Nicomachus of Gerasa, Introduction to Arithmetic — translated by Martin Luther D'Ooge, with studies in Greek arithmetic (1926)
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TB E .olumes of the Uni'lrsity of Michigan
Studies art publishtd by authority of tht
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tTIlnibrtllire of ""bl!llln ilMubitll
HUMANISTIC SERIES
VOLUME XVI
NICOMACHUS OF GERASA
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NICOMACHUS
OF
GERAS~..Lo- ' v
j
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INTRODUCTION
TO
ARITHMETIC
TRANSLATED INTO ENGLiSH
BY
MARTIN LUTHER D'OOGE
WITH STUDIES IN GREEK ARITHMETIC
BY
FRANK EGLESTON ROBBINS
AND
LOUIS CHARLES KARPINSKI
Ntb! l!1arl1
THE MACMILLAN COMPANY
LONDON: MACMILLAN AND COMPANY, LTJI.
19 16
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UNr<OOTY 01' I.OtLIN
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l'RDI"IED Uf 'I'D UNITED STATU or AlaaIC4
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COPYRIGHT, 1926,
By FRANCIS W. KELSEY.
Set up and electrotyped. Published March, 1926.
Non.aob tlnu
J. S. Cuahina: Co. - Berwick 41: Smith Co.
Norwood, M_., U.S.A.
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IN
COM M EMORA TION
OF THE
WNG AND INTIMATE FRIENDSHIP
BETWEEN TWO KINDRED SPIRITS
EDWARD WALDO PENDLETON
MARTIN LUTHER D'OOGE
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PREFACE
PROFESSOR Martin Luther D'Ooge died suddenly on September
12, 1915, leaving unfinished a work on the Introduction to Arithmetic
by Nicomachus. His translation of the Greek text was complete,
but the supporting studies had not been commenced.
As soon as possible after his death, colleagues of Mr. D'Ooge in the
University of Michigan took up the unfinished task, and their work
combined with his appears in this volume. Mr. Karpinski contributed Chapters I, Ill, IV and the greater part of Chapter X of Part
I, together with the first section of Part Ill, Extensions of a Theorem
of i\'icomac/tlts; Mr. Robbins made the final revision of Mr. D 'Ooge's
translation and prepared the rest of the volume. At first it was proposed to present a revised Greek text, but this proved to be impracticable without too great delay.
Sincere thanks are due to Mrs. Edward Waldo Pendleton, whose
generous help made the publication of the volume possible. We are
under much obligation also to our colleagues, who have rendered assistance on many points. A special mention of indebtedness should be
made to the University Editor, Dr. Eugene S. McCartney, for his
scholarly assistance in the preparation of the manuscript for the press.
FRANK EGLESTON RonBINs
LOUIS CHARLES KARPINSKI
ANN ARBOR, MICHIGAN,
September I, 1925_
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CONTENTS
PART I
STUDIES IN GREEK MATHEMATICS
,AGI:
I.
THE SOURCES OF GREEK MATHEMATICS
n.
THE
OF
DEVELOPMENT
COllACKUS
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GR.EEK
THE
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ARITHMETIC
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BEFORE
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16
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THE MAmEMATICAL CONTENT OF THE GREEK MATHEMATICA
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IV.
GREEK ARITHMETICAL NOTATION
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46
66
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THE LIFE OF NICOMACHUS
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VI.
THE \VORKS OF NICOMACHUS.
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VII.
1'lIE PHILOSOPHY OF
N ICOMACHUS .
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88
VIII.
NICOllACHUS'S PHILOSOPHY OF NUMBER
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III
TRANSLATORS AND COMMENTATORS OF NICOMACHUS
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13 8
INTRODUCTION
TO
IX.
X. THE SUCCESSORS OF NICOMACHUS .
XI.
THE
MANUSCRIPTS
ARITHMETIC
XII.
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THE
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THE LANGUAGE AND STYLE OF NICOMACHUS
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PART II
TRANSLATION OF THE INTRODUCTION TO ARITHMETIC
OF NICOMACHUS OF GERASA, THE PYTHAGOREAN
BOOK I
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BOOK II
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PART III
SUPPLEMENTARY AIDS TO THE INTERPRETATION
OF THE INTRODUCTION TO ARITHMETIC
I.
EXTENSIONS OF A THEOREM OF NICOMACHUS
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GLOSSARY OF GREEK TERMS .
