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Full text of "Concepts of Space"




Max Jammer 





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Distributed in Great Britain by 




It is my firm conviction that the study of the history of 
scientific thought is most essential to a full understanding of the 
various aspects and achievements of modern culture. Such under- 
standing is not to be reached by dealing with the problems of 
priority in the history of discoveries, the details of the chronol- 
ogy of inventions, or even the juxtaposition of all the histories of 
the particular sciences. It is the history of scientific thought in 
its broadest perspective against the cultural background of the 
period which has decisive importance for the modern mind., 

The concept of space, in spite of its fundamental role in 
physics and philosophy, has never been treated from such a his- 



torical point of view. To meet this need an attempt has been 
made in the following pages to present the historical develop- 
ment of this concept and its corresponding theories. 

Although the subject has attracted my attention for a long 
time, it was only recently, while lecturing at Harvard University 
that I found at my disposal the necessary documentary material 
for the writing of this book. Since I was careful to confine myself 
to the treatment of "space" as a concept in physics, I had to 
omit many theories of space that are of special interest only 
to the professional philosopher. However, it would have violated 
my principle of broad perspective had I ignored any relevant 
metaphysical or even theological speculations on the subject. 

A presentation of the historical development of a concept does 
not necessarily imply adherence to a strict chronological order 
of discussion. A topical treatment seems to be superior for the 
clear crystallization of the principal ideas involved, all the more 
so when, as in our case, it does not seriously violate the chrono- 
logical order. 

Most sources from which I have drawn my information are 
quoted extensively, some in their original language, but the ma- 
jority in English. I have also supplied abundant bibliographical 
references so that the interested reader can readily check my 
contentions and pursue the study of particular points. 

I am happy to acknowledge publicly my indebtedness to Pro- 
fessor Albert Einstein for the great interest he has manifested 
in this research and for his kind provision of the foreword. I had 
the privilege of discussing with him at the Institute for Advanced 
Study many important issues of the subject. I am also indebted 
to Professor George Sarton, to Professor I. Bernard Cohen, and 
to Professor H. A. Wolfson for their valuable suggestions and 
helpful criticisms in the early stages of the work. Others to 
whom my sincere thanks must be accorded are the staffs of the 
Widener and Houghton Libraries at Harvard University and of 
the Butler Library at Columbia University. Further thanks are 
due to the United States Department of State, for its interest 




in my research, to Professor Alexander Dushkin, and to all my 
colleagues at the Hebrew University with whom I discussed 
various features of the subject. In conclusion the author's grati- 
tude is expressed to the Harvard University Press and in particu- 
lar to its Science Editor, Mr. Joseph D. Elder, for the encourage- 
ment received. 


The Hebrew University 

November 1Q53 










INDEX lgi 



In order to appreciate fully the importance of investiga- 
tions such as the present work of Dr. Jammer one should con- 
sider the following points. The eyes of the scientist are directed 
upon those phenomena which are accessible to observation, upon 
their apperception and conceptual formulation. In the attempt 
to achieve a conceptual formulation of the confusingly immense 
body of observational data, the scientist makes use of a whole 
arsenal of concepts which he imbibed practically with his moth- 
er's milk; and seldom if ever is he aware of the eternally problem- 
atic character of his concepts. He uses this conceptual material, 
or, speaking more exactly, these conceptual tools of thought, as 



something obviously, immutably given; something having an 
objective value of truth which is hardly ever, and in any case 
not seriously, to be doubted. How could he do otherwise? How 
would the ascent of a mountain be possible, if the use of hands, 
legs, and tools had to be sanctioned step by step on the basis 
of the science of mechanics? And yet in the interests of science 
it is necessary over and over again to engage in the critique 
of these fundamental concepts, in order that we may not uncon- 
sciously be ruled by them. This becomes evident especially in 
those situations involving development of ideas in which the 
consistent use of the traditional fundamental concepts leads us 
to paradoxes difficult to resolve. 

Aside from the doubt arising as to the justification for the use 
of the concepts, that is to say, even in cases where this doubt 
is not in the foreground of our interest, there is a purely historical 
interest in the origins or the roots of the fundamental concepts. 
Such investigations, although purely in the field of history of 
thought, are nevertheless in principle not independent of at- 
tempts at a logical and psychological analysis of the basic con- 
cepts. But the limitations to the abilities and working capacity 
of the individual are such that we but rarely find a person who 
has the philological and historical training required for critical 
interpretation and comparison of the source material, which is 
spread over centuries, and who at the same time can evaluate 
the significance of the concepts under discussion for science as 
a whole. I have the impression that Dr. Jammer, through his 
work, has demonstrated that in his case these conditions are in 
great measure satisfied. 

In the main he has limited himself — wisely, it seems to me — 
to the historical investigation of the concept of space. If two 
different authors use the words "red," "hard," or "disappointed," 
no one doubts that they mean approximately the same thing, be- 
cause these words are connected with elementary experiences in 
a manner which is difficult to misinterpret. But in the case of 
words such as "place" or "space," whose relation with psycho- 



logical experience is less direct, there exists a far-reaching un- 
certainty of interpretation. The historian attempts to overcome 
such uncertainty by comparison of the texts, and by taking into 
account the picture, constructed from literature, of the cultural 
stock of the epoch in question. The scientist of the present, how- 
ever, is not primarily trained or oriented as a historian; he is not 
capable of forming nor willing to form his views on the origin 
of the fundamental concepts in this manner. He is more inclined 
to allow his views on the manner in which the relevant concepts 
might have been formed, to arise intuitively from his rudimentary 
knowledge of the achievements of science in the different epochs 
of history. He will, however, be grateful to the historian if the 
latter can convincingly correct such views of purely intuitive 

Now as to the concept of space, it seems that this was preceded 
by the psychologically simpler concept of place. Place is first of 
all a (small) portion of the earth's surface identified by a name. 
The thing whose "place" is being specified is a "material object" 
or body. Simple analysis shows "place" also to be a group of 
material objects. Does the word "place" have a meaning inde- 
pendent of this one, or can one assign such a meaning to it? If 
one has to give a negative answer to this question, then one is 
led to the view that space (or place) is a sort of order of material 
objects and nothing else. If the concept of space is formed and 
limited in this fashion, then to speak of empty space has no mean- 
ing. And because the formation of concepts has always been 
ruled by instinctive striving for economy, one is led quite natu- 
rally to reject the concept of empty space. 

