Produced by Jonathan Ingram, Hagen von Eitzen and the
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[*Transcriber's Note:
The following errors found in the original have been left as is.
Chapter I, 14th paragraph:
drop double quote before 'It is said';
Chapter IV, 1st paragraph:
'so similar than' read 'so similar that';
Chapter IV, table of Hebrew numerals (near footnote 144):
insert comma after 'shemoneh';
Chapter V, table of Tahuatan numerals (near footnote 201):
'tahi,' read 'tahi.';
Same table:
' 20,000. tufa' read '200,000. tufa';
Chapter VI, table of Bagrimma numerals (near footnote 259):
'marta = 5 + 2' read 'marta = 5 + 3';
Same table:
'do-so = [5] + 3' read 'do-so = [5] + 4';
Chapter VII, table of Nahuatl numerals (near footnote 365):
'90-10' read '80-10';
In paragraph following that table:
'+ (15 + 4) × 400 × 800' read
'(15 + 4) × 20 × 400 × 8000 + (15 + 4) × 400 × 8000';
In text of footnote 297:
'II. I. p. 179' read 'II. i. p. 179';
*]
THE MACMILLAN COMPANY
NEW YORK · BOSTON · CHICAGO · DALLAS
ATLANTA · SAN FRANCISCO
MACMILLAN & CO., LIMITED
LONDON · BOMBAY · CALCUTTA
MELBOURNE
THE MACMILLAN COMPANY
OF CANADA, LIMITED
TORONTO
THE NUMBER CONCEPT
ITS ORIGIN AND DEVELOPMENT
BY
LEVI LEONARD CONANT, PH.D.
ASSOCIATE PROFESSOR OF MATHEMATICS IN THE WORCESTER
POLYTECHNIC INSTITUTE
New York
MACMILLAN AND CO.
AND LONDON
1931
COPYRIGHT, 1896,
BY THE MACMILLAN COMPANY.
COPYRIGHT, 1924,
BY EMMA B. CONANT.
All rights reserved--no part of this book may be reproduced in any form
without permission in writing from the publisher.
Set up and electrotyped. Published July, 1896.
Norwood Press
J.S. Cushing Co.--Berwick & Smith Co.
Norwood, Mass., U.S.A.
PREFACE.
In the selection of authorities which have been consulted in the
preparation of this work, and to which reference is made in the following
pages, great care has been taken. Original sources have been drawn upon in
the majority of cases, and nearly all of these are the most recent
attainable. Whenever it has not been possible to cite original and recent
works, the author has quoted only such as are most standard and
trustworthy. In the choice of orthography of proper names and numeral
words, the forms have, in almost all cases, been written as they were
found, with no attempt to reduce them to a systematic English basis. In
many instances this would have been quite impossible; and, even if
possible, it would have been altogether unimportant. Hence the forms,
whether German, French, Italian, Spanish, or Danish in their transcription,
are left unchanged. Diacritical marks are omitted, however, since the
proper key could hardly be furnished in a work of this kind.
With the above exceptions, this study will, it is hoped, be found to be
quite complete; and as the subject here investigated has never before been
treated in any thorough and comprehensive manner, it is hoped that this
book may be found helpful. The collections of numeral systems illustrating
the use of the binary, the quinary, and other number systems, are, taken
together, believed to be the most extensive now existing in any language.
Only the cardinal numerals have been considered. The ordinals present no
marked peculiarities which would, in a work of this kind, render a separate
discussion necessary. Accordingly they have, though with some reluctance,
been omitted entirely.
Sincere thanks are due to those who have assisted the author in the
preparation of his materials. Especial acknowledgment should be made to
Horatio Hale, Dr. D.G. Brinton, Frank Hamilton Cushing, and Dr. A.F.
Chamberlain.
WORCESTER, MASS., Nov. 12, 1895.
CONTENTS.
Chapter I.
Counting 1
Chapter II.
Number System Limits 21
Chapter III.
Origin of Number Words 37
Chapter IV.
Origin of Number Words (_continued_) 74
Chapter V.
Miscellaneous Number Bases 100
Chapter VI.
The Quinary System 134
Chapter VII.
The Vigesimal System 176
* * * * *
Index 211
THE NUMBER CONCEPT: ITS ORIGIN AND DEVELOPMENT.
CHAPTER I.
COUNTING.