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SELECT BIBLIOGRAPHY
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INDEX
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PART I
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CHAPTER I
THE SOURCES OF GREEK MATHEMATICS
is fundamentally associated by modern readers, particularly by scientists and mathematicians, with the art of computation. For the ancient Greeks after Pythagoras, however, arithmetic
was primarily a philosophical study, having no necessary connection
with practical affairs. Indeed the Greeks gave a separate name to the
arithmetic of business, AO'}'W'T£K7}; of this division of the science no
Greek treatise has been transmitted to us. In general the philosophers
and mathematicians of Greece undoubtedly considered it beneath
their dignity to treat of this branch, which probably formed a part of
the elementary instruction of children. The evidence for the existence
of treatises on the fundamental operations is very insecure and vague,
resting upon a passage of Diogenes Laertius 1 and a citation by
Eutocius. 2
So far as the content of the logistic is concerned, our main source
of information is the scholium' on Plato's Charmides, 163 E. This
scholium is undoubtedly based on the lost work of Geminus, although
it may be through the medium of Anatolius.' A passage in Proclus '
which explicitly mentions Geminus touches analogous points.
The scholium is as follows: "Logistic is the theory which deals with
numerable objects and not with numbers j 6 it does not, indeed, consider number in the proper sense of the term, but assumes I to be
unity, and anything which can be numbered to be number (thus in
place of the triad, it employs 3; in place of the decad, 10), and discusses with these the theorems of arithmetic.
ARlTIlMETIC
I ViltU
PhiJosoplwrum, VIII. 12, where a certain Apollodorus is designatro as 6 )"o-Y'fTn,(6r,
which may mean, as Cantor thinks, that he was a RUMnntns/cr.
I In the Com1Mnlary on tlu Measurement of lhe Ci,ck by A,chimedes (in Heiberg, A,dimedis
Opera Omnia cum CommenJa,iis Euctocij, Leipzig, 1881, vol. Ill, p. 302,iine 4), he mentions the
M'YIO'TlKli of a certain Magnus or Magnes.
I Here quoted on the basis of the text given in C. F. Hermann's ~ition of Plato (Leipzig,
1901), vol. VI , p. :lC)O. See also Huitsch, Heronis Aleza1Ulrini Getmut,icorllm cl Stneo",dricorum
ReJiqui4e, Berlin, 1864. pp. 247- 248.
4 Tannery, La GhJ.dtrn Gruqau, Paris, 1881, pp. 48- 49.
I Proclus, In Primum Euclidis EJementorum Libr"", Commaltarii, pp. 38, 1-42, 8 (ed.
Friedlein) .
• Compare the similar distinction made by Aristotle.
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"It treats, then, on the one hand, tbat which Archimedes called
'The Cattle Problem,' and on the other hand, 'melite' and 'phialite'
numbers, the one discussing vials (measures, containers) and the other
flocks; and when dealing with other kinds of problems it has regard
for the number of sensible bodies and makes its pronouncements as
though it were for absolute objects.
"It has for material all numerable objects, and as subdivisions the
so-called Greek and Egyptian methods for multiplication and division,
as well as the summation and decomposition of fractions, whereby it
investigates the secrets lurking in the subject-matter of the problems
by means of the procedure that employs triangles and polygons.
"It has for its aim that which is useful in the relations of life and in
business, although it seems to pronounce upon sensible objects as if
they were absolute,"
The philosophical arithmetic of the Greeks, ap,Op;rrnKT}, of which
the arithmetic of Nicomachus is a specimen, corresponds in a measure
to our number theory; tbe subject was designed for mature students
as a preparation for the study of philosophy, and was not at all intended for children. Arithmetica is, as the name indicates, the study
of that which is implied in number. This branch of arithmetical
science developed along two quite distinct lines. On tbe one hand
we have the rigid, mathematical discussion of the properties of numbers, involving the forms of proof and the rigor of the demonstrational
geometry, which is the great contribution of Greece to science; on the
other hand we have a mystical development, ascribing even magical
powers and life-properties to numbers. This pseudo-science which
employs the results, but not the demonstrations of the rigid science, is
commonly termed arithmology.' Greek arithmetic must be considered, then, from the point of view of the philosopher and theoretical
mathematician, rather than from that of our elementary schools.
Arithmetic was intimately connected by the early Greeks ,,~th both
geometry and music. The treatise on arithmetic by Euclid, as found
in the seventh, eighth, and ninth books of the Elements,' is wholly
from the geometrical standpoint. This point of view is reflected in
many ways in later treatises, that of Nicomachus, for instance, which
considered arithmetic as an independent science.