It is also possible, however, to think in a different way. Into 
a certain box we can place a definite number of grains of rice 
or of cherries, etc. It is here a question of a property of the ma- 
terial object "box," which property must be considered "real" 
in the same sense as the box itself. One can call this property 
the "space" of the box. There may be other boxes which in this 
sense have an equally large "space." This concept "space" thus 



achieves a meaning which is freed from any connection with a 
particular material object. In this way by a natural extension of 
"box space" one can arrive at the concept of an independent 
(absolute) space, unlimited in extent, in which all material ob- 
jects are contained. Then a material object not situated in space 
is simply inconceivable; on the other hand, in the framework 
of this concept formation it is quite conceivable that an empty 
space may exist. 

These two concepts of space may be contrasted as follows: 
(a) space as positional quality of the world of material objects; 
(&) space as container of all material objects. In case (a), space 
without a material object is inconceivable. In case (b), a mate- 
rial object can only be conceived as existing in space; space then 
appears as a reality which in a certain sense is superior to the 
material world. Both space concepts are free creations of the 
human imagination, means devised for easier comprehension of 
our sense experience. 

These schematic considerations concern the nature of space 
from the geometric and from the kinematic point of view, re- 
spectively. They are in a sense reconciled with each other by 
Descartes' introduction of the coordinate system, although this 
already presupposes the logically more daring space concept (b). 

The concept of space was enriched and complicated by 
Galileo and Newton, in that space must be introduced as the 
independent cause of the inertial behavior of bodies if one wishes 
to give the classical principle of inertia ( and therewith the classi- 
cal law of motion) an exact meaning. To have realized this fully 
and clearly is in my opinion one of Newton's greatest achieve- 
ments. In contrast with Leibniz and Huygens, it was clear to 
Newton that the space concept (a) was not sufficient to serve 
as the foundation for the inertia principle and the law of motion. 
He came to this decision even though he actively shared the 
uneasiness which was the cause of the opposition of the other 
two: space is not only introduced as an independent thing apart 
from material objects, but also is assigned an absolute role in 



the whole causal structure of the theory. This role is absolute 
in the sense that space (as an inertial system) acts on all ma- 
terial objects, while these do not in turn exert any reaction on 

The fruitfulness of Newton's system silenced these scruples for 
several centuries. Space of type (b) was generally accepted by 
scientists in the precise form of the inertial system, encompassing 
time as well. Today one would say about that memorable dis- 
cussion: Newton's decision was, in the contemporary state of 
science, the only possible one, and particularly the only fruitful 
one. But the subsequent development of the problems, proceed- 
ing in a roundabout way which no one then could possibly fore- 
see, has shown that the resistance of Leibniz and Huygens, intui- 
tively well founded but supported by inadequate arguments, was 
actually justified. 

It required a severe struggle to arrive at the concept of inde- 
pendent and absolute space, indispensable for the development 
of theory. It has required no less strenuous exertions subsequently 
to overcome this concept — a process which is probably by no 
means as yet completed. 

Dr. Jammer's book is greatly concerned with the investigation 
of the status of the concept of space in ancient times and in the 
Middle Ages. On the basis of his studies, he is inclined toward 
the view that the modern concept of space of type (b), that is, 
space as container of all material objects, was not developed 
until after the Renaissance. It seems to me that the atomic theory 
of the ancients, with its atoms existing separately from each 
other, necessarily presupposed a space of type (b), while the 
more influential Aristotelian school tried to get along without the 
concept of independent (absolute) space. Dr. Jammer's views 
concerning theological influences on the development of the 
concept of space, which he outside the range of my judgment, 
will certainly arouse the interest of those who are concerned with 
the problem of space primarily from the historical point of view. 

The victory over the concept of absolute space or over that 



of the inertial system became possible only because the concept 
of the material object was gradually replaced as the fundamental 
concept of physics by that of the field. Under the influence of 
the ideas of Faraday and Maxwell the notion developed that 
the whole of physical reality could perhaps be represented as a 
field whose components depend on four space-time parameters. 
If the laws of this field are in general covariant, that is, are not 
dependent on a particular choice of coordinate system, then the 
introduction of an independent (absolute) space is no longer 
necessary. That which constitutes the spatial character of reality 
is then simply the four-dimensionality of the field. There is then 
no "empty" space, that is, there is no space without a field. 
Dr. Jammer s presentation also deals with the memorable round- 
about way in which the difficulties of this problem were over- 
come, at least to a great extent. Up to the present time no one 
has found any method of avoiding the inertial system other than 
by way of the field theory. 

Princeton, New Jersey 

Albert Einstein 



Space is the subject, especially in modern philosophy, of 
an extensive metaphysical and epistemological literature. From 
Descartes to Alexander and Whitehead almost every philosopher 
has made his theory of space one of the cornerstones of his 
system. The theory of relativity has led to an enormous increase 
in the literature on space and time. Under the influence of 
logical positivism the physical implications of recent theories of 
space have been recognized, whereas eighteenth- and nineteenth- 
century works were almost completely confined to purely meta- 
physical or psychological considerations. 
Surprising as it may seem, it is a fact that no historical research 


on the concept of space has been published so far that deals with 
the history of the subject from the standpoint of physics. In the 
light of our modern ideas on physical space, such a treatise 
would be of interest not only to the historian of science and 
philosophy, but also to all who share in the great adventure of 
the intellectual progress of mankind. 

It is the purpose of this monograph to show the development 
of the concept of space in the light of the history of physics. On 
the one hand the most important space conceptions in the history 
of scientific thought will be explained and their influence on the 
respective theories of mechanics and physics will be investigated; 
and on the other, it will be shown how experimental and obser- 
vational research — together with theological speculations — af- 
fected the formulation of the corresponding metaphysical founda- 
tions of natural science as far as space is concerned. The theory 
of absolute space, as it finally crystallized in Newtonian mechan- 
ics, will be presented together with the criticism of it by the 
first modern relativists, Leibniz and Huygens. A discussion on 
the final elimination of the concept of absolute space from the 
conceptual scheme of modern physics will bring this monograph 
to its conclusion. 

