George Parker Bidder (1806-1878) stands out as one of the most nota...
### Logarithmic Tables
In the 19th century, the field of engineeri...
In this context, "powers of registration" meant Bidder's capacity t...
In this context the notation &c is an abbreviation for the Latin ph...
The reason Bidder could calculate the logarithm of a number by addi...
To reason why Bidder can make this type of adjustment is because fo...
This is a typo, instead of 8771 it should be 877
There's another typo here, instead of 609 it should be 600. As in: ...
250
POLE
ON
MENTAL
CALCULATION.
[Selected
(Paper
No.
2486.)
“Mental Calculation.
A
Reminiscence
of
the late
Mr.
G.
P.
Bidder, Past-President.”
By
W.
POLE,
F.R.S.,
Hon. Sec. Inst.
C.E.
IN
the year
1855
Mr.
Bidder gave to the Institution1
a
most
interesting description of the wonderful powers of Mental Cal-
culation with which he was endowed, and he made the Address
specially valuable by showing how the methods he used might
be taught in ordinary education, and employed in calculations
generally. He not
only
gave examples
of
his modes of working
complicated
sums
in ordinary arithmetic
(as
applied to weights
and measures, time,
and
money), but also solved more difficult
problems, involving square and cube roots, prime numbers, com-
pound interest and
so
on,
including some special calculations of a
mechanical and engineering nature.
In the course
of
the Address he alluded to still higher feats
as
possible, particularly to the determination of logarithms, which he
appeared to consider the
ne
plus
ultra
of mental calculation. He
said, alluding to the two chief processes
by
which he worked
:-
‘‘Were
my powers
of
registration at all equal
to
the powers of reasoning
or
csecution,
I
should have no difficulty, in an inconceivably short space of time,
in composing
a
voluminous table
of
logarithms; but the power
of
registration
limits the power
of
calculation, and
it
is
only with great labour and stress
of
mind, that montal calculation can be carried on beyond
a
certain extent,.”
The object of the present Paper
is
to put
on
record the fact,
that
Mr.
Bidder subsequently succeeded in performing this
astonishing feat, which at the time he spoke he had thought
beyond his power. He appears
to
have considered, in fact, that
he had, in the above words, given himself a sort
of
challenge,
and after much further study, he declared his competency to cal-
culate, mentally, and
’‘
in
an
inconceivably short space of time,” the
Minutes
of
Proceedings, vol. xv.
p.
255.
The memorandum on which
the prcsent Paper
is
based was prepared during Mr. Bidder’s lifetime, and with
his approval
;
but something happened to delay its publication, and
it
has only
recently come to light again.
Papera.] POLE
ON
NENTAL CALCULATION.
251
logarithm
of
any number, with the accuracy
of
the large published
tables, extending to seven or eight places
of
decimals.
The enormous difficulty
of
this will be evident if any one will
consider how he would set to work to calculate, mentally
and
accurately, the value of
X
in the equation say-
10"
=
369353,
or how he would mentally sum up
a
large number of terms of the
series-
n
being
a
large number whose logarithm is to be found.
Mr. Bidder had often been in communication with the Author of
this Paper
on
the subject
of
mental calculation,
and
towards the
end of his life he communicated this discovery, and requested the
Author to aid him in testing
its
accuracy. For this purpose the
following series
of
prime numbers was selected by the Author,
with the aid of
a
mathematical friend
:-
71, 97, 659,
877,
1297,
6719,
8963, 9973, 115,249, 175,340
290,011, 350,107, 229,847, 369,353.
Each of these was given, separately, to Mr. Bidder, and the
logarithm in answer
was
returned very quickly. The time occu-
pied
on
each varied from half
a
minute to four minutes, being
generally about two minutes.
The logarithms were given in eight places, but the Author had
only Hutton's table of seven places to check them with. The
majority were stated at first quite correctly; but in some cases
errors occurred (generally
of
one figure only), which were dis-
covered
and
corrected
on
simply announcing that the answer was
not exact. The experiment
was
at any rate amply conclusive
as
to the real efficiency
of
the mental power to perform the calculations
for
any
numbers.
After the tests were over Mr. Bidder was good enough to
explain fully the manner in which the calculations were made
;
and
as
this is quite
a
unique incident in mathematical history,
it
may
be worth putting
on
record. Indeed,
Mr.
Bidder wished
it
to be
published
as
an
addendum to his lecture,
and
gave
it
to the Author
with that view.
In
general terms,
it
may be stated that his mode of calculation
was not based
on
any
of the formulas proper for calculating
logarithms in the first instance
;
for probably
no
amount of genius
and
skill would suffice for reducing them to manageable form for
mental calculation, when the given number was large. Neither
252
POLE
ON
MENTAL
CALCULATION.