I
The intimate con-
See Chapter VII, pp. 90 8'., for a discussion of arithmology and of the share of Nicomacbus
in it.
lOur references to the Elements of Euclid will be to the English edition by Sir T. L. Heath,
Tile TmrlUn Books of Euclid's EkrMJUS, three volume!!, Ca.mbridge, 1908 .
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nection between arithmetic and music accounts, in some measure,
for the complete and even tedious discussion of ratios in the Greek
treatises on arithmetic. In consequence, our consideration of the
origins of Greek arithmetic will necessarily touch incidentally not only
the processes of computation of the Greeks, but also geometry, music,
and eveD other sciences, as related to the sciences of the older civilizations.}
For the sources of the early Greek arithmetical sciences we must
look to Egypt and to Babylon, possibly even beyond to India and
China. Evidence of the exchange of ideas between Greece and Egypt,
and between Greece and Babylon, has accumulated so much in recent
years as to show a degree of intimacy long unsuspected.' In the early
centuries of the Christian era, knowledge of Greek astronomy was
carried to India; traces of reciprocal influence in ancient times are
not wanting, although any detailed statement must await more accurate information of the historical development of Hindu learning.
The sciences, biological, physical, and mathematical, as well as the
fine arts and technical arts, are involved in the interchange of ideas
between Orient and Occident, but our interest is centered upon the
mathematical sciences. In this field the Oriental science served primarily as a directive force, determining the topics which for centuries
occupied the attention of Greek mathematicians.
In mathematics and astronomy the early traces of Oriental influence
cover a wide range of ideas, touching at the lower point the simplest
operations of computation and at the upper point the development of
complicated astronomical theories. At the outset we may say that one
extraordinary achievement in mathematics remains undisputedly
Greek in its origin, namely, the development of logical, demonstrative
geometry. Writers 3 who confound with the whole of science the
systematization of the sciences achieved by the Greeks, together with
this process of logical demonstration, entirely mistake the nature of
science and the processes of its progress. Science is concerned with
the problems involved in comprehending the universe in which we live.
Science involves inevitably the knowledge of numbers and form, or
I For more complete discussion of arithmetic and logistic, see Heath, A History of Gruk
J(CIIM/lCQtic.s (Orlord, 19:21), vol. It pp. 13-16.
• F. Cumont, The QrU"tal Religions in Roman Paganism (Chicago, 1911); and Astrology
a1lil Religion among the Greeks and Romans (New York, 1912) ; Milhaud, NOUfJdJt:s &udes .sur
I'Histoirt: dt: la Pt:tISU S,ienljfiq~ (Paris, 191 I), pp. 41- 133 .
• Like John Bumet, Greek PltiJosop"'Y, Part 1, Thale.s to PlaJo (London, 1920), pp. 4 ff.
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mathematics, as well as the sciences of material things and life-processes. This science begins with primitive man, and develops as man
develops.
The processes of computation in Greece were closely allied to those
of Egypt. The abacus with its counters for reckoning, which was
in wide use among the ancient Greeks,1 had its counterpart, according
to Herodotus,' in Egypt. While no trace of any Egyptian abacus has
been found, Plato's statement that in Egypt" systems of calculation
have been actually invented for the use of children" suggests that the
Egyptians may have invented the abacus for the purpose for which it
is now used in our primary schools.
The' Egyptian methods' of multiplication and division, mentioned
in the scholium on Plato's Charmides quoted above, are now known
to us through the preservation and publication of the Ahmes manual,'
an Egyptian arithmetic which dates from about 1700 R.e. Multiplication is effected by repeated doubling. Division is the inverse of
multiplication, effected by doubling and re-doubling the divisor until
the dividend can be obtained by summation of the appropriate doubles.
Thus the product of 27 times 57 is obtained as follows:
I'
57
II4
228
45 6
9 '2
'539
2'
4'
8'
16'
The multiplication of 27 times 57 is treated as 16 + 8 + 2 + I times
57. The accent marks to indicate which numbers are to be summed
appear in the papyrus. Were 1539 to be divided by 27, the same
series of doubles would be written, and the required summands would
be obtained by subtraction from the dividend or by inspection. A
multiplier or quotient involving fractions would be treated in the same
way; thus, to multiply 57 by 27,~, t, the numbers 2~ and 14t, ~ and
t respectively of 57, would appear among the summands to be added.
Multiplication by 10 was sometimes included, without any doubling.
The most distinctive feature of the Egyptian arithmetic is the
Herodotus, 11. 36 ; Aristotle, Constitution of Athens, 68, 3 fI.; Plutarch, Yito Cahmis
Minoris, 70; $extus Empiriclls, Adurms JlalhemaJicos, IX. 194.
t Herodotus, ibid.
