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A HISTORY OF

ELEMENTARY MATHEMATICS

A HISTORY

ELEMENTARY MATHEMATICS

WITH

HINTS ON METHODS OF TEACHING

BY

FLORIAN CAJORI, PH.D.

PROFESSOR OF PHYSICS IN COLORADO COLLEGE

THE MACMILLAN COMPANY

LONDON: MACMILLAN & CO., LTD.

1896

All rights reserved

COPYRIGHT, 1896,

BY THE MACMILLAN COMPANY.

Xortooob

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.

v

VI PREFACE

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

one.

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.

FLORIAN CAJORI.

COLORADO COLLEGE, COLORADO SPRINGS,

July, 1896.

CONTENTS

ANTIQUITY . " 1

NUMBER-SYSTEMS AND NUMERALS ...... 1

ARITHMETIC AND ALGEBRA ........ 19

Egypt . . 19

Greece 26

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

GEOMETRY AND TRIGONOMETRY ....... 43

Egypt and Babylonia 43

Greece . . . ,5 46

Borne 89

MIDDLE AGES .93

ARITHMETIC AND ALGEBRA ........ 93

Hindus . 93

Arabs 103

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

Introduction of Roman Arithmetic Ill

Translation of Arabic Manuscripts . . . . .118

The First Awakening 119

GEOMETRY AND TRIGONOMETRY 122

Hindus 122

Arabs 125

Europe during the Middle Ages 131

Introduction of Roman Geometry 131

Translation of Arabic Manuscripts ..... 132

The First Awakening 134

vii

Vlll CONTENTS

PAGE

MODERN TIMES . . 139

ARITHMETIC 139

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

GEOMETRY AND TRIGONOMETRY . 245

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

A HISTORY OF MATHEMATICS

ANTIQUITY

NUMBER-SYSTEMS AND NUMERALS

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.

V A HISTORY OF MATHEMATICS

" 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.

NUMBER-SYSTEMS AND NUMERALS 3

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.

4 A HISTORY OF MATHEMATICS

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.

NUMBER-SYSTEMS AND NUMERALS 5

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.

6 A HISTORY OF MATHEMATICS

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.

NUMBER-SYSTEMS AND NUMERALS 7

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.

8 A HISTORY OF MATHEMATICS

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

after this as FRIEDLEIN. See also DR. SIEGMUND GUNTHER in MULLER'S

Handbuch der Klassischen Alter tumswissenschaft, Fiinfter Band, 1.

Abteilung, 1888, p. 9.

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

NUMBER-SYSTEMS AND NUMERALS 9

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

10 A HISTORY OF MATHEMATICS

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

forgotten.

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.

NUMBER-SYSTEMS AND NUMERALS 11

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

A HISTORY OF

ELEMENTARY MATHEMATICS

A HISTORY

ELEMENTARY MATHEMATICS

WITH

HINTS ON METHODS OF TEACHING

BY

FLORIAN CAJORI, PH.D.

PROFESSOR OF PHYSICS IN COLORADO COLLEGE

THE MACMILLAN COMPANY

LONDON: MACMILLAN & CO., LTD.

1896

All rights reserved

COPYRIGHT, 1896,

BY THE MACMILLAN COMPANY.

Xortooob

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.

v

VI PREFACE

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

one.

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.

FLORIAN CAJORI.

COLORADO COLLEGE, COLORADO SPRINGS,

July, 1896.

CONTENTS

ANTIQUITY . " 1

NUMBER-SYSTEMS AND NUMERALS ...... 1

ARITHMETIC AND ALGEBRA ........ 19

Egypt . . 19

Greece 26

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

GEOMETRY AND TRIGONOMETRY ....... 43

Egypt and Babylonia 43

Greece . . . ,5 46

Borne 89

MIDDLE AGES .93

ARITHMETIC AND ALGEBRA ........ 93

Hindus . 93

Arabs 103

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

Introduction of Roman Arithmetic Ill

Translation of Arabic Manuscripts . . . . .118

The First Awakening 119

GEOMETRY AND TRIGONOMETRY 122

Hindus 122

Arabs 125

Europe during the Middle Ages 131

Introduction of Roman Geometry 131

Translation of Arabic Manuscripts ..... 132

The First Awakening 134

vii

Vlll CONTENTS

PAGE

MODERN TIMES . . 139

ARITHMETIC 139

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

GEOMETRY AND TRIGONOMETRY . 245

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

A HISTORY OF MATHEMATICS

ANTIQUITY

NUMBER-SYSTEMS AND NUMERALS

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.

V A HISTORY OF MATHEMATICS

" 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.

NUMBER-SYSTEMS AND NUMERALS 3

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.

4 A HISTORY OF MATHEMATICS

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.

NUMBER-SYSTEMS AND NUMERALS 5

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.

6 A HISTORY OF MATHEMATICS

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.

NUMBER-SYSTEMS AND NUMERALS 7

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.

8 A HISTORY OF MATHEMATICS

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

after this as FRIEDLEIN. See also DR. SIEGMUND GUNTHER in MULLER'S

Handbuch der Klassischen Alter tumswissenschaft, Fiinfter Band, 1.

Abteilung, 1888, p. 9.

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

NUMBER-SYSTEMS AND NUMERALS 9

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

10 A HISTORY OF MATHEMATICS

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

forgotten.

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.

NUMBER-SYSTEMS AND NUMERALS 11

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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