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Call No. * Accession No. (4 ( 

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This book should be returned on or before the date 
last rharked below. ' 

George Sarton 

Ancient Science 
and Modern Civilization 

Euclid and His Time 

Ptolemy and His Time 

The End of Greek Science and Culture 

HARPER TORCH BOOKS /Science Library 


New York 


Copyright 1954 by University of Nebraska Press 
Printed in the United States of America 

Reprinted by arrangement with 
University of Nebraska Press 

All rights in this book are reserved. 
No part of the book may be used or reproduced 
in any manner whatsoever without written per- 
mission except in the case of brief quotations 
embodied in critical articles and reviews. For 
information address Harper & Brothers 
49 East 33rd Street, New York 16, N. Y. 

First HARPER TORCHBOOK edition published 1959 






Ancient Science 
and Modern Civilization 


This book reproduces the full text of the three Montgomery 
Lectures which it was my privilege to deliver at the University 
of Nebraska, in Lincoln, on April 19, 21, 23, 1954. 

In spite of their name the "lectures" were not read but 
spoken; the essential of the spoken and written texts is the 
same, but there are naturally considerable differences in the 
details. The spoken text is to the written one with its ex- 
planatory footnotes like a fresco to a miniature. That must 
be so, because people cannot listen as accurately as they can 
read. I have explained my views on this subject many times, 
last in the preface to my Logan Clendening Lecture on Galen 
of Pergamon (University of Kansas Press, Lawrence, Kansas, 

As mechanical progress discourages the printing of Greek 
type, it has become necessary to transcribe the Greek words 


in our alphabet as exactly as possible. The diphthongs are 
written as in Greek with the same vowels (e.g., ai not ae, ei not 
i, oi not oe), except ou, which is written u to conform with 
English pronunciation (by the way, the Greek ou is not a real 
diphthong but a single vowel sound) . The omicron is always 
replaced by an o, and hence the Greek names are not Latin- 
ized but preserve their Greek look and sound. There is really 
no reason for giving a Latin ending to a Greek name when one 
is writing not in Latin but in English. Hence, we write 
Epicures not Epicurus (the two u's of the Latin word repre- 
sent different Greek vowels) . We indicate the differences be- 
tween the short vowels epsilon and omicron and the long 
ones eta and omega, as we have just done in their names. 
Hence, we shall write Heron, Philon, but some names have 
become so familiar to English readers that we must write them 
in the English way. We cannot help writing Plato instead of 
Platon and Aristotle instead of Aristoteles, etc. For more de- 
tails, see my History of Science, p. xvii. 

Indications such as (III-2 B.C.) or (II-l) after a name 
mean two things: (1) the man flourished in the second half of 
the third century before Christ or in the first half of the second 
century; (2) he is dealt with in my Introduction. 

Harvard University 
Cambridge, Massachusetts 


(first half of third century B.C.) 


HAT has ancient science to do with modern 
civilization? one might ask. Very much. Modern civilization 
is focused upon science and technology, and modern science 
is but the continuation of ancient science; it would not exist 
without the latter. For example, Euclid flourished in Alexan- 
dria more than twenty-two centuries ago, and yet he is still 
very much alive, and his name is equated with that of 
geometry itself. What has happened to him happens to every 
man whose name was equated with that of a thing; the thing 
is known but the man himself is forgotten. When I was a 
child, the table of multiplication was called the Table of 
Pythagoras, but the teacher did not tell us who Pythagoras 
was; perhaps she did not know it herself; she would have been 


a very wise person if she did. Pythagoras was simply a common 
name to us like sandwich, mackintosh or macadam. Thus it 
was wrong to say that Euclid is very much alive today; geome- 
try is but not he. His name is very often on our lips, but who 
was he? The purpose of my first lecture is to explain that, but 
no man ever lives in a social vacuum, and to bring him back to 
life we must, first of all, describe his environment. This is 
something important which many historians of science shame- 
fully neglect; it is foolish to speak of great men of science 
without trying to explain their personality and their genius, 
neither of which can be understood outside of the social en- 
vironment wherein they developed. 


In the first volume of my History of Science, I have described 
ancient science down to the end of Hellenic days. Euclid stands 
at the beginning of a new age, absolutely different in many 
respects from the preceding, and generally called the Hellenis- 
tic Age. The word Hellenistic is well chosen, it suggests Hel- 
lenism plus something else, foreign to it, Egyptian and oriental. 

The break between those two ages, one of the greatest revo- 
lutions or discontinuities in history, was caused by Alexander 
the Great (IV-2 B.C.), who conquered a great part of the world 
within twelve years, from 334 to his death in 323 at the ripe 
age of thirty-three. As his armies were Greek, he carried Greek 
culture into the very heart of Asia; it has been said that he 
Hellenized Western Asia, but it could be said as well that he 
helped to orientalize Eastern Europe. Many cities were 
founded by him and bear a name derived from his, Alexandria, 
some as far East as Sogdiana beyond the Oxus, or India 


Superior beyond the Indus. By far the most important was 
the one founded soon after his conquest of Egypt in 331. 

The Greeks called that city Alexandreia he pros Aigypto 
(Latin, Alexandria ad Aegyptum) and rightly so, because it 
stood at the edge of Egypt and was very different from it. It 
is as if we said that Hong Kong is near China. The com- 
parison is useful, because, just as in Hong Kong the over- 
whelming majority of the inhabitants are Chinese, we may as- 
sume that in Alexandria the majority were native Egyptians. 
The ruling class was Macedonian or Greek; as the city became 
more prosperous, it attracted a great diversity of foreigners, 
Ethiopians or Abyssinians and other Africans who came down 
the Nile; Asiatics, primarily Jews, but also Syrians, Persians, 
Arabs, Hindus. Alexandria soon became (and has remained 
throughout the ages) one of the most cosmopolitan cities of the 
world. Its harbor was and has remained the largest of the East- 
ern Mediterranean Sea. 

This suggests another comparison, which I find very help 
ful, with New York. Alexandria's relationship to Athens in 
ancient times was comparable to New York's relationship to 
London. If one considers the speed of communication then 
and now, the distances, Alexandria-Athens and New York-Lon- 
don were about the same; New York was an offspring of 
Europe, just as much as Alexandria. Finally, its cosmopoli- 
tanism, and especially its Jewishness, make of it the American 
Alexandria. The main difference is that New York is es- 
sentially American, while Alexandria was definitely a Greek 

Alexander died in Babylon in the middle of June 323 and 
soon afterwards one of his closest companions, a Macedonian 


called Ptolemaios, son of Lagos, 1 became the governor or 
master of Egypt; in 304 he proclaimed himself king and found- 
ed the Ptolemaic dynasty, which lasted until 30 B.C. for three 
centuries. Ptolemaios I Soter must have been a man of con- 
siderable genius; not only was he the founder of a dynasty, but 
he was a patron of science and arts and wrote what was per- 
haps the best history of Alexander the Great. When he died 
in 283/2, he was succeeded by his son, Ptolemaios II Philadel- 
phos, who ruled until 246 and completed his father's task. 
The Alexandrian Renaissance was mainly accomplished by 
these two kings within the first half of the third century; I 
introduced them both, because it is not always possible to 
separate their achievements. 

In order to create the new civilization in Alexandria, they 
needed the help of other Greeks, not only soldiers and mer- 
chants but also intellectuals of various kinds, administrators, 
philosophers, teachers, poets, artists and men of science. Before 
dealing with Euclid, it is well to speak of some of them. 

In the first place, we shall speak of the architects, for to 
build a new town in the Greek style such were needed. The 
Greeks were great town builders and did not allow the new 
cities to grow at random. The planning of Alexandria was 
intrusted by Alexander or more probably by the first Ptole- 
maios to Deinocrates of Rhodes, who was perhaps the most 

1 The kings of that dynasty are often called Ptolemy; I prefer to use 
the original Greek form Ptolemaios (plural, Ptolemaioi) , however, re- 
serving the English form Ptolemy for a more illustrious person and one 
of far greater international significance, the astronomer Ptolemy (II- 1) , to 
whom my second lecture will be devoted. Hence, there will be no am- 
biguity; when I write Ptolemy, the astronomer is meant, while Ptolemaios 
is only a king. 


eminent architect of his time. He it was who designed the 
new temple of Artemis at Ephesos, and he had conceived the 
idea of cutting one of the peaks of Mt. Athos in the shape of 
a gigantic statue of Alexander. The other architect, Sostratos 
of Cnidos, built a lighthouse on a little island in the harbor. 
The island was called Pharos, and therefore the lighthouse 
received the same name. 2 It was the earliest lighthouse to be 
definitely known and described. A tower of about four 
hundred feet high, it could be seen over the plains or the sea 
from a long distance. It became so famous that it was generally 
listed as one of the seven wonders of the world. 

The pharos was an outstanding symbol of Alexandrian 
prosperity; two institutions, the Museum and the Library, 
illustrated the greatness of Alexandrian culture. 

There had been museums before in Greece, because a 


museum was simply a temple dedicated to the Muses, the nine 
goddesses of poetry, history and astronomy, but this museum 
was a new kind of institution which was so noteworthy that its 
name was preserved and has been incorporated into many 
languages. The meaning has changed, however, and museums 
all over the world are primarily buildings containing exhibi- 
tions of art, archaeology, natural history, etc. A certain 
amount of teaching and research is connected with the best of 
them; yet the Alexandrian exemplar was very different. If we 
had to describe its function in modern language, we would 
say that the Museum of Alexandria was primarily an institute 
for scientific research. It probably included dormitories for 
the men of science, their assistants and disciples, assembly 

2 Later the name was given to any lighthouse; it was transcribed with 
the same meaning in Latin and many Romance languages (L. farus, F. 
phare, It. and Sp. faro, Port, farol or pharol, etc.) . 


rooms, roofed colonnades for open-air study or discussion, 
laboratories, an observatory, botanical and zoological gardens. 
The Museum did not include all these features at the begin- 
ning, but like every institution, it grew in size and com- 
plexity as long as it was actually flourishing. Its scientific de- 
velopment owed much to its royal patrons and even more to 
Straton, who had been a pupil of Theophrastos. Straton was 
called to Alexandria by the first Ptolemaios (c. 300); we may 
call him the real founder of the Museum for he brought to it 
the intellectual atmosphere of the Lyceum, and it was thanks 
tc him that it became not a school of poetry and eloquence, 
but an institute of scientific research. Straton was so deeply 
interested in the study of nature that he was nicknamed ho 
physicos, the physicist. Under the distant influence of Aristotle 
and the closer one of his own master, he realized that no 
progress is possible except on a scientific basis and he stressed 
the physical (vs. the metaphysical) tendencies of the Lyceum. 
He remained in Egypt many years, perhaps as many as twelve, 
or even more, being finally recalled to Athens when Theo- 
phrastos died in 288; he was appointed president or head- 
master of the Lyceum (the third one) and directed it for about 
eighteen years (c. 288-c. 270). It is pleasant to think of the 
Museum being organized by an alumnus of the Lyceum, who 
later became its very head. 

Much was done at the Museum during the first century of 
its existence. Mathematical investigations were led by Euclid, 
Eratosthenes of Gyrene, who was first to measure the size of 
the earth and did it with remarkable precision, Apollonios of 
Perga, who composed the first textbook on conies. Another 
contemporary giant, Archimedes, flourished in Syracuse, but 
he may have visited Alexandria and he was certainly influenced 



by its mathematical school. The astronomical work was equally 
remarkable. Alexandria was an ideal place for astronomical 
syncretism; Greek, Egyptian and Babylonian ideas could mix 
freely, in the first place, because there were no established tradi- 
tions, no "vested interests" of any kind, and secondly, because 
representatives of various races and creeds could and did 
actually meet. Astronomical observations were made by Aris- 
tyllos and Timocharis, and a little later by Conon of Samos; 
the last-named used and discussed Babylonian observations of 
eclipses. Meanwhile, another Samosian, Aristarchos, was not 
only making observations of his own but defending theories 
of such boldness that he has been called "the Copernicus of 

The anatomical investigations carried through in the 
Museum were .equally bold and fertile. Herophilos of dial- 
cedon might be called the first scientific anatomist. He was 
flourishing under Ptolemaios Soter, and it was probably he who 
devised the ambitious program of anatomical research, an 
elaborate survey of the human body on the basis of dissections. 
As this was done systematically for the first time, the men in 
charge were bound to make as many discoveries as an ex- 
plorer who would happen to be the first to visit a new con- 
tinent. Herophilos was the main investigator and the catalogue 
of his observations is so long that it reads like the table of 
contents of an anatomical textbook. He obtained the help of 
another Greek, somewhat younger than himself, Erasistratos of 
Ceos, who continued the anatomical survey and paid more 
attention to physiology. It was claimed by Celsus (1-1) and 
by church fathers who were eager to discredit pagan science 
that the Alexandrian anatomists were not satisfied with the 
dissection of dead bodies but obtained permission to dissect 


the bodies of living men, in order to have a better understand- 
ing of the functioning of the organs. The story as told by 
Celsus is plausible. We must bear in mind that the sensi- 
bility of the ancients was less keen than ours and that the 
Alexandrian anatomists were not hindered by religious or 
social restrictions. As far as we know, medicine was not in- 
cluded in the Museum program of research. It is possible that 
Straton or Herophilos decided that medicine was too much of 
an art to reward purely scientific research; the time was not 
yet ripe for "experimental medicine." 

Much of the work accomplished in mathematics, astronomy, 
mathematical geography, anatomy and physiology was analy- 
tical. With the exception of Euclid's Elements, the men of 
science wrote what we would call monographs, such as would 
be published today not in independent books but in journals. 
This reminds us of the cardinal fact that the Alexandrian 
Renaissance was a complete renaissance. At the beginning, I 
remarked that the discontinuity and the revolution following 
it were created by Alexander the Great. There is another 
aspect of this which deserves emphasis. A deeper discontinuity 
had been caused in the time of Alexander's youth by another 
Macedonian but a greater man than himself, his tutor, Aris- 
totle. One ought to say Aristotle the Great and Alexander the 
Less. Aristotle was a philosopher, a man of science, an 
encyclopaedist who tried to organize and to unify the whole of 
knowledge. Considering his time and circumstances, his 
achievements are astounding and many of the results attained 
by him kept their validity for two thousand years. The con- 
quests of Alexander were ephemeral; those of Aristotle were 
durable and exceedingly fertile. After the master's death, his 



disciples in Athens and even more so those of Alexandria 
realized that the best way, nay, the only way of improving the 
Aristotelian synthesis was by means of analysis. 

As opposed to the fourth century in Athens, the Alexandrian 
Renaissance was a period of analysis and research. This is an 
outstanding example of one of the fundamental rhythms of 
progress: analysis, synthesis, analysis, synthesis, and so on in- 

Of the two leading institutions the one of greater interest 
to historians of science is the Museum. But it is probable that 
the Library was an integral part of the Museum (even as every 
research institute has a library of its own) ; both institutions 
were included in the royal city or enclosure; both were royal 
institutions, in 'the same way that they would be government 
ones today, for the king was the state, and everything done for 
the public good was done at the royal initiative and expense or 
not at all. The Museum and its Library were public utilities. 

An elaborate study of the Library has recently been pub- 
lished by Dr. Parsons, 3 who has put together all the documents 
available, but in spite of his zeal and ingenuity, our knowledge 
of it is still very fragmentary. Many questions are still un- 
answerable. The first organizer as well as the first collector 
was almost certainly Demetrios of Phaleron, who worked hand 
in glove with the first king and was probably clever enough 
to give his royal patron the feeling of being the real creator. 
Dr. Parsons gives us a list of the "librarians" beginning with 
Demetrios and ending with the eighth one, Aristarchos of 

Edward Alexander Parsons, The Alexandrian Library, Glory of the 
Hellenic World. Its Rise, Antiquities and Destruction (New Yoik, Elsevier, 
1952; Isis 43, 286). 



Samothrace (in 145 B.C.), which is very interesting in spit* 
of the many conjectures which are implied. The mair 
conclusion that one can draw from it is that the perioc 
of creative activity of the Library lasted only one and a 
half centuries (otherwise we would know the names of latei 
librarians) ; this period was also that of greatest commercial 
prosperity. After the second century B.C., the Library de 
clined and fell into somnolence. At the time of its climax, il 
had been exceedingly rich. It may have contained 400,OOC 
"rolls." But it is impossible to be sure, not only because the 
sources are lacking but also because the counting of rolls and 
books is not as simple an operation, nor the total result as 
determined, as one might think. It was not by any means the 
earliest library, but it was by far the largest one of antiquit) 
and found no equal perhaps until the tenth century when ver) 
large collections of books became available in the Muslim 
world, both East in Baghdad and West in Cordova. 4 By the 
middle of the third century, the Library of Alexandria was 
already so large that the creation of a new library, or call it 
"branch" library, was found to be necessary. This was the 
Serapeion, which earned some fame of its own, especially in 
Roman times. 

The Library suffered many vicissitudes. It may have been 
damaged (or many books lost) in 48 B.C., when Caesar was 
obliged to set fire to the Egyptian fleet in the harbor nearby. 
A few years later, in 40, Anthony is said to have given to 

* For the Baghdad libraries, see their catalogue, Fihrist al-'ulum, written 
in 987 (see my Introduction to the History of Science [? vols., Baltimore 
Carnegie Institution of Washington, 1927-48] 1, 662) ; the Cordova library 
was gathered mainly by the caliph al-Hakam II, who died in 976 (Intro. 1, 
658). It is curious that these two libraries date from the same time (X-2). 



Cleopatra the library of Pergamon, but did that really hap- 
pen? At the time of the Jewish historian Joseph (1-2) both 
libraries were still very rich. Decadence was rapid during the 
second century and there is good reason for believing that 
many books (as well as other things) were taken to Rome. 
Under Aurelian (Emperor, 270-75) the Museum and the 
mother Library ceased to exist; the Serapeion then became the 
main Library and the last refuge of pagan culture. In 391, 
Theophilos (Bishop of Alexandria, 385-412), wishing to put 
an end to paganism, destroyed the Serapeion; it is possible, 
however, that the destruction was not complete and that many 
books could be saved in one way or another. Not a great many, 
however, if we believe Orosius' account of c. 416. When the 
Muslims sacked Alexandria in 646, it is claimed that they des- 
troyed the Library; that can only mean that they destroyed the 
little that was left of it. The story of the great library, if it 
could be told with precision, would be a history of the de- 
cadence and fall of Alexandrian (pagan) culture. This cannot 
be done, but it is certain that the climax was long past before 
the age of Christ. 

Let us return to its golden days. The Library was the main 
center of information for every department, but for the 
humanities it was much more than that: it was the brain and 
heart of every literary and historical study. The astronomers 
observed the heavens and measured the Earth, the anatomists 
dissected human bodies. But the primary materials of his- 
torians and philologists were in the library books and nowhere 


The librarians had not as easy a task as their colleagues 
of today, who deal almost exclusively with printed books, each 
of which is a very tangible object. The first technical librarian, 



Zenodotos of Ephesos, had to identify the rolls and put to- 
gether those which belonged together, for example, the rolls 
of the Iliad and Odyssey. He was, in fact, the first scientific 
editor of those epics. The same process had to be followed 
for all the rolls; they had to be investigated, one by one, identi- 
fied, classified and finally edited as much as possible; it was 
necessary to establish the text of each author and to de- 
termine canons the Homeric one, the Hippocratic, etc. In 
other words, Zenodotos and his followers were not only li- 
brarians, but philologists. Callimachos of Gyrene, poet and 
scholar, came to Alexandria before the middle of the third 
century and was employed in making a catalogue of the library, 
the Pinaces, which was the earliest work of its kind. 5 It was 
very large, for it filled 120 rolls. Would that it had been pre- 
served! Our knowledge of ancient literature, 'chiefly but not 
exclusively Greek, would have been much greater than it is. 
Indeed, a great many of the books which were available to 
Alexandrian scholars have long ceased to exist; we often know 
the names of the authors and the titles of the lost books; in 
some favorable cases, extracts have been transmitted to us in 
other books; in exceptional cases, the whole books have been 

Many historians used the Library of Alexandria; one of the 
first to do so, perhaps, was the first king when he composed the 
life of Alexander. A curious case was that of Manethon, who 

B Some lists of Sumerian writings are considerably older but very short 
(see my A History of Science: Ancient Science through the Golden Age of 
Greece [Cambridge, Harvard University Press, 1952] I, 96). Whenever a 
large number of tablets was kept together, some kind of list may have 
proved necessary, but such lists were so rudimentary as compared with 
Callimachos' catalogue raisonnd that the term catalogue as applied to 
them is figurative. 



wrote Annals of Egypt in Greek on the basis of Egyptian docu- 
ments (whether these existed in the Library or in Temples 
cannot be ascertained). The great geographer Eratosthenes who 
was Librarian (the only man of science to hold that position, 
but he was also a distinguished man of letters) realized the 
essential need of historical research, scientific chronology. 
When one deals with a single country, say, Egypt, a precise 
dynastic history such as Manethon had tried to produce may 
be sufficient, but when one has to study many countries, one 
must be able to correlate their national chronologies, and this 
is not possible unless one has a chronological frame applying 
to all of them. The first such frame had been imagined by the 
Sicilian Timaios, who suggested using the Olympic games as 
references. Those games had become international events in 
the Greek-speaking world and were of such importance that 
we may assume that foreigners would attend them occasionally; 
they occurred every fourth year from 776 on and hence might 
provide an international scale. 6 It is not clear whether Timaios 
was ever in touch with the historians of the Museum, and 
whether Eratosthenes improved his invention. The Olympic 
scale was introduced too late (beginning of the third century 
B.C.) to remain long in use, because the rulers of the Western 
world replaced it by another scale (A.U.C., from the founda- 
tion of Rome in 753 B.C.), and it was completely superseded 
in the course of time by the Christian and the Muslim 

The numbering of the games began with those of 776, but many had 
occurred before. A list of the Olympic winners has been preserved by 
Eusebios (IV-1); it extends from 776 B.C. to 217 A.D., almost a millenium 
(994 years). The Olympic era was used only by a few scholars, such as 
Polybios (II-l B.C.) and Castor of Rhodes (1-1 B.C.); the Greek cities 
continued to date events with reference to their own magistrates and, 
moreover, they used different calendars. 



eras. 7 The point to bear in mind is that scientific chronology 
began in Alexandria; Eratosthenes' interest in it is comparable 
to his interest in geographical coordinates which are of the 
same necessity in a two-dimensional continuum (a spherical 
surface) as fixed dates along the line of time. 