, Eisenlohr, Ein matlumaJiscMS Hondblle" der alien Acgypter (Papyrus Rhind des British
Musellm ). Leipzig, r&)l; T. Eric Peet, The R},inJ MaJhcmaJical Papyrus, London, 1923,
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restriction of the operations with fractions to unit fractions, i.e. those
having one as the numerator j I is the single exception. Thus t was
written as i t, and t as t -i,,; the juxtaposition indicates that the
fractions are to be summed. Now the Greek symbolism for fractions
includes special devices for writing such unit fractions, together with
a separate symbol for the fraction f.' The little that we know of
ancient computation, supported by definite indications of later documents, shows the intimate connection between the Greek and Egyptian
methods of treating fractions. Thus, Euclid has a special term for
a unit fraction,' while in the works of Hero of Alexandria and Diophantus series of unit fractions in true Egyptian form are common. 3
Furthermore, in the aritltmetica the superparticular is definitely connected with the notion of a unit fraction.
Mention has been made of the 'summation and decomposition of
fractions.' In the absence of any treatise on logistic from the classical
period, the meaning of the scholiast's phrase is revealed to us only
by later documents. A Greek papyrus of the eighth century A.D.,
found at Akhmim in Egypt,' includes unit fractions entirely after the
manner of the Abmes manual. Tbe products of ~, t, t, i, t, !, t, and
by the integers from I to 10, and by the tens to 9" are written in
tenns of unit fractions. A fragment of the same nature appears in the
ancient Egyptian manual, giving the product of ~ by~,!, i, i, t, +.
iT, and i, as well as the products separately, of ! by ~,and i by I, by
i, .,\, t, and iT, and i by t. The distribution problems by Abmes of
I, 2,3,6, 7,8, and 9 loaves of bread among ten people are arithmetically analogous.
Undoubtedly we have here the' decomposition' process into unit
fractions; this also appears in the introductory material of the Egyptian manual wherein the fractions having 2 as a numerator and odd
numbers to 99 as denominators are resolved into unit fractions. ri
Some of the same numerical operations are found also in two letters
n
Heath, DiopluJnlus of Alexandria, A Stud)' in the Hislory of Gruk Algebra (s«ond edition,
Cambridge, 1910) , pp. 44- 47.
2 Elements, Book VI, definition 3: "A number is a part of a number, the less of the greater,
when it measures the greater; but parts when it does not measure it." Heath adds (vol. H,
p. 280) that "by the expression parIs (~ip,), the plural of ~) Euclid denotes what we should
call a proper fraction."
J Heath, DiopMntus, p. 46 .
• J. Baillet. Le Papyrus Mathlmatiql~ d'Akhmim, AUmoires P'lblilcs par les MeMes tU la
Mission Archtologique Fran,aise all CairtJ (Paris, 1&)2). T. IX, pp. I- &] .
• Similar fractions in Michigan Papyrus, No. 621; described by L. C. Karpinski, lsis , vol. V
(1922), pp. 20-25, with facsimile. Tbe text was published by F. E. Robbins, ClassioJl Philology.
vol. XVIII (1923) , pp. 328-333.
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of the fourteenth century written by the Greek monk Nicolas Rhabdas
Atarvasda of Smyrna, who gives further the reverse process of 'summation' of unit fractions into ordinary ones.' Doubtless it was early
recognized that for multiplication and division by a series of unit
fractions the combination of the set into a single common fraction was
desirable. Nicolas explains the process of combination.
Europe continued to employ the unit fractions for many centuries.
Leonard of Pisa in the thirteenth century includes in his famous
Liber Abbaci a table for decomposition into unit fractions,' and employs
them frequently. The Arabs 3 and the Hindus, too, used Egyptian
methods, although not exclusively, in their discussions of fractions,
and traces of the Egyptian process of multiplication are preserved to
this day among the Russian peasants.
Plato makes a statement about Egyptian mathematics whicb shows
not only his own respect for Egyptian methods of instruction, but
also brings to light certain Egyptian problems which may have had to
do with the problems on containers, 'phialite numbers,' mentioned in
the scholium on the Charmides already cited. Plato says (Laws, 819) :
U
All freemen, I conceive, should learn as much of these various
disciplines as every child in Egypt is taught when he learns his alphabet.
In that country, systems of calculation have actually been invented
for the use of children, which they learn as a pleasure and amusement.