Among the speculative questions which arise in connection with the study of
arithmetic from a historical standpoint, the origin of number is one that
has provoked much lively discussion, and has led to a great amount of
learned research among the primitive and savage languages of the human
race. A few simple considerations will, however, show that such research
must necessarily leave this question entirely unsettled, and will indicate
clearly that it is, from the very nature of things, a question to which no
definite and final answer can be given.
Among the barbarous tribes whose languages have been studied, even in a
most cursory manner, none have ever been discovered which did not show some
familiarity with the number concept. The knowledge thus indicated has often
proved to be most limited; not extending beyond the numbers 1 and 2, or 1,
2, and 3. Examples of this poverty of number knowledge are found among the
forest tribes of Brazil, the native races of Australia and elsewhere, and
they are considered in some detail in the next chapter. At first thought it
seems quite inconceivable that any human being should be destitute of the
power of counting beyond 2. But such is the case; and in a few instances
languages have been found to be absolutely destitute of pure numeral words.
The Chiquitos of Bolivia had no real numerals whatever,[1] but expressed
their idea for "one" by the word _etama_, meaning alone. The Tacanas of the
same country have no numerals except those borrowed from Spanish, or from
Aymara or Peno, languages with which they have long been in contact.[2] A
few other South American languages are almost equally destitute of numeral
words. But even here, rudimentary as the number sense undoubtedly is, it is
not wholly lacking; and some indirect expression, or some form of
circumlocution, shows a conception of the difference between _one_ and
_two_, or at least, between _one_ and _many_.
These facts must of necessity deter the mathematician from seeking to push
his investigation too far back toward the very origin of number.
Philosophers have endeavoured to establish certain propositions concerning
this subject, but, as might have been expected, have failed to reach any
common ground of agreement. Whewell has maintained that "such propositions
as that two and three make five are necessary truths, containing in them an
element of certainty beyond that which mere experience can give." Mill, on
the other hand, argues that any such statement merely expresses a truth
derived from early and constant experience; and in this view he is heartily
supported by Tylor.[3] But why this question should provoke controversy, it
is difficult for the mathematician to understand. Either view would seem to
be correct, according to the standpoint from which the question is
approached. We know of no language in which the suggestion of number does
not appear, and we must admit that the words which give expression to the
number sense would be among the early words to be formed in any language.
They express ideas which are, at first, wholly concrete, which are of the
greatest possible simplicity, and which seem in many ways to be clearly
understood, even by the higher orders of the brute creation. The origin of
number would in itself, then, appear to lie beyond the proper limits of
inquiry; and the primitive conception of number to be fundamental with
human thought.
In connection with the assertion that the idea of number seems to be
understood by the higher orders of animals, the following brief quotation
from a paper by Sir John Lubbock may not be out of place: "Leroy ...
mentions a case in which a man was anxious to shoot a crow. 'To deceive
this suspicious bird, the plan was hit upon of sending two men to the watch
house, one of whom passed on, while the other remained; but the crow
counted and kept her distance. The next day three went, and again she
perceived that only two retired. In fine, it was found necessary to send
five or six men to the watch house to put her out in her calculation. The
crow, thinking that this number of men had passed by, lost no time in
returning.' From this he inferred that crows could count up to four.
Lichtenberg mentions a nightingale which was said to count up to three.