The identification of texts and their establishment opened 
the door to every branch of philology, in the first place, gram- 
mar. Not only was grammar needed to determine the sense 
of a text without ambiguity but in a polyglot city like Alex- 
andria it became necessary for the teaching of Greek to for- 
eigners. Erastosthenes was the first man to call himself 
philologist (philologos). Aristophanes of Byzantion (II-l B.C.) 
and Aristarchos of Samothrace (II-l B.C.) were the first gram- 
marians stricto scnsu. 8 Both were librarians of the Museum, 
Aristophanes from 195 to 180, Aristarchos from c. 160 to 143 
(or 131?). The earliest Greek grammar extant was composed 

7 To summarize: 
Ol. 1.1 776 B.C. 
01. 2.1 77Ii B.C. 

I I.C. 1 753 B.C. = Ol. 6.4. 

B.C. 1 = 753 U.C. - 01. 194.4. 

A.D. 1 754 U.C. = Ol. 195.1. 

To make matters worse, a new Olympiad era was introduced by 
Hadrian; it began when he dedicated the Olympieion in Athens: New Ol. 
1= Ol. 227.3 = U.C. 884 A.D. 131. 

8 Philology and in particular grammar are bound to occur when differ- 
ent languages are used simultaneously, e.g., in the Mesopotamian and 
Anatolian world (History of Science 1, 67) . In Greece proper, it developed 
relatively late, because the language spoken in educated circles was rela- 
tively pure and homogeneous. Nevertheless, grammar was a child of logic 
and some grammatical functions were bound to be discovered as soon as 
one attempted the logical analysis of any sentence (History of Science 1, 
257, 579, 602). 

* According to Parsons' list (p. 60), they were the sixth and eighth 



by another Alexandrian, Dionysios Thrax (II-2 B.C.) . The 
masterpieces of Greek literature were written before 300 B.C., 
the first grammar almost two centuries later. The fact that 
the Hellenistic age witnessed the development of grammar as 
well as the development of anatomy is a natural coincidence. 
They were the fruits of the same analytical and scientific 
mentality, applied in the first place to the language and, in 
the second, to the body of man. 

Euclid has long been waiting for us, and it is high time 
that we return to him; yet, a few words should be said of the 
most astonishing philological achievement of his time, the 

The name will explain itself in a moment. According to 
the story told in Greek by the Jew Aristeas, 10 Demetrios of 
Phaleron explained to King Ptolemaios II the need for trans- 
lating the Torah into Greek. It is a fact that the large and 
influential Jewish colony of Alexandria was losing its com- 
mand of the Hebrew language; on the other hand, a Greek 
version of the Torah might interest some of the Gentiles. The 
king sent two ambassadors to the High Priest Eleazar in 
Jerusalem, asking for Hebrew rolls of the Old Testament and 
for six representatives of each tribe. The royal demand was 
obeyed and seventy-two Jewish scholars were soon established 
in the Pharos island and started their translation of the Holy 
Scriptures. The translation might have been called Septuagin- 
ta duo, but the second word was dropped. Aristeas' story was 

director-librarians, the eighth being the last. The list is tentative and 
suggests many objections, yet it is useful. 

10 For more details, see the excellent edition and translation of the letter 

of Aristeas to Philocrates by Moses Hadas (New York, Harper, 1951; Isis43, 



embellished by later writers; the details of it do not matter. 
The Torah was actually translated into Greek during the 
third century. Other books of the Old Testament were trans- 
lated later, many of them in the second century B.C., the last 
one, Qoheleth (Ecclesiastes) not until about 100 A.D. 11 

This Greek translation of the Old Testament is very im- 
portant, because it was made upon the basis of a Hebrew text 
more ancient than the Hebrew text which has been transmitted 
to us. 12 Hence, any student of the Old Testament must know 
Greek as well as Hebrew. 

11 The original text of Qoheleth was produced very late, say, in the 
period 250-168. This accounts for the exceptional lateness of its translation. 
It was probably prepared c. 130 by Aquila, the Christianized disciple of 
R. Akiba ben Joseph. It is not really a part of the Septuagint but of the 
Version of Aquila (Intro. 1, 291). Practically the whole of the O.T. was 
translated into Greek before the Christian era, and the name Septuagint 
should be restricted to these pre-Christian versions. 

12 It was believed that the Hebrew scrolls discovered by Beduins in 
1947 in a cave along the western shore of the Dead Sea included earlier 
readings than those reproduced in the Hebrew Bible. Isaiah and Habakkuk 
scrolls and other fragments already deciphered do not sustain that belief, 
for they do not seem to have a closer connection with the Septuagint text 
than the Masoretic text has. The dating of those scrolls is very difficult, 
but the arguments from paleography, archeology, radio-carbon testing, and 
historical background would appear to fix the Mishnaic period at least 
as well as any others. If more precision were desired, one might perhaps 
say that the scrolls date from the century following the destruction of the 
Second Temple and of the Jewish State in 70 A.D. Incidentally, the radio- 
carbon dating is very inconclusive, because according to that method the 
piece of linen used for wrapping dates from the period 33 A.D. 200. 
There is already an abundant literature on the many problems raised by 
those scrolls. For general information, see Harold Henry Rowley, The 
Zadokite Fragments and the Dead Sea Scrolls (Oxford, Blackwell, 1952). 
The writing of this footnote was made possible by the kindness of Abraham 
A. Neuraan, president of Dropsie College in Philadelphia (letter dated 
30 Nov., 1953). 



The ancient Greeks had hardly paid any attention to the 
queer people living in Palestine so near to their own colonies. 
In Hellenistic times, this situation was reversed, because Greeks 
and Jews were sharing the same environment in Egypt. This 
was carried so far that Hellenistic scholars actually helped the 
tradition of the Hebrew Scriptures. 


And now, at last, let us consider Euclid 13 himself. We can 
visualize very clearly his environment, the people and things 
surrounding him, but who was he himself? 

Unfortunately, our knowledge of him is very limited. This 
is not an exceptional case. Mankind remembers the tyrants, 
the successful politicians, the men of wealth, but it forgets its 
true benefactors. How much do we know about Shakespeare? 
I shall tell you all we know about Euclid, and that will not 
take very long. 

The places and dates of birth and death are unknown. He 
was probably educated in Athens and, if so, received his mathe- 
matical training at the Academy; he flourished in Alexandria 
under the first Ptolemaios and possibly under the second. Two 
anecdotes help to reveal his personality. It is said that the 
king (Ptolemaios I) asked him "if there was in geometry any 
shorter way than that of the Elements, and he answered that 
there was no royal road to geometry." This is an excellent 
story, which may not be true as far as Euclid is concerned 
but has an eternal validity. Mathematics is "no respecter of 

18 His name reads Eucleides, but it would be pedantic to use it in- 
stead of Euclid, a proper name which has attained the dignity of a com- 
mon name in our language. It is for the same reason (fear of pedantry) 
that I shall write Ptolemy when speaking of the astronomer. 



persons." The other anecdote is equally good. "Someone who 
had begun to read geometry with Euclid when he had learned 
the first theorem asked him, 'But what shall I get by learning 
these things?' Euclid called his slave and said, 'Give him an 
obol, since he must gain out of what he learns.' " 

Both anecdotes are recorded relatively late, the first by 
Proclos, the second by Stobaios, both of whom lived in the 
second half of the fifth century; they are plausible enough 
and traditions of that homely kind would be tenacious. 

Euclid was not officially connected with the Museum; 
otherwise the fact would have been recorded. But if he 
flourished in Alexandria, he was necessarily acquainted with 
the Museum and the Library. As a pure mathematician, how- 
ever, he did not need any laboratory and the manuscripts in 
his own possession might have made him independent of the 
Library. The number of manuscripts which he needed was 
not considerable; a good student might easily copy the needed 
texts during his school years. A mathematician does not need 
many collaborators; like the poet, he does his best work alone, 
very quietly. On the other hand, he may have been teaching a 
few disciples; this would have been natural and is confirmed by 
Pappos' remark that Apollonios of Perga (III-2 B.C.) was 
trained in Alexandria by Euclid's pupils. 

Euclid himself was so little known that he was confused 
for a considerable time with the philosopher, Euclid of 
Megara, 14 who had been one of Socrates' disciples (one of the 

14 1 failed to devote a special note to him in my Introduction; he is 
simply referred to in a footnote (/, 153) ; thus was an old tradition re- 
versed. For a long time, Euclid of Alexandria was overshadowed by Euclid 
of Megara; now the latter tends to be forgotten, because he is eclipsed by 
the only Euclid whom everybody knows, the mathematician. 



faithful who attended the master's death) , a friend of Plato's 
and the founder of a philosophical school in Megara. The 
confusion began very early and was confirmed by the early 
printers until late in the sixteenth century. The first to correct 
the error in an Euclidean edition was Federigo Commandino 
in his Latin translation (Pesaro, 1572). 

Euclid is thus like Homer. As everybody knows the Iliad 
and the Odyssey, so does everybody know the Elements. Who 
is Homer? He is the author of the Iliad. Who is Euclid? He is 
the author of the Elements. 

The Elements is the earliest textbook on geometry which 
has come down to us. Its importance was soon realized and 
thus the text has been transmitted to us in its integrity. It is 
divided into thirteen books, which may be described briefly 
as follows: 

Books I to VI: Plane geometry. Book I is, of course, 
fundamental; it includes the definitions and postulates and 
deals with triangles, parallels, parallelograms, etc. The con- 
tents of Book II might be called "geometrical algebra." Book 
III: Geometry of the circle. Book IV: Regular polygons. Book 
V: New theory of proportion applied to incommensurable as 
well as commensurable quantities. Book VI: Applications of 
the theory to plane geometry. 

Books VII to X: Arithmetic, theory of numbers. Numbers 
of many kinds, primes or prime to one another, least common 
multiples, numbers in geometrical progression, etc. Book X, 
which is Euclid's masterpiece, is devoted to irrational lines, all 
the lines which can be represented by an expression, such as 

wherein a and b are commensurable lines. 



Books XI-XIII: Solid geometry. Book XI is very much like 
Books I and VI extended to a third dimension. Book XII ap- 
plies the method of exhaustion to the measurement of circles, 
spheres, pyramids, etc. Book XIII deals with regular solids. 

Plato's fantastic speculations had raised the theory of 
regular polyhedra to a high level of significance. Hence, a good 
knowledge of the "Platonic bodies" 15 was considered by many 
good people as the crown of geometry. Proclos (V-2) suggested 
that Euclid was a Platonist and that he had built his geome- 
trical monument for the purpose of explaining the Platonic 
figures. That is obviously wrong. Euclid may have been a 
Platonist, of course, but he may have preferred another phi- 
losophy or he may have carefully avoided philosophical im- 
plications. The theory of regular polyhedra is the natural 
culmination of solid geometry and hence the Elements could 
not but end with it. 

It is not surprising, however, that the early geometers who 
tried to continue the Euclidean efforts devoted special atten- 
tion to the regular solids. Whatever Euclid may have thought 
of these solids "beyond mathematics" they were, especially 
for the neo-Platonists, the most fascinating items in geometry. 
Thanks to them, geometry obtained a cosmical meaning and 
a theological value. 

Two more books dealing with the regular solids were added 
to the Elements, called books XIV and XV and included in 
many editions and translations, manuscript or printed. The 
so-called "Book XIV" was composed by Hypsicles of Alex- 
andria at the beginning of the second century B.C. and is a 

15 For a discussion of the regular polyhedra and of the Platonic 
aberrations relative to them, see my History of Science (1, 438-39) . 



work of outstanding merit; the other treatise "Book XV" is of 
a much later time and inferior in quality; it was written by a 
pupil of Isidores of Miletos (the architect of Hagia Sophia, 
c. 532) . 

To return to Euclid and especially to his main work, the 
thirteen books of the Elements, when judging him, we should 
avoid two opposite mistakes which have been made repeatedly. 
The first is to speak of him as if he were the originator, the 
father of geometry. As I have already explained apropos of 
Hippocrates, the so-called "father of medicine," there are no 
unbegotten fathers except Our Father in heaven. If we take 
Egyptian and Babylonian efforts into account, as we should, 
Euclid's Elements is the climax of more than a thousand years. 
One might object that Euclid deserves to be called the father 
of geometry for another reason. Granted that many dis- 
coveries were made before him, he was the first to build a 
synthesis of all the knowledge obtained by others and himself 
and to put all the known propositions in a strong logical order. 
That statement is not absolutely true. Propositions had been 
proved before Euclid and chains of propositions established; 
moreover, "Elements" had been composed before him by Hip- 
pocrates of Chios (V B.C.), by Leon (IV-1 B.C.), and finally 
by Theudios of Magnesia (IV-2 B.C.). Theudios' treatise, with 
which Euclid was certainly familiar, had been prepared for 
the Academy, and it is probable that a similar one was in use 
in the Lyceum. At any rate, Aristotle knew Eudoxos' theory 
of proportion and the method of exhaustion, which Euclid 
expanded in Books V, VI and XII of the Elements. In short, 
whether you consider particular theorems or methods or the 
arrangement of the Elements, Euclid was seldom a complete 



innovator; he did much better and on a larger scale what other 
geometers had done before him. 

The opposite mistake is to consider Euclid as a "textbook 
maker" who invented nothing and simply put together in bet- 
ter order the discoveries of other people. It is clear that a 
schoolmaster preparing today an elementary book of geometry 
can hardly be considered a creative mathematician; he is a 
textbook maker (not a dishonorable calling, even if the pur- 
pose is more often than not purely meretricious), but Euclid 
was not. 

A good many propositions in the Elements can be ascribed 
to earlier geometers, but we may assume that those which 
cannot be ascribed to others were discovered by Euclid him- 
self; and their number is considerable. As to the arrangement, 
it is sale to assume that it is to a large extent Euclid's own. He 
created a monument which is as marvelous in its symmetry, 
inner beauty and clearness as the Parthenon, but incomparably 
more complex and more durable. 

A full proof of this bold statement cannot be given in a 
few paragraphs or in a few pages. To appreciate the richness 
and greatness of the Elements one must study them in a well 
annotated translation like Heath's. It is not possible to do 
more, here and now, than emphasize a few points. Consider 
Book I, explaining first principles, definitions, postulates, ax- 
ioms, theorems and problems. It is possible to do better at 
present, but it is almost unbelievable that anybody could have 
done as well twenty-two centuries ago. The most amazing 
part of Book I is Euclid's choice of postulates. Aristotle was, 
of course, Euclid's teacher in such matters; he had devoted 
much attention to mathematical principles, had shown the un- 



avoidability of postulates and the need for reducing them to a 
minimum; 16 yet, the choice of postulates was Euclid's. 

In particular, the choice of postulate 5 is, perhaps, his 
greatest achievement, the one which has done more than any 
other to immortalize the word "Euclidean." Let us quote it 
verbatim: 17 

... if a straight line falling on two straight lines make the 
interior angles on the same side less than two right angles, 
the two straight lines if produced indefinitely meet on that 
side on which the angles are less than two right angles." 

A person of average intelligence would say that the pro- 
position is evident and needs no proof; a better mathematician 
would realize the need of a proof and attempt to give it; it 
required extraordinary genius to realize that a proof was 
needed yet imppssible. There was no way out, then, from 
Euclid's point of view, but to accept it as a postulate and go 

The best way to measure Euclid's genius as evidenced by 
this momentous decision is to examine the consequences of it. 
The first consequence, as far as Euclid was immediately con- 
cerned, was the admirable concatenation of his Elements. The 
second was the endless attempts which mathematicians made 
to correct him; the first to make them were Greeks, like 
Ptolemy (II- 1) and Proclos (V-2) ; then Muslims, chiefly the 
Persian, Nasir al-dln al-TusI (XIII-2), the Jew, Levi ben Ger- 
son (XIV-1) , and finally "modern" mathematicians, like John 

ia Aristotle's views can be read in Heath's Euclid (1, 117 ff., 1926) or in 
his posthumous book, Mathematics in Aristotle (Oxford, Clarendon Press, 
1949; Isis 41, 329). 

17 For the Greek text and a much fuller discussion of it than can be 
given here, see Heath's Euclid (1, 202-20). See also Roberto Bonola, Non- 
Euclidean Geometry (Chicago, 1912; Horus 154). 



Wallis (1616-1703), the Jesuit father, Gerolamo Saccheri (1667- 
1733) , of San Remo in his Euclides ab omni naevo vindicatus 
(1733), the Swiss, 18 Johann Heinrich Lambert (1728-77), and 
the Frenchman, Adrien Marie Legendre (1752-1833). The 
list could be lengthened considerably, but these names suffice, 
because they are the names of illustrious mathematicians repre- 
senting many countries and many ages, down to the middle 
of the last century. The third consequence is illustrated by the 
list of alternatives to the fifth postulate. Some bright men 
thought that they could rid themselves of the postulate and 
succeeded in doing so, but at the cost of introducing another 
one (explicit or implicit) equivalent to it. For example, 

"If a straight line intersects one of two parallels, it will 
intersect the other also." (Proclos) 

"Given any figure there exists a figure similar to it of 
any size." (John Wallis) 

"Through a given point only one parallel can be drawn 
to a given straight line." (John Playfair) 

"There exists a triangle in which the sum of the three 
angles is equal to two right angles." (Legendre) 

"Given any three points not in a straight line there ex- 
ists a circle passing through them." (Legendre) 

"If I could prove that a rectilinear triangle is possible 
the content of which is greater than any given area, I 

would be in a position to prove perfectly rigorously the 
whole of geometry." (Gauss, 1799). 

All these men proved that the fifth postulate is not neces- 
sary if one accepts another postulate rendering the same ser- 

18 Yes, Swiss (Isis 40, 139). 


vice. The acceptance of any of those alternatives (those 
quoted above and many others) would, however, increase the 
difficulty of geometrical teaching; the use of some of them 
would seem very artificial and would discourage young stu- 
dents. It is clear that a simple exposition is preferable to one 
which is more difficult; the setting up of avoidable hurdles 
would prove the teacher's cleverness and his lack of common 
sense. Thanks to his genius, Euclid saw the necessity of this 
postulate and selected intuitively the simplest form of it. 
There were also many mathematicians who were so blind that 
they rejected the fifth postulate without realizing that an- 
other was taking its place. They kicked one postulate out of 
the door and another came in through the window without 
their being aware of it! 

The fourth consequence, and the most remarkable, was 
the creation of non-Euclidean geometries. The initiators have 
already been named, Saccheri, Lambert, Gauss. Inasmuch as 
the fifth postulate cannot be proved, we are not obliged to 
accept it, and if so, let us deliberately reject it. The first to 
build a new geometry on an opposite postulate was a Russian, 
Nikolai Ivanovich Lobachevskii (1793-1850), who assumed 
that through a given point more than one parallel can be 
drawn to a given straight line or that the sum of the angles 
of a triangle is less than two right angles. The discovery of a 
non-Euclidean geometry was made at about the same time 
by a Transylvanian, Janos Bolyai (1802-60). Sometime later, 
another geometry was outlined by a German, Bernhard Rie- 
mann (1826-66), who was not acquainted with the writings 
of Lobachevskii and B61yai and made radically new assump- 
tions. In Riemann's geometry, there are no parallel lines and 
the sum of the angles of a triangle is greater than two right 



angles. The great mathematical teacher Felix Klein (1847- 
1 925) showed the relationship of all those geometries. Euclid's 
geometry refers to a surface of zero curvature, in between Rie- 
mann's geometry on a surface of positive curvature (like the 
sphere) and Lobachevskii's applying to a surface of negative 
curvature. To put it more briefly, he called the Euclid geome- 
try parabolic, because it is the limit of elliptic (Riemann's) 
geometry on one side and of the hyperbolic (Lobachevskii's) 
geometry on the other. 

It would be foolish to give credit to Euclid for pan- 
geometrical conceptions; the idea of a geometry different from 
the common-sense one never occurred to his mind. Yet, when 
he stated the fifth postulate, he stood at the parting of the ways. 
His subconscious prescience is astounding. There is nothing 
comparable to it in the whole history of science. 