[Selected
was
it
a matter of simple memory, for
it
would be equally
impossible for any one to recollect the logarithms, to eight places,
of
hundreds of thousands
of
numbers.
It
was, however, founded
on
a power of memory, of a more
limited range. Mr. Bidder's process may be said briefly
to
be
this
:
having stored in his memory the logarithms of a few
simple numbers, he was able, by his wonderful skill in dealing
with figures, to make use of them, mentally, to calculate accurately
the logarithms of any other numbers however large.
A
brief
description will give an idea of the manner in which this was
done.
In
the
first
place, as to the mnemonic bases used. He knew by
heart the logarithms of a great many
small prime numbers,
nearly
all, he said, under
100,
and some few above.
Secondly, it will be easily understood that these would enable
him
to
calculate the logarithms of any large numbers which were
nmltiples of those he knew. Thus for the logarithm of 63 he had
only to add together the logarithms of
7
and
9
;
for the logarithm
of 3567, he had only to sum up the logarithms of
29,
41,
and 3,
and
so
on.
In
carrying out this procesg he had an almost miraculous power
of
seeing, as it were intuitively, what factors would divide any
large given number not a prime. Thus if he was given the
number
17,861,
he would instantly remark
it
was
=
337
X
63
;
or
he
would see
as
quickly that
1659
was
=
79
X
7
X
3.
He could
not, he said, explain how he did this,
it
seemed like a natural
instinct to him.
These two qualifications, therefore-the knowledge of the logar-
ithms of certain small primes, and the power of reducing large
compound numbers to their component factors-constituted the
foundation
on
which Mr. Bidder's operations proceeded. They
would suffice for determining the logarithms of a great many
numbers by simply adding together the logarithms of their com-
ponent factors, which he could
do
mentally with the greatest ease.
But
it
still remains to be explained how he treated the case of
large primes, and this is really the great interest of his process.
His first endeavour was to find some multiple number
cery near
the number given, differing from
it
only
in
the last place of
figures, and if possible only by unity. For example, being given
the number
1051,
he would easily find the logarithm
of
1050
(=
30
X
7
X
5)
to which he would then have to make an addition
for the difference
of
1.
This addition would be very important
(for the accuracy aimed at), as
it
would affect the last five figures
Papers.] POLE
ON
'MENTAL CALCULATION.
253
of the logarithm. The operation is difficult
and
complicated, and
the manner
of
dealing with
it
was really the key to Mr. Bidder's
success.
It
is
due to him, therefore, to explain
it
somewhat fully.
His
method consisted in the use,
as
a
basis,
of
the folIowing
Table, which he knew by heart (only seven places are given-here,
he himself used eight)
:-
For
an
Addition
to
any
Number
m
of
n
-
100
-
.
1000
-
* *
l0000
n
-
n
..
n
100000
There
must
be
added
to
its
Logarithm.
.
Log.
1-01
=
0.0043214
.
Log.
1.001
=
0.00043407
.
Log.
1.0001
=
0.0000434
Log.
1.00001
=
0.0000043
Suppose therefore, as in the above case, there is to be added to
the logarithm a sum corresponding
to
an addition to the
number
of
-
(=
I).
Mr.
Bidder would take the proportion
n
1050
1
-
:
0
0004341
:
*
--
:
0
*
0004134.
This calculation would pass
1000
*
1050
1
through his mind instantaneously,
and
thus he
would
get, by
mental addition-
Log.
30
. . .
=
1.4771213
,,
5
.
.
.
=
0.6989700
,,
7
.
. .
=
0.8450980
Addition
for
the
1
=
0.0004134
Log.
of
1051 .
.
=
3.0216027
But
it
would sometimes be necessary to
add
a
larger proportion
than above mentioned,
say
between
-
and
-
For these he
adopted mentally
a
kind
of
proportionate sliding scale. Thus, for
the number
601,
he would see that
--
would require
0*0007i32,
so
that-
n
n
100
1000.
n
600
Log.
600
.
.
.
=
2.7781513
Addition
for
the
1
=
0'0007232
Log.
of
601
.
.
=
2'7788745
254
POLE ON MENTAL CALCULATION. [Selected
He preferred that the differences should be less than
~-
1000'
and
to attain this result he would often multiply the given
number by some other.
For
example, with the number
8771,
instead of taking
it
as-
he multiplied
it
by
13,
which gave
11,401
=
(609
X
19)
+
1
and
he found the logarithm thus-
?a
(73 X 12)
+
1
Log.
600
.
,
.
=
2.7781512
..
19
. .
.
