T H E J O U R N A L OF I N D U S T R I A L A N D E N G I N E E R I N G C H E M I S T R Y
June, 1 9 2 0
flow instruments of giving true instead of kinematic viscosity. 4-111 conclusion we wish t o emphasize the importance of expressing viscosities in absolute units. The additional time required is slight and its expenditure is well repaid; not only are all results made of permanent value and directly comparable, being entirely independent of the instrument employed, but any possible confusion between the manufacturer and the trade, which often arises due t o the use of a multiplicity of empirical standards, is entirely avoided. THE RAPID PREPARATION OF TABLES By J. C. Witt PITTSBURGH, PENNSYLVANIA Received January 2, 1920
I n commercial and manufacturing control laboratories, certain determinations and tests are made a great many times, and the calculation of results becomes tiresome. T o lessen this work, various types of computing apparatus, charts, and factor tables are employed. The desirability and usefulness of a table are often recognized long before a chemist has the time t o prepare one or can bring himself t o begin the considerable amount of uninteresting work necessary. Logarithm tables or some type of slide rule may be used. The former, however, requires considerable time, and the latter is often insufficiently accurate for chemical calculations. TABLE I dA
E2
E3
d3-0.5248
b2
ba 2.624
2.478
a
a
+ di _.,.
dr-0.5321
CI
ds-0.5394
+ 3dA . . a + 4dA .. . a + 5 d ~ ..
.... ..,.
a
,
,
,
ce
de-0.5467
,
C8
ds--J0.5613 CO
d~-O.5686 C10
d10+0.5759
,
+ idA . . I . a + 8dA ,.,. a + 9dA . . a
,,
A
bi
= 2332
ci a
=
d,
=
+ 2d1 .,..
,
c4
a
+ dA a + 2 d ~ .. . a
dz-0.5175
dC
bi 2.332 C,
di-0.5102
dB
B
dD
dE
b4
bi 2.916
2.770
a f 3d1 a
.,..
0
.1
+ 4d1
1
.... .... ....
....
2
....
3
....
4
....
5
....
....
.... ....
....
....
....
7
....
....
8
....
.... ....
....
9
....
....
....
10
D
E
c
6
0.5102
= bici = 1.18979
d,,
d,,, dA di
- bi - 0.146 (for index line) ClO = -= 0.0073 (for index column) 10 - 1 b5
5 - 1
c1
= drd,, = =
= 0,00107 bid,, = 0.01702 cid, = 0.07449
cipal labor consists in multiplying one series of numbers b y another number or series of numbers. Also 1
Copyrighted 1920 b y J. C. Witt.
591
i t very frequently happens t h a t one or both series are in the form of arithmetical progressions. When the numbers fulfill these conditions, the table may be prepared with little labor. There is nothing new about the principles involved, but their application enables one t o prepare a table, accurate t o any desired number of decimal places, much more rapidly t h a n by t h e use of logarithms, and t o check easily the correctness of the results. The greater the number of terms, the more rapidly, relatively, a table can be prepared, because the preliminary work is independent of the size of the table. To explain the procedure, we may refer t o Table I which is similar t o one the writer recently had occasion t o prepare. Suppose i t is desired t o multiply 2 . 3 3 2 , 2 . 4 7 8 , 2 . 6 2 4 , 2 . 7 7 0 , and 2 . 9 1 6 , respectively, by 0 . 5 1 0 2 , 0 . 5 1 7 5 , 0 . 5 2 4 8 , 0 . 5 3 2 1 , 0 . 5 3 9 4 , 0 . 5 4 6 7 , 0 . j 5 4 0 , 0.5613, 0 . 5 6 8 6 , and 0 . 5 7 5 9 , respectively, carrying out the
results t o the fourth decimal place. Designate the first b5, and the second series series by bl, bP, by cl, c2, . . . . . c10. Draw the outline of the table and set down the numbers in the index line and in the index column. The index line is a progression, of which the common difference,
. . . . .
d,, is
2.916-2.332 5-1
= 0 . 1 4 6 ; and the index column is a
progression, of which the common difference, dII, is 0 . 5 7 59-0.5 I 0 2 = 0.0073. Let the first, or least, term 10-1
of the table be a. We shall discuss two cases; ( I ) when only one series is in progression, and ( 2 ) when both series are in progression. O N E SERIES I N PROGRESSION-Let US consider t h a t the index column of Table I is in progression but t h a t the index line is not. The terms of line I are calculated by performing the indicated multiplication and carrying the results out t o a t least one more decimal place t h a n is required in the finished tab1e.l Now each column of the table will be a progression, of which the first term is known. The common difference of each column is equal t o the product of the index of t h a t column and the common difference of the index column. For example dA = bld,,, dB = btd,,, etc. I t may be seen t h a t if the common difference of the index column is one, the common difference of each column will be the same as the index of t h a t column. The remaining terms of each column may then be found by successively adding the common difference of the column t o t h e first term. I n case the index line of a table is in progression and the index column is not, the table may be prepared by first finding the terms in the first and last columns, and then computing the remaining terms by use of the common difference of each line. The only systematic error which arises in substituting this method of calculation for multiplication is due t o the fact t h a t in many cases the exact common
1 During the process of preparing the table all the terms and common differences should be carried to at least one more decimal place than is reauired in the finished table. The extra olace is drouued - . when the table is completed.