They have to distribute apples and garlands, apportioning the same
number either to a larger or smaller number of persons. . .. Another
mode of amusing them is by taking vessels of gold, and brass, and
silver, and the like, and mingling them or distributing them without
mingling; as I was saying, they adapt their amusement to the numbers
in common use, and in this way make more intelligible to their pupils
the arrangements and movements of annies and expeditions, and in
the management of a household they make people more useful to
themselves, and more wide awake; and again in measurements of
things which have length, and breadth, and depth, they free us from
that ludicrous and disgraceful ignorance of all things which is natural
to man."
Tannery, Notices sur les Dna Leu,es ArithmtJiquts tk Nicolas Rhabd4s, Notius et
prl.rails de Matlwcrus tk la Bihliol/teqxe NaliotJQk (Paris, 1886), vol. XXXII, pp. 121-252.
I Scnui de LetmaTdo Pisano, published by B. BODcompagni, Illiberabbati (Rome, 1857), vol. I,
p. 79· '
• See Karpinski, TIu Algebra of Abu Ka"," Slwja bm As14".. in Bibliatluca M aI~
(third series), vo!. XII, pp. 5:1-54.
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The problems of the Ahmes papyrus on the distribution of loaves
of bread among ten people,' and the problem to which we shall recur
on the distribution of 100 loaves of bread according to the terms of an
arithmetical series, are certainly suggestive of the type of problems of
distribution to which Plato had reference. The following problem'
in the Egyptian papyrus is doubtless one of the type dealing with
containers ('phialite'): "I pour (from my container) three times;
I add t and!; I fill it up. What part of the measure have I?" In
the Greek anthology are found a series of problems on the distribution
of apples and nuts, and problems on the weights of bowls, which
involve linear equations in one and two unknown quantities. 3
The intimate connection between Greek logistic and Egyptian arithmetic can hardly be seriously questioned. So far as Greek arithmetica
is concerned here again we find that the Greeks were inspired by their
Oriental predecessors. The available Babylonian and Egyptian
documents in the exact sciences are as yet extremely limited; our
present infonnation is more or less accidental, and by no means comprehensive. So far as early Egyptian mathematical science is concerned, we are largely dependent upon the Abmes papyrus. But these
few surviving documents give indications of development along many
different lines of mathematical thought. Their content is, as we have
already partially indicated, quite in harmony with the Greek traditions
concerning Egyptian and Babylonian science. In view of this correspondence and of further definite indications of real progress in
mathematical thinking among the Egyptians, we are warranted in
giving some credence to the Greek traditions concerning Oriental
science which are not yet confirmed by indigenous evidence.
Arithmology is closely related to the occult sciences, astrology,
alchemy, and magic. While alchemy is undoubtedly a comparatively
late development, the Oriental source of its theories is unquestioned.'
Between the industrial arts of Egypt and Babylon and the development of theories of alchemy, there is an intimate connection, as
Berthelot has shown. Furthermore this authority even asserts that
Thales may have taken from Babylonian myths his theory that water
is the material cause of all things.'
I Eisen1ohr, 0/1. cU .• pp. 71-74; Peet, 0/1. cil., pp. 78-79.
1 Eisenlohr, 0/1. cil., pp. 63--65; Peet, 0/1. dJ., pp. 70-72.
'See Heath, A History of Greek A1aJJumaJics, vol. I, pp. 15-16; vol. 11, pp. 441-443;
Tannery, DWpltanti Alexandrini, Opera omnia, vol. 11 (Leipzig, 1895), pp. 43-72.
'BertheIot, Lu Originu tk l'Aldimu (Paris, 1885), Chapter Ill.
• 01. &il., p. 251.
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So far as the origin of the signs of the zodiac and the star-symbols is
concerned, Oriental and Occidental contributions are not separable,
and the same is true of other scientific ideas.' Astrology was born
and bred in the temples of the Babylonians. The desire to forecast
the future and equally the desire to establish a connection between the
marvels of the beautiful heavens of the East and the events on the
mundane sphere resulted in the cultivation of astrology. The devotion
to the art constituted the first scientific study of the stars.' "The
observations which the priests of the ancient Orient gathered with
indefatigable patience inspired the first physical and astronomical
discoveries, and just as in the period of scholasticism, the occult
sciences [astrology and magic] led to the exact sciences. But these,
by making evident later the vanity of the marvellous illusions by
which they were nourished, destroyed the foundations of astrology
and magic to which they owed their birth.'"
The observations of the Egyptian and, more particularly, of the
Babylonian astronomers, furnished a mass of material which was
used by the Greeks.' Ptolemy and Hipparchus utilized the observations and the computations of the Chaldeans, mentioning specifically
certain eclipses observed j 6 Theon of Smyma discusses the different
types of treatment of astronomical problems by Egyptians as compared with the Babylonians;' Diodorus Siculus notes that both the
Egyptian priests and the Chaldeans were skilled in the prediction of
eclipses.' As scientific observers of celestial phenomena the Babylonians compare favorably with the greatest of the Greek astronomers.