Newton's conception of absolute space is based upon a syn- 
thesis of two heterogeneous elements. One of these elements is 
rooted in the emancipation of space from the scholastic substance- 
accident scheme, a scheme which was finally abandoned by the 
Italian natural philosophers of the Renaissance. The other element 
draws on certain ideas that identify space with an attribute of 
God. These ideas appear to go back to Palestinian Judaism of the 
first century. They were adopted by Jewish mystical philosophy, 
and, with the spread of cabalistic teachings to Western Europe, 
they found an especially fruitful soil in seventeenth-century Eng- 
land. Under the influence of Henry More, an ardent scholar of 
cabalistic lore, Newton thought it necessary and expedient to 
make these theological ideas an integral part of his theory of 



space. We have, therefore, two more or less independent in- 
tellectual developments reaching back to antiquity and coming 
together in Newton's theory of absolute space. 

Accordingly, our treatise dealing with the historical develop- 
ment of the concept of physical space 1 is not one continuous 
narrative, but is interrupted for the purpose of tracing the 
theological influence. So the first chapter expounds the concept 
of space from earliest antiquity until toward the close of Hellen- 
istic science; the second chapter deals with the theological in- 
fluences down to the time of Henry More; the third chapter 
resumes the subject of the first chapter; the fourth chapter deals 
exclusively with Newton's concept of space and Leibniz's and 
Huygens's criticism of it; the last chapter shows the post-Newto- 
nian development of the concept of space and its final elimina- 
tion in modern physics. In presenting the subject great care has 
been given to an accurate documentation of the material. 

As far as pre-Newtonian and Newtonian physics are con- 
cerned we can confine our discussion to the concept of space, 
space and time being completely heterogeneous and noninter- 
dependent 2 entities, although connected by the concept of 
motion. 3 Historically and psychologically, a discussion of space 
is preferable to that of time, since most probably the category 
of space preceded that of time as an object of consciousness. 
Language proves this assumption: qualifications of time, as 
"short," or "long," are taken from the vocabulary of spatial con- 
cepts. We say "thereafter" and not the more logical "thenafter"; 
'For an exact definition of this concept, see R. Carnap, "Der Raum. Ein 
Beitrag zur Wissenschaftslehre," Katdstudien, Erganzungsheft No. 56 
\ 19-^2/. 

* In the Galilean transformation of classical mechanics, f = t , that is, the 
transformed time variable is independent of the space variable. 
/„ ,? P omte <* ° u * by C. A. Brandis in his Griechisch-romische Philosophie 
^Berlin, 1835), vol. 1, pp. 4 i 3 , 4 i 5) Zeno of Elea seems to have been the 
nrst who emphasized this connection between space and time. Cf . Locke's 

— „ — lmlA , u „„ uciwccu suaue ana time. i_.r. l_,ockes 

tssay concerning human understanding (London, 1785), vol. 1, pp. 149 
I- * " 'JS° measure moti °n space is as necessary to be considered as 

time . . . They are made use of, to denote the position of finite real Be- 
ings m respect one to another in those uniform oceans of Duration and 



"always" means "at all times"; we even speak of a "space" or an 
"interval" of time: "before" means etymologically "in front of." 
In this respect the Semitic languages are especially instructive, 
a fact pointed out by Ignaz Goldziher. 4 The Hebrew word for 
"before" is "lifney," which originally means "to the face of," "to 
the front of"; many other words, for instance "Kedem," "aharey," 
show clearly the trend from spatial to temporal qualifications. 
As a matter of fact, this trend can be recognized already in 
the ancient Sumerian expression danna, which was originally a 
measure of length and later signified a certain fraction of the day 
(unit of time). 5 Modern psychology undoubtedly allows more 
concreteness to the concept of space than to the concept of time. 
If we remember that it was only late in the Middle Ages that 
the role of time as the fundamental variable parameter in physical 
processes was clearly understood, we can justify our concentra- 
tion on the concept of space, at least as far as early theories of 
space are concerned. 

But we are fully aware of the fact that since Leibniz's pro- 
found analysis of the concepts of space and time the notion of 
time has often been held to precede the notion of space in 
the construction of a philosophical system. The direction of the 
flow of time was thought to be determined by the causal inter- 
connection of phenomena. Space, then, was only the order of 
coexisting data. "Spatium est ordo coexistendi," said Leibniz 
in his Initio, return metaphysial, a surprisingly modern analysis 
of our concepts of space and time. Yet for the purposes of the 
present work, which deals on the whole with the history of 
the classical concept of space, we may ignore these results and 
treat the concept of space as the elementary and primary notion. 

'Ienaz Goldziher, Mythology among the Hebrews (London, 1877). 

•O Neueebauer, "Untersuchungen zur Geschichte der antiken Astrono- 
mie, III," Quellen und Studien zur Geschichte der Mathemattk (Springer, 
Beriin, 1938), part B, vol. 4, p. 193- 




Modern physics on the whole — if we neglect certain 
relativistic theories — qualifies space as continuous, isotropic, 
homogeneous, finite, or infinite, in so far as it is not a pure 
system of relations. Not all of these qualities, however, are 
accessible to sense perception. They are the result of a long 
and continuous process of abstraction which had its beginning 
in the mind of primitive man. Philological, archaeological, and 
anthropological research shows clearly that primitive thought 
was not capable of abstracting the concept of space from the 
experience of space. To the primitive mind, "space" was merely 
an accidental set of concrete orientations, a more or less ordered 


multitude of local directions, each associated with certain emo- 
tional reminiscences. This primitive "space," as experienced and 
subconsciously formed by the individual, may have been co- 
ordinated with a "space" common to the group, the family or 
the tribe. Certain astronomical or meteorological events, such as 
sunrise and sunset, storms and floods, no doubt endowed cer- 
tain directions with values of common importance. "Mesopota- 
mian astrology evolved a very extensive system of correlations 
between heavenly bodies and events in the sky and earthly 
localities. Thus mythopoeic thought may succeed no less than 
modern thought in establishing a coordinated spatial system; but 
the system is determined, not by objective measurements, but 
by an emotional recognition of values." 1 It can be shown that 
even with the introduction of conventional standards of measure- 
ment in early urban society, lengths, areas, and volumes were 
not conceived in abstracto as purely spatial extensions. To be 
sure, measurement leads eventually to generalization and ulti- 
mately to abstract thinking. Ignoring the color, design, and tex- 
ture of the object to be measured, human thought begins by 
"abstraction" to concentrate on the idea of pure extension and 
unqualified space. However, it must not be supposed that this 
was a simple and short process. Archaeology shows that the early 
abstractions were limited by practical interests. The ancient Su- 
merian unit of area — incidentally also the unit of weight — was 
the be or "grain." This designation indicates clearly that areal ex- 
tension was in those times conceived from the aspect of the 
quantity of seed necessary for the sowing of the area in question, 
which means, in the final analysis, from the anthropocentric 
aspect of the labor involved. 