T H E JOURNAL O F INDUSTRIAL A N D ENGINEERING CHEMISTRY
592
extra decimal place is dropped a t t h e completion of the table, in rare cases a term will differ from the value found by multiplication by -I I in the last place. Since the last decimal place is usually not exactly correct even when the term has been found by multiplication, however, the true error due t o the substitution of addition for multiplication is less t h a n the apparent error. For example, suppose in calculating the terms for a 4-place table, a term is 0.45934 by multiplication, and 0.45935 by the short method. This will be set down in the first case as 0.4593 and in the second case as o.45g4-an apparent error of +I. However, the error in the first case is -0.00004 and in the second, +0.00005. Therefore, the true error caused b y the short method is only O.OOOOI, which is negligible. The error may be made less t h a n any given value by increasing the number of decimal places of the terms, during the process of preparing the table. Any accidental error is easily found, except in the very unlikely case of compensation of errors. For example, in finding the terms of a given column, the last term as found by adding the common difference is compared with the same term as found b y multiplication, and if the difference between t h e two results is greater than A 5 in the last decimal place, an accidental error exists some place in t h a t column. It often saves time, t o find all the terms of the last line by multiplication a t the beginning so t h a t the accuracy of each column may be checked as completed. B O T H S E R I E S I N PROGRESSION-In this case, t h e method of preparation may be still further shortened. Aside from checking the accuracy of t h e results, i t is necessary t o prepare only one term, u,by multiplication. After performing two other multiplications, the remainder of the work may be done by addition alone. When both the index line and the index column are in progression, then every column and every line will be in progression. I n finding the terms of the first line, a is the first term of the progression. T h e common difference is the product of t h e index of the line and the common difference of the index line, or, dl = cld, = 0.5102 X 0.146 = 0.07449. By adding this quantity successively t o a, all the terms of the first line may be found. As has been explained, the common differences for the columns are as follows: d , = bld,,dB = bzd,,, etc. But bz = bl d,, bS = bl 2 d,, etc. Therefore, we may consider these common differences themselves as in progression (line 0). The first term The common difference of the is bld,, = 0 . 0 1 7 0 2 . The series is therefore bld,,, series is d,d,,, or d,,,. bid,, d,,,, bld,, 2 d,,,, etc. Now, having t h e first term and the common difference for each column, the table may be readily completed. The common differences for the lines are also in progression (column 0). The first term is c l d , and the common difference is d,,,. T o check the results for accidental errors, may be calculated by multiplication or by adding a multiple of the common difference of t h a t line t o the first term in the line.
+
+
+
+
Vol.
12,
No. 6
The systematic errors are likely t o be greater t h a n in the first case. There is a n error due t o the common difference of line 0,of line I , and t o the common difference of each column. Of course these will either combine or compensate each other, depending on t h e signs. I n starting a table, especially a large one, i t i s well t o check the last term in line 0 , line I , and Column E, t o see how much the maximum error will be in each case. If i t is greater t h a n desired, the number of decimal places may be increased, or in the case of a table with 50 t o I O O columns, the true values, line o and line I , may be calculated by multiplication for every 5 or I O columns and t h e remaining terms of those lines filled in by use of t h e common differences. After several tables have been prepared by this method, many ways of looking for accidental and regular errors will present themselves. I t is of interest, though of no practical importance, t h a t any term, t , may be found by the following formula, in which n~ is t h e number of the column in which the terrp is sought, diminished by one, and nz is the number of the line: diminished by one: t =
u
+ n l c l d , + nzbld,, + fllnzdIII
I n case the common difference of the index line and t h a t of the index column are the same quantity, d , this formula will become 1 = a (nlc1 nzbl)d mnzd2.
+
+
+
The method of preparation may be further illustrated by Table 11, where the numerical relations are so simple they may b e seen a t a glance. We shall
* x 3
4
12 4
I
U
B
c
8 x d
7
2
$ 8 8+12= s 20 20 12 E 32 8 32 f 12 = 44 11 14 44 12 56 Column A
+
+
TABLE I1 1~2 + 6 = 18
18+6= 24
--Index 8
6 8 + 4 =
24f630
30+6= 36
10 16+4=
12 20+4=
Line----
12+4=
Line
412
416
4 2o
4 24
1
30
40
50
60
2
48
64
80
96
3
66
88
110
132
4
84
112
140
168
5
B
C
D
E
consider only the second case, where both series are in progression. The index line is 4, 6, 8, IO, 12, and t h e index column is 2 , 5 , 8, 11, 14. It may be readily seen t h a t the common difference of t h e index line is 2 , t h e common difference of the index column is 3, and the first (and least) term of the table is 8. Now the common difference of line I is equal t o the index of t h a t line times the common difference of the index line, or 2 X 2 = 4. Starting with 8, we add 4 successively, obtaining 8, 1 2 , 16, 2 0 , 24 for the terms of line I . I n the same way the common differenceof Column A is 4 X 3 = 12. By adding successively the product of t h e common difference of the index line and t h a t of t h e index column (2 X 3 = 6) we obtain 1 2 , 18, 24, 3 0 , and 36 as the common differences of Columns A, B, C, D, and E, respectively. Now having the first term and t h e common difference of each column, the other terms of each column may be rapidly found.