Every day he gave it three mealworms, one at a time. When it had finished
one it returned for another, but after the third it knew that the feast was
over.... There is an amusing and suggestive remark in Mr. Galton's
interesting _Narrative of an Explorer in Tropical South Africa_. After
describing the Demara's weakness in calculations, he says: 'Once while I
watched a Demara floundering hopelessly in a calculation on one side of me,
I observed, "Dinah," my spaniel, equally embarrassed on the other; she was
overlooking half a dozen of her new-born puppies, which had been removed
two or three times from her, and her anxiety was excessive, as she tried to
find out if they were all present, or if any were still missing. She kept
puzzling and running her eyes over them backwards and forwards, but could
not satisfy herself. She evidently had a vague notion of counting, but the
figure was too large for her brain. Taking the two as they stood, dog and
Demara, the comparison reflected no great honour on the man....' According
to my bird-nesting recollections, which I have refreshed by more recent
experience, if a nest contains four eggs, one may safely be taken; but if
two are removed, the bird generally deserts. Here, then, it would seem as
if we had some reason for supposing that there is sufficient intelligence
to distinguish three from four. An interesting consideration arises with
reference to the number of the victims allotted to each cell by the
solitary wasps. One species of Ammophila considers one large caterpillar of
_Noctua segetum_ enough; one species of Eumenes supplies its young with
five victims; another 10, 15, and even up to 24. The number appears to be
constant in each species. How does the insect know when her task is
fulfilled? Not by the cell being filled, for if some be removed, she does
not replace them. When she has brought her complement she considers her
task accomplished, whether the victims are still there or not. How, then,
does she know when she has made up the number 24? Perhaps it will be said
that each species feels some mysterious and innate tendency to provide a
certain number of victims. This would, under no circumstances, be any
explanation; but it is not in accordance with the facts. In the genus
Eumenes the males are much smaller than the females.... If the egg is male,
she supplies five; if female, 10 victims. Does she count? Certainly this
seems very like a commencement of arithmetic."[4]
Many writers do not agree with the conclusions which Lubbock reaches;
maintaining that there is, in all such instances, a perception of greater
or less quantity rather than any idea of number. But a careful
consideration of the objections offered fails entirely to weaken the
argument. Example after example of a nature similar to those just quoted
might be given, indicating on the part of animals a perception of the
difference between 1 and 2, or between 2 and 3 and 4; and any reasoning
which tends to show that it is quantity rather than number which the animal
perceives, will apply with equal force to the Demara, the Chiquito, and the
Australian. Hence the actual origin of number may safely be excluded from
the limits of investigation, and, for the present, be left in the field of
pure speculation.
A most inviting field for research is, however, furnished by the primitive
methods of counting and of giving visible expression to the idea of number.
Our starting-point must, of course, be the sign language, which always
precedes intelligible speech; and which is so convenient and so expressive
a method of communication that the human family, even in its most highly
developed branches, never wholly lays it aside. It may, indeed, be stated
as a universal law, that some practical method of numeration has, in the
childhood of every nation or tribe, preceded the formation of numeral
words.
Practical methods of numeration are many in number and diverse in kind. But
the one primitive method of counting which seems to have been almost
universal throughout all time is the finger method. It is a matter of
common experience and observation that every child, when he begins to
count, turns instinctively to his fingers; and, with these convenient aids
as counters, tallies off the little number he has in mind. This method is
at once so natural and obvious that there can be no doubt that it has
always been employed by savage tribes, since the first appearance of the
human race in remote antiquity. All research among uncivilized peoples has
tended to confirm this view, were confirmation needed of anything so
patent. Occasionally some exception to this rule is found; or some
variation, such as is presented by the forest tribes of Brazil, who,
instead of counting on the fingers themselves, count on the joints of their
fingers.[5] As the entire number system of these tribes appears to be
limited to _three_, this variation is no cause for surprise.
The variety in practical methods of numeration observed among savage races,
and among civilized peoples as well, is so great that any detailed account
of them would be almost impossible. In one region we find sticks or splints
used; in another, pebbles or shells; in another, simple scratches, or
notches cut in a stick, Robinson Crusoe fashion; in another, kernels or
little heaps of grain; in another, knots on a string; and so on, in
diversity of method almost endless. Such are the devices which have been,
and still are, to be found in the daily habit of great numbers of Indian,
negro, Mongolian, and Malay tribes; while, to pass at a single step to the
other extremity of intellectual development, the German student keeps his
beer score by chalk marks on the table or on the wall. But back of all
these devices, and forming a common origin to which all may be referred, is
the universal finger method; the method with which all begin, and which all
find too convenient ever to relinquish entirely, even though their
civilization be of the highest type. Any such mode of counting, whether
involving the use of the fingers or not, is to be regarded simply as an
extraneous aid in the expression or comprehension of an idea which the mind
cannot grasp, or cannot retain, without assistance. The German student