Hesiod's "chaos," 2 which may be taken as the earliest poetical 
expression of the idea of a universal space, is mixed with emotion; 

1 H. Frankfort, H. A. Frankfort, J. A. Wilson, and T. Jacobson, The 
intellectual adventure of ancient man (University of Chicago Press, Chi- 
cago, 1946), p. 20. 3 

"Hesiod, Theogony, 116. Cf. Deichmann's objection to Zellers interpreta- 
tion in Carl Deichmann, Das Problem des Raumes in der griechischen 
Philosophie bis Aristoteles (Halle a. S., 1893). 


the very word "chaos," derived from the Greek root cha- (cha- 
skein, chainein), implies as "yawning," "gaping," an idea of 
terror and fright. To what extent such poetical-mystical concepts 
have been conditioned by early folklore and myth (such as the 
Aditi lore of the Arians ) is a matter that falls outside the scope 
of this monograph. 

Space as a subject of philosophical inquiry appears very early 
in Greek philosophy. According to Aristotle, 8 numbers were 
accredited with a kind of spatiality by the Pythagoreans: "The 
Pythagoreans, too, asserted the existence of the void and declared 
that it enters into the heavens out of the limitless breath — 
regarding the heavens as breathing the very vacancy — which 
vacancy 'distinguishes' natural objects, as constituting a kind of 
separation and division between things next to each other, its 
prime seat being in numbers, since it is this void that delimits 
their nature." Spatial vacancies were necessary to guarantee the 
discreteness of individual numbers in the Pythagorean geometri- 
zation of number. Space here has not yet any physical implica- 
tions apart from serving as the limiting agent between different 
bodies. In early Pythagorean philosophy this kind of "space" is 
still called pneuma apeiron and only occasionally kenon (void). 
The concept of space is still confounded with that of matter. 
As J. Bumet says: "The Pythagoreans, or some of them, certainly 
identified 'air with the void. This is the beginning, but no more 
than the beginning, of the conception of abstract space or exten- 
sion." 4 Only later on is this confusion cleared up by Xutus and 
Philolaus. 5 In Simplicius 6 we find that Archytas, the Pythagorean, 
already had a clear understanding of this abstract notion, since, 
as related by Eudemus, he asked whether it would be possible 
at the end of the world to stretch out one's hand or not. Un- 
fortunately, Archytas' work on the nature of space is lost except 

* Aristotle, Metaphysics, 1080 b 33. 

*J. Burnet, Early Greek philosophy (London, 1914), p. 51. 
"P. Tannery, Revue philosophique 20 (1885), 389. 
•Simplicius, Physics, 108 a. 

''In Aristotelis categorias commentarium (ed. Carolus Kalbfleisch; Berolini, 
1907), p. 13. 



for a few fragments to be found in Simplicius' Commentaries, 7 
according to which Archytas composed a book on our subject. 
Archytas distinguishes between place (topos), or space, and 
matter. Space differs from matter and is independent of it. 
Every body occupies some place, and cannot exist unless its 
place exists. "Since what is moved is moved into a certain place 
and doing and suffering are motions, it is plain that place, in 
which what is done and suffered exists, is the first of things. 
Since everything which is moved is moved into a certain place, 
it is plain that the place where the thing moving or being moved 
shall be, must exist first. Perhaps it is the first of all beings, since 
everything that exists is in a place and cannot exist without a 
place. If place has existence in itself and is independent of 
bodies, then, as Archytas seems to mean, place determines the 
volume of bodies." 8 A characteristic property of space is that 
all things are in it, but it is never in something else; its sur- 
roundings are the infinite void itself. Apart from this metaphysical 
property, space has the physical property of setting frontiers or 
limits to bodies in it and of preventing these bodies from be- 
coming indefinitely large or small. It is also owing to this con- 
straining power of space that the universe as a whole occupies 
a finite space. To Archytas, space is therefore not some pure 
extension, lacking all qualities or force, but is rather a kind of 
primordial atmosphere, endowed with pressure and tension and 
bounded by the infinite void. 

The function of the void, or of space, in the atomism of 
Democritus is too well known to need any elaboration here. But 
it is of interest to note that according to Democritus infinity of 
space is not only inherent in the concept itself, 9 but may be 
deduced from the infinite number of atoms in existence, since 
these, although indivisible, have a certain magnitude and exten- 
sion, even if they are not perceptible to our senses. Democritus 
himself seems not to have attributed weight to the atoms but to 
have assumed that as a result of constant collisions among them- 

* Ibid., p. 357- 

'Aristotle, De caelo, III, 2, 300 b. 


selves they were in motion in infinite space. It was only later, 
when an explanation of the cause of their motion was sought, 
that his disciples introduced weight as the cause of the "up and 
down" movements (Epicurus). If Aristotle says that Democritus' 
atoms differed in weight according to their size, one has to as- 
sume — in modern words — that it was not gravitational force 
but "force of impact" that was implied. This point is of some 
importance for our point of view, since it shows that in the first 
atomistic conception of physical reality space was conceived 
as an empty extension without any influence on the motion of 

However, there still remains one question to be asked: Was 
space conceived by the atomists of antiquity as an unbounded 
extension, permeated by all bodies and permeating all bodies, or 
was it only the sum total of all the diastemata, the intervals that 
separate atom from atom and body from body, assuring their 
discreteness and possibility of motion? The stress laid time and 
again by the atomists on the existence of the void was directed 
against the school of Parmenides and Melissus, according to whom 
the universe was a compact plenum, one continuous unchanging 
whole. "Nor is there anything empty," says Melissus, "for the 
empty is nothing and that which is nothing cannot be." Against 
such argument Leucippus and Democritus maintained the exist- 
ence of the void as a logical conclusion of the assumption of the 
atomistic structure of reality. But here the void or the empty 
means clearly unoccupied space. The universe is the full and the 
empty. Space, in this sense, is complementary to matter and is 
bounded by matter; matter and space are mutually exclusive. 
This interpretation gains additional weight if we note that the 
term "the empty" (kenon) was used often as synonymous with 
the word "space"; the term "the empty" obviously implies only the 
unoccupied space. Additional evidence is furnished by Leucip- 
pus' explicit use of the adjective "porous" (manon) for the 
description of the structure of space, which indicates that he had 
in mind the intervals between particles of matter and not un- 
bounded space. Although Epicurus' recurrent description of the 




universe as "body and void" seems also to confirm this interpre- 
tation, we find in Lucretius, who bases himself on Epicurus, 
a different view. In general, Lucretius' complete and coherent 
scheme of atomistic natural philosophy is the best representation 
of Epicurean views. As far as the problem of space is concerned, 
Lucretius emphasizes in the first book of De rerum natura the 
maxim: "All nature then, as it exists, by itself, is founded on two 
things: there are bodies and there is void in which these bodies 
are placed and through which they move about." 10 