It would be unwise to claim too much for Euclid. The fact 
that he put at the beginning of the Elements a relatively small 
number of postulates is very remarkable, especially when one 
considers the early date, say, 300 B.C., but he could not 
fathom the depths of postulational thinking any more than he 
could fathom those of non-Euclidian geometry. Yet he was 
the distant forerunner of David Hilbert (1862-1911) even as he 
was Lobachevskii's spiritual ancestor. 19 

Enough has been said of Euclid the geometer, but one 
must not overlook other aspects of his genius as mathematician 
and physicist. To begin with, the Elements does not deal 

19 For details, see Florian Cajori, History of Mathematics (2nd ed., 
326-28, 1919); Cassius Jackson Keyser, The Rational and the Superrational 
(pp. 136-44, New York, Scripta Mathematica, 1952; Isis 44, 171) . 



simply with geometry but also with algebra and the theory o 

Book II might be called a treatise on geometrical algebra. 
Algebraical problems are stated in geometrical terms and 
solved by geometrical methods. For example, the product of 
two numbers a, b is represented by the rectangle whose sides 
have the lengths a and b\ the extraction of a square root is 
reduced to the finding of a square equal to a given rectangle, 
etc. The distributive and commutative laws of algebra are 
proved geometrically. Various identities, even complicated 
ones, are presented by Euclid in a purely geometrical form. 

or (a + b) 2 + (b - a) 2 = 2 (a 2 + 6 2 ) 

This might seem to be a step backward as compared with 
the methods of Babylonian algebra, and one wonders how 
that could happen. It is highly probable that the clumsy 
symbolism of Greek numeration was the fundamental cause 
of that regression; it was easier to handle lines than Greek 

At any rate, Babylonian algebraists were not acquainted 
with irrational quantities, while Book X of the Elements (the 
largest of the thirteen books, even larger than Book I) is de- 
voted exclusively to them. Here again, Euclid was building 
on older foundations, but this time the foundations were 
purely Greek. We may believe the story ascribing recognition 
of irrational quantitites to early Pythagoreans, and Plato's 
friend Theaitetos (IV-1 B.C.) gave a comprehensive theory of 
them as well as of the five regular solids. There is no better 



illustration of the Greek mathematical genius (as opposed tc 
the Babyonian one) than the theory of irrationals as explained 
by Hippasos of Mctapontion, Theodoros of Gyrene, Theaitetos 
of Athens and finally by Euclid. 20 It is impossible to say just 
how much of Book X was created by Theaitetos and how much 
by Euclid himself. We have no choice but to consider that 
book as an essential part of the Elements, irrespective of its 
origin. It is divided into three parts, each of which is pre- 
ceded by a group of definitions. A number of propositions 
deal with surds in general, but the bulk of the book investi- 
ga'es the complex irrationals which we would represent by the 

V (Va V&) 

wherein a and b are commensurable quantities. These irra- 
tionals are divided into twenty-five species, each of which is 
discussed separately. As Euclid did not use algebraical sym- 
bols, he adopted geometrical representations for these quanti- 
ties and his discussion of them was geometrical. Book X was 
much admired, especially by Arabic mathematicians; it re- 
mains a great achievement but is practically obsolete, for such 
discussions are futile from the point of view of modern 

Books VII to IX of the Elements might be called the first 
treatise on the theory of numbers, one of the most abstruse 
branches of the mathematical tree. It would be impossible to 
summarize their contents, for the summary would be almost 

20 For Hippasos', Theodores', and Theaitetos' contributions see my 
History of Science (pp. 282-85, 437) . 



meaningless unless it covered a good many pages. 21 Let me 
just say that Book VII begins with a list of twenty-two defini- 
tions, which are comparable to the geometrical definitions 
placed at the beginning of Book I. Euclid sets forth a series 
of proportions concerning the divisibility of numbers, even and 
odd numbers, squares and cubes, prime and perfect numbers, 

Let us give two examples. In IX, 36 he proves that if 
p=l + 2 + . . . -f 2 n is prime, 2 n p is perfect (that is, equal to 
the sum of its divisors). In IX, 20 we are given an excellent 
demonstration that the number of prime numbers is infinite. 

The demonstration is so simple and our intuitive feeling 
ad hoc so strong that one would readily accept other proposi- 
tions of the same kind. For example, there are many prime 
pairs, that is, prime numbers packed as closely as possible 
(2n + 1, 2n + 3, both primes, e.g., 11, 13; 17, 19; 41, 43) . As 
one proceeds in the series of numbers, prime pairs become 
rarer and rarer, yet one can hardly escape the feeling that the 
number of prime pairs is infinite. The proof of that is so 
difficult, however, that it has not yet been completed. 22 

In this field again, Euclid was an outstanding innovator, 
and the few mathematicians of our own days who are trying 
to cultivate it recognize him as their master. 

21 The Greek text of Books VII to IX covers 116 pages in Heiberg's 
edition (2, Leipzig, 1884) and the English translation with notes, J50 pages 
in Heath's second volume. 

32 Charles N. Moore offered a proof in 1944, but that proof was shown 
to be insufficient (Horus 62). The incredible complexity of the theory of 
numbers can be appreciated by looking at its History written by Leonard 
Eugene Dickson (3 vols., Carnegie Institution, 1919-23; Isis 3, 446-48; 4, 
107-08: 6, 96-98). For prime pairs, see Dickson, 7, 353, 425, 438. 



Thus far, we have spoken only of the Elements, but Euclid 
wrote many other works, some of which are lost; these works 
deal not only with geometry but also with astronomy, physics, 
and music. The genuineness of some of those treatises is 
doubtful. For example, two treatises on optics are ascribed to 
him, the Optics and the Catoptrics. 23 The first is genuine, the 
second probably apocryphal. We have the text of the Optics, 
and we have also a review of both treatises by Theon of Alex- 
andria (IV-2). The Optics begins with definitions or rather 
assumptions, derived from the Pythagorean theory that the 
r?ys of light are straight lines and proceed from the eye. Euclid 
then explains problems of perspective. The Catoptrics deals 
with mirrors and sets forth the law of reflection. It is a 
remarkable chapter of mathematical physics which remained 
almost alone of its kind for a very long period of time, but 
does it date from the third century B.C., or is it later, much 

The tradition concerned with the fifth postulate has al- 
ready been referred to; it can be traced from the time of the 
Elements to our own. That is only a small part of it, however. 
The Euclidean tradition, even if restricted to mathematics, is 
remarkable for its continuity and the greatness of many of its 
bearers. The ancient tradition includes such men as Pappos 
(III-2) , Theon of Alexandria (IV-2) , Proclos (V-2), Marines of 

" French translation by Paul Ver Eecke, l/Optitjiie ct la Catoptriqne 
(Bruges, 1938; Isis 30, 520-21). This includes French versions of the 
Catoptrics and of the two texts of the Optics, the original one and the 
ene edited by Theon of Alexandria (IV-2). English translation of the 
original text of the Optics by Harry Edwin Burton (Journal of the Optical 
Society of America 3$ [1945], 357-72). 



Sichem (V-2), Simplicios (VI- 1). It was wholly Greek. Some 
Western scholars, such as Censorinus (1II-1) and Boethius 

(VI- 1) , translated parts of the Elements from Greek into Latin, 
but very little remains of their efforts and one cannot speak 
of any complete translation, or of any one covering a large 
part of the Elements. There is much worse to be said; various 
manuscripts circulated in the West until as late as the twelfth 
century which contained only the propositions of Euclid 
without demonstrations. 24 The story was spread that Euclid 
himself had given no proofs and that these had been supplied 
only seven centuries later by Theon of Alexandria (1V-2). One 
could not find a better example of incomprehension, for if 
Euclid had not known the proofs of his theorems he would not 
have been able to put them in a logical order. That order is 
the very essence and the greatness of the Elements, but 
medieval scholars did not see it, or at least did not see it until 
their eyes had been opened by Muslim commentators. 

The Muslim study of Euclid was begun by al-Kindi (IX- 1) , 
if not before (al-Kiridi's interest, however, was centered upon 
the optics, and in mathematics it extended to non-Euclidean 
topics such as Hindu numerals), and Muhammad ibn Musa 

(IX- 1). The Elements were first translated into Arabic by 
al-Hajjaj ibn Yusuf (IX- 1); he made a first translation for 
Harun al-Rashid (caliph, 786-809), then revised it for al- 
Ma'mun (caliph, 813-33) . During the 250 years which fol- 
lowed, the Muslim mathematicians kept very close to Euclid, 
the algebraist as well as the geometer, and published other 
translations and many commentaries. Before the end of the 
ninth century, Euclid was translated and discussed in Arabic 

24 Greek and Latin editions of the propositions only, without proofs, 
were printed from 1547 to as late as 1587. 



by al-Mahanl, al-Nairlzi, Thabit ibn Qurra, Ishaq ibn Hunain, 
Qusta ibn Luqa. A great step forward was made in the first 
quarter of the tenth century by Abu 'Uthman Sa'Id ibn 
Ya'qub al-Dimishqi, who translated Book X with Pappos' com- 
mentary (the Greek of which is lost) , 25 This translation in- 
creased Arabic interest in the contents of Book X (classification 
of incommensurable lines), as witnessed by the new translation 

of Nazlf ibn Yumn (X-2), a Christian priest, and by the com- 
mentaries of Abu Ja't'ar al-Khazin (X-2) and Muhammad ibn 

'Abd al-Baqi al-Baghdadi (XI-2). My Arabic list is long yet 
very incomplete, because we must assume that every Arabic 
mathematician of this age was acquainted with the Elements 
and discussed Euclid. For example, Abu-1-Wafa' (X-2) is said 
to have written a commentary on Euclid which is lost. 

We may now interrupt the Arabic story and return to the 
West. Western efforts to translate the Elements directly from 
Greek into Latin had been ineffective; it is probable that their 
knowledge of Greek diminished and dwindled almost to noth- 
ing at the very time when their interest in Euclid was in- 
creasing. Translators from the Arabic were beginning to ap- 
pear and these were bound to come across Euclidean manu- 
scripts. Efforts to Latinize these were made by Hermann the 
Dalmatian (XII-1), John O'Creat (XII-1) , and Gerard of 
Cremona (XII-2) , but there is no reason to believe that the 
translation was completed, except by Adelard of Bath (XII- 
l). 26 However, the Latin climate was not so favorable to 


The Arabic text of Abu 'Uthman was edited and Englished by Wil- 
liam Thomson, with mathematical introduction by Gustav Junge (Harvard 
Semitic Series 8, Cambridge, 1930; Isis 16, 132-36). 

26 The story is simplified for the sake of brevity; for details, see Marshall 
Clagett, "The medieval Latin translations from the Arabic of the Elements 



geometrical research in the twelfth century as the Arabic cli- 
mate had proved to be from the ninth century on. Indeed, we 
have to wait until the beginning of the thirteenth century to 
witness a Latin revival of the Euclidean genius, and we owe 
that revival to Leonardo of Pisa (XI1I-1), better known under 
the name of Fibonacci. In his Practica gcometriae, written in 
1220, Fibonacci did not continue the Elements, however, but 
another Euclidean work on the Divisions of figures, which is 
lost. 27 

In the meanwhile, the Hebrew tradition was begun by 
Judah ben Solomon ha-Kohen (XIII-1) and continued by 
Moses ibn Tibbon (XIII-2), Jacob ben Mahir ibn Tibbon 
(XIII-2), and Levi ben Gerson (XIV-1) . The Syriac tradition 
was illustrated by Abu-1-Faraj, called Barhebraeus (XIII-2), 
who lectured on Euclid at the observatory of Maragha in 1268; 
unfortunately, the beginning of the Syriac tradition was also 
the end of it, because Abu-1-Faraj was the last Syriac writer 
of importance; after his death, Syriac was gradually replaced 
by Arabic. 

The golden age of Arabic science was also on the wane, 
though there remained a few illustrious Euclidians in the 
thirteenth century, like Qaiar ibn abl-l-Qasim (XIII-1), Ibn 
al-Lubudl (XIII-1), Nasjr al-dm al-Tusi (XIII-2), Muhylal-dm 
al-Maghribi (XIII-2), Qutb al-din al-Shlrazi (XIII-2), and even 
some in the fourteenth century. We may overlook the late Mus- 

with special empasis on the versions of Adelard of Bath" (Isis 44, 16-42, 
1953) . 

27 The text of that little treatise peri diaireseon was restored as far as 
possible by Raymond Clare Archibald on the basis of Leonardo's Practica 
and of an Arabic translation (Intro, 1, 154-55). 



lim and Jewish mathematicians, however, for the main river 
was now flowing in the West. 

Adelard's Latin text was revised by Giovanni Campano 
(XIII-2) and Campano's revision was immortalized in the 
earliest printed edition of the Elements (Venice, 1482). The 
first edition of the Greek text was printed in Basel, 1533, and 
the princeps of the Arabic text, as edited by Nair al-dm al- 
Tusi, was published in Rome in 1594. 

The rest of the story need not be told here. The list of 
Euclidean editions which began in 1482 and is not ended yet 
is immense, and the history of the Euclidean tradition is an 
essential part of the history of geometry. 

As far as elementary geometry is concerned, the Elements of 
Euclid is the only example of a textbook which has remained 
serviceable until our own days. Twenty-two centuries of 
changes, wars, revolutions, catastrophies of every kind, yet it 
still is possible and profitable to study geometry in Euclid! 


Standard edition of the Greek text of all the works, with 
Latin versions, Euclidis opera omnia ediderunt J. L. Heiberg 
et H. Menge (8 vols., Leipzig, 1883-1916; supplement 1899) . 

Sir Thomas L. Heath. Euclid's Elements in English (3 
vols., Cambridge, 1908), revised edition (3 vols., 1926; I sis 10, 
60-62) . 

Charles Thomas-Stanford. Early Editions of Euclid's Ele- 
ments (64 pp., 13 pi., London, 1926; Isis 10, 59-60). 



(second century A.D.) 



.GNORANT people think of "antiquity" or of 
the "Middle Ages" as if each of these periods were something 
homogeneous and unchanging, and they would put everything 
concerning ancient science (or medieval science) in a single 
box, just as if all these things were of the selfsame kind. That 
is very silly. The one thing which one might concede is that 
the change is faster now than it was in the past, but much of 
the increasing speed is superficial. 

What we call classical antiquity, if we count it from Homer 
to Damascios, is a period of about fourteen centuries; if we 
count the length of American civilization in the same way 



(that is, leaving out, in both cases, the prehistoric times which 
are ageless), it has lasted about four centuries. Thus, the first 
of these periods is more than thrice longer than the second. 
And yet should one put the whole of American culture in a 
single bag, as if the whole of it were the same kind of biscuit? 
Certainly not. 

There was incredible variety in ancient times, even within 
a single century, but there were also traditions which continued 
across the ages and are very helpful to us as guiding threads. 
For example, from Euclid's time on, there appeared in each 
century some mathematicians who continued Euclid's ideas 
or discussed them. 

By the second century after Christ more than three cen- 
turies had elapsed since the beginning of the Hellenistic Age, 
and the world was exceedingly different from what it had been. 
The dilfercnce was not due so much to Christianity, which was 
still unfelt except by a small minority of people, and re- 
mained inoperative as an influence. The philosophical climate 
continued to be dominated by Stoicism. The political world, 
however, was absolutely different. 


Let us consider a little more closely the world in which 
Ptolemy lived. It is probable that he was born in Egypt and 
flourished in Alexandria, but Egypt had been a Roman pro- 
vince since 30 B.C. The Greek chaos and the wars between 
Alexander's successors had been finally ended by Roman 
power. That new world was very imperfect in many ways 
but, for the first time in many centuries, there was a modicum 
of international order, law and peace. The second century was 
the end of the golden age of the Roman empire; it was de- 



cidedly the golden age of Roman science, but the best of 
Roman science was really Greek. 

It was Ptolemy's privilege to live under some of the best 
emperors, the Spaniard Trajan (ruled 98-1 17), who built roads, 
libraries, bridges across the Danube and the Tagus; Hadrian 
(ruled 117-38), also a great builder in Athens, Rome and 
Tivoli; Antoninus Pius (ruled 138-61) and, perhaps, Marcus 
Aurelius (ruled 161-80) ; these last two, not only great em- 
perors, but good men. When one speaks of "pax romana," one 
has in mind chiefly the forty-four years covered by the rules 
of Hadrian and Antoninus, and apropos of the rules of 
Antoninus and Marcus Aureiius, a stretch of almost equal 
duration, Gibbon declared: "Their united reigns are possibly 
the only period of history in which the happiness of a gieat 
people was the sole object of government." 1 

The most significant thing about the Roman empire, from 
the intellectual point of view, was its bilingualism. Every 
educated man in the West was supposed to know two langu- 
ages, Greek as well as Latin. By this time, the second century 
after Christ, the golden age of Latin literature was already 
past and yet the top culture of the West was Greek, not Latin. 
Greek was the language of science and philosophy; Latin the 
language of law, administration and business. Hadrian knew 
Greek very well and had created in Rome a college of arts 
which was called Athenaeum, 2 in honor of the goddess, 

1 Decline and Fall, chap. 3. In Bury's illustrated edition, 1, 84. 

2 The name Athenaeum has become a common name in almost every 
European language. Every government high school in Belgium is called 
athene'e. In English and other languages the word is used to designate 
a literary or scientific association or club. It is one of the words which 
remind us every day of our debt to antiquity, the others being academy, 
lyceum, museum. 



Athene, the city of Athens (which Hadrian loved) and Greek 
culture. Marcus Aurelius wrote his famous Meditations in 
Greek. In spite of the prestige attained by such writers as 
Lucretius, Cicero, Virgil and Seneca, and of the scientific books 
written in Latin by Vitruvius, Celsus, Frontinus and Pliny, the 
language of science was still predominantly Greek. It is true 
that the two greatest scientists of the age were born in the 
Orient, Ptolemy in Egypt and Galen in the province of Asia, 
and neither of them would have been able to write in Latin, 
even if they had wished to do so. But why should anyone 
write artificially in an inferior language, if he was able to 
write naturally in a superior one? 

Any Roman of the second century who was intellectually 
ambitious had to learn Greek; the result was obtained chiefly 
with the help of Greek tutors or by years of "graduate study" 
in Athens, Alexandria or any other city in the eastern pro- 
vinces. The situation can be compared to another closer to 
us. When Frederick the Great was king of Prussia (1740-86), 
he would speak German to his soldiers or servants, but French 
was the language of polite conversation; memoirs sent to the 
Berlin Academy had to be written in French or Latin, not in 
German, to be published. 

The world in which Ptolemy lived was a Roman world, 
whose intellectual ideas were still predominantly Greek. 


The two outsanding men of science of the second century 
were Ptolemy, in the first half, and Galen in the second. They 
were two giants of the most genuine kind; the kind of giants 
who do not become smaller as the centuries pass but greater 
and greater. One cannot consider Ptolemy without evoking 



his predecessor, Hipparchos of Nicaia, who flourished in the 
Hellenistic Age, 3 almost three centuries before him. It is 
strange to think of two men separated by so large a barrier- 
three centuriesyet working as if the second were the im- 
mediate disciple of the first. 

Hipparchos' works are lost, and it is possible that their 
loss was partly the result of the fact that Ptolemy's great book 
superseded them and made them superfluous. In some in- 
stances, Ptolemy's debt to his predecessor is acknowledged or 
is made clear in other ways. What we know of Hipparchos we 
know almost exclusively from Ptolemy, who quotes him often, 
sometimes verbatim. 4 Nevertheless, in the majority of cases, it 
is impossible to say whether the real inventor was the older 
or the younger man. 

In what follows we shall not bother too much about that, 
and Ptolemy's achievements will be described as if they were 
exclusively or mainly his own. After all, that is the method 
which one cannot help following in discussing the achieve- 
ments of almost any ancient scientist. 

Euclid is mainly known as a mathematician, and his fame 
is based upon the Elements; Ptolemy's personality was far 
more complex and two of his books, the Almagest and the 
Geography, remained standard textbooks in their fields for at 
least fourteen centuries. 

The comparison of Ptolemy with Euclid is a very useful 
one, because the fact that their books superseded earlier ones 
was essentially due to the same causes. Ptolemy, like Euclid, 

8 Hipparchos flourished in Rhodes from 146 to 127 and perhaps also 
from 161 to 146 in Alexandria. 

4 See the index nominum in Heiberg's edition (1907) , 3 (called II), 
pp. 275-77. 



was an excellent expositor or teacher; while their predecessors 
had written monographs or short treatises, they wrote very 
large ones of encyclopaedic nature and did it in the best order 
and with perfect lucidity. Both men combined an extra- 
ordinary power of synthesis and exposition with genius of the 
highest potential. The earlier treatises which had been the 
foundation of their own were soon judged to be incomplete 
and obsolete and the scribes ceased to copy them; thus, they 
were not only superseded but dropped out of existence. 


It is tempting to compare Ptolemy with Euclid, two giants 
who shared the distinction of composing textbooks which 
would remain standard books in their respective fields for 
more than a thousand years. They are singularly alike in their 
greatness and in their loneliness. We know their works ex- 
ceedingly well, but they themselves are practically unknown. 

Ptolemy's biography is as empty as Euclid's. We do not 
even know when and where he was born and died. It has 
been said, very late (fourteenth century) that he was born in 
Ptolemai's Hermeiu, a Greek city of the Thebais. 5 That is 
possible. He was probably a Greek Egyptian or an Egyptian 
Greek; he made astronomical observations in Alexandria or 
in Canopos nearby from 127 to 151 (or 141?); according to 
an Arabic story, he lived to be seventy-eight; according to Suidas 
(X-2), he was still alive under Marcus Aurelius (emperor 161- 
80); we may conclude that he was probably born at the end of 
the first century. 

B Upper Egypt, he ano chora. Ptolemais Hermeiu was on the site of 
the Egyptian village al-Minshah. 



As to his character, we have a glimpse of it in the Prooimion 

(or preface) to the Almagest, addressed to his friend Syros. 6 

That preface is a noble defense of mathematics and especially 

of celestial mechanics. Another glimpse, indirect, is given in 

an early epigram: 

I know that I am mortal and ephemeral, but when 1 
scan the crowded circling spirals of the stars 1 do no longer 
touch the earth with my feet, but side by side with Zeus I 
take my fill of ambrosia, the food of the gods. 

This epigram is included in the Greek Anthology (IX, 
577) , bearing Ptolemy's name; this does not prove Ptolemy's 
authorship but is a good witness of him, like a portrait. The 
poet saw him as a man lifted up far above other men by his 
lofty purpose and equanimity. 


Out of many books of his, and of his two great classics, 
the best known is the Almagest. Its curious name will be ex- 
plained later when we discuss the Ptolemaic tradition. At pre- 
sent, let us take it for granted as most people do. The original 
Greek title he tnathcmatike syntaxis means the Mathematical 
Synthesis. It was really a treatise of astronomy but astronomy 
was then a branch of mathematics; one is reminded of another 
classic which was published more than eighteen centuries later, 
Newton's Mathematical Principles of Natural Philosophy. 

Ptolemy's astronomy, like Hipparchos', was based upon 
observations, his own and those of Greek and Babylonian pre- 

a Syros, otherwise unknown, must have been a very good friend of 
Ptolemy, for the latter appeals to him thrice, "o Syre," at the beginnings 
of Books I and VII and at the end of Book XIII; that is the beginning, 
the middle and the very end of the Almagest. 



decessors. Hipparchos had used various instruments, e.g., a 
celestial sphere and an improved diopter, and Ptolemy had 
perhaps added new instruments or improved the older ones. 
In this case, as in most cases, it is impossible to separate the 
achievements of both men and to say whether the meridian 
circle, the astrolabon organon, the parallactic instrument and 
the mural quadrant were invented by Ptolemy or improved 
by him or completely invented by Hipparchos. The history 
of instruments, we should remember, is one of the best ap- 
proaches to the understanding of scientific progress, but it is 
full of difficulties; each instrument is developed gradually; 
none is created in one time for all time by a single man. 7 Their 
main task, however, as they undersood it, was not so much 
the taking and recording of observations, but the mathematical 
explanation of the facts which those observations revealed, 
and their synthesis. Therefore, the Almagest of Ptolemy, like 
the Principia of Newton, was primarily a mathematical book 
and its original title, Mathematical Syntaxis, was adequate. 

