Florian Cajori.

A history of elementary mathematics, with hints on methods of teaching online

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J, S. Gushing & Co. - Berwick & Smith
Norwood Mass. U.S.A.

"THE education of the child must accord both in mode
and arrangement with the education of mankind as consid-
ered historically ; or, in other words, the genesis of knowledge
in the individual must follow the same course as the genesis I
of knowledge in the race. To M. Comte we believe society i
owes the enunciation of this doctrine a doctrine which we
may accept without committing ourselves to his theory of
the genesis of knowledge, either in its causes or its order." *
If this principle, held also by Pestalozzi and Froebel, be
correct, then it would seem as if the knowledge of the
history of a science must be an effectual aid in teaching
that science. Be this doctrine true or false, certainly the
experience of many instructors establishes the importance
of mathematical history in teaching. 2 With the hope of
being of some assistance to my fellow-teachers, I have pre-
pared this book and have interlined my narrative with
occasional remarks and suggestions on methods of teaching.
No doubt, the thoughtful reader will draw many useful

1 HERBERT SPENCER, Education : Intellectual, Moral, and Physical.
New York, 1894, p. 122. See also R. H. QUICK, Educational Reformers,
1879, p. 191.

2 See G. HEPPEL, u The Use of History in Teaching Mathematics,"
Nature, Vol. 48, 1893, pp. 16-18.



lessons from the study of mathematical history which are
not directly pointed out in the text.

In the preparation of this history, I have made extensive
use of the works of Cantor, Hankel, Unger, De Morgan, Pea-
cock, Gow, Allman, Loria, and of other prominent writers
on the history of mathematics. Original sources have been
consulted, whenever opportunity has presented itself. It
gives me much pleasure to acknowledge the assistance ren-
dered by the United States Bureau of Education, in for-
warding for examination a number of old text-books which
otherwise would have been inaccessible to me. It should
also be said that a large number of passages in this book
are taken, with only slight alteration, from my History of
Mathematics, Macmillan & Co., 1895. Some parts of the
present work are, therefore, not independent of the earlier

It has been my privilege to have my manuscript read by
two scholars of well-known ability, Dr. G. B. Halsted of
the University of Texas, and Professor F. H. Loud of Colorado
College. Through their suggestions and corrections many
infelicities in language and several inaccuracies of statement
have disappeared. Valuable assistance in proof-reading has
been rendered by Professor Loud, by Mr. P. E. Doudna,
formerly Fellow in Mathematics at the University of Wis-
consin, and by Mr. F. K. Bailey, a student in Colorado
College. I extend to them my sincere thanks.

July, 1896.





Egypt . . 19

Greece 26

Home ........... 37


Egypt and Babylonia 43

Greece . . . ,5 46

Borne 89



Hindus . 93

Arabs 103

Europe during the Middle .Ages . . . . . Ill

Introduction of Roman Arithmetic Ill

Translation of Arabic Manuscripts . . . . .118

The First Awakening 119


Hindus 122

Arabs 125

Europe during the Middle Ages 131

Introduction of Roman Geometry 131

Translation of Arabic Manuscripts ..... 132

The First Awakening 134






Its Development as a Science and Art ..... 139

English Weights and Measures 167

Else of the Commercial School in England. . . . 179
Causes which Checked the Growth of Demonstrative Arith-
metic in England ........ 204

Eeforms in Arithmetical Teaching 211

Arithmetic in the United States 215

"Pleasant and Diverting Questions" 219

ALGEBRA ........... 224

The Renaissance 224

The Last Three Centuries 234


Editions of Euclid. Early Researches ..... 245

The Beginning of Modern Synthetic Geometry . . . 252

Modern Elementary Geometry . . . . . . 256

Modern Synthetic Geometry ...... 257

Modern Geometry of the Triangle and Circle . . 259

Non-Euclidean Geometry 266

Text-books on Elementary Geometry . . . .275




NEARLY all number-systems, both ancient and modern, are
based on the scale of 5, 10, or 20. The reason for this it is
not difficult to see. When a child learns to count, he makes
use of his fingers and perhaps of his toes. In the same way
the savages of prehistoric times unquestionably counted on
their fingers and in some cases also on their toes. Such is
indeed the practice of the African, the Eskimo, and the South
Sea Islander of to-day. 1 This recourse to the fingers has
often resulted in the development of a more or less extended
pantomime number-system, in which the fingers were used as
in a deaf and dumb alphabet. 1 Evidence of the prevalence of
finger symbolisms is found among the ancient Egyptians,
Babylonians, Greeks, and Romans, as also among the Euro-
peans of the middle ages : even nojv nearly all Eastern nations
use finger symbolisms. The Chinese express on the left hand