scores his reckoning with chalk marks because he might otherwise forget;
while the Andaman Islander counts on his fingers because he has no other
method of counting,--or, in other words, of grasping the idea of number. A
single illustration may be given which typifies all practical methods of
numeration. More than a century ago travellers in Madagascar observed a
curious but simple mode of ascertaining the number of soldiers in an
army.[6] Each soldier was made to go through a passage in the presence of
the principal chiefs; and as he went through, a pebble was dropped on the
ground. This continued until a heap of 10 was obtained, when one was set
aside and a new heap begun. Upon the completion of 10 heaps, a pebble was
set aside to indicate 100; and so on until the entire army had been
numbered. Another illustration, taken from the very antipodes of
Madagascar, recently found its way into print in an incidental manner,[7]
and is so good that it deserves a place beside de Flacourt's time-honoured
example. Mom Cely, a Southern negro of unknown age, finds herself in debt
to the storekeeper; and, unwilling to believe that the amount is as great
as he represents, she proceeds to investigate the matter in her own
peculiar way. She had "kept a tally of these purchases by means of a
string, in which she tied commemorative knots." When her creditor
"undertook to make the matter clear to Cely's comprehension, he had to
proceed upon a system of her own devising. A small notch was cut in a
smooth white stick for every dime she owed, and a large notch when the
dimes amounted to a dollar; for every five dollars a string was tied in the
fifth big notch, Cely keeping tally by the knots in her bit of twine; thus,
when two strings were tied about the stick, the ten dollars were seen to be
an indisputable fact." This interesting method of computing the amount of
her debt, whether an invention of her own or a survival of the African life
of her parents, served the old negro woman's purpose perfectly; and it
illustrates, as well as a score of examples could, the methods of
numeration to which the children of barbarism resort when any number is to
be expressed which exceeds the number of counters with which nature has
provided them. The fingers are, however, often employed in counting numbers
far above the first decade. After giving the Il-Oigob numerals up to 60,
Müller adds:[8] "Above 60 all numbers, indicated by the proper figure
pantomime, are expressed by means of the word _ipi_." We know, moreover,
that many of the American Indian tribes count one ten after another on
their fingers; so that, whatever number they are endeavouring to indicate,
we need feel no surprise if the savage continues to use his fingers
throughout the entire extent of his counts. In rare instances we find
tribes which, like the Mairassis of the interior of New Guinea, appear to
use nothing but finger pantomime.[9] This tribe, though by no means
destitute of the number sense, is said to have no numerals whatever, but to
use the single word _awari_ with each show of fingers, no matter how few or
how many are displayed.
In the methods of finger counting employed by savages a considerable degree
of uniformity has been observed. Not only does he use his fingers to assist
him in his tally, but he almost always begins with the little finger of his
left hand, thence proceeding towards the thumb, which is 5. From this point
onward the method varies. Sometimes the second 5 also is told off on the
left hand, the same order being observed as in the first 5; but oftener the
fingers of the right hand are used, with a reversal of the order previously
employed; _i.e._ the thumb denotes 6, the index finger 7, and so on to the
little finger, which completes the count to 10.
At first thought there would seem to be no good reason for any marked
uniformity of method in finger counting. Observation among children fails
to detect any such thing; the child beginning, with almost entire
indifference, on the thumb or on the little finger of the left hand. My own
observation leads to the conclusion that very young children have a slight,
though not decided preference for beginning with the thumb. Experiments in
five different primary rooms in the public schools of Worcester, Mass.,
showed that out of a total of 206 children, 57 began with the little finger
and 149 with the thumb. But the fact that nearly three-fourths of the
children began with the thumb, and but one-fourth with the little finger,
is really far less significant than would appear at first thought. Children
of this age, four to eight years, will count in either way, and sometimes
seem at a loss themselves to know where to begin. In one school room where
this experiment was tried the teacher incautiously asked one child to count
on his fingers, while all the other children in the room watched eagerly to
see what he would do. He began with the little finger--and so did every
child in the room after him. In another case the same error was made by the
teacher, and the child first asked began with the thumb. Every other child
in the room did the same, each following, consciously or unconsciously, the
example of the leader. The results from these two schools were of course
rejected from the totals which are given above; but they serve an excellent
purpose in showing how slight is the preference which very young children
have in this particular. So slight is it that no definite law can be
postulated of this age; but the tendency seems to be to hold the palm of
the hand downward, and then begin with the thumb. The writer once saw a boy
about seven years old trying to multiply 3 by 6; and his method of
procedure was as follows: holding his left hand with its palm down, he
touched with the forefinger of his right hand the thumb, forefinger, and
middle finger successively of his left hand. Then returning to his
starting-point, he told off a second three in the same manner. This process
he continued until he had obtained 6 threes, and then he announced his
result correctly. If he had been a few years older, he might not have
turned so readily to his thumb as a starting-point for any digital count.