The Almagest is divided into thirteen books. The first two 
are introductory, explaining astronomical assumptions and 
mathematical methods. Ptolemy proves the sphericity of the 
Earth and postulates the sphericity of the heavens and their 
revolution around the Earth immobile in the center. He dis- 
cusses and redetermines the obliquity of the ecliptic. The main 
mathematical method is trigonometry, for Ptolemy realized 
that spherical geometry and graphical means were incon- 
venient and insufficient. In this he was not independent of 

7 For general considerations on instruments, see Maurice Daumas, Les 
instruments scientifiques aux XVII* et XVIHe sitcles (Paris, 1953: Isis 44, 
391). Daumas deals with late instruments, but many of his remarks apply 
just as well to the ancient ones. 



Hipparchos but, in addition, he was privileged to stand upon 
the shoulders of Menelaos of Alexandria. 

The trigonometry is explained in chapters numbered 11 
and 13 in Heiberg's edition. Every distance on the sphere is 
an angular one; the measurement of angles is replaced by the 
consideration of the chords subtending the corresponding 
arcs. 8 The circle is divided into 360 and the diameter into 
120 parts. Ptolemy used sexagesimal numbers in order to 
avoid the embarrassment of fractions (that is the way he put 
it, Almagest I, 10). Thus, each of the 60 parts of the radius 
was divided into sixty small parts, and these again were 
divided into sixty smaller ones. 9 A table of chords was com- 
puted for every half degree, from to ISO , 10 each chord be- 
ing expressed in parts of the radius, minutes and seconds. The 
size of some chords (sides of regular polygons) could be de- 
rived easily from Euclid; the size of others was obtained, thanks 

8 Later on, Arabic astronomers inspired by Hindu ones replaced the 
chords by sines and other ratios, but the purpose of Ptolemaic (Hipp- 
archian) trigonometry was the same as ours. Assuming the radius to be 
the unit, 

chord a = 2 sin (a/2) 
sin a (i/ 2 ) chord (2a) . 

9 In Latin, the small parts were called partes minutae primae, and the 
smaller ones, partes minutae secundae. Our words minutes and seconds 
were stupidly derived from the first adjective in the first expression and 
from the second in the second. 

10 Ptolemy's table of chords, as given in the Almagest (I, 11), is thus 
like a table of sines for every quarter degree from 1 to 90. The sines 
which could be obtained from his table would be correct to 5 places. The 
table allowed him to determine pi with remarkable precision. Let us 
assume that the length of the circumference is very close to 360 times the 
chord of 1, each of which measures 1 part 2'50". Pi is the ratio of ihe 
circumference to the diameter, or 360/120 (1 part 2'50") = 3 parts 8'30" 
= 3.14166 (real value 3.14159...). 



to Ptolemy's theorem about quadrilaterals inscribed in a circle; 
that theorem enabled one to find the chord of a sum of angles. 
Opposite the value of each chord in the table is given 1/30 of 
the excess of that chord over the preceding one; this 1/30 is 
expressed in minutes, seconds and thirds; this would enable 
one to compute the chords for every minute of angle. Ptolemy 
understood the meaning of interpolations and approximations; 
his correct appreciation of them was one of the bases of applied 

The table of chords is followed by a geometrical argument 
leading to the calculation of the relations of arcs of the equa- 
tor, ecliptic, horizon and meridian, and tables ad hoc. The 
same kind of discussion is continued in Book II with reference 
to the length of the longest day at a given latitude. 

Book III deals with the length of the year and the motion 
of the Sun, Ptolemy using epicycles and eccentrics (the first of 
which certainly and the second probably invented by Apol- 
lonios of Perga, II 1-2 B.C.) . 

Book IV. Length of the month and theory of the Moon. 
This contained what is supposed to be one of his discoveries 
(as distinguished from those of Hipparchos), the second in- 
equality of the Moon called evection. He fixed the amount of 
it at 119'30", and accounted for it in terms of eccentrics and 
epicycles and of a small oscillation (prosneusis) of the epicycle. 
This is a good example of mathematical ingenuity. 11 

Book V. Construction of the astrolabe. Theory of the Moon 

11 The evection, caused by the Sun's attraction, depends upon the 

alternate increase and decrease of the eccentricity of the Moon's orbit; 

the eccentricity is maximum when the Sun is crossing the line of the apses 

(syzygies) and minimum at the quadratures. The value of the evection 

is about 1 15', and its period, about li/& year. 



ontinued. Diameters of the Sun, Moon, Earth's shadow, dis- 
ance of the Sun, dimensions of the Sun, Moon and Earth. 

Book VI. Solar and lunar eclipses. 

Book VII- VIII. Stars. Precession of the equinoxes. The 
able of stars covers the end of VII and the beginning of VIII. 
The rest of VIII describes the Milky Way and the construction 
>f a celestial globe. 

Books IX-XIII. Planetary motions. This is perhaps the 
nost original part of the Almagest, because Hipparchos had 
lot been able to complete his own synthesis of planetary sys- 
ems. Book IX deals with generalities, such as the order of the 
>lanets according to their distances from the Earth and periods 
>f revolution; then with Mercury. Book X Venus; XI Jupiter 
nd Saturn; XII Stationary points and retrogressions, greatest 
longations of Mercury and Venus; XIII Motions of planets 
n latitude, inclinations and magnitudes of their orbits. 

In short, the Almagest was a survey of the astronomical 
Lnowledge available about 150 A.D., and that knowledge was 
lot essentially different from that attained in 150 B.C. It is 
mpossible to discuss the details of it without discussing the 
vhole of ancient astronomy. Let us consider a few points. 

First the Almagest defined what we call the "Ptolemaic 
ystem," that is, the solar system centered upon the Earth. 
Allowing Hipparchos, Ptolemy rejected the ideas of Arist- 
irchos of Samos (III-l B.C.) , who had anticipated the Coper- 
lican system; Hipparchos and Ptolemy rejected those ideas 12 
>ecause they did not tally sufficiently well with the observa- 
ions. Their objections were of the same nature as Tycho 

12 They even rejected the geoheliocentric system of Heracleids of 
'ontos (IV-2 B.C.). The Ptolemaic system was completely geocentrical. 



Brahc's at the end of the sixteenth century; a sufficient agree- 
ment between observations and the Aristarchian or Copernican 
ideas became possible only when Kepler replaced circular 
trajectories by elliptic ones (1609). The methodic excellence 
of the Almagest caused the supremacy of the Ptolemaic system 
until the sixteenth century, in spite of abundant criticisms 
which became more and more acute as observations increased 
in number and precision. 

One might say that Hipparchos and Ptolemy were back- 
ward in two respects, because they rejected the heliocentrical 
Ideas of Aristarchos and the ellipses of Apollonios, but such 
a conclusion would be very unfair. Men of science are not 
prophets; they see a little further than other men but can 
never completely shake off the prejudices of their own en- 
vironment. As heliocentricity did not lead to greater simplicity 
or precision, their rejection of it was justified. 

The Catalogue of Stars is the earliest catalogue which has 
come down to us. It includes 1,028 stars and gives the longi- 
tude, latitude and magnitude of each. It was largely derived 
from Hipparchos' 13 catalogue of c. 130 B.C.; Ptolemy left the 
latitudes unchanged but added 240' to every longitude in 
order to take the precession into account. The precession of 
the equinoxes had been discovered by Hipparchos on the basis 
of earlier observations, Babylonian and Greek. The precession 
amounts to little more than one degree per century; 14 con- 

18 Hipparchos had listed not many more than 850 stars giving the 
latitude, longitude and magnitude of each. 

14 Hipparchos assumed that the precession amounted to 45" or 46" 
a year, which would add up to l.3 in a century; Ptolemy corrected that 
to 36", which is exactly 1 a century. The real value is 50."25, equivalent 
to l.4 a century. Hipparchos was closer to the truth than Ptolemy. 



sidering the observational means of the ancient astronomers, 
it is clear that they could not discover it without the knowledge 
of stellar longitudes antedating their own by many centuries. 

Before abandoning Ptolemaic astronomy, a few words 
must be said of the methods of projection, orthographic and 
stereographic, in spite of the fact that they are not explained 
in the Almagest but in separate monographs. 15 It is possible 
that both methods were invented by Hipparchos; at any rate, 
Ptolemy's explanation of them is the earliest available. 

Both methods were needed to solve a fundamental problem, 
the representation of points or arcs of the spherical surface 
of heaven 16 upon a plane (or map) . In the Analemma 
method, the points and arcs were projected orthogonally 
upon three planes mutually at right angles, the meridian, 
hori/on and prime vertical; this method was used chiefly to 
find the position of the Sun at a given hour. The method of 
the Planisphaerium was what is now called stereographic pro- 
jection. Every point of the sphere is represented by its pro- 
jection upon the equator from the opposite pole (the northern 
hemisphere was projected by Ptolemy from the south pole). 
This particular system of projection had very remarkable and 
useful properties, of which Ptolemy was aware though he did 


The orthogonal projection is explained in the Analemma (meaning 
taking-up, Aufnahme, also sundial), and the stereographic in the Plani- 
sphaerium, both lost in Greek but preserved in Latin translations from 
the Arabic. Latest editions, by J. L. Heiberg, in the Ptolemaei Opera (2, 
187-223, 225-59, 1907). The second was translated into German by }. 
Drecker (Isis 9, 255-78, 1927), who summarized the tradition of the 
Planisphaerium in his preface. 

16 All the stars and planets were supposed, for geometrical purposes, to 
move on a single sphere. That was all right; if a star was not on the 
sphere, its central projection on it was considered; the angular distances 
remained the same. 



not give general proofs of them. The projection of all circles 
are circles (with the apparent exception of circles passing 
through the pole which are projected as straight lines). The 
stereographic projection is the only one which is both con- 
formal and perspective, 17 Ptolemy could not have known that 
unicity, but he had made a good study of projections and was 


Ptolemy's geographical treatise or guide (geographice 
hyphegesis) is almost as important as the Almagest. It covered 
the whole of mathematical geography, just as the Almagest 
covered the whole of mathematical astronomy, and it in- 
fluenced geography as deeply and as long as the Almagest in- 
fluenced astronomy. During fourteen centuries, at least, the 
Almagest was the standard book, or call it the Bible, of 
astronomy, while the Geography was the Bible of geography. 
The name Ptolemy meant geography to geographers and 
astronomy to astronomers. 

The Geography was composed after the Almagest, say, after 
150. It was divided into eight books and was restricted to 
mathematical geography and to all the information needed for 
the drawing of accurate maps. His knowledge was derived 
mainly from Eratosthenes, Hipparchos, Strabon (1-2 B.C.), and, 
above all, from Marinos of Tyre (II- 1), whom he praised yet 

17 A conformal projection is one in which the angles between two in- 
tersected curves are the same in projection. A perspective projection is one 
in which there is a 1 to 1 correspondence between any point on the sphere 
and its projection on the plane. 

The first to prove that the stereograph ical projections of spherical 
circles are circles was Jordanus Nemorarius (XIII- 1). 



We know Marines only through Ptolemy, who paid a very 
moving tribute to him in chapter 5 of Book I and referred to 
him many times; we may be sure that he quoted Marines 
fairly, even when he disagreed with him. The relationship of 
Ptolemy to Marinos is very much like his relationship to 
Hipparchos, the great difference being that Marinos flourished 
not Jong before Ptolemy, 18 while Hipparchos was three cen- 
turies distant. 

Ptolemy put together the geographical contributions of 
his predecessors and his own and thus created the first general 
treatise on geography. He was not interested in physical and 
human geography as Strabon and Pliny were, and it is not fair 
to reproach him for not having dealt with subjects which 
did not concern him. 

Book I discusses generalities, the size of the Earth and of 
the known world, methods of cartographic projection, etc. 
Books II to VII are systematic descriptions of the world in the 
form of tables giving the longitudes and latitudes of places, 
for every country of which he had sufficient knowledge. 
Ptolemy (or Marinos) was the first to speak of longitudes 
and latitudes (mecos, platos) as we do, meaning the distance 
in longitude or latitude to a zero circle. Some 8,000 places, 
"remarkable cities" (poleis episemoi), rivers, etc., are listed. 
The identification of many of those places is very difficult, if 
not impossible, in spite of abundant investigations by scholars 
very familiar with the regions concerned. The world which he 
tried to describe extended roughly from 20 S to 65 N and 


Ptolemy called him (Geography 1, 6) "the latest of our age" 
(hystatos te ton cath' hemas) which is not quite clear; he does not say 
that he knew him personally. Hence, Marinos was a late predecessor, 
how late? Hipparchos was also, in some respects, a late predecessor. 



from the Canary Islands at the extreme west to some 180 
eastward from them. The tables made it possible to draw 
maps wherein every item would be placed at its proper latitude 
and longitude; such maps were probably included in the proto- 
type manuscripts, because there are definite references to them 
in Book VIII, which is a kind of astronomical epilogue. The 
earliest manuscripts that have come to us are considerably 
later, say, thirteenth century, but may represent a tradition 
going back to Ptolemy and Marinos. 

Ptolemy's intentions were excellent, but their realization 
very imperfect. He was right in believing that in order to pro- 
duce an accurate map, one must first prepare a net of meridians 
and parallels, and his method of projection was distinctly 
superior to Marinos'. When the net is ready, one may easily 
mark upon it as many places as possible, the coordinates of 
which are known. So far so good, but the map will be true 
only if those coordinates have been established by astrono- 
mical methods. Unfortunately, very few latitudes were cor- 
rectly determined and no longitudes at all (the means were 
lacking). His coordinates were computed on the basis of dead 
reckonings, itineraries, traveller's tales and very few scientific 
observations. His theory of projection was very much better 
than the data to be projectedl The net itself was insufficient, 
because his estimate of the Earth's size was inaccurate and be- 
cause his first meridian was wobbly. 

The central degree of latitude was our 36 (Gibraltar, 
Rhodos) and that was convenient. The prime meridian was 
drawn through the Fortunate Islands (Canaries plus Madeiras); 
thus all the longitudes would extend only on the east side of 0. 
Unfortunately, the relationship of that first meridian to the 



continent was very inaccurate. As to the size of the Earth, 
Ptolemy had preferred the estimate of Poseidonios (1-1 B.C.) 
to that far more correct one of Eratosthenes (III-2 B.C.). 19 His 
estimate of the length of the Eurasian continent was much 
exaggerated, 180 instead of 130. This would eventually in- 
crease the hopes of Columbus and early circumnavigators but 
was poor geography. 

There is not much point in criticizing his views of the 
unknown part of the world, for such views could only be worth- 
less guesses. For example, his rejection of the circumambient 
ocean 20 was not more arbitrary than its acceptance by earlier 

The tradition of every Greek text is open to doubts be- 
cause the earliest manuscripts that have come down to us are 
always many centuries late. In the case of the Geography, the 
difficulties are much increased by the necessity of considering 
two traditions which may have concurred or not, the literary 
tradition and the cartographic one. I am willing to accept the 
conclusions of one of the greatest scholars, Father Joseph 
Fischer, S. J., 21 who devoted the best part of his life to that 
subject that the maps which have come down to us in the 
earliest manuscripts (none earlier than the thirteenth century, 

19 According to Eratosthenes, the circumference of the Earth was 252,- 
000 stadia; according to Poseidonios, 180,000 stadia. This might be the 
same measurement, if the stadia used in both cases were in the ratio 
20/21. If Eratosthenes' stadia were 10 to a mile, then his measurement 
equalled 37,495 km. (close to the real value 40,120 km.). For details, see 
Aubrey Diller, "Ancient Measurements of the Earth" (Isis 40 [1949], 6-9) . 

20 The Homeric views of the circumambient ocean were probably of 
Phoenician origin. However far the Phoenicians might sail, they were 
always stopped by the ocean. Herodotos was alone in his scepticism about 
it (History of Science, pp. 138, 186, 310, 510, 526). 

81 Joseph Fischer, S. J. (1858-1944). See Isis (37, 183). 



eleven centuries later than the lost prototypes) go back, even 
as the text, to Ptolemy or even to Marines (it is hardly pos- 
sible to distinguish between these two). The production of a 
world map was Ptolemy's definite aim; 22 he may have failed to 
produce it himself, and later maps, by Agathodaimon of 
Alexandria or others, may have been graphical representations 
of the tables. Certain knowledge is out of the question, but 
I prefer to share Father Fischer's confidence than Bagrow's 
hypercriticism. 23 

On the Ptolemaic maps, meridians are drawn for every 
5 and marked so in the margin, but parallels are established 
according to the length of the longest day (for every quarter- 
hour difference). In the Geography (I, 23), there is a table 
giving lengths of day with corresponding latitudes. 24 This 
part of the tradition goes back to the Eratosthenian concept of 
climata: zones of the Earth's surface at such a distance from 
each other that the average length of the longest day differs 
by half an hour from the one to the other. There were seven 
such climata, because there was no room for more in the 
known world, ranging from a longest day of thirteen hours in 

23 Geography (1,2, 2). Text quoted in Greek and Latin in Isis (20, 269). 
28 Leo Bagrow, The Origin of Ptolemy's Geographia (Stockholm, 1046; 

Isis 37, 187) . According to Bagrow, the text of the Geography is a late 
Byzantine compilation (say, tenth or eleventh century) and the maps, as 
we have them, are later than the text, say, thirteenth century. Such claims 
can be neither proved nor disproved. 

24 There is a similar table in the Almagest (XII, 6) wherein the lati- 
tudes are expressed with more precision in degrees and minutes. In the 
Geography, they are expressed in degrees and Egyptian fractions. Thus, 
to 13 hours correspond in the Almagest lat. 1627', in the Geography 
16 1/3 1/12 (=1625'). Aubrey Diller, "The Parallels on Ptolemaic 
Maps" (Isis 33, 5-7, 19*t). 

54 . 


Meroe (in Nubia, lat. 17 N) to one of sixteen hours at the 
Borysthenes (Dnieper). 

Ptolemy was aware of the imperfection of his knowledge 
and of the indetermination of his data, but the tabular form 
obliging him to state for each place definite latitudes and 
longitudes gave an impression of far greater exactness than 
was warranted, and his followers' assumption of the correctness 
of those numbers was the cause of many errors. 

The knowledge of the world revealed in the Geography is 
often inaccurate, but its extent and diversity are, nevertheless, 
astonishing. Consider, for example, the data relative to equa- 
torial Africa, the Upper Nile and the equatorial mountains 
(Lunae Mons, Geography IV, 8) . This is the more remark- 
able, if one bears in mind the confusion of ideas which still 
obtained as late as the third quarter of the last century. 25 


In speaking of Euclid's Optics I remarked that he dealt 
with a few phenomena in a geometrical way. Two optical 
treatises are ascribed to Ptolemy; one, entitled in Latin Pto- 
lomei de speculis, has been restituted to Heron of Alexandria, 
who flourished possibly before Ptolemy; the other, called 
Ptolemy's Optics, has come down to us in a Latin version 
made from the Arabic in 1154 by Eugene of Palermo (XII-2). 26 

This second treatise, the only one which we need consider 
here, is divided into five books, but Book I and the end of 

86 Intro. (3, 1158-60). 

26 Heron, wrongly placed in my Introduction (/, 208), flourished after 
62 and before 150 (Isis 30, 140; 32, 263-66). Latin-German edition of De 
speculis by Wilhelm Schmidt (Heronis opera, 2, 301-65, 1900). Gilberto 
Govi, L'ottica di Tolomeo de Eugenio (Torino, 1885) . Lejeune is prepar- 
ing a new edition of this text. 



Book V arc lost. Such as it has come to us, it is very different 
from Euclid's work, being physical and even psychological, for 
Ptolemy tried to explain vision in concrete sensual terms. His 
effort was understandable but premature, for the anatomical 
and physiological knowledge of the eye was still utterly in- 
sufficient. 27 

Books III and IV deal with catoptrics and constitute the 
most elaborate study of mirrors which has come down to us 
from antiquity. Book V deals with refraction and includes 
a table of refraction from air to water which is remarkable 
enough to be reproduced here. 28 

first real 

i r difference value of r error 















6 30' 



















4 30' 





27 Albert Lejeune, "Les tables de retraction de Ptolme" (Annales de 
la Societe scientifique de Bruxelles 60 [1946], 93-101) ; "Les lois de la 
reflection dans 1'Optique de Ptolemee" (L'antiquite classique 15 [1947], 
241-56; Isis 39, 244); Euclide et Ptolemee. Deux stades de I'optique 
geometrique grecque (Louvain, 1948) , Isis 40, 278). 

* 8 Figures as given by Lejeune, 1946 (p. 94). 



That table is unique in classical literature, and it astonished 
historians of physics so much that they took it too readily at 
its face value. Ptolemy's study of refraction was spoken of as 
the most remarkable experimental research of antiquity. I am 
sorry to have to confess that I helped to diffuse that judg- 
ment, 29 which has proved since to be erroneous; or, to put it 
otherwise, Ptolemy's results are still very remarkable but in an 
unexpected way. 

Looking at the first differences in column 3, one im- 
mediately sees that they form an arithmetical series, the differ- 
ence between two successive terms being i/g . Now, can that be 
the result of observations? (Note the observational errors in 
the last column.) It is certain that Ptolemy made some ob- 
servations with care; he did not continue them, however, but 
generalized them prematurely, and built his table a priori. 
Lejeune has suggested that he may have been misled by early 
Greek authorities or by Babylonian examples. Constancy of 
second differences may be noticed in polygonal numbers and 
some tables of the Sun show that Chaldean astronomers had 
tried to account for the Sun's irregular speed by constant 
second differences. 