Further than this, the evidence of their ability to use the data intelligently is indisputable. The determination of the period and mean
motion of the moon, the determination of the lengths of the seasons
and of the year, the determination of the period of eclipses and the
periods (ePMmerides) of the planets, and a host of minor deductions
were derived by the scientists of the Orient from their data. The most
Franz Boil, Splroera, Nt14t griuhische Texte UM Unlt:Tsllchungen IIur' Gesc/rUNte dct SIn'1Jbilder (Leipzig, 1903), p. 461.
J Cumont, Lt.s Religions Orienlales dons le PaganisnK Romoin (Paris, 1999).
See also his
Astrology and Religion among (he Gruks and Romans (New York, 1912) .
I Cumont, La Religions Orientales dons le Paganismf] Romo;", p. 2354 Heath, AdsJarchus DJ Somas, The AndmJ Copernicus, A History of Greek Astronomy to
A.ristarchus (Oxford, 1913), Chapter Ill.
• Pto!emy, Composition MaJhlmaliqlle de Claude PtoUmle (Almagest), cd. Halma (Paris,
1813). vol. I, Book IV, pp. 216, 244-247. 267 .
• Theon of Smyrna, p. 177,9 ff., HiIler.
7 Diodorus Siculus, Bibliotheca Historica, I. 50; H·30.
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notable advance in astronomy in Babylon was undoubtedly made
during the period in which the science was making real progress in
Greece; indebtedness was mutual, but independent scientific progress
on both sides is incontestably established.
To deny to Babylon, to Egypt, and to India their part in the development of science and scientific thinking is to defy the testimony of the
ancients,' supported by the discoveries of modem authorities.' The
efforts which have been made to ascribe to Greek influence the science
of Egypt, of later Babylon, of India, and later of the Arabs, do not add
to the glory that was Greece. How could the Babylonians of the
golden age of Greece have taken over the developments of Greek
astronomy? This would have been possible only if they had arrived
at a stage of development in astronomy which would have enabled
them properly to estimate and appreciate the work which was to be
absorbed. There has never been any question concerning the nature
and origin of such feeble beginnings of science as are found among the
American Indians. As regards the Babylonians, the Hindus, and the
civilization of Europe in the time of Alexander the Great and up to
600 A.D., the problem is entirely different. These are peoples who had
reached approximately the same stage of development. The admission that Greek astronomy immediately affected the astronomical
theories of Babylon and India carries with it the implication that this
science had attained somewhat the same level in these countries as in
Greece. Without serious questioning we may assume that a significant part of the science of Babylon and Egypt that was developed
during the times which we think of as Greek was indigenous. Nor do
we thereby detract from the real greatness of Greece. The Hellenic
civilization remains as an integral and vital part of all civilization,
and not as something apart.
Turning to the arilhmetica proper, we may first inquire as to the
Egyptian attempts at systematization of the science. The Ahmes
manual in itself is evidence of a noteworthy step in this direction,
since it establishes the fact that the body of ideas which we now group
under the name t mathematics' was recognized as a separate field by
Hippnrchus and Ptolemy, Theon of Smyma, and Diodorus. as cited above; Herodotus, 11.
109 i Berosus, fragg. I] fr. in C. MUller, Fragmtnta Histori((wlfm GrauOflfm , vol. 11 , pp. 509 f.;
Clemens Alexnndrinus, Stromata, 11. 4; PHny, NOll/ralis llistorifJ, VI. 121; VII . 193.
! Heath, Berthelot, Boil, Cumont, as cited above;
K ugler, Die babyloniuhe AI ondrechnung
(Freiburg, 1900), pp. 50-51; 203-211; Epping, Astronom;sches aus Babylo" (Freiburg, 18&),
pp. J 83- 190_
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the Egyptians. While no definitions of number, as such, have as yet
been found, lamblichus informs us that Thales gave the classical
definition of it as a collection of units, and the definition of the unit,
arithmetically, as one of a group, "following the custom of the Egyptians with whom he studied." 1 Furthermore, the distinction between
even and odd, fundamental in the arithmetica, is implicit in the Egyptian manual. For example, the first part of the work is devoted to a
table for the conversion into unit fractions of fractions with odd denominators from 5 to 99 and with 2 as numerator. This table in and
of itself marks real progress in systematization.