1 L. L. CONANT, "Primitive Number-Systems," in Smithsonian Re-
port, 1892, p. 584.


" all numbers less than 100,000 ; the thumb nail of the right
hand touches each joint of the little finger, passing first up
the external side, then down the middle, and afterwards up
the other side of it, in order to express the nine digits ; the
tens are denoted in the same way, on the second finger ; the
hundreds on the third ; the thousands on the fourth ; and ten-
thousands on the thumb. It would be merely necessary to
proceed to the right hand in order to be able to extend this
system of numeration." l So common is the use of this finger-
symbolism that traders are said to communicate to one another
the price at which they are willing to buy or sell by touching
hands, the act being concealed by their cloaks from observa-
tion of by-standers.

Had the number of fingers and toes been different in man,
then the prevalent number-systems of the world would have
been different also. We are safe in saying that had one more
finger sprouted from each human hand, making twelve fingers
in all, then the numerical scale adopted by civilized nations
would not be the decimal, but the duodecimal. Two more
symbols would be necessary to represent 10 and 11, respec-
tively. As far as arithmetic is concerned, it is certainly to be
regretted that a sixth ringer did not appear. Except for the
necessity of using two more signs or numerals and of being
obliged to learn the multiplication table as far as 12 x 12, the
duodecimal system is decidedly superior to the decimal. The
number twelve has for its exact divisors 2, 3, 4, 6, while ten has
only 2 and 5. In ordinary business affairs, the fractions , -|, J,
are used extensively, and it is very convenient to have a base
which is an exact multiple of 2, 3, and 4. Among the most
zealous advocates of the duodecimal scale was Charles XII.

1 GEORGE PEACOCK, article "Arithmetic," in Encyclopaedia Metropoli-
tana (The Encyclopaedia of Pure Mathematics), p. 394. Hereafter we
shall cite this very valuable article as PEACOCK.


of Sweden, who, at the time of his death, was contemplating
the change for his dominions from the decimal to the duo-
decimal. 1 But it is not likely that the change will ever be
brought about. So deeply rooted is the decimal system that
when the storm of the French Eevolution swept out of exist-
ence other old institutions, the decimal system not only
remained unshaken, but was more firmly established than
ever.. The advantages of twelve as a base were not recognized
until arithmetic was so far developed as to make a change
impossible. "The case is the not uncommon one of high
civilization bearing evident traces of the rudeness of its origin
in ancient barbaric life." 2

Of the notations based on human anatomy, the quinary and
vigesimal systems are frequent among the lower races, while
the higher nations have usually avoided the one as too scanty
and the other as too cumbrous, preferring the intermediate
decimal system. 3 Peoples have not always consistently
adhered to any one scale. In the quinary system, 5, 25, 125,
625, etc., should be the values of the successive higher units,
but a quinary system thus carried out was never in actual use :
whenever it was extended to higher numbers it invariably ran
either into the decimal or into the vigesimal system. " The
home par excellence of the quinary, or rather of the quinary-
vigesimal scale, is America. It is practically universal among
the Eskimo tribes of the Arctic regions. It prevailed among
a considerable portion of the North American Indian tribes,
and was almost universal with the native races of Central and

1 CONANT, op. cit., p. 589.

2 E. B. TYLOR, Primitive Culture, New York, 1889, Vol. I., p. 272. In
some respects a scale having for its base a power of 2 the base 8 or 16,
for instance, is superior to the duodecimal, but it has the disadvantage
of not being divisible by 3. See W. W. JOHNSON, "Octonary Numera-
tion," Bull. N. T. Math. Soc., 1891, Vol. I., pp. 1-6.

8 TYLOK, op. cit., Vol. I., p. 262.


South America." * This system was used also by many of the
North Siberian and African tribes. Traces of it are found in
the languages of peoples who now use the decimal scale ; for
example, in Homeric Greek. The Eoman notation reveals
traces of it; viz., I, II, ... V, VI, ... X, XI, XV, etc.