The ancients did not yet understand the supremacy of ob- 
servations as we do and used observational results rather as 
indicators justifying the formulation of a theory, even as guide- 
posts help travellers to find the right path. Before judging 
them too severely, we should remember that their observational 
means were generally so poor that the results of observations 
could not possibly have with them the same authority as 
they have with us. 

29 Intro. (1,274). 



As Ptolemy was unfamiliar with sines, one could not expect 
him to discover the law of refraction, 30 but it is interesting to 
examine his results from that hindsight point of view. Let us 
call the angles of incidence and refraction enumerated in his 
table a and b. The average ratio sin a/sin b is 1.311, with an 
average error of 0.043; the ratio a/b, however, is 1.42 with an 
average error of 0.044. 31 Hence, Ptolemy's results, as given in 
his table, would not have enabled him to find the constancy 
of sin a/sin b; that is, he would have risked, instead, finding 
tne constancy of a/b; he would have found a wrong law in- 
stead of the true one. 

At any rate, Ptolemy understood very clearly the fact that 
a ray of light is deviated when it passes from one medium into 
another of different density (as we would put it), and he ex- 
plained the error caused by refraction in astronomical observa- 
tions. It is disturbing, however, to find no mention of atmos- 
pheric refraction in the Almagest; we must conclude that the 
Optics was written by Ptolemy after the Almagest^ 2 or that it 
was written by somebody else. The subject was not tackled 
again until much later, by Ibn al-Haitham (XI-1); for the first 
accurate determinations one had to wait until Tycho Brahe 

80 The law was discovered by Willebrord Snel in 1618; published again 
by Descartes in 1637. 

81 The figures quoted are taken from Ernst Gerland, Geschichte der 
Physik (p. 124, Miinchen, 1913; Isis 1, 527-29). 

82 1 prefer the first hypothesis. Having discovered refraction, Ptolemy 
could conceive the idea of atmospheric refraction. This is maximum at the 
horizon (almost 35') and creates phenomena (e.g., at sunset or sunrise) 
which must or may puzzle the intelligent observer. A knowledge of re- 
fraction (cataclasis) , even atmospheric refraction, is ascribed also to 
Cleomedes, who may be posterior to Ptolemy, in spite of my having classi- 
fied him tentatively under (1-1 B.C.). 



(1580), Kepler (1604) , and the first Cassini, Jean Dominique 
(c. 1661). 


Among the various other works ascribed to Ptolemy, I must 
select for discussion his astrological treatise, in spite of the 
fact that many men of science would refuse to consider it. 33 
Two astrological books bear his name, the Tetrabiblos (Quad- 
ripartituni) and the Carpos (Fructus);^ according to the con- 
sensus of scholarly opinion, the first is genuine, the second 
apocryphal. These two books have been transmitted together 
in Greek and other languages, in manuscript and printed tradi- 
tions, but for our purpose, it will suffice to consider the first. 

Many scholars have claimed that the same man could not 
possibly be the author of the rational Almagest and of the 
Tetrabiblos, which is chockful of irrational assumptions. They 
forget that astrology was the scientific religion of Ptolemy's day. 
At a time when the old mythology had become untenable, the 
sidereal religion had gradually taken its place in the minds of 
men who were loyal to pagan traditions as well as scientifically 
minded. Stemming from Greek astronomy and Chaldean 
astrology, it was a compromise between the popular religion 
and monotheism; the concept of sidereal immortality which 
it fostered reconciled astronomy with religion; it was a kind 

33 1 have claimed repeatedly that if we would understand ancient 
science and culture we must take the errors and superstitions into account 
as well as the progressive achievements. See, e.g., my History of Science 
(1952), p. xiii. 

34 Fructus is the translation of Carpos, but the Latin title more com- 
monly used is Centiloquium, referring to the fact that that booklet is a 
collection of a hundred aphorisms. The author was probably a court 
astrologer who flourished after Ptolemy and before Proclos (V-2) . 



of scientific pantheism indorsed by men of science as well as 
by philosophers, especially by neo-Platonists and Stoics. 

We now realize that such a compromise, however useful 
it may have been in a period of confusion and distress, was 
very dangerous; there was a fatal ambiguity in the astrological 
creed, in that it claimed to be science and religion at the same 
time. It was a poor application of good science, and the 
religious side of it had the weakness of any superstition. There 
has never been a better example of pseudo-science and pseudo- 
religion. Yet, it prospered for a few centuries in the religious 
vacuum caused by the repudiation of the old mythology. It 
would be very unfair to blame Ptolemy for having failed to 
understand eighteen hundred years ago what many of our own 
contemporaries have not yet understood today. The ambi- 
guities obtaining between rational knowledge and creed are 
still cultivated by pragmatists, by Christian scientists, and other 
sectarians who handle religion and science in the way thimble- 
riggers cause balls or peas to vanish or reappear. 

The Tetrabiblos is dedicated to the same Syros whom 
Ptolemy called upon thrice in the Almagest. What is more 
convincing, its style is similar to that of the Almagest. It is a 
great pity, however, that Ptolemy wrote it, because the prestige 
of his name was fully exploited, and the fame of the Tetra- 
biblos was not only equal to that of the Almagest, but much 

In his excellent book on Hellenistic Civilization** Pro- 
fessor Tarn has developed the view that the triumph of 
astrology was assured when Hipparchos and Ptolemy rejected 

38 First published in 1927; I quote from the third edition revised by 
W. W. Tarn and G. T. Griffith (pp. 298, 348, London, Arnold, 1952). 



the heliocentric system of Aristarchos. That theory does not 
hold water. In the first place, the postulates of astrology are 
independent of whether the Sun or the Earth is the center of 
our planetary system; in the second place, astrology did not 
stop after the acceptance of the Copernican system but con- 
tinued to grow lustily. Kepler himself drew horoscopes. Our 
country is leading the world in astronomy, and we have every 
right to be proud of that, but if we be honest, we cannot accept 
praise for our astronomers without accepting full blame for 
our astrologers. There are more astrologers than astronomers 
in America and some of them, at least, earn considerably more 
than the latter; the astrological publications are far more 
popular than the astronomical; almost every newspaper has 
an astrological column which has to be paid for and would not 
be published at all if a large number of people did not want it. 

Astrology was perhaps excusable in the social and spiritual 
disarray of Hellenistic and Roman days; it is unforgivable 
today. The professional astrologers of our time are fools or 
crooks or both, and they ought to be restrained, but who will 
do it? Astronomers are too busy with their own work and find 
it unnecessary to castigate obvious errors; they do not want to 
get into trouble, for in a trial ignorant judges or jurymen 
might decide that astrologers have as much right to express 
their views as the astronomers. And yet to ignore a contagious 
disease is the worst way of dealing with it. If one wishes to cure 
it, one must first throw light upon it and show it for what it is. 

Superstitions are like diseases, highly contagious diseases. 
We should be indulgent to Ptolemy, who had innocently ac- 
cepted the prejudices endemic in his age and could not foresee 
their evil consequences, but the modern diffusion of astro- 
logical superstitions deserves no mercy, and the newspaper 



owners who do not hesitate to spread lies for the sake of money 
should be punished just as one punishes the purveyors of 
adulterated food. 

To return to the Tetrabiblos Ptolemy refers to the Alma- 
gest in his general introduction and explains that the Almagest 
is a mathematical book dealing with matters which can be 
demonstrated, while the new book deals with matters which 
are less tangible and highly conjectural, yet deserve to be in- 
vestigated. One has the impression that in his old age, when 
Ptolemy had completed his scientific work, he applied himself 
to mcia-asironomy and tried to justify as well as he could the 
astrological prejudices of his time, prejudices which he fully 
shared. The first chapters constitute an apology for divination 
and particularly for astrology. Granted the almost universal 
beliel in divination, divination by the stars and planets seemed 
less irrational, "more scientific," than divination by means of 
birds, entrails, dreams or other omina. Ptolemy added that the 
possibility and occurrence of error should not discourage the 
astrologer any more than they discourage the pilot or the 
physician (I, 2) . 

The Tetrabiblos is a compilation of Chaldean, Egyptian 
and Greek folklore and of earlier writings, especially those of 
Poscidonios, 37 which is so complete and so well-ordered that it 

36 The original title seems to have been Mathematike tetrabiblos 
syntaxis, which was strangely enough the same title as that of the Almagest, 
plus the neutral word tetrabiblos. That title was erroneous and mis- 
leading, for the Tetrabiblos is definitely not a mathematical treatise. 
Some MSS are entitled Ta pros Syron apotelesmatika (Prognostics de- 
dicated to Syros). Prognostics was a correct title and meaningful. The 
most common title, however, is Tetrabiblos, which means "four books," 
and is as cryptic as Centilo^uium. 

87 Poseidonios is not named in the Tetrabiblos, but Franz Boll has 
shown, in his Studien iiber Claudius Ptolemdus (Leipzig, 1894) that the 



remained a standard work until our own days. In that it was 
even more successful than the Almagest, for the simple reason 
that astronomy being a science was bound to develop and 
change, while modern astrology is essentially the same as the 
ancient one. Superstitions may change but do not progress; 
in fact, they do not change much, because they are exceedingly 
conservative. The Almagest is published anew from time to 
time for scholarly purposes, but has no practical value; on 
the other hand, new editions of the Tetrabiblos are issued for 
the guidance of practising astrologers. 38 

The contents of the four books of the Tetrabiblos may be 
roughly described as follows: I. Generalities concerning astrol- 
ogy and the planets. Beneficent and maleficent planets, mas- 
culine and feminine ones, diurnal and nocturnal, etc. II. Catho- 
lic astrology, astrological geography and ethnography. Prog- 
nostications of a general kind, applying to races, countries, 
cities or to catastrophies which affect many men at the same 
time, such as wars, famines, plagues, earthquakes, floods, or 
the weather, seasons and climes (latitudes). III. Genethlia- 
logical prognostications relative to individuals. IV. Fortune. 
Astrological aspects of material fortune, personal dignity (axi- 
oma), degree of activity, marriage, children, friends and ene- 
mies, foreign travel, quality of death, various periods of life. 
In Robbins' Greek-English edition (Loeb Library) , the four 

author of Tetrabiblos used the lost writings of Poseidonios, especially for 
what concerns the defense of astrology and astrological ethnography 
(Book II) . In many geographical details, Tetrabiblos and the Geography 
do not agree, but it does not follow that the authors of those two works 
are different. 

88 An English edition published for the astrological market in Chicago, 
1936, was reviewed in Isis (35, 181). 



books cover, respectively, 116, 104, 152 and 87 pages; and the 
whole Greek text extends to 230 pages. 

One cannot read the whole of that treatise or a part of it 
without being terribly dismayed. If Ptolemy was really the 
author of it, it is a thousand pities, but that only shows that 
he was a man of his clime and time. Even the greatest genius 
cannot transcend all those limitations at once. 


We shall outline only the tradition of his three most famous 
works, the Almagest, the Cosmography, and the Tetrabiblos. 


The Greek tradition was solidly established from the be- 
ginning and it was kept alive by the commentaries of a series 
of illustrious mathematicians, Pappos (1II-2), Theon of Alex- 
andria (IV-2), Hypatia (V-l) and Proclos (V-2) . The book 
entitled Mathcmatike syntaxis was often called Megale syntaxis 
(the great collection) or even Megiste synlaxis (the very great 

The importance of the Arabic tradition is symbolized by the 
common name Almagest which combines the Arabic article 
with the Greek adjective megiste. Arabic mathematicians were 
acquainted with the book very early, for it was translated into 
Arabic by an unknown scholar at the insistence of the illus- 
trious wazir, Yahya ibn Khalid ibn Barmak (Joannes the Bar- 
mecide), who lived from 738 to 805; it was translated again 
in 829, on the basis of a Syriac version, by al-Hajjaj ibn Yusuf 
(IX- 1) and a third time by Ishaq ibn Hunain (IX-2), and 
Ishaq's translation was corrected by Thabit ibn Qurra (IX-2). 
Further editions and adaptations were prepared by such 



eminent men as Abu-1-Wafa' (X-2) and Nair al-din al-Tusi 

Meanwhile, the Arabic geographers had produced astro- 
nomical treatises which were not translations of the Almagest, 
yet were profoundly indebted to it. The first of these treatises 
was the one by al-Farghani (IX-1) which became in the ori- 
ginal Arabic and in Latin and Hebrew versions one of the 
main sources of Ptolemaic astronomy until the Renaissance. 
The same can be said of al-Battam's treatise (IX-2) , but though 
it was far superior to al-Fargham's, it was less popular. More- 
over, since al-Battani was a greater mathematician and a more 
original mind than al-Farghani, he modified the Ptolemaic 
tradition more deeply. 

Not only was it possible to read the Almagest in Arabic, and 
the treatises of al-Farghani and al-Battani which were derived 
from it, but the Muslim astronomers worked so well that they 
were soon able to criticize Ptolemy's ideas. As the astronomical 
observations were more numerous and more precise, it became 
increasingly difficult to reconcile them with the theories. The 
philosopher, Ibn Bajja (Avempace, XII-1), expressed the 
difficulties and this was soon done with more authority by Jabir 
ibn Aflah (XII-1) in his treatise called Isldh al-magisti, (the 
Correction of the Almagest). Other Muslims, the philosopher 
Ibn Tufail (XII-2) and his disciple al-BitrujI (XII-2) thought 
of solving the difficulties by rejecting Ptolemy's eccentrics and 
epicycles and reverting to the earlier theory of homocentric 
spheres which had been endorsed by Aristotle himself. After 
the twelfth century, the vicissitudes of astronomical theory 
were largely the result of a protracted struggle between the 
followers of Ptolemy and those of Aristotle. 39 

39 For more details, see Intro. 2, 16-19; 3, 110-37, 1105-21. 



Within the twelfth century, the Almagest as well as the 
treatises of Alfraganus and Albategnius 40 became all of them 
available in Latin. Alfraganus was first translated by John of 
Seville (XII-1) in 1134, then again by Plato of Tivoli (XII-1). 

The Almagest was translated from Greek into Latin, in 
Sicily, c. 1160, and from Arabic into Latin by Gerard of 
Cremona (XII-2) in Toledo in 1175. Such was the prestige of 
the Arabic source or of the Toledo academy that the indirect 
translation of 1175 displaced the direct one of 1160. 

Gerald did not simply translate the Almagest, but he trans- 
lated the Isjah al-majisti as well, before 1187 41 (that is, when 
Jabir's work was still a novelty in Muslim circles) . 

The Hebrew translations were a little slower in appearing; 
they belong to the thirteenth century. The summary of the 
Almagest written by Ibn Rushd (Averroes, XII-2), the Arabic 
text of which is lost, was translated into Hebrew by Jacob 
Anatoli (XIII- 1), and the same translated also, c. 1232, al- 
Fargham's treatise from Latin and Arabic into Hebrew. Moses 
ibn Tibbon (XI1I-2) translated into Hebrew al-Bitruji in 1259 
and Jabir ibn Aflah in 1274. 

Finally, we may mention, for the sake of curiosty, the 
Syriac summary of the Almagest written by Abu-1-Faraj in 1279; 
this was probably the redaction of his lectures on the subject 
delivered at Maragha between 1272 and 1279. 

In short, during the medieval period, every astronomer, 
whether Jewish, Christian or Muslim, might be assumed to be 
familiar with Ptolemaic astronomy, directly or indirectly; we 
might even say that every one was a Ptolemaist with few, if 

40 Meaning al-Fargham (IX-1) and al-Battani (IX-2). 

41 1187 is the year of Gerard's death in Toledo. Jabir (Latin, Geber) 
died about the middle of the twelfth century. 



any, qualifications. The history of medieval astronomy is a 
history of Ptolemaic ideas and of a growing discontent with 
them. The difficulties could not be solved with cinematical 
expedients, nor could they be solved by replacing the Sun in 
the center instead of the Earth. The main stumbling block 
was the notion that celestial trajectories must be circular (or 
combinations of circles) and that block was removed only by 
Kepler as late as 1609. 

The history of the Ptolemaic tradition includes the history 
of astronomical tables, all of which were ultimately derived 
from those of the Almagest. 

One more aspect of the Ptolemaic tradition must be in- 
dicated, however. The Almagest consecrated the use of sexa- 
gesimal fractions, and obstructed the natural extension of 
decimal numbers to decimal fractions; or to put it otherwise, 
it discouraged the use of decimal submultiples in the same 
manner that decimal multiples were used. The superiority of 
decimal fractions was well explained for the first time by the 
Fleming Simon Stevin in 1585, and their exclusive use has not 
been obtained to this day. 

With the slowness of progress, or the persistance of Ptole- 
maic errors, the geocentric error was not proved until 1543 by 
Copernicus, the sexagesimal error not until 1585 by Stevin, 
the circular error not until 1609 by Kepler. 

The first printed edition of Ptolemaic astronomy was al- 
Fargham's treatise as Latinized by John of Seville (XII-1), 
Compilatio astronomica (Ferrara, 1493. Klebs no. 51. Facsimiles 
of both sides of first leaf, Osiris 5, 141) . 

The Epitoma in Almagestum by Regiomontanus (XV-2) 
was printed three years later (Venice, 1496. Klebs no. 841.1. 
Facsimile of title page, Osiris 5, 162) . 

So much for the incunabula. 



The first printed editions of the Almagest are the following: 
Latin version from the Arabic by Gerard of Cremona, Toledo, 
1175, edited by Peter Liechtenstein (Venice, 1515). 

Latin version from the Greek by George of Trebizond, 
1451, edited by Luca Gaurico (Venice, Junta, 1528) . 

First Greek text, edited by Simon Grynaeus, made upon 
the Bessarion manuscript once used by Regiomontanus (Basel, 
Walderus, 1538). Facsimile of title page (I sis 36, 256). 

The following indications may be of interest. 

First printed edition of al-Battam (IX-2), in the Latin 
translation by Plato of Tivoli (XII- 1) (Nurnberg, Joh. Pet- 
reius, 1537). Splendid Arabic-Latin edition by C. A. Nallino 
(3 vols., Milano, 1899-1907). 

First printed editions of the Isldh al-majistl of Jabir ibn 
Aflah (XII-1) as Latinized by Gerard of Cremona before 1187 
(Nurnberg, Joh. Petreius, 1534). 

First printed edition of al-Bitruji (XII-2), as Latinized 
by Qalonymous ben David in 1528-29 (Venice, Junta, 1531). 
The tradition of this text is curious. It was translated from 
Arabic into Latin by Michael Scot in 1217, 42 from Arabic into 
Hebrew by Moses ibn Tibbon in 1259, from Hebrew into 
Latin by Qalonymos. 

To these printed texts a good many others could be added, 
even if one restrict oneself to the pre-Copernican period (pre- 
1543). It will suffice to mention the many editions of the 
Sphaera mundi by Joannes de Sacrobosco (XIII- 1) , which was 
slavishly derived from al-Farghani and al-Battani. There are 
thirty-one separate incunabular editions of the Sphaera, plus 
many others in combination with other texts. 43 

48 Michael Scot's translation was recently edited by Francis J. Carmody 
(Berkeley, Calif., 1952; Isis 44, 280-81) . 

48 For Sacrobosco, see Klebs (nos. 874, 875). Lynn Thorndike, The 
Sphere and its Commentators (Chicago, 1949; Isis 40, 257-63). 




The early tradition of the Cosmography is not by any 
means as well known as that of the Almagest. We have already 
explained that in this case it does not suffice to consider the 
text, but there is also a cartographic tradition which is very 

The Cosmography was known in Syriac circles; witness 
a chapter of the Syriac Chronicle of 569, and the Hexaemeron 
of Jacob of Edessa (VII-2) . Much was added to it by Muslim 
geographers, such as al-Khwarizmi (IX- 1), al-Battam (IX-2), 
and many others, East and West. 

The Latin translation of the Greek text was made by 
Giacomo d'Angelo (Jacobus Angelus) in 1409. 

The growing popularity of the Cosmography in the fifteenth 
century is well illustrated by the number of incunabula. While 
there was no incunabula edition of the Almagest (excepting 
Regiomontanus' Epitoma of 1496), there were seven of the 
Cosmographia (Klebs no. 812). The first was issued by Her- 
mann Liechtenstein (Vicenza, 1475); the first with maps by 
Lapis (Bologna, 1477) ; 44 facsimile copy of the edition of 1477 
(Klebs no. 812.2) by Edward Lynam: The First Engraved 
Atlas of the World (26 maps, Jenkintown, George H. Beans, 

The first Greek edition was prepared by no less a person 
than Erasmus (Basel, Froben and Episcopius, 1533) . 

44 Not 1462 as printed by mistake in its colophon (Osiris 5, 103). First 
and last page of the first edition, 1475 (Osiris 5, 134-35) . 




The Tetrabiblos must have been a popular book in Greek 
circles, because astrological fancies and other aberrations 
flourished more and more as the old culture was decaying, but 
the ancient tradition is obscure. An introduction to it is 
ascribed to Porphyries (III-2), a paraphrase to Proclos (V-2), 
and there is an anonymous commentary which might also be 
by the latter. That is not much to go by. 45 

TheTetrabiblos was one of the earliest Greek books to be 
translated into Arabic, under al-Mansur (VIII-2), the second 
'Abbasi caliph (754-75), the founder of Baghdad the trans- 
lator being Abu Yahya al-Batriq (VIII-2). Al-Batriq's version 
was commented upon by 'Umar ibn al-Farrukhan (IX- 1), and 
by Ahmad ibn Yusuf (1X-2). The Tetrabiblos was translated 
again by Hunain ibn Ishaq (IX-2) and this translation was 
commented upon by 'All ibn Ridwan (XI- 1); 'All's commen- 
tary was much used by astrologers. 

Another translation by Ibrahim ibn al-Salt (date un- 
known) corrected by Thabit ibn Qurra (IX-2) and (or) Hun- 
ain ibn Ishaq was Latinized by Plato of Tivoli (XII- 1) and 
was the first Ptolemaic work to be translated into Latin. A 
new Latin translation was made in 1206 by an unknown 
scholar. The Tetrabiblos and 'All ibn Ridwan's commentary 
upon it were translated into Spanish, perhaps by Judah ben 
Moses (XIII-2), for Alfonso el Sabio (XIII-2) , and from 
Spanish into Latin by Aegidius of Thebaldis soon after 1256. 