The decad, which is prominent in the Pythagorean arithmetic, also
receives, in a way, particular attention in the Ahmes papyrus, for 10
appears over and over again in the problems of the Egyptian manual.'
Attention to arithmetical and geometrical series was given both
in early Babylon and in early Egypt. The single reference which we
have, as yet, to the arithmetical and geometrical series in Babylon
is found in a moon tablet 3 deciphered by Hincks. This gives the
geometric series S,lO, 20, 40, 80 followed by the arithmetical series,
Ba, 96, I12, 128, . . . 240.
In the Egyptian manual we have much more than the simple appearance of arithmetical and geometrical series. The discussion of
arithmetical and geometrical progressions reveals an unexpected
familiarity with rules which we now express by algebraical formulas, a
familiarity which has not received adequate appreciation. The essential points of the two formulas which we have for the nth term and
the sum of the arithmetical series, a, a + d, a + 2 d, a + 3 d, .. "'
appear from the problems to have been familiar to the Egyptians.
Comparatively intricate problems are handled with the ease and intimacy born of long acquaintance.
The problem numbered 40 by Eisenlohr reads: "To distribute 100
loaves of bread among 5 people so that t of the (total of the) first three
equals that of the last two. What is the difference?" The solution
shows that it is understood that the loaves are to be distributed in
arithmetical progression.
"Following instructions, the difference si," is the next somewhat
cryptical suggestion of the manual. I hold that this reference implies
NicMMCii .A.rill.,,,etic4,,. lnl,od"dimrem LiJJ6, p. to, 8 (Piste1li).
I EivnJohr, ojJ. ciI., 208, 211, 216, 217. 218, 219, et passim.
I 1,.
I T&e Likrory GauJk, Aug. Sf 18S4. with reference to Tablet K 90 of the British Museum.
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definite rules of procedure in such problems, leading to the difference
st, if unity be taken as tbe first term, under the conditions proposed.
Our common procedure, in analytical solution of this problem, leads
to the result, d = st a or d = st if a is I. Even if the method of
arriving at this value for d be that of 'false position,' the procedure
which, being adaptable to similar problems, arrives definitely and
surely at the complete solution of the proposed problem must be
regarded as scientific.
From this point the solution follows the lines of previous problems.
With I as the first term and st as the difference, the terms are I, 6t,
12, 17t, and 23, having 60 as a sum. To complete this to the required
100 loaves there must be added 40, or i of 60.
After it has been noted
that this is the case, there is added to each of the numbers in the discovered series i of itself, a process that gives Ii, la! t, 20, 2,*, and
38! as the series fulfilling the required conditions.
A second problem involving an arithmetical series is entitled" Instructions for the difference in distribution." The solution opens
with the phrase, "If you are told," which was later adopted by Arabic
mathematicians, and is not uncommon even today. "If you are told,
[distribute] 10 measures of grain to 10 people so that the difference in
[the amount received by] each person as compared with the next one is
t of a measure of grain. I take the mean, one measure. I subtract
I from 10, leaving 9.
I take t of the difference,
and take it nine
times. This gives t
which I add to the mean. From this take
away t measure for each person in order to arrive at the goal. FollowIH
It
It"\-, I.,\-, t t t -h, t t -h,
ing instructions: It
t t -h, t -h, t t .,\-, together 10." The solution of this problem as
given by the Egyptian manual should be compared step by step with
the solution by the ordinary procedure with the formulas of our elementary algebra; the close correspondence is too striking to be
regarded as wholly accidental.
No one could ask that the ancient Egyptians should have modern
formulas with a literal symbolism, for this advance was not made in
Europe until the end of the sixteenth century of the Christian era.
The similarity in method is, however, highly significant, revealing a
development in analytical thinking that is not equalled for many
centuries. In effect, we have in these problems the first term of an
arithmetical series regarded as a function of the common difference,
under given conditions, and the last term as a function of the mean
n,
n,
n,
n, n,
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and the difference. This is true functional thinking whose like is
hardly met again until Archimedes.
The single illustration of a geometrical series confirms the implications of the solutions found in the problems involving arithmetical
series. The text is extremely concise, and possibly mutilated:
"A ladder
together
scribe
2
2,801
5,602
4
11,204
mouse
19,60 7
sheaf
•
gram
together
I
cat
7
49
343
2,401
16, 80 7
19, 60 7."
At the right we have the summation of the series 7, 49, 343, 2,401, and
16,807 by actual addition; at the left we have the summation of the
same series 7 X 2,801, with the multiplication effected in the usual
manner. Now our formula for the summation of this series gives
75 7
7 -
I
.