It is curious that the quinary should so frequently merge
into the vigesimal scale ; that savages should have passed from
the number of fingers on one hand as an upper unit or a stop-
ping-place, to the total number of fingers and toes as an upper
unit or resting-point. The vigesimal system is less common
than the quinary, but, like it, is never found entirely pure. In
this the first four units are 20, 400, 8000, 160,000, and special
words for these numbers are actually found among the Mayas
of Yucatan. The transition from quinary to vigesimal is
shown in the Aztec system, which may be represented thus,
1,2,3, 4, 5, 5 + 1, ... 10,10 + 1, ... 10 + 5, 10 + 5 + 1, ... 20,
20 + 1, ... 20+10, 20 + 10 + 1, ... 40, etc. 2 Special words
occur here for the numbers 1, 2, 3, 4, 5, 10, 20, 40, etc. The
vigesimal system flourished in America, but was rare in the
Old World. Celtic remnants of one occur in the French words
quatre-vingts (Ax 20 or 80), six-vingts (6 x 20 or 120), quinze-
vingts (15 x 20 or 300). Note also the English word score in
such expressions as three-score years and ten.

Of the three systems based on human anatomy, the decimal
system is the most prevalent, so prevalent, in fact, that accord-
ing to ancient tradition it was used by all the races of the
world. It is only within the last few centuries that the other

1 COKANT, op. cit., p. 592. For further information see also POTT,
Die quindre und vigesimale Zahlmethode bei Volkern aller Welttheile,
Halle, 1847 ; POTT, Die Sprachverschiedenheit in Europa an den Zahl-
wortern nachyewiesen, sowie die quindre und vigesimale Zahlmethode,
Halle, 1868.

- TTLOR, op. cit., Vol. I., p. 262.


two systems have been found in use among previously unknown
tribes. 1 The decimal scale was used in North America by
the greater number of Indian tribes, but in South America it
was rare.

In the construction of the decimal system, 10 was suggested
by the number of fingers as the first stopping-place in count-
ing, and as the first higher unit. Any number between 10
and 100 was pronounced according to the plan &(10) + a(l),
a and b being integers less than 10. But the number 110
' might be expressed in two ways, (1) as 10 x 10 + 10, (2) as
11 x 10. The latter method would not seem unnatural.
Why not imitate eighty, ninety, and say eleventy, instead of
hundred and ten f But upon this choice between io x 10 -f- 10
and 11 x 10 hinges the systematic construction of the number
system. 2 Good luck led all nations which developed the
decimal system to the choice of the former; 3 the unit 10
being here treated in a manner similar to the treatment of the
lower unit 1 in expressing numbers below 100. Any num-
ber between 100 and 1000 was designated c(10) 2 + 6(10) -f- a,
a, 6, c representing integers less than 10. Similarly for num-
bers below 10,000, d(10) 3 + c(10) 2 + fc(lO) 1 + o(10); and simi-
larly for still higher numbers.

Proceeding to describe the notations of numbers, we
begin with the Babylonian. Cuneiform writing, as also the
accompanying notation of numbers, was probably invented

1 CONANT, op. cit., p. 588.

2 HERMANN HANKEL, Zur Geschichte der Mathematik in Alterthum
und Mittelalter, Leipzig, 1874, p. 11. Hereafter we shall cite this brilliant
work as HANKEL.

3 In this connection read also MORITZ CANTOR, Vorlesungen uber
Geschichte der Mathematik, Vol. I. (Second Edition), Leipzig, 1894,
pp. 6 and 7. This history, by the prince of mathematical historians of
this century, will be in three volumes, when completed, and will be cited
hereafter as CANTOK.


by the early Sumerians. A vertic&l wedge Y stood for
one, while ^ and y^- signified 10 and 100, respectively.
In case of numbers below 100, the values of the separate sym-
bols were added. Thus, <?T for 23, < < < for 30. The

signs of higher value are written on the left of those of lower
value. But in writing the hundreds a smaller symbol was
placed before that for 100 and was multiplied into 100. Thus,

< y - signified 10 x 100 or 1000. Taking this for a new
unit,' ^ ^ f ^~ was interpreted, net as 20 x 100, but as
10 x 1000. In this notation no numbers have been found as
large as a million. 1 The principles applied in this notation
are the additive and the multiplicative. Besides this the
Babylonians had another, the sexagesimal notation, to be
noticed later.