The Greek text of the paraphrase was published with a preface by 
Philip Melanchthon (Basel, J. Oporinus, 1554), a Greek-Latin edition of 
the two other texts by Hieronymus Wolf was published a few years later 
(Basel, Petreius, 1559). 



Still another Latin translation was prepared, c. 1305, by Simon 
de Bredon (XIV-1) . Etc. 

The Latin version from the Arabic was printed very early. 
There are two separate incunabula, the first by Ratdolt (Ven- 
ice, 1484) and the second by Locatellus (Venice, 1493), plus 
many included in other incunabula editions (Klebs no. 814). 

There were also Latin versions from the Greek, one being 
mentioned by Henry Bate of Malines (XIII-2) in 1281. The 
first edition of the Greek text, by Joachim Camerarius was 
printed by J. Petreius of Nurnberg in 1535, and reprinted by 
Joannes Oporinus at Basel in 1553. Both editions included 
Latin translations from the Greek, the first by Camerarius and 
the second by Philip Melanchthon; both included also the 
Carpos in Greek and Latin. 

An English translation by the Dublin quack, John Whalley, 
was printed in London in 1701 and again in 1786. Another 
English translation by J. M. Ashmand in London in 1822 was 
reprinted there in 1917 and in Chicago in 1936 (his 35, 181). 

Two critical editions of the Greek text were published in- 
dependently in 1940, the one by Franz Boll and Aemilia Boer 
in the Opera omnia of Ptolemy (III, 1, Teubner, Leipzig) 
and the other by Frank Egleston Robbins, with an English 
version, in the Loeb Classical Library (reprinted in 1948; 

There are thus three English versions of the Tetrabiblos. 
Until 1952, this was the only Ptolemaic text which could be 
read in our language. Horresco referens! (Isis 44, 278). 




1 . Complete Works 

Opera quae extant omnia. Edited by J. L. Heiberg (Teub- 
ner, Leipzig, 1898 f.). Vol. I in 2 vols., Almagest (1903). Vol. II, 
Opera astronomica minora (1907). Vol. Ill, 1, Tetrabiblos 
edited by Franz Boll and Aemilia Boer (1940). 

This is all in Greek, except when the Greek text is lost. 

2. The Almagest 

The standard edition is Heiberg's in the Opera omnia (Vol. 
I in 2 vols, 1898-1903). The Greek-French edition by the Abbe" 
Nicolas B. Halma with notes by J. B. J. Delambre is very con- 
venient (2 vols., Paris, 1813-16). Facsimile reprint of smaller 
size (Paris, Hermann, 1927) . 

German translation by Karl Manitius derived from the 
Heiberg text (2 vols., Leipzig, 1912-13). 

An English translation by Catasby Taliaferro is included in 
Great Books of the Western World (XVI, 1-478, Chicago, 
1952; 7sw 44, 278-80). 

Christian H. F. Peters and Edward Ball Knobel. Ptolemy's 
Catalogue of Stars. A revision of the Almagest (208 pp., 
Carnegie Institution of Washington, 1915; Isis 2, 401) . 

3. The Geography 

Ptolemaei Geographiae Codex Urbinas Graecus 82. Edited 
by Joseph Fischer and Pius Francus de Cavalieri (4 vols., 
Leiden, Brill, 1932). For fuller description and review, see 
Isis 20, 266-70). This includes an elaborate study of Ptolemy 
and his Geography by Father Fischer with indices (Tomus 
prodromus, pars prior, 624 pp.). 

Traite de geographic traduit pour la premiere fois du grec 
en francais sur les MSS de la Bibliotheque du Roi par l'abb 
Halma (quarto 214 pp., Paris, 1828) , not seen. 

Geography of Ptolemy, translated into English by Edward 
Luther Stevenson (folio, 183 pp., 29 pi., New York Public 



Library, 1932; his 20, 270-74; 22, 533-39) . No index. Imperfect 

Let us hope that the edition of the Greek text being pre- 
pared for the Opera Omnia will soon appear. Thus far, we 
have no better edition of the Greek text than that of Carolus 
Miiller: Ptolemaei Geographia (2 vols., Paris, Firmin Didot, 
1883-1901), with Latin translation; but it is incomplete (stop- 
ping at vol. V, cap. 19), and hence lacks an index. 

For an index, one must refer to the old Greek edition of 

C. F. A. Nobbe (Ed. stereotype, 3 vols., Leipzig, Tauchnitz, 

1843-45) , or the old Nomenclator which the Fleming Abraham 

Ortelius (1527-98) added to his Theatrum orbis terrarum 

(Antwerp, 1579) and later editions and also published separate- 


Two bibliographies may be added. Henry Newton Stevens: 
Ptolemy's Geography. A Brief Account of all the Printed Edi- 
tions down to 1730 (62 pp., London, Stevens and Stiles, 1908). 
William Harris Stahl: Ptolemy's Geography (86 pp., New York 
Public Library) . This is especially useful to find studies de- 
voted to the Ptolemaic account of specific regions, say, Sicily 
or Ceylon. 

4. Alia 

For editions of the Optics or Tetrabiblos, see chapters 7-8 
above devoted to these books. For additional bibliography, 
see my Introduction (1 , 274-78) and the Critical Bibliographies 
of Isis, section II- 1. 



(from c. 300 to 529) 


CAVING out of the question prehistoric times, 
which cannot be measured, Greek culture begins with Homer 
(say, in the ninth or eighth centuries); Greek science begins a 
little later with Thales and Pythagoras (sixth century). My 
first lecture on Euclid (c. 300 B.C.) dealt with a relatively late 
stage of Greek culture, the so-called Hellenistic. In order to 
deal with Ptolemy in my second lecture, we had to make a 
jump of more than four centuries; we shall now consider a 
period which begins 150 years and ends 350 years later. This 
illustrates once more the length of ancient Greek culture, its 
duration and its inexhaustible variety. The Roman world of 
Ptolemy was very different from the Alexandrian of Euclid 



and the world which I shall try to evoke here is again extremely 

The Roman Empire and Christianity were born at about 
the same time. By the beginning of the fourth century the 
Roman Empire was going down rapidly while Christianity 
was going up, and we witness the symbiosis of the old Pagan 
moving slowly to his death and of the Christian youth prepar- 
ing to live and conquer. 

This lecture will be divided into three parts: Greek mathe- 
tics, Greek medicine, and the philosophical and religious back- 
ground. My reason for speaking of the background in the last 
part instead of the first will be apparent later on. 


Ptolemy's gigantic efforts were followed by a lull of more 
than a century. So much so that when the next great 
mathematician appeared he felt obliged to prepare a sum- 
mary of earlier books, under the title Mathematical Collection 
(synagoge). This mathematician was Pappos of Alexandria. 
According to a scholion (marginal note) in an old manuscript, 
he lived under Diocletian (emperor, 284-305) and therefore it 
is tempting to consider him a man of the third century like 
the algebraist Diophantos, 1 but according to Canon Rome 2 
Pappos' commentary on the Almagest was probably written 
after 320 and the Mathematical Collection even later. Pappos 
wrote various commentaries on Euclid and Ptolemy, but his 

1 That was done in my Introduction, where Pappos was placed in the 
time of Diophantos (III-2) . It would have been better, perhaps, to place 
him in (IV-1) (Intro. 3, ix) . Pappos would seem to be half-way between 
Diophantos and Theon of Alexandria. 

a Adolphe Rome: "Sur la date de Pappus" (Annales de la Soctttt 
scientifique de Bruxelles, serie A [1927], 46-48), Isis 11, 415-16. 



main work is the Synagoge already mentioned, of which a 
great part has come down to us. It is divided into eight books; 
we have everything except Book I and chapters 1 to 13 of 
Book II, the preface to IV, and perhaps the end of VIII. It is 
difficult to analyze it because it is devoted to a multiplicity of 
mathematical subjects and combines for most of them old and 
new. Pappos was not a teacher like Euclid or Ptolemy but a 
learned man who was familiar with the whole of Greek mathe- 
matics and tried to summarize it in his own peculiar way. He 
was a good commentator because he was on a level with his 
greatest predecessors and was able to add ingenious theorems 
and problems of his own, but he was not very methodical. As 
far as we understand the general composition of his Synapoge, 
he had taken notes on the mathematical classics, invented and 
solved new problems, and then classified them in eight books. 
Each book is preceded by general reflections which give to that 
group of problems its philosophical, mathematical and his- 
torical setting. These prefaces are of deep interest to historians 
of mathematics and, therefore, it is a great pity that three of 
them are lost (the prefaces to Books I, II and IV) . They may 
turn up some day in an Arabic version. 

The following notes will indicate roughly the contents of 
the Synagoge, book by book. 

Book II (chapters 14-16). Commentary on Apollonios* 
method for the writing of large numbers in terms of powers of 
myriads (10,000 n ) and for operating with them. 

Book III. History of the problem: to find two mean pro- 
portionals in continued proportion between two given straight 
lines. Classification of geometrical problems in three classes 
(1) plane, (2) solid, (3) those requiring the use of higher 
curves for their solution. Curious propositions suggested by 



the paradoxes of Erycirios (otherwise unknown). How to ir 
scribe the five regular solids in a given sphere. 

Book IV. Extension of the Pythagorean problem on th 
square of the hypotenuse. Circles inscribed in the arbelc 
(semicircular knife used by cobblers) , commentary on a boo 
of Archimedes (lost in Greek, preserved in Arabic) . Discussio: 
of Archimedes' spiral, Nicomedes' conchoid, the quadratri> 
spherical spiral. Trisection of any angle, etc. This include 
a method of integration (for the spiral) different from that c 

Book V. Isoperimetry, derived from Zenodoros (II- 1 B.C/ 
The delightful preface refers to the bees whose cells are bui] 
with a great regularity and a marvelous economy of space 
Pappos did not deal only with plane problems; he also state* 
that the sphere has the greatest volume for a -given surface. 

Book VI, mainly astronomical, being inspired by some c 
the authors of the "little astronomy," Autolycos (IV-2 B.C. 
Aristarchos (III-l B.C.), Euclid (III-l B.C.), Theodosios (I- 
B.C.) and Menelaos (1-2) . 3 

Book VII is by far the longest book of the collection; th 
longest books next to it are III, IV and V, but VII is almos 
as long as these three books put together. It is also the mos 
important for historians, because it discusses a good many los 
books of Aristaios (IV-2 B.C.), Euclid, Apollonios, an 

3 The "little astronomy" or ho micros astronomumenos (topos) w; 
so called perhaps by contrast with the megale syntaxis. Many of the* 
writings were transmitted together (in the same manuscripts) to Grec 
readers and later to Arabic ones. The Arabic collection, including th 
Greek texts plus some original Arabic ones, was called Kitdb al-mutt 
ivassitdt bain al-handasa wal-hai'a, The middle books between geometi 
and astronomy (Intro. 2, 1001) . 




A C B 


If P, Q, R are the centers of these circles, d lt d 2 , d a their diameters, P lf P 2 , 
p 8 the distances of the centers to the base line, then p a = d^a = 2d 2 p a = 
3d 8 p 4 = etc. 

From Heath: Manual of Greek Mathematics, Oxford, 1931, p. 442; I sis 
16, 450. 



Eratosthenes. 4 According to its own title, it contains the 
lemmata of the "solved locus" (ho topos analyomenos), and is 
a kind of textbook on geometrical method for advanced stu- 
dents. It was dedicated (as well as Book VIII) to Pappos' son 
Hermodoros. After a preface wherein he defines and explains 
analysis and synthesis, he deals with each of those ancient 
treatises, stressing one point or another. For example, we find 
in it the famous Pappos' problem: "given several straight lines 
in a plane, to find the locus of a point, such that when straight 
lines are drawn from it to the given lines at a given angle, 
the products of certain of the segments shall be in a given 
ratio to the product of the remaining ones." This problem is 
important in itself, but even more so because it exercized 
Descartes' mind and caused him to invent the method of co- 
ordinates explained in his Geometric (1637). Think of a seed 
lying asleep for more than thirteen centuries and then helping 
to produce that magnificent flowering, analytical geometry. 
Another proposition was the seed of the centrobaric method; 
it proves a theorem equivalent to the Guldin theorem: "if a 
plane closed curve rotates around an axis, the volume created 
by its revolution will be equal to the product of the area by 
the length of the path of its center of gravity." The Jesuit 
father Paul Guldin published that theorem more clearly in 
1640. 8 

4 No less than twelve treatises in 33 books, most of them by Euclid 
(3 treatises in 6 books) and Apollonios (7 treatises in 20 books). 

6 The Guldin anticipation in Pappos is imperfect and perhaps an inter- 
polation; it does not occur in all the manuscripts. Guldin expressed the 
theorem very clearly for the first time but his proof is incomplete. The 
first complete proof was given by his adversary, Bonaventura Cavalieri, in 



Another problem bearing Pappos' name is not in his Col- 
lection, however. Given a point A on the bisectrix of a given 
angle, to draw through A, a segment a ending on both sides of 
the angle. This problem had an extraordinary fortune, prob- 
ably because of this singularity: it leads to an equation of the 
4th degree and yet can be solved with ruler and compass. 6 

The most astonishing part of that Book VII has not yet 
been mentioned. Dealing with the lost treatise of Apollonios 
on the determinate section (diorismene tome), Pappos ex- 
plains the involution of points. 

The final Book VIII is mechanical and is largely derived 
from Heron of Alexandria. Following Heron, Pappos dis- 
tinguished various parts of theoretical mechanics (geometry, 
arithmetic, astronomy and physics) and a practical or manual 
part. This book may be considered the climax of Greek me- 
chanics and helps us to realize the great variety of problems 
to which the Hellenistic mechanicians 7 addressed themselves. 
Many needs had to be filled: the moving of heavy bodies, war 
engines for offensive or defensive purposes, machines for the 
pumping of water, automata and other gadgets for the use of 
wonder workers; water clocks and moving spheres. Pappos was 
interested in practical problems, such as the construction of 
toothed gearings and of a cylindrical helix (cochlias) acting 
upon the teeth of a wheel, but he was even more concerned 
with mathematical methods, such as the finding of two mean 
proportionals between two given lines, the determination of 

A thick volume was devoted to it by A. Maroger: Le Probleme de 
Pappus et Ses Cent Premieres Solutions (Paris, Vuibert, 1925) , reviewed 
in Revue Generate des Sciences (37, 338) . 

7 Pappos knew them chiefly through Heron; Philon is quoted only a 
few times, Ctesibios not at all. 



centers of gravity, the drawing of a conic through five given 
points. The mathematician in him was so keen that he was 
trying to solve theoretical problems such as this one: how to 
fill the area of a circle with seven equal regular hexagons. 

If Book VIII is the climax of Greek mechanics, we may say 
as well that the whole Collection is a treasury and to some 
extent the culmination ol Greek mathematics. Little was 
added to it in the Byzantine age and the Western world, hav- 
ing lost its knowledge of Greek together with its interest in 
higher mathematics, was not able to avail itself of the riches 
which Pappos had put together. The ideas collected or in- 
vented by Pappos did not stimulate Western mathematicians 
until very late, but when they finally did, they caused the 
birth of modern mathematics analytical geometry, projective 
geometry, centrobaric method. That birth or rebirth, from 
Pappos' ashes, occurred within four years (1637-40). Thus was 
modern geometry connected immediately with the ancient one 
as if nothing had happened between. 

Pappos was the greatest mathematician of the final period 
of ancient science, and no one emulated him in Byzantine 
times. He was the last mathematical giant of antiquity. Never- 
theless, he was followed by a very distinguished group of 
mathematicians, so numerous indeed that it will not be pos- 
sible to speak of each of them except in the briefest mannef. 
Serenos of Antinoopolis (IV-1) was another Egyptian Greek, 
hailing from the city in Middle Egypt which Hadrian had 
founded in memory of the beautiful Antinoos, drowned in the 
Nile in 122. We must assume that Serenos studied or flourished 
in Alexandria, and in any case he was in touch with the Alex- 
andrian school, the greatest mathematical school of his age as 
well as the one which was nearest to him. He wrote a com- 



mentary on the Conies of Apollonios and two original treatises 
on the sections of cylinders and cones. 

And now let us consider two other illustrious Alexandrians, 
father and daughter, Theon (IV-2) and Hypatia (V-l), both 
teachers in the Museum. Theon edited Euclid's Elements and 
wrote a very elaborate commentary on the Almagest. He com- 
pleted Ptolemy's establishment of sexagesimal fractions; Hy- 
patia revised her father's commentary on Books III and follow- 
ing of the Almagest and she may be responsible for a new 
method of sexagesimal division closer to the Babylonian than 
her father's, but it is impossible to know exactly what belongs 
to either of them. Her own commentaries on Apollonios, 
Ptolemy's Canon (chronology) and Diophantos are all lost, 
but she is immortalized by the grateful letters of Synesios of 
Gyrene (V-l) 8 and, above all, by her martyrdom in 415. She 
enjoys the double honor of being the first female mathe- 
matician and one of the first martyrs of science. 

After Hypatia's death, there was a lull in the mathematical 
(pagan) school of Alexandria, and no wonder. The next 
leaders belong to the following century, Ammonios and Philo- 
ponos. Ammonios, son of Hermias (VI-1), studied under 
Proclos in Athens, but he restored the school of Alexandria 
and, judging by the merit of some of his pupils, must have 

8 Synesios of Gyrene (c. 370 c. 11'}) was converted in middle age (c. 
407) and became soon afterward bishop of Ptolemais (410), one of the 
five cities of Cyrenaica (Pentapolis). Of his letters 159 have been pre- 
served, dating from 394 to 413; seven are addressed to Hypatia covering the 
same lapse of time. In letter 15, he asked her to have a baryllion (a kind 
of hydrometer) made for him. This is the first description of that instru- 
ment in literature, but its use is such an obvious application of Archime- 
dian hydrostatics that some Hellenistic mechanician had probably ii/vcntcd 
it long before the fifth century. 



been a great teacher. He divided mathematics into four 
branches: arithmetic, geometry, astronomy, music a division 
which became in the Latin world the quadrivium. 9 His dis- 
ciple, Joannes Philoponos (VI-1), 10 was primarily a philosopher, 
but he wrote the earliest treatise on the astrolabe and a com- 
mentary on the arithmetic of Nicomachos. 

Now, let us return to Athens. When it had become a pro- 
vincial city of the Roman empire, its schools were eclipsed by 
the Museum, yet it continued to be the sacred metropolis of 
Hellenism. Political and commercial power had withdrawn, 
but philosophy had remained. Yet, it must be admitted that 
by the end of the fourth century, only one of the four main 
schools was really alive. We cannot name the headmasters or 
leaders of the Aristotelian, Stoic and Epicurean schools. Only 
in the Academy was the succession (diadoche) of the head- 
masters preserved. Let us name them for the sake of curiosity: 
Priscos (c. 370), Plutarchos, son of Nestorios 11 (d. 431), Syrianos 
of Alexandria (V-l), Domninos of Larissa (V-2) , Proclos the 
Successor (V-2), Marines of Sichem (V-2) , Isidores of Alex- 
andria, Hegias, Zenodotos, and finally Damascios (VI-1). 

The word quadrivium was introduced by Ammonios' Latin con- 
temporary, Boetius (VI-1) , but the idea is considerably older. It was 
adumbrated by Archytas of Tarentum (IV- 1 B.C.), for whom see my 
History of Science (pp. 434, 440, 521) . 

10 Joannes Philoponos is identical with John the Grammarian (Intro. 1, 
421, 480) . He was a Jacobite Christian and one of the very greatest per- 
sonalities of his age (Isis 18, 447). 

11 The decadence of the age is illustrated by the fact that this Plutarchos 
was called the Great! Plutarchos of Athens is now almost unknown. When 
referring to his illustrious namesake, Plutarchos of Chaironeia (1-2), I 
shall call the latter "Plutarch," because he belongs to world literature. 
Plutarchos' daughter, AsclSpigeneia, was a "femme savante," the Athenian 
contemporary and counterpart of the Alexandrian Hypatia. 



This list suggests two remarks. First, it is probably com- 
plete 12 and thus proves a modicum of continuity, but the fact 
that many of the diadochoi are all but unknown is an ominous 
sign. Who were Priscos, Hegias and Zenodotos? As to the last 
head of the Academy, we do not even know his personal name, 
for Damascios simply means the Damascene. Second, an 
analysis of the list would show that the schools of Athens and 
Alexandria were relatively close to each other. Ammonios was 
a pupil of Proclos and a teacher of Damascios; it is a regular 
chasse-croise. Alexandrians would study in Athens and Athen- 
ians in Alexandria. Two at least of the diadochoi of the 
Academy, Syrianos and Isidores, were Alexandrians. 

It is clear that the Academy had ceased to be a high mathe- 
matical school. The majority of the teachers and students 
were interested only in Neoplatonic arithmetic, that is, number 
mysticism. However, Domninos of Larissa tried to react 
against that and to revive the Euclidian theory of numbers. 
Proclos was by far the greatest headmaster in the last century 
of the Academy's existence. He was of Lycian origin 13 but born 
at Byzantion; he studied in Alexandria, but too late to drink at 
the sources of Hypatia's wisdom; he went back to Athens and 
was head of the Academy until his death in 485. He has been 
called "the Hegel of Neoplatonism" by people who wanted 
to praise him as much as possible; he was certainly more in- 
fluential as a philosopher than as an astronomer or a mathe- 
matician. Yet we owe him gratitude for his introduction to 
Ptolemaic astronomy and his commentary on Book I of the 

18 Ten headmasters seem enough to cover a period of 150 years. 

18 In our list of the last ten headmasters of the Academy, only seven are 
of known origin; six of these came from Egypt or Western Asia; only one 
(Plutarchos) was Athenian. Simplicios also came from the Near East. 