The indifference manifested by very young children gradually disappears,
and at the age of twelve or thirteen the tendency is decidedly in the
direction of beginning with the little finger. Fully three-fourths of all
persons above that age will be found to count from the little finger toward
the thumb, thus reversing the proportion that was found to obtain in the
primary school rooms examined.
With respect to finger counting among civilized peoples, we fail, then, to
find any universal law; the most that can be said is that more begin with
the little finger than with the thumb. But when we proceed to the study of
this slight but important particular among savages, we find them employing
a certain order of succession with such substantial uniformity that the
conclusion is inevitable that there must lie back of this some well-defined
reason, or perhaps instinct, which guides them in their choice. This
instinct is undoubtedly the outgrowth of the almost universal
right-handedness of the human race. In finger counting, whether among
children or adults, the beginning is made on the left hand, except in the
case of left-handed individuals; and even then the start is almost as
likely to be on the left hand as on the right. Savage tribes, as might be
expected, begin with the left hand. Not only is this custom almost
invariable, when tribes as a whole are considered, but the little finger is
nearly always called into requisition first. To account for this
uniformity, Lieutenant Gushing gives the following theory,[10] which is
well considered, and is based on the results of careful study and
observation among the Zuñi Indians of the Southwest: "Primitive man when
abroad never lightly quit hold of his weapons. If he wanted to count, he
did as the Zuñi afield does to-day; he tucked his instrument under his left
arm, thus constraining the latter, but leaving the right hand free, that he
might check off with it the fingers of the rigidly elevated left hand. From
the nature of this position, however, the palm of the left hand was
presented to the face of the counter, so that he had to begin his score on
the little finger of it, and continue his counting from the right leftward.
An inheritance of this may be detected to-day in the confirmed habit the
Zuñi has of gesticulating from the right leftward, with the fingers of the
right hand over those of the left, whether he be counting and summing up,
or relating in any orderly manner." Here, then, is the reason for this
otherwise unaccountable phenomenon. If savage man is universally
right-handed, he will almost inevitably use the index finger of his right
hand to mark the fingers counted, and he will begin his count just where it
is most convenient. In his case it is with the little finger of the left
hand. In the case of the child trying to multiply 3 by 6, it was with the
thumb of the same hand. He had nothing to tuck under his arm; so, in
raising his left hand to a position where both eye and counting finger
could readily run over its fingers, he held the palm turned away from his
face. The same choice of starting-point then followed as with the
savage--the finger nearest his right hand; only in this case the finger was
a thumb. The deaf mute is sometimes taught in this manner, which is for him
an entirely natural manner. A left-handed child might be expected to count
in a left-to-right manner, beginning, probably, with the thumb of his right
hand.
To the law just given, that savages begin to count on the little finger of
the left hand, there have been a few exceptions noted; and it has been
observed that the method of progression on the second hand is by no means
as invariable as on the first. The Otomacs[11] of South America began their
count with the thumb, and to express the number 3 would use the thumb,
forefinger, and middle finger. The Maipures,[12] oddly enough, seem to have
begun, in some cases at least, with the forefinger; for they are reported
as expressing 3 by means of the fore, middle, and ring fingers. The
Andamans[13] begin with the little finger of either hand, tapping the nose
with each finger in succession. If they have but one to express, they use
the forefinger of either hand, pronouncing at the same time the proper
word. The Bahnars,[14] one of the native tribes of the interior of Cochin
China, exhibit no particular order in the sequence of fingers used, though
they employ their digits freely to assist them in counting. Among certain
of the negro tribes of South Africa[15] the little finger of the right hand
is used for 1, and their count proceeds from right to left. With them, 6 is
the thumb of the left hand, 7 the forefinger, and so on. They hold the palm
downward instead of upward, and thus form a complete and striking exception
to the law which has been found to obtain with such substantial uniformity
in other parts of the uncivilized world. In Melanesia a few examples of
preference for beginning with the thumb may also be noticed. In the Banks
Islands the natives begin by turning down the thumb of the right hand, and
then the fingers in succession to the little finger, which is 5. This is
followed by the fingers of the left hand, both hands with closed fists
being held up to show the completed 10. In Lepers' Island, they begin with
the thumb, but, having reached 5 with the little finger, they do not pass
to the other hand, but throw up the fingers they have turned down,
beginning with the forefinger and keeping the thumb for 10.[16] In the use
of the single hand this people is quite peculiar. The second 5 is almost
invariably told off by savage tribes on the second hand, though in passing
from the one to the other primitive man does not follow any invariable law.