An insight into Egyptian methods of notation was obtained
through the deciphering of the hieroglyphics by Champollion,

Young, and others. The numerals are I (1), (H (10), (J (100),
S (1000), f (10,000), ^ (100,000), X (1,000,000), .Q.
(10,000,000). The sign for one represents a vertical staff;
that for 10,000, a pointing finger ; that for 100,000, a burbot ;
that for 1,000,000, a man in astonishment. No certainty has
been reached regarding the significance of the other symbols.
These numerals like the other hieroglyphic signs were plainly
pictures of animals or objects familiar to the Egyptians,
which in some way suggested the idea to be conveyed. They
are excellent examples of picture-writing. The principle in-
volved in the Egyptian notation was the additive throughout.

Thus, (3 /H) I would be 111.

1 For fuller treatment see MORITZ CANTOR, Mathematische Beitrage
zum Kulturleben der Volker, Halle, 1863, pp. 22-38.


Hieroglyphics are found on monuments, obelisks, and walls
of temples. Besides these the Egyptians had hieratic and
demotic writings, both supposed to be degenerated forms of
hieroglyphics, such as would be likely to evolve through pro-
longed use and attempts at rapid writing. ^The following are
hieratic signs : 1

10 20 30 40 50 60 70 80 90

100 200 1000 9000

Since there are more hieratic symbols than hieroglyphic,
numbers could be written more concisely in the former. The
additive principle rules in both, and the symbols for larger
values always precede those for smaller values.

About the time of Solon, the Greeks used the initial letters
of the numeral adjectives to represent numbers. These signs
are often called Herodianic signs (after Herodianus, a Byzan-
tine grammarian of about 200 A.D., who describes them).
They are also called Attic, because they occur frequently in
Athenian inscriptions. The Phoenicians, Syrians, and Hebrews
possessed at this time alphabets and the two latter used letters
of the alphabet to designate numbers. The Greeks began to
adopt the same course about 500 B.C. The letters of the Greek
alphabet, together with three antique letters, r, 9, , and the

1 CANTOR, Vol. L, pp. 44 and 45. The hieratic numerals are taken
from Cantor's table at the end of the volume.


symbol M, were used for numbers. For the numbers 1-9 they
wrote a, /?, y, 8, c, s, , ry, ; for the tens 10-90, t, K, X , /u,, v, ,
o, TT, 9 , for the hundreds 100-900, p, cr, T, v, <, x? ^> w > 7D ; f or
the thousands they wrote ,a, ,/?, ,y, ,8, ,e, etc.; for 10,000, M;
for 20,000, M; for 30,000, ]&, etc. The change from Attic to
alphabetic numerals was decidedly for the worse, as the
former were less burdensome to the memory. In Greek gram-
mars we often find it stated that alphabetic numerals were
marked with an accent to distinguish them from words, but
this was not commonly the case ; a horizontal line drawn over
the number usually answered this purpose, while the accent

generally indicated a unit-fraction, thus 8' = J. 1 The Greeks

applied to their numerals the additive and, in cases like M

for 50,000, also the multiplicative principle.

In the Eoman -notation we have, besides the additive, the
principle of subtraction. If a letter is placed before another
of greater value, the former is to be subtracted from the latter.
Thus, IV = 4, while VI = 6. Though this principle has not
been found in any other notation, it sometimes occurs in numer-
ation. Thus in Latin duodeviginti = 2 from 20, or 18. 2 The
Eoman numerals are supposed to be of Etruscan origin.

Thus, in the Babylonian, Egyptian, Greek, Eoman, and
other decimal notations of antiquity, numbers are expressed
by means of a few signs, these symbols being combined by
addition alone, or by addition together with multiplication
or subtraction. But in none of these decimal systems do we
find the all-important principle of position or principle of

1 DR. G. FRIEDLEIN, Die Zahlzeichen und das Elementare Eechnen
der Griechen und Earner, Erlangen, 1869, p. 13. The work will be cited
Handbuch der Klassischen Alter tumswissenschaft, Fiinfter Band, 1.
Abteilung, 1888, p. 9.

2 CANTOR, Vol. I., pp. 11 and 489.


local value, such as we have in the notation now in use.
Having missed this principle, the ancients had no use for a
symbol to represent zero, and were indeed very far removed
from an ideal notation. In this matter even the Greeks and
Romans failed to achieve what a remote nation in Asia, little
known to Europeans before the present century, accomplished
most admirably. But before we speak of the Hindus, we
must speak of an ancient Babylonian notation, which, strange
to say, is not based on the scale 5, 10, or 20, and which,
moreover, came very near a full embodiment of the ideal
principle found wanting in other ancient systems. We refer
to the sexagesimal notation.