Elements. That commentary is of considerable value for the 
history of Euclid's sources; much of the information which is 
thus conveyed to us was derived from the lost works of two 
Rhodians, Eudemos (IV-2 B.C.) and Geminos (1-1 B.C.). 
Without Proclos, our knowledge of ancient geometry would be 
considerably poorer than it is. 

Marinos of Sichem wrote a preface to Euclid's Data (ex- 
ercises of geometry), but Damascios did not write the "XVth 
book of Euclid" ascribed to him. 

The greatest mathematician who flourished at Athens in 
the sixth century has not yet been named, for he was not a 
headmaster of the Academy, that is, Simplicios (VI-1) . His 
Aristotelian commentaries contain many items of mechanical 
and astronomical interest and he composed a commentary on 
Euclid I. The Cilician Simplicios and the Egyptian Philoponos 
were the outstanding men of science of their age. 

One last remark about the Academy. From the end of the 
third century, it was the only philosophical school left in 
Athens, but that was at the price of its own integrity. The 
Academy had ceased for centuries to be Platonic; not only was 
its prevailing philosophy Neoplatonic but it gave hospitality 
to other philosophies and was ready to discuss them all and to 
syncretize them. Syrianos, Proclos, Marinos wrote commen- 
taries on Aristotle; Simplicios wrote one on Epictetos. 

In addition to the mathematical schools of Alexandria and 
Athens there was also in the first half of the sixth century a 
new school in Constantinople, illustrated by Isidores of Miletos 
and his pupil Eutocios of Ascalon, but their main activities 
were probably posterior to the closing of the Academy. 14 The 

14 And hence outside of the scope of this lecture. The same may be 
said of Philoponos and Simplicios. 



Constantinopolitan mathematicians were probably Christians, 
not so any of the others, except Philoponos, who was a Mono- 

We have spoken of a dozen mathematicians. Instead of 
considering the tradition of each of them, we shall restrict our- 
selves to five, Pappos, Serenos, Theon, Hypatia and Proclos. 

The tradition of Pappos is exceptional in that it involves 
Armenian literature, for Moses of Chorene (V-l), who had been 
educated in Alexandria, wrote in Armenian a Geography 
which was based on Pappos' lost work ad hoc. The commen- 
tary on the Almagest was amplified by Theon; his commentary 
on the Elements of Euclid was used by Proclos and Eutocios. 
The part of it devoted to Book X, lost in Greek, was preserved 
in the Arabic version of Abu 'Uthman al-Dimishqi (X-l) . 
Abu-1-Wafa' (X-2) derived his knowledge of the solid poly- 
hedra from Pappos' Collection. 

The first Greek edition of the Almagest (Basel, J. Walderus, 
1538) 15 included Pappos' commentary to Book V. 

The first printed edition of the Collection was the Latin 
translation from the Greek by Federigo Commandino (Pesaro, 
Hier. Concordia, 1588), reprinted in Venice, 1589, and Bolog- 
na, 1660. The first complete edition of the Greek text appeared 
only three centuries later; it was admirably prepared by 
Friedrich Hultsch (3 vols., Berlin, 1876-78) , 16 

William Thomson: The Commentary of Pappos on Book X 
of Euclid's Elements, Arabic text and translation (Cambridge, 
Harvard, 1930; Isis 16, 132-36). 

16 Facsimile of title page in Isis 36, 256. 

16 Hultsch's edition was a model followed by later editors of Greek 
mathematical texts such as Heiberg. For Friedrich Hultsch (1833-1906), 
see Tannery, Mdmoires 75, 243-317; Isis 25, 57-59) . 



Adolphe Rome: "Pappus, Commentaire sur les livres 5 ct 
6 dc 1'Alniageste" (Studi c testi 54, Vatican, 1931; Isis 19, 381), 
Greek text. 

Paul Ver Eecke: Pappus. La Collection mathematique (2 
vols., Bruges, 1933; Isis 26, 495), French translation. 

The early tradition of Sercnos was mixed up with the 
Apollonian tradition in both Greek and Arabic. The first 
printed text was the Latin version which Federigo Com- 
rnandino published in his Apollonios (Bologna, Alex. Benatius, 
1566). The first Greek edition was included in the splendid 
Greek-Latin edition of Apollonios by Edmund Halley (Ox- 
ford, 1710). New Greek-Latin edition by J. L. Heiberg (Leip- 
zig, 1896). French translation by Paul Ver Eecke (208 pp., 
Bruges, 1929; Isis 15, 397). 

Theon's commentary on the Almagest, as revised by his 
daughter Hypatia, was known to the Byzantine mathematicians 
Nicolaos Cabasilas (XIV-2) and Theodoros Meliteniotes (XIV- 
2). It was included in the first Greek edition of the Almagest 
(Basel, 1538). A new Greek edition, with French translation, 
was begun by Nicolas Halma (Paris, 1813-16). An exemplary 
edition of the Greek text was begun by Adolphe Rome in 
1936; thus far, it extends to Books I-IV (Vatican, 1936-1943; 
Isis 28, 543; 36, 255) ; the continuation is being prepared by 
his disciple, Joseph Mogenet. 

Proclos ,was far more popular as a philosopher, theologian, 
and even as a physicist than as a mathematician, and the tra- 
dition of his many writings is very complex. We shall consider 
here only his mathematical work. Isaac Argyros (XIV-2) re- 
vised his commentary on the arithmetic of Nicomachos. His 
commentary on Euclid, Book I, was first printed in Greek in 
Simon Gryneus' Greek edition of Euclid (Basel, Hervagius, 
1533). Latin editions were prepared by Franciscus Barocius 
(Padova, Gratiosus Perchacinus, 1560) and by Federigo Com- 
mandino with Euclid (Pesaro, 1572). Critical Greek edition by 
Gottfried Friedlein (515 pp., Leipzig, 1873). French transla- 
tion by Paul Ver Eecke (396 pp., Bruges, 1948; Isis 40, 256). 



The tradition of the final mathematical achievements of 
Hellenism is curious in at least two respects. In the first place, 
it hardly involved the Arabic detour, except in the case of 
Pappos. Their rediscovery was largely due to Byzantine 
scholars and later to Renaissance ones, with the result that 
Greek printed editions were anterior to the Latin ones, except 
in Serenos' case. As far as the Latin tradition is concerned, 
the lion's share was done by Federigo Commandino of Urbino 
(1509-75), especially if one considers that he was the first to 
publish Pappos' Collection, the influence of which upon later 
mathematicians was considerable. 


For the sake of simplicity it will be best to deal with only 
one physician, the greatest of this age, 17 Oribasios (IV-2), and 
we call him Byzantine rather than Greek or Hellenistic be- 
cause he was a physician to the Byzantine court in Con- 
stantinople. Oribasios was born in Pergamon like his pre- 
decessor Galen (II-2), of whose fame he was the main artisan. 
His greatest work was a medical encyclopedia, latricai syna- 
gogai, of such immense size that only one third of it has come 
down to us; the original extended to seventy books. 18 It is of 
great value for historians, for it has helped to preserve a good 
many earlier medical texts which would have been lost other- 
wise; its numerous quotations are always referred to their 

1T Aetios of Amida (VI-1), Justinian's archiater, comes just after its 
end. For a general view of Byzantine medicine, see Isis 42, 150, or my 
Philadelphia lectures (1954). 

" We have only Books I to XV, XXI-XXII, XXIV-XXV, XLIV-LI, with 
lacunas a total of less than 27 books. 



authors. Oribasios was befriended by Prince Julian, 19 became 
his physician, and was almost the only person to whom the 
latter revealed his apostasy. In 355, when Julian was made a 
Caesar and sent to Gaul, he took Oribasios with him. During 
his brief rule (361-63), he appointed him quaestor of Con- 
stantinople and charged him to go to Delphoi in order to con- 
sult the oracle and possibly revive its glory; that undertaking 
ended in failure 20 but Julian did not take it ill and con- 
tinued to favor his physician. He encouraged him to write his 
medical encyclopedia, and when he started on his last cam- 
paign against Persia, Oribasios went with him and was with 
him at Antiocheia and at the moment of his death on the 
battlefield on 26 June, 363. It is clear that Oribasios shared 
the pagan faith of his master. This is sufficiently proved by 
the facts already mentioned but also by the persecution which 
he suffered after his protector's death. The Christian emperors 
who followed Julian the Apostate, Valens and Valentinian, 

"Julian, born in Constantinople in 331, was but a few years younger 
than Oribasios, born c. 325. While he was wintering in Paris, 358-59, 
Julian wrote to Oribasios, then in Vienne, a letter the terms of which 
prove their intimacy. 

90 According to Georgios Cedrenos (flourished, eleventh /twelfth cen- 
tury) , author of a world chronicle from the Creation to 1057, the oracle 
of Apollon gav this answer: 

"Tell the king, on earth has fallen the glorious dwelling, 

"And the watersprings that spake are quenched and dead. 

"Not a cell is left the God, no roof, no cover, 

"In his hand the prophet laurel flowers no more." 

(Swinburne's version in The Last Oracle). The sacred oracle foretold 
the end of paganism! 

If one wishes to understand how the Pythian prophetess functioned, 
he should read Herbert William Parke, History of the Delphic Oracle 
(Oxford, 1939; Isis 35, 250). A similar institution is still functioning today 
in Tibet and was observed and described by Heinrich Harrer, Seven 
Years in Tibet (pp. 180-82, London. 1953) . 



confiscated Oribasios' estates and drove him into exile. Ori- 
basios flourished for a time at the court of barbarian (Gothic?) 
kings and distinguished himself so well that he was recalled to 
Constantinople, c. 369. His goods were restituted to him and 
he was permitted to continue his medical practice and writing. 
He died c. 400. 

He is a good example of the transition between paganism 
and Christianity. It is possible that he had been brought up as 
a Christian even as Julian was, but that under the latter's 
ascendency his pagan feelings 21 were revived. According to 
Eunapios (V-l) , he studied medicine under Zenon of Cypros, 22 
and sat at the latter's feet at the Museum together with Magnos 
of Antiocheia, the latrosophist. Both Zenon and Magnos were 
pagans. Julian died too young (at thirty-two) to recant; Ori- 
basios lived until- he was about seventy-five; we may safely as- 
sume that he became a Christian again and died as such, for 
paganism was no longer acceptable either in the empire or in 
the barbarian kingdoms. His son Eustathios, to whom his 
Synopsis is dedicated, was a Christian and a friend of St. Basil 

The purpose of Oribasios' Medical Collection is so well ex- 


The word feelings is the correct one, for the main cause of attach- 
ment to paganism was not rational but sentimental, the love of the 
ancient cult and liturgy. The situation is similar to that of Catholics who 
become Protestants, but in the course of time cannot bear any longer the 
loss of sacramental aids and of the sacred liturgy and music, and return 
to their original faith. 

as Zenon was eventually driven out of the Museum by Georgios of 
Cappadocia (Arian bishop of Alexandria, 356-61) but reinstated by 
Julian. The founder of Stoicism, Zenon of Cition (IV-2 B.C.) is some- 
times called Z6non of Cypros, but there can be no confusion between two 
men separated by seven centuries. 



plained at the beginning of it that it is best to quote his own 

Autocrator lulian, I have completed during our stay 
in Western 23 Gaul the medical summary which your 
Divinity had commanded me to prepare and which I have 
drawn exclusively from the writings of Galen. After having 
praised it, you commanded me to search for and put to- 
gether all that is most important in the best medical books 
and all that has contributed to attain the medical purpose. 
I gladly undertook that work, being convinced that such a 
collection would be very useful. ... As it would be super- 
fluous arid even absurd to quote from the authors who have 
written in the best manner and then again from those who 
have not written as carefully, I shall take my materials ex- 
clusively from the best authors, without omitting anything 
which I first obtained from Galen, and I shall adapt my 
own compilation to the fact of his superiority; Galen used 
the best methods and the most exact definitions, because he 
follows the Hippocratic principles and opinions. I shall 
adopt the following order: hygiene and therapeutics, man's 
nature and structure; conservation of health and its restora- 
tion, diagnosis and prognosis; correction of diseases and 
symptoms, etc. 

My rough translation of the preface tells us the essential: 
Julian was really Oribasios' patron and animator, and Galen 
was the main source, to which every other source was sub- 
ordinated. Galen's perfection was ascribed partly to the ex- 
cellence of his own source, Hippocrates. Oribasios' references 
to Galen are innumerable and his praise of him so frequent 

13 Western Gaul as opposed to Eastern Gaul or Galatia in Anatolia, 
with which Oribasios and Julian were more familiar. As Oribasios com- 
pleted his summary in Gaul, we may assume that part of it at least was 
written in Paris. 



and emphatic that it established Galen's superiority as a kind 
of medical dogma. 

The books of the Synagogai which have come down to us 
are Book I, 1-65, II, 1-27, Plant foods. II, 28-58, Animal 
foods; 59-69, Milk, cheese, honey, horse flesh and flesh of 
other solipeds, generalities. Ill, Various kinds of foods, 
divided according to their physiological properties. IV, 
Preparation of various kinds of food. V, Beverages. VI, 
Physical exercises. VII, 1-22, Bloodletting. VII, 23-26, VIII, 
Purgatives, diuretics, emetics, hemagogues. IX, 1-20, Air, 
climates of various localities. IX, 21-55, External remedies, 
such as fomentations, cataplasms, poultices, embrocations, 
cupping. X, 1-9, Water, sand and air baths. X, 10-42, Ex- 
ternal remedies. XI-XIII, Materia medica (copied verbatim 
from Dioscorides but in alphabetical order). XIV-XV, 
Simple drugs. XVI (only a short fragment), Composite 
drugs. (XVI-XX lost.) XXI. Elements and temperaments. 
XXII, Generation (XXIII lost.) XXIV, Internal organs, 
from the brain to the sexual parts. XXV, Anatomical 
nomenclature, Bones and muscles (57 chapters), Nerves 
and vessels (4 chapters) . 

XLIV, Inflammations, tumors, abscesses, fistulae, gang- 
rene, erysipelas, herpes, boils. XLV, Tumors. XLVI, Frac- 
tures. XLVII, Dislocations. XLVIII, Slings and bandages. 
XLIX, Apparatus used to reduce luxations. L, Genito- 
urinary troubles, Hernias. LI, Ulcers. (LII-LXX are lost.) 

These books plus fragments from the lost ones were 
edited in Greek and French by Ulco Cats Bussemaker and 
Charles Victor Daremberg in four thick volumes (Paris, 
1851-62). Two more volumes of the same magnificent edi- 



tion were published posthumously by Auguste Molinier. 
Vol. 5 (1873) contains Oribasios' Synopsis' 24 (medical sum- 
mary) in nine books dedicated to his son Eustathios, and 
his Euporisla (Remedia parabilia, home medicine) in four 
books dedicated to Eunapios, plus ancient Latin versions 
of the Synopsis and Latin additions to the Greek text. Vol. 
6 (1876) contains more ancient Latin versions of the 
Synopsis and Euporista and an elaborate index to the six 

It is well nigh impossible to assess the intrinisic merits of 
such a bulky legacy as Oribasios' is. It gives us a clear idea of 
the medical experience available in the second half of the 
fourth century; that experience and knowledge were essentially 
of pagan origin, and we may call Oribasios the last of the 
pagan doctors as well as the first of the Byzantine age. 

The Oribasios tradition was triple Latin, Greek and 
Arabic. The Latin versions edited by Molinier (1873-76) go 
back, some of them, to the sixth century; the earliest were 
made in Ravenna during the Ostrogothic period (489-554) ; 
others were made in the seventh and eighth centuries. These 
Latin versions have transmitted to us parts of the text lost in 
the original Greek. They were made when Oribasios was re- 
latively modern and when relations between the Latin and 
Greek worlds were still frequent. 

The main tradition was Greek, however; the other By- 
zantine physicians Aetios of Amida (VI-1), Alexandros of 

24 Would this be a revised edition of the summary which Oribasios 
completed for Julian in Gaul before the compilation of his Synagogai? 
See Oribasios' preface quoted above. 



Tralleis (Vl-2) , Paulos of Aigina (VI I- 1), etc., were to some 
extent dependent upon it. 

The Arabic tradition, instead of being anterior to the Latin 
and the basis of it, was much posterior. The only Arabic 
versions of Oribasios were made by 'Isa ibn Yahya (IX-2) and 
perhaps by Stephanos, son of Basileios (IX-2). The Arabs paid 
more attention to Aetios, Alexandros, and especially to Paulos 
than to Oribasios, and even more to the latter's sources, Hip- 
pocrates and Galen. Galen's extraordinary fame was built 
up gradually by Oribasios, by the other Byzantine physicians, 
by the Arabic ones and by Latin doctors of the thirteen century 
and later; it reached its natural culmination during the 

There are no incunabula editions, but a number of Latin 
editions appeared in the sixteenth century. Most of them were 
restricted to a part of his writings but Giovanni Battista 
Rasario attempted to publish the Opera omnia (Basel, Is- 
ingrinius, 1557) ; reprinted in Paris, 1567. Greek editions were 
fewer in number in the sixteenth century, partial and small. 
The largest of the early Greek-Latin editions (Books I to XV 
of the Collection) was prepared by Christian Friedrich de 
Matthaei and published by the Imperial University of Moscow 
(1808) . The first complete edition of the Greek text (as com- 
plete as it could be) was the Greek-French edition of Busse- 
maker, Darembergand Molinier (6 vols., Paris, 1851-76), which 
has already been mentioned, because it is the most convenient. 
A more critical edition of the Greek text is included in the 
Corpus Medicorum Graecorum, Part VI, the Opera Omnia 
edited by Joannes Raeder (1926-33). General indices are being 
prepared by M. Haesler; in the meanwhile, the Greek-French 
edition is indispensable. 




The reader may be astonished by the fact that most of the 
men of science of whom I have spoken were pagans (or were 
pagans most of the time) and exclaim, "How could that be 
after more than three or four centuries of missionary efforts?" 
The situation was extremely complex. 25 Philosophical teach- 
ing continued; that teaching was essentially pagan, restricted 
to Neoplatonism and mixed up with various forms of mystic- 
ism. Stoicism was very strong but was also befouled with 

The old mythology had become untenable, but the 
mysteries, cults and liturgies were still popular among all 
classes. As far as the educated and sophisticated people were 
concerned, the myths were treasured only as a form of national 
poetry but had been otherwise replaced by the astral religion, 
which favored astrological delusions and was in turn fostered 
by them. This was much too learned and too objective for the 
common men and women who craved a living faith and a 
religion which was personal, emotional, and colorful. Those 
cravings were satisfied in varying degrees by a number of 
oriental religions, 26 of which Christianity was for a long time 
the least conspicuous. The development of Christianity, early 
and late, is one of the mysteries of the world; it is the sacred 
mystery in the highest sense. The events which guided the 
Church and caused its final triumph in the face of innumerable 
calamities are so incredible, or call them miraculous, that 

25 The following discussion concerns only the Greek world, and this 
means southeastern Europe and the Near East. 

86 Masterly account of them by Franz Cumont, Les religions Orientates 
dans le Paganisme Romain (4th ed., Paris, Geuthner, 1929; Isis 15, 271). 



Christian apologists have used them as clinching proofs of the 
truth and superiority of their faith. 

One of the most astonishing factors is the pre-eminence in 
the earliest times of the poorest people, those who were despis- 
ed and downtrodden. The men who had the least amount of 
social influence were the main agents of the revolution which 
changed the whole world. It was only later arid very gradually 
that men of substance joined the catechumens. That story is 
so well known that I need not repeat it here. Let us make a 
big jump to the time which we are now contemplating. It 
was beautifully introduced by a woman of humble parentage, 
the daughter, it is said, of an innkeeper, Helene, who became 
the mistress of Constantios, a Roman officer. A child was 
born to them at York, c. 274, named Constantine, and the 
parents were then duly married, but when Constantios was 
elevated to the Caesarship in 292, he was obliged to put her 
aside in order to marry one who was more respectable. Con- 
stantios Chloros was emperor from 305 to 306, his son Con- 
stantine the Great, from 306 to 337. 

Constantine was the first emperor to support Christianity. 
In 313, he issued the Edict of Milan, securing toleration for 
the Christians throughout the empire, and the official recog- 
nition of Christendom occurred soon afterward. By 324, 
Christian monograms became prominent on the coinage. Con- 
stantine moved his capital away from Rome which was still 
a stronghold of paganism and established it in 326 on the site 
of Byzantion; the new city was called after himself, Con- 
stantinople, inaugurated in 330 and dedicated to the Holy 
Virgin. Constantine was called the Great; he was really a 
little man, but he saw visions and took momentous decisions; 
he caused the political success of Christianity and the relega- 



tion of paganism, and he elaborated the comprehensive and 
absolute authority of the Autocrator in church and state. His 
many sins and crimes were washed away when he was bap- 
tized by Eusebios of Caisareia (IV-1) not long before his death, 
which occurred near Nicomedeia in 337; he was buried in his 
own city, Constantinople. 

It is possible that Constantine called his mother to the 
imperial court in or after 306, and that after his own conver- 
sion to Christianity in 312 he converted her (it is also said 
that it was she who converted him) . Various crimes committed 
by Constantine were probably the cause of her vow, when 
already eighty years old, to make a pilgrimage to the Holy 
Land. She accomplished the pilgrimage and discovered the 
True Cross in Jerusalem, on the third of May, 326. 27 She died 
not long afterwards, say in 327 or 328 (in Rome?); the places 
of death and burial are not known. She never was an em- 
press, even for a short time, but was eventually canonized 

After Constantirie's death in 337, his three sons ordered the 
murder of other members of the imperial family, but two of 
his nephews, the brothers Gallos and Julian were spared. The 
younger one, Julian, who interests us more deeply, vvas born 
in Constantinople in 331. After his mother's untimely death, 
he was put under the care of Eusebios, bishop of Nicomedeia, 28 

"The feast of the Invention of the Cross (Inventio S. Cruets) is 
celebrated on May 3. It is given far more importance by the Orthodox 
churches than by the Catholic or Anglican. 