He marks 6 with either the thumb or the little finger. Probably the former
is the more common practice, but the statement cannot be made with any
degree of certainty. Among the Zulus the sequence is from thumb to thumb,
as is the case among the other South African tribes just mentioned; while
the Veis and numerous other African tribes pass from thumb to little
finger. The Eskimo, and nearly all the American Indian tribes, use the
correspondence between 6 and the thumb; but this habit is by no means
universal. Respecting progression from right to left or left to right on
the toes, there is no general law with which the author is familiar. Many
tribes never use the toes in counting, but signify the close of the first
10 by clapping the hands together, by a wave of the right hand, or by
designating some object; after which the fingers are again used as before.
One other detail in finger counting is worthy of a moment's notice. It
seems to have been the opinion of earlier investigators that in his passage
from one finger to the next, the savage would invariably bend down, or
close, the last finger used; that is, that the count began with the fingers
open and outspread. This opinion is, however, erroneous. Several of the
Indian tribes of the West[17] begin with the hand clenched, and open the
fingers one by one as they proceed. This method is much less common than
the other, but that it exists is beyond question.
In the Muralug Island, in the western part of Torres Strait, a somewhat
remarkable method of counting formerly existed, which grew out of, and is
to be regarded as an extension of, the digital method. Beginning with the
little finger of the left hand, the natives counted up to 5 in the usual
manner, and then, instead of passing to the other hand, or repeating the
count on the same fingers, they expressed the numbers from 6 to 10 by
touching and naming successively the left wrist, left elbow, left shoulder,
left breast, and sternum. Then the numbers from 11 to 19 were indicated by
the use, in inverse order, of the corresponding portions of the right side,
arm, and hand, the little finger of the right hand signifying 19. The words
used were in each case the actual names of the parts touched; the same
word, for example, standing for 6 and 14; but they were never used in the
numerical sense unless accompanied by the proper gesture, and bear no
resemblance to the common numerals, which are but few in number. This
method of counting is rapidly dying out among the natives of the island,
and is at the present time used only by old people.[18] Variations on this
most unusual custom have been found to exist in others of the neighbouring
islands, but none were exactly similar to it. One is also reminded by it of
a custom[19] which has for centuries prevailed among bargainers in the
East, of signifying numbers by touching the joints of each other's fingers
under a cloth. Every joint has a special signification; and the entire
system is undoubtedly a development from finger counting. The buyer or
seller will by this method express 6 or 60 by stretching out the thumb and
little finger and closing the rest of the fingers. The addition of the
fourth finger to the two thus used signifies 7 or 70; and so on. "It is
said that between two brokers settling a price by thus snipping with the
fingers, cleverness in bargaining, offering a little more, hesitating,
expressing an obstinate refusal to go further, etc., are as clearly
indicated as though the bargaining were being carried on in words.
The place occupied, in the intellectual development of man, by finger
counting and by the many other artificial methods of reckoning,--pebbles,
shells, knots, the abacus, etc.,--seems to be this: The abstract processes
of addition, subtraction, multiplication, division, and even counting
itself, present to the mind a certain degree of difficulty. To assist in
overcoming that difficulty, these artificial aids are called in; and, among
savages of a low degree of development, like the Australians, they make
counting possible. A little higher in the intellectual scale, among the
American Indians, for example, they are employed merely as an artificial
aid to what could be done by mental effort alone. Finally, among
semi-civilized and civilized peoples, the same processes are retained, and
form a part of the daily life of almost every person who has to do with
counting, reckoning, or keeping tally in any manner whatever. They are no
longer necessary, but they are so convenient and so useful that
civilization can never dispense with them. The use of the abacus, in the
form of the ordinary numeral frame, has increased greatly within the past
few years; and the time may come when the abacus in its proper form will
again find in civilized countries a use as common as that of five centuries
ago.
In the elaborate calculating machines of the present, such as are used by
life insurance actuaries and others having difficult computations to make,
we have the extreme of development in the direction of artificial aid to
reckoning. But instead of appearing merely as an extraneous aid to a
defective intelligence, it now presents itself as a machine so complex that
a high degree of intellectual power is required for the mere grasp of its
construction and method of working.