Here we find, in contrast to the early Greek atomism, a clear 
and explicit expression of the idea that bodies are placed in the 
void, in space. With Lucretius, therefore, space becomes an 
infinite receptacle for bodies. Lucretius' proof for the unbounded- 
ness of space, resembling Archytas' argument mentioned earlier, 11 
runs as follows: "Now since we must admit that there is nothing 
outside the sum, it has no outside, and therefore is without end 
and limit. And it matters not in which of its regions you take 
your stand; so invariably, whatever position any one has taken 
up, he leaves the universe just as infinite as before in all direc- 
tions. Again, if for the moment all existing space be held to be 
bounded, supposing a man runs forward to its outside borders 
and stands on the utmost verge and then throws a winged javelin, 
do you choose that when hurled with vigorous force it shall ad- 
vance to the point to which it has been sent and fly to a distance, 
or do you decide that something can get in its way and stop it? 
for you must admit and adopt one of the two suppositions; either 
of which shuts you out from all escape and compels you to grant 
that the universe stretches without end." 12 

10 T. Lucreti Cari, De rerum natura (trans, by Munro; Cambridge, 1886), 
vol. 3, p. 23. The original Latin text is: 

. . . nam corpora sunt et inane, 
haec in quo sita sunt et qua diversa moventur. 

— Liber I, 420. 
a See p. 8. 
M Reference 10. 



This argument, and in particular the idea of a man placed at 
the supposed boundary of space stretching out his hand or throw- 
ing a spear, is a recurrent idea in the history of natural philos- 
ophy. In fact, an illustration of this kind is to be expected. We 
find it in Richard of Middleton's writings 13 in the fourteenth 
century (perhaps with reference to Simplicius' Physics 108 a), 
still before the rediscovery of the De rerum natura in 1418 
bv Poggio- We also find it as late as in Locke's Essay concern- 
ing human understanding (1690), where the question is asked 
"whether if God placed a man at the extremity of corporeal 
beings, he could not stretch his hand beyond his body." 14 

Lucretius adduces a further argument for the infinitude of 
space which reveals an important physical aspect of the atom- 
istic theory: If space were not infinite, he claims, all matter 
would have sunk in the course of past eternity in a mass to the 
bottom 15 of space and nothing would exist any more. This remark 
shows clearly that Lucretius, in the wake of Epicurus, conceived 
space as endowed with an objectively distinguished direction, 
the vertical. It is in this direction in which the atoms are racing 
through space in parallel lines. According to Epicurus and Lucre- 
tius, space, though homogeneous, is not isotropic. 

Although the idea of a continuous homogeneous and isotropic 
space, as we see, seems to have been too abstract even for the 
theoretically minded atomists, it has been justly pointed out 16 
that their conception of the noncorporeal existence of a void 
introduced a new conception of reality. Indeed, it is a strange 
coincidence that the very founders of the great materialistic 
school in antiquity had to be "the first to say distinctly that a 
thing might be real without being a body." 

The first clear idea of space and matter as belonging to dif- 

M See chapter 3, reference 46. 

"John Locke, An essay concerning human understanding, book II, 13, 
21; see, for example, the edition by A. S. Pringle-Pattison (Clarendon 
Press, Oxford, 1950), p. 102. 

"Ad imum. Liber I, 987. 

M J. Burnet, Early Greek philosophy (London, ed. 3, 1920), p. 389. 



ferent categories is to be found in Gorgias. 17 Gorgias first 
proves that space cannot be infinite. For if the existent were in- 
finite, it would be nowhere. For were it anywhere, that wherein 
it would be, would be different from it, and therefore the exist- 
ent, encompassed by something, ceases to be infinite; for the 
encompassing is larger than the encompassed, and nothing can 
be larger than the infinite; therefore the infinite is not anywhere. 
Nor on the other hand, can it be encompassed by itself. For in 
that case, that wherein it is found would be identical with that 
which is found therein, and the existent would become two 
things at a time, space and matter; but this is impossible. The 
impossibility of the existence of the infinite excludes the possi- 
bility of infinite space. 

Plato, who, according to Aristotle, was not satisfied, as his 
predecessors were, with the mere statement of the existence of 
space, but "attempted to tell us what it is," 18 develops his the- 
ory of space mainly in Timaeus. The upshot of the rather obscure 
exposition of this dialogue, as interpreted by Aristotle, 19 and in 
modern times by E. Zeller, 20 is that matter — at least in one 
sense of the word — has to be identified with empty space. Al- 
though "Platonic matter" was sometimes held to be a kind of 
body lacking all quality (Stoics, Plutarch, Hegel) or to be the 
mere possibility of corporeality (Chalcidius, Neoplatonists), crit- 
ical analysis seems to show that Plato intended to identify the 
world of physical bodies with the world of geometric forms. 
A physical body is merely a part of space limited by geomet- 
ric surfaces containing nothing but empty space. 21 With Plato 
physics becomes geometry, just as with the Pythagoreans it be- 
came arithmetic. Stereometric similarity becomes the ordering 
principle in the formation of macroscopic bodies. "Now the 

"Sexti Empirici opera, "Adversus dogmaticos" (ed. H. Mutschmann; 
Leipzig, 1912-14), vol. 2, p. 17- 
"Aristotle, Physics, 209 d. 
u Ibid., 203 a, 209 b. 

" E. Zeller, Die Philosophie der Griechen (Leipzig, 1869-1879), vol. 2. 
* Plato, Timaeus, 55 ff. 