28 Not to confuse Eusebios of Nicomedeia (d. 343) with Eusebios of 
Caisareia (c. 265-340) , the historian, he who baptized Constantine the 
Great in extremis. They were close contemporaries and both attended 
the Council of Nicaia (325). Julian refers to the latter in his Letter to 
the Galilaeans. 



one of the most active defenders of Arianism. When Eusebios 
died in 343, Julian was sent by the emperor to a castle in the 
highlands of Cappadocia, where he remained six years in 
solitary confinement. When his elder brother, Gallos, was 
appointed Caesar in 35 1, 29 Julian was permitted to return to 
Constantinople, where he continued his Hellenic and Chris- 
tian studies. Soon afterwards, he was sent to Nicomedeia, 
where he acted as lector (anagnostes) in the local churches, 
yet was friendly with the sophist Libanios, whose lectures he 
had been forbidden to attend. A little later, he went to 
Pergamon, then to Ephesos to commune with Maximos, Neo- 
platonic wonderworker and theurgist (Ihaiirnaturgos, thcurgos), 
and it was probably in that sacred city that his apostasy was 
completed. Julian was initiated to Mhhraism :m about the 
year 352, for he. wrote in one of his letters that he had been a 
Christian until his twentieth year; 31 his apostasy was kept 
secret, however, tor ten years. The confusion of his mind is 
shown by ihe fact that being in Athens in 355, he followed lec- 
tures of the Christian teacher Prohairesios (St. Gregory Nazian- 
zen and St. Basil being possibly among his classmates) and yet 
was initiated to the Eleusinian mysteries. In the same year, 
355, he was raised to the ranV of Caesar in Milano and then 
ordered to Gaul to drive out the German invaders; in the 

20 Gallos did not enjoy the Caesarship very long, for he was executed 
by imperial order in 354. 

30 The Persian god Mithras had been identified with Helios, Sol 
invictus. Joseph Bidez has shown that Mithraist influences had been 
operating in Julian's family, beginning with his grandfather, Constantios 
Chloros. Therefore, Julian fancied that he was a descendant of Helios. 
This helps to understand his apostasy. J. Bidez, "Julien 1'Apostat" (Revue 
de ['instruction publique 57 [1914], 97-125, Bruxelles). 

31 Letter 47 to the Alexandrians, 434 D (Loeb ed., 3, 149) . 



course of that campaign he was able to redeem some 20,000 
Gallic prisoners. Julian proved himself to be a good soldier, 
a clever general and a capable administrator; he did so well 
indeed that the emperor took umbrage at him and tried, in 360, 
to withdraw part of his army, but the soldiers raised Julian 
on their shields and nominated him their emperor. In January 
361, he attended the feast of the Epiphany in Vienne (on the 
Rhone) , then moved his army across Europe. During his 
passage through Naisos 32 in the same year he addressed to the 
Roman Senate and to the peoples of Sparta, Corinth and 
Athens manifestoes proclaiming the revival of the Hellenic 
religion. The rival emperor, Constantios, died and Julian 
entered Constantinople as sole emperor at the very end of 
the year. In the following year (362), he began his fateful 
campaign against the Persians and was killed on the battle- 
field, somewhere east of the Tigris, on 26 June 363, at the age 
of thirty-two. 

Julian had been all his life, with increasing fervor, an 
enthusiastic lover of Hellenism; he was initiated into vari- 
ous Greek and oriental mysteries, but as soon as he found 
himself a soldier in the field he gave his full devotion to 
Mithras, who was the favorite god of the Roman legions. On 
4 February 362, he proclaimed religious freedom 83 and ordered 
the restoration of the temples. He showed friendliness to the 
Jews, restored Jerusalem to them and permitted them to re- 
build the "Temple of the most high God"; the building had 

88 Naisos or Nissa, Nish in eastern Yugoslavia, the very birthplace of 
Constantine the Great in 306. 

** Julian's edict of toleration of 362 was the counterpart of Constantine's 
edict of half a century before (313) , but Constantine asked freedom of 
religion for the Christians and Julian for the pagans. Constantine's edict 
was slanted against the pagans, Julian's against the Christians. 



soon to be stopped, however, because of the earthquakes of 
the winter 362-63 and of the Persian war. Julian tried to be, 
if not impartial, at least tolerant, but as resistance to his 
proselytism increased, he became impatient and more and 
more intolerant. He gave special privileges to the pagans and 
withdrew those which Christians had enjoyed. The main 
troubles were caused by his efforts to suppress or restrict 
Christian education. He would have liked to avoid violence, 
but the old pagans who had never been Christians except in 
name, if at all, as soon as they escaped Christian persecution, 
naturally abused their new freedom and began their own 
destruction of men and properties. One of their outstanding 
victims was Georgios of Cappadocia, 34 the Arian bishop of 
Alexandria, against whom great savings of hatred had ac- 
cumulated because of his own persecutions. He ventured to 
build a new church upon the ruins of a Mithraion and in- 
furiated the populace; he was murdered and his body ignomin- 
iously handled by the crazy mob. This happened on 24 
December 361, that is, on the eve of the Mithriac feast, Natalis 
invicti, now replaced by our Christmas. 

34 In the Decline and Fall (chap. 23), Gibbon speaks very harshly of 
him, concluding, "The odious stranger, disguising every circumstance of 
time and place, assumed the mask of a martyr, a saint, and a Christian 
hero, and the infamous George of Cappadocia has been transformed into 
the renowned St. George of England, the patron of arms, of chivalry and 
of the garter." Gibbon confused two different martyrs, Catholic and 
Arian. St. George of England or George the Martyr, probably an officer 
in Diocletian's army, was beheaded at Nicomedeia in 303, when Arian ism 
did not yet exist (Areios began to teach his doctrine c. 318). George of 
Cappadocia was an Arian; it is interesting to note that Julian seems to 
have had more to do with Arians, as friends or adversaries, than with 



As soon as Julian heard of this atrocious murder, he wrote 
two letters (from Constantinople, January 362), one to the 
Alexandrians to rebuke them mildly (he gave them "an advice 
and arguments," paraincsin cai logus), the other to the Prefect 
of Egypt, demanding Georgios' library, which he had had 
occasion to use in his youth. This second letter does not con- 
tain a word of regret or of blame for the murders. It is 

It is clear that in the end Julian's mind was distorted by 
violent anti-Christian prejudices, yet he was, or had been, a 
very intelligent man of superior morality. This is remarkable, 
if one remembers the terrible vicissitudes of his life. 35 

The last words ascribed to him, necicecas Galilaie (Thou 
hast conquered, o Galileian), are legendary and paradoxical, 
for he died at the head of an army which must have included 
many Christian soldiers. The defeat of a Byzantine army by 
Persian barbarians was a defeat for the empire which was still, 
in spite of Julian's apostasy, a Christian empire. 

Bibliography of Julian 

Greek-Latin edition of Julian's works, Qiiae extant omnia 
by Petrus Martinius and Carolus Cantoclarus, i.e., Pierre 
Martini and Charles de Chanteclair (4 parts in 1 vol., Paris, 
Duvallius, 1583). 

The works of Julian were edited in Greek by Friedrich 
Carl Hertlein (2 vols., Teubner, Leipzig, 1875-76), in Greek 

35 The vicissitudes of Julian's life were so strange and momentous 
that they soon became legendary. Richard Forster, "Kaiser Julian in der 
Dichtung alter und neuer Zeit" (Studien zur vergleichenden Literaturge- 
schichte 5, 1-120, Berlin, 1905). As to the modern literature inspired by 
Julian's fate, it will suffice to recall the names of Voltaire, Alfred de Vigny, 
Ibsen and Merezhkovski. 



and English by Mrs. Wilmer Cave Wright 36 (Loeb Library, 
3 vols., 1913-23); in Greek and French by Joseph Bidez (Assoc. 
Guillaume Bude, Paris, 1924 if., Isis 7, 534) . 

For the very interesting Syriac legend, see Georg Hoffmann, 
Julianas der Abtriinnige, Syrische Erzdhlungen (Leiden, 1880). 
Richard J. H. Gottheil: "A selection from the Syriac Julian 
romance, with complete glossary in English and German" 
(Semitic Study Series, no. 7, 112 pp., Leiden, 1906). Sir Her- 
mann Gollancz, Julian the Apostate, now translated lor the 
first time from the Syriac original (the only known manuscript 
in the British Museum, edited by Hoffmann of Kiel) (264 pp., 
London, 1928). 

It is impossible to know how much the Greek people were 
influenced by Julian's apostasy. How many of them were un- 
regenerated pagans, how many converted ones, how many born 
Christians? How many temples had continued to function, 
openly or secretly, before Julian's rule? How many churches 
or monasteries were closed during it? The rule was too short 
to do irreparable harm. 

The period of Julian's life was one of great theological 
activity because of the existence of various heresies. Not only 
that, but one of the heresies, Arianism, was orthodoxy itself 
during the greatest part of that time. It was condemned by 
the Council of Nicaia, 87 325, then again by the Council of 
Constantinople, 381; yet after the death of Constantine in 337, 
it became the orthodox doctrine and remained so, roughly, 
until 378. To be more precise, out of the fifty-six years 

3e Professor in Bryn Mawr, died 1951 (Isis 43, 368). 

87 Nicaia (= Nice, Isnik) was not far from Nicomedea, so often men- 
tioned above. These were the two leading cities of Bithynia, disputing the 
title of metropolis. Nicomedeia is at the east end of the Propontis (Sea of 
Marmara), Nicaia at the east end of Lake Ascania, south of Nicomedeia. 



separating the first two councils of the Church, forty were 
years of Arian ascendency. Ulfilas, apostle of the Goths, was 
consecrated bishop by Eusebios of Nicomedeia in 341, during 
the Arian supremacy, and therefore the Gothic and other 
Germanic tribes remained Arian for centuries. 

Yet, the Catholic doctrine was very ably defended by the 
Nicene and post-Nicene Fathers of the Church. Of the ten 
generally mentioned, 38 no less than nine lived or began to live 
during Julian's life. They are St. Athanasios of Alexandria 
(d. 373), St. Basil of Cappadocia (d. 379), St. Gregory of 
Nazianzos (d. 389), St. Gregory of Nyssa (d. 395) , St. Ambrose 
of Treves (d. 397), St. Epiphanios of Palestine (d. 403), St. 
John Chrysostom of Antioch (d. 407) , St. Jerome of Dalmatia 
(d. 420), St. Augustine of Tagaste (d. 430). (The tenth one, 
St. Cyril of Alexandria, was born only in 376, many years 
after Julian's death; we shall come across him presently) . All 
of these Fathers were Greek, except three of them, Ambrose, 
Jerome and Augustine. Julian was well acquainted with at 
least three of the Fathers, Athanasios, Basil and Gregory 
Nazianzen. Athanasios was the main opponent of Arianism 
from the beginning, and his life is the best symbol of the 
ecclesiastical vicissitudes of that turbulent age. He was bishop 
of Alexandria for forty-seven years, but spent about twenty 
years away from his see, being exiled or driven into hiding 
five times. We have recalled above that at the time of Julian's 
accession, the very see of Alexandria was held by an Arian 
bishop, Georgios of Cappadocia (bishop of Alexandria from 
356 to 361) . 

It is noteworthy that in spite of the fact that the Empire 


E.g., in my Introduction (3, viii). 


had become Christian soon after 313, the pagan schools con- 
tinued to function, chiefly, the Academy of Athens and the 
Museum of Alexandria. The Christians had their own schools, 
but none had yet obtained a prestige comparable to that which 
the pagan institutions enjoyed. In Alexandria, an ambitious 
Christian school, the Didascaleion, had been made illustrious 
by Clement of Alexandria (150-220) and Origen (IIM), but 
it is doubtful whether it still flourished in the end of the fourth 
century. The Museum, however, was thriving, and we have 
already spoken of two illustrious teachers, Theon and his 
daughter, Hypatia, the leading mathematicians of their time. 
St. Cyril, who became bishop of Alexandria in 412, decided to 
put an end to pagan and Jewish learning. He persecuted the 
Jews and drove them out of the city. It was din ing his rule 
that Hypatia was murdered by a Christian mob in 415. She 
was dragged into a Christian church, entirely divested and her 
body torn to pieces. Cyril died in 444, was canonized by Leo 
XIII and proclaimed a Doctor of the Church. 39 

Julian's apostasy and Hypatia's martyrdom are two drama- 
tic events of very great significance, but we must be careful 
not to misunderstand them as has been done repeatedly by 
anti-clerical writers. Neither of them was a champion of free 
thought. Julian was a Mithraist and a passionate defender of 
Hellenism; his revival of paganism was a very queer one be- 
cause it involved oriental religions of which the ancient Greeks 
knew but little or nothing. He was a pagan mystic who ignored 

80 St. Cyril of Alexandria (376-444) should not be confused with his 
elder contemporary, St. Cyril of Jerusalem (c. 315-86) , who was Patriarch 
of Jerusalem in 350, but was driven out by the Arians; he was permitted 
to return to Jerusalem only in 379, and died there in 386. He took part 
in the Council of Constantinople in 381. 



the best part of rational Hellenism. It would not be fair to 
reproach him for his neglect of Greek science, but even in the 
field of morality, he was not well acquainted with the best 
thought or did not understand it. He admired equally Alex- 
ander the Great and Marcus Aurelius but was very remote 
from both; his Persian campaign may have been inspired by 
the first, but he never tried to continue Marcus' effort. He 
liked virtue but lacked Marcus' passion for it, his deep kindness 
and sanctity. 

As to Hypatia, she was a Neoplatonist, not in any sense a 
free thinker. She was very superior to Julian in that she 
loved science more than myths; as a scientist, she was bound 
to strive for objectivity and precision, while Julian was a man 
of letters, a mystic and a mythomaniac. Socrates might be 
called a martyr of freedom of thought; she was rather a martyr 
of science, the first, or one of the first, known to us. 

To understand fairly the attitude of both of them, one 
must realize that in their time the defense of Hellenic tradi- 
tions was the best rearguard action against Christian advance; 
they were not so much anti-Christian as passionately Greek. 

Jn this period of transition and spiritual travail, Hellen- 
ism tried to take a religious form, and Christianity, a philoso- 
phical one; Christianity was struggling hard to establish an 
ecumenical orthodoxy against heretical distortions. They 
could not meet, however, because it was impossible to accept 
Christian doctrines without Christian faith, and the Greeks 
were unwilling to abandon their mythological poetry, which 
was the very core of Hellenism. 

The educated pagans and the Christians were equally 
capable of enthusiasm and ecstasy but their theological con- 
ceptions were utterly incompatible. 



The general situation in the fourth and fifth centuries 
was this. Whatever scientific work was done in the Greco- 
Roman world was done chiefly, if not exclusively, by pagans. 
In spite of Greek and oriental cults, the Church was gaining 
ground steadily, but its unity was jeopardized by schisms. 

The fundamental progress of the Church, without which 
no later progress would have been possible, was due to the 
generous faith of the humbler people. This is the best ex- 
ample throughout the ages of the essential goodness of the 
masses. By and by, men of substance joined the little men, and 
finally the princes and rulers came in, but the Christian em- 
perors were seldom good men; none was as good as Antoninus 
Pius or Marcus Aurelius. In other words, even after Con- 
stantine's recognition, the Church continued to be saved and 
vindicated by saints and by men and women who were poor 
and weak rather than rich and powerful. 

As soon as Christianity was officially recognized in or soon 
after 813, it was necessary to define the creed with greater 
precision, and this was the source of endless difficulties. The 
definition of each dogma was bound to instigate alternatives 
in the minds of sophisticated theologians, quarrelsome and 
vain, jealous of their spiritual authority. It was extremely 
difficult, if not impossible, to reconcile on rational grounds 
the notions of monotheism and Trinity; what were the rela- 
tions of Jesus Christ to God and to man? Areios began to 
preach c. 318 that God is absolutely unique and separate and 
he denied the eternity and divinity of Christ. This heresy was 
received so favorably by many clerks that Constantine was 
compelled to summon the first Council at Nicaia in 325 in 
order to discuss it and to push it out. The Nicene Creed re- 
jected Arianism. In spite of that, Arianism enjoyed consider- 



able popularity, was countenanced by emperors until 378, and 
remained the orthodox doctrine of the Teutonic tribes for 
centuries. It is very remarkable that that heresy, the first great 
one, was so bold that the sixteenth century Socinianism and 
later Unitarianism may be considered as stemming from it. 

Arianism was condemned again by the second Council in 
Constantinople in 381 and from that time on was driven out 
of the Byzantine orthodoxy. New heresies diverged from the 
accepted dogmas as to the nature of Christ in two opposite 
directions. The orthodox view was, then and now, that there 
are two natures in Christ (human and divine) but one person. 
The followers of the Syrian priest, Nestorios (V-l) , claimed 
that there are in Christ two natures and two persons. Eutyches, 
archimandrite of a monastery near Constantinople, fought the 
Nestorians so hard that he fell into the opposite error. He 
created the heresy named after him Eutychianism and later 
Monophysitism. Eutyches claimed that the divine and the 
human are so blended in the person of Christ as to constitute 
but one nature; Christ is of two natures but in one nature. 
The Monophysites declared more bluntly that there is in 
Christ but one nature and one person. 

These Christological differences went very close to rending 
apart the seamless coat of Christ. The various kinds of Chris- 
tians hated one another more than they hated the infidels. The 
Nestorian heresy was condemned by the third Council, in 
Ephesos, 431; the fourth Council, in Chalcedon, 451, anathe- 
matized the Eutychians as well as the Nestorians. 

Condemnations and curses were rapidly enforced by eccles- 
iastical and lay officials, and the final result was that many 
good men were either killed or banished. We may assume that 
men who prefer to abandon their homes and business and 



suffer all the rigors of poverty and exile rather than recant or 
dissemble their religious thoughts that such men must be 
exceptionally brave and good. The empire impoverished itself 
to the profit of foreign countries. The Arians had been driven 
westward; the Monophysites swarmed out into Syria and into 
Egypt; the Nestorians emigrated eastward, and the school 
of Edessa was their main center until it was closed by the em- 
peror Zenon the Isaurian in 489. This caused a further dis- 
persion of them; the Nestorian seat was in Seleuceia-Ctesiphon 
in 498, in Baghdad in 762. They swarmed all over Asia as far 
as the Pacific Ocean. 

There was a medical school in Edessa, and the Nestorians 
found themselves there in a scientific community. They trans- 
lated many Greek books of philosophy and science into Syriac 
and these Syrian books were later translated into Arabic. The 
"scientific road" from Alexandria to Baghdad passed through 
Edessa. 40 Thus would be completed in the fulness of time a 
remarkable cycle. Greek science was born in Asia Minor, then 
flourished in Greece proper, chiefly in Athens, then in Alex- 
andria, and back to Asia, Pergamon, Constantinople, Edessa, 

The move from Athens to Alexandria was due to political 
causes, that from Egypt and Greece to Asia very largely to re- 
ligious ones. Every persecution is a centrifugal force. The 
"good Christians" drove the Arians, Nestorians, Eutychians 

40 It may be that when the school of Edessa (modern Urfa) was closed 
in 489 some of the Nestorians took refuge at Jundlshapur in Khuzistan, 
where a medical school was functioning; some of the pagans may have 
resorted to the same place, which became a center of dispersion of 
Greek culture in the Near East (Intro. 1, 435). JundishSpur is at a con- 
siderable distance east of Baghdad, however. 



further and further away and thus helped to diffuse Greek 
science in the Asiatic world. 

We have dealt so long with Christian sects that the reader 
might forget the existence of pagans. There were still pagans, 
especially among the least educated and the best educated 
people. There were undoubtedly pagans (pagani, "rustics" 
in the isolated places, and on the other hand, the "intellect- 
uals," the outstanding philosophers, were reluctant to accept 
Christianity and reject Hellenism. This was especially true 
of those who were privileged to teach in the Academy of 
Athens, which became, as it were, a center of resistance to the 
new religion. Therefore, Justinian closed it in 529. 

This is a fateful date, which I consider to be the best 
symbol of the end of an age. The same year witnessed the 
foundation of Monte Cassino by St. Benedict (Vl-1). Seven 
teachers of the Academy escaped to the court of the king of 
Persia, Chosroes, and remained there a few years until a treaty 
of peace enabled them to return. 

As to the empire itself, a part of its strength and of its vir- 
tue was drained away by each persecution; some of the best 
men were driven into exile some of the worst rose to the surface. 

The final transition from paganism to Christianity was 
difficult enough. It implied conflict of loyalties, the destruc- 
tion of vested interests and the precarious establishment of 
new ones. Moreover, the process was reversed during Julian's 
reign. The situation was enormously aggravated, however, 
by profound discords within the new Christian world. The 
Arians went up and down, the Nestorians and the Mono- 
physites were relentlessly persecuted. By the beginning of the 
Sixth Century the Byzantine empire was weakened in many 



ways, chiefly because it had lost the good will of its own sub- 
jects. The persecution of heterodoxies had been continued too 
long, too many good people had been driven into sulkiness 
and resentment or even into exile. Refugees carried Greek 
science to the East and helped to prepare intellectual weapons 
outside the Christian world, weapons which would soon be 
used against it. 

The Byzantine empire had finally become orthodox in 
fact as well as in name, but it was totering; its material im- 
proverishment was great, the spiritual one extreme. The time 
would soon be ripe for Arabic conquests and no dike would 
be strong enough to resist the Islamic flood. 

Modern science is the continuation and fructification of 
Greek science and would not exist without it. Our lectures 
suggest another conclusion, however, which is more timely 
today than it ever was. 

Intolerance and persecution are self-defeating. The hunger 
for knowledge and the search for truth can never be eradicated; 
the best that persecution can do is to drive out non-conformists. 
In the end this will be a loss not for humanity but for their 
own country. The refugees carry wisdom and knowledge from 
one place to another and mankind goes on. 

Greek scholars were driven out of the Greek world and 
helped to develop Arabic science. Later the Arabic writing 
was translated into Latin, into Hebrew, and into our own 
vernaculars. The treasure of Greek science, most of it at least, 
came to us through that immense detour. We should be grate- 
ful not only to the inventors, but also to all the men thanks 
to those courage and obstinacy the ancient treasure finally 
reached us and helped to make us what we are. 



the History of Science at Harvard. 


self ntif ic precision 'and philosophical depth/' THOUGHT