=
1.2787536
Addition for
1
-
-
Oe0~.~fN28)
.
=
0*0000381
Log. 11401. .
.
=
4.0569429
Deduct log.
13.
.
=
1.1139433
Log.
877
. .
.
=
2.9429996
Similarly to
find
the logarithm
of
97
(which he could not
remember at once), he multiplied
it
by
33,
giving
3,201
(=
100
X
2j
+
1)
which 'he worked in the same. way, saying that
it
was
more certain than
96
+
1,
and quite as easy to him.
The following statement will illustrate how these principles
were,?pplied for each
of
the test numbers given him
:-
71
given by memory
;
97
already explained
;
877
already .explained
;
G59
=
(3
X
2
X
11)
-
l
;
1297
=
6719
=
(G4
X 105)
-
1;
8963
=
(27 X
4
X
83)
-
l;
(400
X
107)
+
1.
33
2
(9
X
41)
+
1
9973
=
37
'
115249
=
(25
X 461)
-
1
;
[and
461
=
(23x 20)
+
11
[and
7014
=
7000
+
141
175349
=
(25
X
7014)
-
1;
229847
=
230000
-
153;
290011
=
(29
X
10000)
+
11;
350107
=
(7
X
50000)
+
107;
369353
=
(9
X
41000)
+
369
-
16.
Papers.]
POLE
OR
KENTAL CALCULATION.
255
In these latter cases the calculation of the excess
was
more
complicated, thus-
In the case of
369,353
he
first
added for a
T&
part and then
deducted for the
16-a
good example of his readiness in devising
the best way of performing the calculation.
It
is
worth mentibning that Mr. Bidder declared that the
logarithms he knew by heart had been all calculated mentally;
he had never written them down, nor looked for information into
a
Table for many years.
Although this great feat
is
probably far in advance of any other
mental calculation
on
record,
it
will be seen that, when thus
analysed,
it
is explained on the same principles as are given in
Mr. Bidder’s lecture of
1855,
and which may be classified as
follows
:-
1.
A
good
memory for retaining certain standard numbers for
reference.
2.
The power of performing the ordinary simple arithmetical
operations of multiplication, division, proportion,
&C.,
on
large
numbers with great facility, quickness,
and
accuracy.
3.
A
remarkable intuitive detection of multiple numbers, and an
instantaneous perception of the factors forming them.
4.
The power of what Mr. Bidder calIed “registration,”
i.e.,
of
keeping a string of successive processes
and
results clear in the
mind, and working accurately upon them.
5.
The power of devising instantly the best mode of performing
a
complicated problem, as regards facility, quickness, and certainty,
All these were undoubtedly the result of natural gifts; the
second, third, and fourth, special and phenomenal. The
fifth
Mr. Bidder considered
to
be largely improved by practice
and
experience.
It
is obvious that such powers cannot be taught to even
the most promising and intelligent pupils
;
but undoubtedly the
processes described may furnish useful hints for improving the
power
of
practical calculation.
KOTE.
Nr.
Bidder’s son, Mr.
G.
P.
Bidder,
&.C.
(himself an accomplished
mathematician), has favoured the Author with the following
256
POLE
ON
MENTAL CALCULATION.
[Selectcd
remarks on the above memorandum. They show the truth
of
the
assertion,
so
confidently made by his father, that many of the
principles
of
labour-saving in calculation which he devised
could
be taught
and
made useful.
“
1
remember very well my father’s fondncss for calculating logarithms, and
the facility with which he performed the operation.
We
often discussed the
subject, and in consequcnce my own attention was drawn
to
it, and
I
eventually
became able to calculate them mentally without much trouble, though not
at
a
speed comparable with his, nor with such accuracy. Morcover,
I
always con-
tented myself with six places of decimals.
‘‘
Of course the great desideratum is to devise a method which relieves the
mind, as far as possible, of the burthen
of
performing and registering long’
calculations. The details of my father’s method, as described in the Paper, arc
new to me, and
I
think
it
may be interesting to compare
it
with my own, which,
although in its main lines very similar, yet differs somewhat in form; for having
obtained from him an idea of the general principles he adopted,
I
arrived at
my
own details independently. My method for numbers not large was as follows
:-
“I
knew by heart the logarithms of
2,3,7,
and
11,
and also the modulus
0,4343.
My rule then was to select some multiple of the prime
of
the form
m
+
R,
where
m
is
a
multiple of
2,
3,
7,
and
11,
and
IZ
was very small,
usually
=
1.
Then
loglo
(m
+
11)
=
m
+
log,,
(1
+;>.
But loglo
(l
+
i)
=
0.4343
log.
c
1
+
-
,
(
2
and logc
(l
+
:)
=
i-
4
(y)’
m
+
j-
(?)I,
tn
$C.
