°) above it, on T1 (Tangent scale). We then move the cursor to 21.07 on the sine scale S and find the value for ς
(10.92) above it, on CI.
Thus = (10.17 + i 3·92) = 10·92/21·07°
and with ρ x ϕ = arc ϕ, (arc ϕ =0·368) we have = 10·92/21·07° = 10.92 e i 0.368
Example for the use of the T2 scale: ς = 192 - i 256
Place C 10 above D 256, move cursor to CI 192. On T2 we obtain the angular value 53.1°. Then place C 1 above D 256 and move the cursor.line to 5 53.1. On CI we obtain the result 320.
As the numb is in the IVth Quadrant, the angular value must be negative.
The multiplication of complex numbers is carried out in accordance with the equation:
Ξ · ς = X · e i ϕ· Y · e i ψ= XY · e i (ϕ+ψ) = XY / ϕ+ψ
Examples: (1 + 2i) · (3 + 1i) = 2,236 · e i · 1,107 · 3,162 · e i·0,316 = 2,236/63.4° · 3,162 /18,42° = 7,07/81.9° = 7,07 · e i·1,423
Calculations with the Root Scales W
The advantage of these scales is the increased degree of accuracy with which calculations can be carried out on the usual handy 10" standard model. It is above all for the main calculations that
they are used. Operations with the root scales differ to some extent from the method to which we have so far become accustomed but can be mastered according to a basic rule and with a short
period of practice. It should be noted that the root scales are sub-divided on the 20 inch graduation length.
Multiplication
(I) When the setting is carried out with the black index-1 (or index-10), the product is found on that scale of the body of the slide rule which is adjacent to the second factor.
(II) When the setting is carried out with the red index mark, the product is read on that scale of the body of the slide rule which is opposite to the second factor.
Examples for (I): 1.635 x 5.365 = 8.77
809 x 1.414 = 1144
Solution: Black Index-1 (W1'-1) above W1-1·635.
Solution: Black Index-l0 (W2'-10) underneath W2-809.
Cursor line above W2'-5·365; find result (8.77) on W2.
Cursor line above W1-1·414; find result (1144) on W1.
file:///C|/SlideRules/WebPage/a-to-z/fc283n/english/fc-283n-eng.htm (13 of 20) [03/03/2001 13:28:27]
Faber Castell 2/83N (and 62/83N) instructions in English
777 x 66·3 = 515; 5·165 x 0.2265 = 11699
Examples for practice: 236 x 4.06 = 958; 2.34 x 0·409 = 0.957
50.45 x 4·64 = 234.1. Solution: , Red index-mark under W2
Examples for (II): 1·804 x 7.73 = 13·95
Solution: Red Index-mark above W1-1·804; cursor line above W2'-7·73; at the same time find the 50.45; cursor line above W2'-4·64; at the same time find the result (234.1) underneath the cursor
result (13.95) underneath the cursor line, on the opposite scale W3.
line on the opposite scale W1.
Examples for practice: 14·78 x 0·945 = 13·97; 29.4 x 123·6 = 3634
0.395 x 0.5