Nurse of Becoming, being made watery and fiery and receiving 
the characters of earth and air, and qualified by all the other 
affections that go with these, had every sort of diverse appear- 
ance to the sight; but because it was filled with powers that 
were neither alike nor evenly balanced, there was no equipoise 
in any region of it; but it was everywhere swayed unevenly and 
shaken by these things, and by its motion shook them in turn. 
And they, being thus moved, were perpetually being separated 
and carried in different directions; just as when things are shaken 
and winnowed by means of winnowing-baskets and other instru- 
ments for cleaning corn, the dense and heavy things go one 
way, while the rare and light are carried to another place and 
settle there. In the same way at that time the four kinds were 
shaken by the Recipient, which itself was in motion like an 
instrument for shaking, and it separated the most unlike kinds 
farthest apart from one another, and thrust most alike closest 
together; whereby the different kinds came to have different 
regions, even before the ordered whole consisting of them came 
to be." 22 Physical coherence, or, if one likes, chemical affinity, 
is the outcome of stereometric formation in empty space, which 
itself is the undifferentiated material substrate, the raw material 
for the Demiurgus. The shaking and the winnowing process 
characterizes space with a certain stratification and anisotropy 
which is manifested physically in the difference between the 
layers of the elements. Geometric structure is the final cause of 
what has been called "selective gravitation," where like attracts 

In accordance with certain ideas expressed by the Pythagorean 

Philolaus, 23 Plato conceived the elements as endowed with 

definite spatial structures: 24 to water he assigned the spatial 

structure of an icosahedron, to air of an octahedron, to fire 

of a pyramid, and to earth of a cube. Earth, in Plato's view, 

"Ibid., 52 d; F. M. Cornford, Plato's cosmology (Harcourt, Brace, New 
York, 1937), p . 198. 

" Zeller, Philosophie der Griechen, vol. 1, p. 376. 
"Plato, Timaeus, 56. 

i 4 



owing to its cubical form, is the most immovable of the four 
having the most stable bases. It is only natural, therefore «£ 

£ etbedded T " ^T ° f *" "^ ^ * ^ 
it is embedded m layers of the other elements of space according 

to then: increasing movability. The varieties of the four ^element! 
and their gravitational behavior are due to differences^ 
form and size, or, in the final analysis, are due to differences fa 
form and size of the elementary triangles of which thTSane 
surfaces are formed. As much as matter is reduced to spacT 
physics is reduced to geometry. P ' 

ttis identification of space and matter, or, in the words of 
later pseudo-Platonic teachings, of ^dimensionality and matter 
hac la great influence on physical thought during 7 the S 

log ' S Ugh AriSt0tle ' S ° rgan0n ^ *• ^dard textt 
n L £ TT 5 ^ SUCCCeded ^ *"***>'» Physics only 

r n wbuS to T e a ; d vague lan ^ e ° f ** ««««• 

contr buted to preventing the concept of space from becoming 
a subject of strict mathematical research. Greek mathem!^ 
regards the geometry of space. Plato himself, for whom sotd 
boche, and the* eby were of fundamen ^ - -1 d 

*e formulation of his philosophy, lamented the ne^ecZ f 

Min^Td ^7^ lD *• ^^ he Refill 
fading to discuss solid geometry when listing the e^ential sub- 
jects for instruction. So we read: 

and C chC: te!%£^£T'r e *«™^ attractiveness 

Socrates: Yes. 
4^S ^ *" "' «" >™ *°° l a*""* »=* «nd then y m 

"Plato, Republic, 528. 



it, and mentioned next after geometry, astronomy, which deals with 
the movements of solids. 26 

Aristotle's theory of space is expounded chiefly in his Catego- 
ries and, what is of greater relevance for our purpose, in his 
Physics. In the Categories Aristotle begins his short discussion 
with the remark that quantity is either discrete or continuous. 
"Space," belonging to the category of quantity, is a continuous 
quantity. "For the parts of a solid occupy a certain space, and 
these have a common boundary; it follows that the parts of 
space also, which are occupied by the parts of the solid, have 
the same common boundary as the parts of the solid. Thus, not 
only time, but space also, is a continuous quantity, for its parts 
have a common boundary." ^ "Space" here is conceived as the 
sum total of all places occupied by bodies, and "place" (topos), 
conversely, is conceived as that part of space whose limits coin- 
cide with the limits of the occupying body. 28 

In the Physics Aristotle uses exclusively the term "place" (to- 
pos), so that strictly speaking the Physics does not advance a 
theory of space at all, but only a theory of place or a theory or 
positions in space. However, since the Platonic and Democritian 
conceptions of space are unacceptable to the Aristotelian system 
of thought, and since the notion of empty space is incompatible 
with his physics, Aristotle develops only a theory of positions in 
space, with the exclusion of the rejected conception of general 

For our purpose, Aristotle's theory of places is of greatest 
pertinence not only because of its important implications for 
physics, but also because it was the most decisive stage for the 
further development of space theories. In our treatment we shall 
adhere as much as possible to Aristotle's original terminology 
and use the term "place." 

■9i£™9ir° '"ST""?"' °" P "™ D ° h8m ' mMm ' d " """* (p " is - 



In Book IV of the Physics Aristotle develops on an axiomatic 
basis a deductive theory of the characteristics of place. Place 
is an accidens, having real existence, but not independent exist- 
ence in the sense of a substantial being. Aristotle's four primary 
assumptions regarding our concept are as follows: "(1) That the 
place of a thing is no part or factor of the thing itself, but is 
that which embraces it; (2) that the immediate or "proper" 
place of a thing is neither smaller nor greater than the thing 
itself; (3) that the place where the thing is can be quitted by it, 
and is therefore separable from it; and lastly (4): that any and 
every place implies and involves the correlatives of 'above' and 
"below,' and that all the elemental substances have a natural 
tendency to move towards their own special places, or to rest 
in them when there — such movement being 'upward' or 'down- 
ward,' and such rest 'above' or 'below.' " 29 It is this last assump- 
tion that makes space a carrier of qualitative differences and 
furnishes thereby the metaphysical foundation of the mechanics 
of "natural" motion. Starting from these assumptions, Aristotle 
proceeds by a lucid process of logical elimination 30 to his famous 
definition of "place" as the adjacent boundary of the contain- 
ing body. By this definition the concept became immune to all 
the criticisms that were designed to show the logical inconsistency 
of former definitions, as, for instance, Zeno's famous epicheirema 
(Everything is in place; this means that it is in something; but if 
place is something, then place itself is in something, etc.). In 
fact, this "nest of superimposed places" is mentioned as an 
argument against the existence of a kind of dimensional entity — 
distinct from the body that has shifted away when the encircled 
content is taken out and changed again and again, while the 
encircling continent remains unchanged. 