The Babylonians used this chiefly in the construction of
weights and measures. The systematic development of the
sexagesimal scale, both for integers and fractions, reveals a
high degree of mathematical insight on the part of the early
Sumerians. __The notation has been found on two Babylonian
tablets. One of them, probably dating from 1600 or 2300 B.C.,
contains a list of square numbers up to 60 2 . The first seven
are 1, 4, 9, 16, 25, 36, 49. We have next 1.4 = 8 2 , 1.21 = 9 2 ,
1.40 = 10 2 , 2.1 = II 2 , etc. This remains unintelligible, unless
we assume the scale of sixty, which makes 1.4 = 60 + 4,
1.21 = 60 + 21, etc. The second tablet records the magnitude
of the illuminated portion of the moon's disc for every day
from -new to full moon, the whole disc being assumed to con-
sist of 240 parts. The illuminated parts during the first five
days are the series 5, 10, 20, 40, 1.20(= 80). This reveals
again the sexagesimal scale and also some knowledge of
geometrical progressions. From here on the series becomes an
arithmetical progression, the numbers from the fifth to the
fifteenth day being respectively, 1.20, 1.36, 1.52, 2^8, 2.24, 2.40,
2.56, 3.12, 3.28, 3.44, 4. In this sexagesimal notation we have,
then, the principle of local value. Thus, in 1.4 {= 64), the 1 is


made to stand for 60, the unit of the second order, by virtue of
its position with respect to the 4. In Babylonia some use was
thus made of the principle of position, perhaps 2000 years
before the Hindus developed it/ This was at a time when
Romulus and Remus, yea even Achilles, Menelaus, and Helen,
were still unknown -to history and song. But the full develop-
ment of the principle of position calls for a symbol to represent
the absence of quantity, or zero. Did the Babylonians have
that ? Ancient tablets thus far deciphered give us no answer ;
they contain no number in which there was occasion to use a
zero. Indications so far seem to be that this notation was a
possession of the few and was used but little. While the sexa-
gesimal division of units of time and of circular measure was
transmitted to other nations, the brilliant device of local value
in numerical notation appears to have been neglected and

What was it that suggested to the Babylonians the number
sixty as a base ? It could not have been human anatomy as in
the previous scales. Cantor 1 and others offer the following
provisional reply : At first the Babylonians reckoned the year
at 360 days. This led to the division of the circumference of
a circle into 360 degrees, each degree representing the daily
part of the supposed yearly revolution of the sun around
the earth. Probably they knew that the radius could be
applied to the circumference as a chord six times, and that
each arc thus cut off contained 60 degrees. Thus the
division into 60 parts may have suggested itself. When
greater precision was needed, each degree was divided into 60
equal parts, or minutes. In this way the sexagesimal notation
may have originated. The division of the day into 24 hours,
and of the hour into minutes and seconds on the scale of 60,

i Vol. I., pp. 91-93.


is due to the Babylonians. There are also indications of a
knowledge of sexagesimal fractions,* such as were used later
by the Greeks, Arabs, by scholars of the middle ages and of
even recent times.

Babylonian science has made its impress upon modern civili-
zation. Whenever a surveyor copies the readings from the
graduated circle on his theodolite, whenever the modern man
notes the time of day, he is, unconsciously perhaps, but
unmistakably, doing homage to the ancient astronomers on
the banks of the Euphrates.

The full development of our decimal notation belongs to
comparatively modern times. Decimal notation had been in
use for thousands of years, before it was perceived that its
simplicity and usefulness could be enormously increased by the
adoption of the principle of position. To the Hindus of the
fifth or sixth century after Christ we owe the re-discovery of
this principle and the invention and adoption of the zero, the
symbol for the absence of quantity. Of all mathematical dis-
coveries, no one has contributed more to the general progress
of intelligence than this. While the older notations served
merely to record the answer of an arithmetical computation,
the Hindu notation (wrongly called the Arabic notation)
assists with marvellous power in performing the computation
itself. To verify this truth, try to multiply 723 by 364, by
first expressing the numbers in the Eoman notation; thus,
multiply DCCXXIII by CCCLXIV. This notation offers
little or no help; the Romans were compelled to invoke
the aid of the abacus in calculations like this.

Very little is known concerning the mode of evolution of the
Hindu notation. There is evidence for the belief that the
Hindu notation of the second century, A.D., did not include

i CANTOR, Vol. I., p. 85. X

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