‘‘
This
I
found quite practicable mentally for primes not very large.
“For
very large numbers (not having my father’s great power of seizing
instantaneously upon component factors),
I
adopted another method, which
I
can
best illustrate by an example, which recently
I
worked out mentally, namely,
to find the logarithm of
724871.
“The method depends on the use
of
the powers of
1*1,1~01,1~001, &C.,
of
which the logarithms are assumed to be known.
It
is not difficult to make
out
that-
724871
=
72
X
(1.001)6
X
(1.0001)
7
X
(1.00001)4.5.
“Adding the logarithms together, the result
is
obtained,
*860261.
The
mental strain
is
much less than would at first sight appear.
‘‘
GEORGE
P.
BIDDER.”
To reason why Bidder can make this type of adjustment is because for small changes in the input of a logarithm, the change in the log's output is approximately proportional to the percentage change in the input. Since:
$$\frac{{d}}{{dx}} (\log_{10}(x)) = \frac{{1}}{{x \ln(10)}}$$
In the paper, *n* stands for a small change in the input number (the number for which the log is being calculated), and 1/1050 represents the proportional change to the input number 1050 when it is increased by 1. Given that Bidder had memorized a table of logarithmic values for small adjustments, he knows that the change in the log output for an input change of 1 in 1000 is 0.0004341. Therefore to know the adjustment for a change of 1 in 1050, Bidder would use the proportion 1/1000: 0.0004341 :: 1/1050: x, solving for x to find the adjustment. The result of this calculation would be mentally added to the logarithm of 1050 to get the logarithm of 1051.
There's another typo here, instead of 609 it should be 600. As in:
$$11,401 = (600 \times 19) + 1$$
In this context the notation &c is an abbreviation for the Latin phrase "et cetera", which is used to indicate that the established pattern continues indefinitely. In this case the next term of the series would be:
$$\frac{1}{7}\left(\frac{n - 1}{n + 1}\right)^7$$
In this context, "powers of registration" meant Bidder's capacity to keep a string of successive processes and results clear in his mind, allowing him to work accurately upon them. If we were to use a digital computer analogy, we could think of it as his "RAM".
This is a typo, instead of 8771 it should be 877
The reason Bidder could calculate the logarithm of a number by adding the logarithms of its factors is based on one of the fundamental properties of logarithms, namely the product rule. The product rule states that the logarithm of a product is equal to the sum of the logarithms of the individual factors.
$$\log(xy) = \log (x) + \log (y)$$
So, in the case of calculating the logarithm of 63, which can be factored as 7 * 9, Bidder could add the logarithm of 7 to the logarithm of 9 to get the logarithm of 63:
$$ \log (63) = \log (7) + \log (9) $$
### Logarithmic Tables
In the 19th century, the field of engineering, like many other disciplines, relied heavily on the use of logarithmic tables for calculations involving multiplication, division, exponentiation, and root extraction.
Logarithms greatly simplified these calculations, as they have a property where the multiplication of numbers can be transformed into an addition of their logarithms, and division can be transformed into subtraction. Similarly, exponentiation becomes multiplication, and root extraction becomes division when using logarithms. These transformations allowed engineers to perform complex calculations more accurately and swiftly.
However, even using logarithmic tables required the time-consuming task of looking up logarithms and antilogarithms, then performing the addition or subtraction, and then looking up the result. As a result computations could still be slow and prone to errors, particularly if the lookup was done incorrectly or if the table itself had errors, which was not uncommon.
In this context, Bidder's ability to calculate logarithms in his head was nothing short of extraordinary. It gave him the capability to perform complex calculations rapidly and accurately, without needing to rely on extensive tables or spend time on manual lookups.
*Logarithmic Table*
George Parker Bidder (1806-1878) stands out as one of the most notable human calculators in history. Born to a stonemason, George began to display his extraordinary mathematical abilities around the tender age of five. After receiving initial counting lessons from his elder brother, he self-taught arithmetic, mastering the art of performing complex calculations mentally even before he could write. His father would showcase George's abilities at local fairs, earning him the nickname "The Calculating Boy".
Coming from a humble background, one might have assumed that formal education was beyond Bidder's reach. However, his remarkable calculating skills soon attracted the attention of benefactors who sponsored his education, paving his way to study engineering at the University of Edinburgh. This opportunity led him to achieve substantial success and wealth as a civil engineer. He was exceptionally capable at appearing before Parliamentary Committees to gain approval of projects because his prodigious ability at mental calculation made him highly effective in presenting his plans and spotting errors in the plans of others.
*George Parker Bidder*