Further, this "replacement" of the content of a vessel by 
another content reveals that place is something different from 

"Aristotle, Physics, 211 a, trans, by P. H. Wicksteed and F. M. Cornford 
(Loeb Classical Library; Harvard University Press, Cambridge, 1929), vol. 
1, p. 303. 

*Ibid., 211b. 



its changing contents and so proves the reality of space. Of 
great importance from our point of view is a passage in Aristotle's 
Physics in which space is likened (using a modern expression) 
to a field of force: "Moreover the trends of the physical elements 
(fire, earth, and the rest) show not only that locality or place 
is a reality but also that it exerts an active influence; for fire 
and earth are borne, the one upwards and the other downwards, 
if unimpeded, each towards its own place,' and these terms — 
up' and 'down' I mean, and the rest of the six dimensional direc- 
tions — indicate subdivisions or distinct classes of positions or 
places in general." 31 

The dynamical field structure, inherent in space, is conditioned 
by the geometric structure of space as a whole. Space, as defined 
by Aristotle, namely, as the inner boundary of the containing 
receptacle, is, so to speak, a reference system which generally is 
of very limited scope. The place of the sailor is in the boat, the 
boat itself is in the river, and the river is in the river bed. This 
last receptacle is at rest relative to the earth and therefore also 
to the universe as a whole, according to contemporary cosmology. 
For astronomy, with its moving spheres, the reference system has 
to be generalized still further, leading to the finite space of 
the universe limited by the interior boundary of the outermost 
sphere, which itself is not contained in any further receptacle. 
This universal space, of spherical symmetry, has as its center the 
center of the earth, to which heavy bodies move under the dy- 
namic influence intrinsic to space. It is natural for us, who have 
read Mach and Einstein, to raise the question whether the ge- 
ometric aspect of this dynamical "field structure" depends on 
the distribution of matter in space or is completely independent 
of mass. Aristotle anticipated this question and tried to show that 
the dynamics of natural motion depends on spatial conditions 

It might be asked, since the center of both (i.e., the earth and the 
universe) is the same point, in which capacity the natural motion of 
a Ibid., 208 b; Loeb edition, p. 279. 




heavy bodies, or parts of the earth, is directed towards it; whether as 
center of the universe or of the earth. But it must be towards the 
center of the universe that they move, seeing that light bodies like 
fire, whose motion is contrary to that of the heavy, move to the 
extremity of the region which surrounds the center. It so happens 
that the earth and the universe have the same center, for the heavy 
bodies do move also towards the center of the earth, yet only in- 
cidentally, because it has its center at the center of the universe." 32 

This description is suggestive of the electrostatic field that exists 
between a small charged sphere enclosed by another sphere 
at a different potential. As is well known, the field itself may be 
nonspherically symmetric, as in the case of an excentric position 
of the inner sphere, which corresponds to the earth when shifted 
from the center of the universe, although the lines of force leave 
the surface of the enclosed body in a normal direction. To 
Aristotle, such a distortion seemed to be absurd; his world is a 
world of order and symmetry. 

The directional tendencies of the elemental particles are 
possible only because of the difference in the conditions of the 
place in which they move from the conditions of the place to 
which they move. It is clear, therefore, that it is not a kind of 
buoyancy (corresponding to Archimedes' principle) that causes 
the motion of heavy or light bodies. For in this case the dynam- 
ical field structure would be mass-dependent. However, although 
these tendencies are independent of the distribution of mass, 
they are dependent on the very existence of matter. A void, con- 
ceived by Aristotle as the privation of all conceivable properties, 
cannot by its very definition be something differentiated direc- 
tionally. It is well known how Aristotle exploited this argument 
in his repudiation of the void. 

In conformity with the rejection of a vacuum, Aristotle insists 
repeatedly that the containing body has to be everywhere in 
contact with the contained. Polemizing against the Pythagorean 
doctrine of spatial vacancies, Aristotle offers a psychological 
explanation of the origin of such "gap" theories. "Because the 

M Aristotle, De caelo, II, 14, 296 b; Loeb edition, p. 243. 



encircled content may be taken out and changed again and 
again, while the encircling continent remains unchanged — as 
when water passes out of a vessel — the imagination pictures a 
kind of dimensional entity left there, distinct from the body that 
has shifted away." 33 But, he holds, to suppose that this "interval" 
is the place or space of the contained would inescapably lead 
to serious inconsistencies. He argues that on the basis of such 
a "gap" theory "place" would have to change its "place" and an 
ascending series of orders of spaces would be involved. Thus, 
when carrying a vessel of water from one place to another, one 
has to carry about also the "interval" and a transport of space in 
space is implied. His second objection is based on the assertion 
that transporting a vessel full of water means changing the place 
of the whole but not the places of its parts. According to Sim- 
plicius, Aristotle's line of thought seems to have used the fol- 
lowing reductio ad absurdum: On the basis of an "interval" 
theory every part of water has to have its own place, since a 
transport of a vessel of water is accompanied by a rotation or 
wave disturbance of the liquid, which is possible only if the parts 
can shift from one interval to another. However, matter is in- 
definitely divisible and the number of such intervals must con- 
sequently be unlimited even for the smallest quantity of water. 
It follows that the volume, the sum total of all these intervals, 
being a sum of an infinite series, is infinitely great. 

While expounding the inadequacy of these "interval" theories, 
Aristotle, on his part, ignores the fact that his very insistence on 
the all-over contact of the two distinct surfaces of the container 
and the contained must necessarily lead to a serious inconsistency 
between his own space theory on the one hand and his cosmology 
and theology on the other. For if the interior concave surface of 
the sphere of one planet is everywhere in contact with the convex 
surface of the sphere of another, then obviously the "fifth body," 
the substance of which the heavens are made, is not continuous, 
a conclusion that is contrary to the results of his cosmological 

"Aristode, Physics, IV, 211 b 15; Loeb edition, p. 309. 



doctrines as presented in De caelo. 3i Simplicius, who noticed this 
inconsistency, tried to avoid it by maintaining that all celestial 
spheres extend to one common center which coincides with the 
center of the earth. But obviously, Simplicius' solution of the 
problem is not only a theory ad hoc, but is also incompatible 
with the principles of Aristotelian physics which explicitly rejects 
the interpenetrability of different bodies. 35 

It should be noted that Aristotle's remarks in the Categories 
indicate a different way of attacking the problem of space. Here 
space seems to be some kind of continuous extension; it is given 
no strict definition, and, what is more important for our point of 
view, it has no physical implications for Aristotle's natural philos- 
ophy or that of his successors. 