, or 7 bmes 2,801.
1
Some three thousand years after Ahmes an Italian mathematician
of prominence, Leonard of Pisa, includes in his arithmetic the same
series with one further term. He effects the solution in precisely
the two ways selected by his Egyptian predecessor. In India, too,
powers of 7 received special attention. The words, or illustrations,
which accompany the numbers suggest the nursery rhyme concerning
the old woman going to St. Ives.
So far as geometrical formulas are concerned the Egyptians had
definite methods for finding the area of triangles, rectangles, trapezoids,
and circles, as well as for finding the volume of cylinders.' Recently
an Egyptian document has been published which gives the expression
for the volume of a truncated pyramid' Further there have been
found in Egyptian papyri problems concerned with determining the
sides of a rectangle when the area and the ratio of the sides are given. 3
All of this material connects directly with the geometry of Greece.
Iamblichus asserts that Pythagoras brought the harmonical progresI Eisenlohr, op. cil., pp. 75-1I5.
1 B. Touraeff. The Volume of tile Tmncalcd Pyrami4 in Egyptian Malhematics, A.ncitnl
Egypt, 1917. pp. 100-102; Karpinski, An Egyptian Malhemoticai PapyrtlS in Moscow, Scierra,
vol. 57. 1923. pp. 528-529.
I H. Schack-Schackenhurg. Der Berliner Papyrus 6619. bilschrif' fur iigyptisclu Sprache.
vol. XXXVIII, pp. 135-140, and vol. XL, pp. 65-66.
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sion to Greece from Babylon, but no Babylonian record confirming the
statement has yet been found.
The squares and cubes of numbers also received particular attention
in Babylon, and tables of squares and cubes have been found. In
Egypt the relation 3' + 4' = 5' appears to have been used in the laying
out of right angles by means of a stretched rope. Democritus, an able
Greek mathematician of the fifth century B.e., boasts: "So far as the
laying out of lines is concerned, no one has surpassed me, not even
the rope-stretchers of ancient Egypt." 1 Michael Psellus of the
eleventh century mentions the Greek equivalents of the Egyptian
names of the higher powers, first power to twelfth power; it is supposed that the statement is based upon the lost commentary upon the
Arithmetic of Diophantus by Hypatia. Again the reference, although
not confirmed by available Egyptian material, adds to the probability
of mathematical developments in addition to those with which we
happen, almost accidentally, to be familiar.
The mystical element in Greek arithmetic is undoubtedly also of
Oriental origin.
It must be remembered that at Babylon a number
was a very different thing from a figure. Just as in ancient times and,
above all, in Egypt, the name had a magic power, and ceremonial
words formed an irresistible incantation, so here the number possesses
an active force, the number is a symbol, and its properties are sacred
attributes." 2 This attitude we see occasionally in Nicomachus, and
to a more pronounced degree in later mediaeval times.
The purpose of this introduction is to show the Oriental inspiration
and origin of many of the Greek developments in mathematics. The
assertion, which has been seriously made by Burnct,3 that all science
is Greek in its origin, is shown to be not at all in accordance with the
facts. The well-established tradition' of Babylonian and Egyptian
influence upon the science of early Greece is confirmed by a mass of
self-supporting evidence, naturally not confined to one branch of
science, which has been illustrated above with particular reference to
arithmetic. Greece retains the right to enjoy the profound admiration
of the world of science, but the Orient, also, must be credited with
•
contributions worthy of note.
j/
Clement of Alexandria (cd. Potter), p. 357.
2 Cumont, Astrology and Religion among tht Greeks and Romans (New York, 19I1), p. 30.
I John Burnet, Greek Philosophy, Pari!, Thalfs to Plato (London, IQI4) , pp. 4 11.
t Bretschneider. DU Geomtir;e und die Geometer i'or Eukiides (Leipzig, 18iO), pp. 3-35'
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CHAPTER II
THE DEVELOPMENT OF THE GREEK ARITHMETIC BEFORE
NICOMACHUS
ONLY a slight acquaintance with Greek mathematics is necessary
to convince one that the Introduction to Arithmetic of Nicomachus is
but a restatement of facts which were common property not only in
Nicomachus's own generation but even long before him, and that,
except for the few unimportant propositions the discovery of which
our author with pardonable pride claims for himself, the book is largely
unoriginal.' This naturally leads to the inference that the Introduction must be closely connected with other mathematical treatises,
which served as the fountains whence Nicomachus drew his supply.
Because so little remains of this literature, it is difficult to demonstrate
the hypothesis in detail; few, however, will ques