It is evident that space as an accident of matter is, according 
to Aristotle, finite, matter being itself finite. 

Space, here, means the sum total of all places. The idea of a 
finite physical space, thus understood, is not as absurd today as 
it must have appeared fifty years ago, when physics acknowl- 
edged solely the conception of an infinite Euclidean space and 
when a finite material universe could but be conceived as an 
island, so to speak, in the infinite ocean of space. It is perhaps not 
wholly unjustified to suggest a comparison between the notion of 
physical space in Aristotle's cosmology and the notion of Einstein's 
"spherical space" as expounded in early relativistic cosmology. 
In both theories a question of what is "outside" finite space is 
nonsensical. Furthermore, the idea of "geodesic lines," deter- 
mined by the geometry of space, and their importance for the 
description of the paths of material particles or fight rays, suggest 
a certain analogy to the notion of "natural places" and the paths 
leading to them. The difference is, of course, that in Einstein's 
theory the geometry of space itself is a function of the mass- 
energy distribution in accord with the famous field equations, 
and is not Euclidean but Riemannian. 

** Aristotle, De caelo, I, 3, 270 a et seq., Loeb edition, p. 21. 
* Aristotle, Physics, IV, 209 a 7; Loeb edition, p. 283. 



Although until the fourteenth century Aristotle's and Plato's 
conceptions were the prototypes, with only minor changes, of all 
theories of space, yet these conceptions were the object of con- 
stant attack, mostly on metaphysical grounds. Aristotle's pupil 
Theophrastus criticizes the master's theory 36 and speaks of the 
possibility of a motion of space, of the incomprehensibihty of the 
universe as not being in space, and comes to the conclusion 37 
that space is no entity in itself but only an ordering relation that 
holds between bodies and determines their relative positions. 
Like a biologist who dissects an animal and considers one organ 
in relation to another, so Theophrastus views space as a system 
of interconnected relations. 

Concerned as we are with the problem of space in its implica- 
tions for physics, we may disregard the few original contributions 
of the Epicureans, Skeptics, and other schools. We should, how- 
ever, mention in this connection the important deviation of the 
Stoics from the traditional Aristotelian conception of the cosmos. 
Continuity, which for Aristotle was a purely geometric property 
of coherent matter, became with the Stoa a physical principle, 
an agent responsible for the propagation of physical processes 
through space. It is by this inner connection, manifested as a 
tension (tonos) in its active state, that distant parts of the uni- 
verse are able to influence each other, thereby turning the cosmos 
into one field of action. The void, being incorporeal and there- 
fore lacking all continuity, necessarily precludes all sense per- 
ception and so cannot exist inside the world. 38 

This elaboration of the Aristotelian idea of tendencies per- 
meating the continuous plenum is an important generalization in 
two respects; in the variety of phenomena envisaged, and in their 
extension beyond the sublunar world (for instance, Posidonius' 
discovery of the moon's "influence" on the tides, which was re- 
garded as an ostensible proof of the reality of this transmitting 

M Simplicius, Physics, 141. 
"Ibid., 141, 149. 

88 Cleomedes, "De motu circulari corporum caelestium libri duo," in J. ab 
Arnim, ed., Stoicorum veterum fragmenta, II, 546 (Leipzig, 1905), p. 172. 

: ' 



agent connecting even celestial with mundane phenomena [Chry- 
sippus]). The range of activity of the propagating tensions is 
the whole material universe (holon) as distinct from the "AH" 

In order to explain this important distinction we have to refer 
to the changed definition of "space." In general, the Stoics ac- 
cepted not Aristotle's definition of space, as the containing sur- 
face of the encircling body, but his discarded alternative, that 
is, the dimensional extension lying between the points of the 
containing surface. This change enabled the Stoics to maintain 
the existence of a void outside the material universe, whereas the 
material universe was conceived as an island of continuous matter 
surrounded by an infinite void. Needless to say, this infinite void 
lacked all qualities and differentiations, and, being thus com- 
pletely indeterminate, it could not act in any way on the matter 
surrounded by it. 39 Hence the position of bodies was not deter % 
mined by any properties of the void, but by their own nature. 
With no reason to move as a whole, the material world rests im- 
movable in the infinite void. To the Stoics it made no sense to 
speak of the center of the "All"; on the other hand, the center of 
the material universe was a clear concept, cosmologically and 
physically well founded. Criticizing this doctrine, the Peripatetics 
raised the following question: If the material world is really sur- 
rounded by an infinite void, why does it not become dissipated 
and lost in the course of time? *° The answer is now clear: the 
different parts of the material world are connected, not, as Aris- 
totle thought, by an exterior continent, an upper sphere which 
forces the parts to stay together, like samples in a box, but by 
an internal cohesion (hexis), which is only another aspect of the 
tension mentioned before. It is this binding force that holds the 
world together, and the void without force of its own can do 
nothing to loosen it. In the void there is no "up" or "down" or 

"Ibid., II, 173, 176 (pp. 49. 5i)- 
"Ibid., II, 540. 



any other direction or dimension. 41 In other words, it is isotropic, 
bare of any qualities whatever. As for space in the material uni- 
verse, the Stoics adhere to the traditional Aristotelian doctrine. 
It was owing chiefly to these Stoic controversies that the problem 
of space could no longer be considered as one simple question, 
but had to appear under the form of two different considerations: 
space and void. 

As we have tried to show in this chapter, space was conceived 
by classical Greek philosophy and science at first as something 
inhomogeneous because of its local geometric variance (as with 
Plato), and later as something anisotropic owing to directional 
differentiation in the substratum (Aristotle). It is perhaps not too 
conjectural to assume that these doctrines concerning the nature 
of space account for the failure of mathematics, especially ge- 
ometry, to deal with space as a subject of scientific inquiry. Per- 
haps this is the reason why Greek geometry was so much con- 
fined to the plane. It may be objected that "space" according to 
Aristotle is "the adjacent boundary of the containing body" and 
so by its very definition is only of a two-dimensional character. 
But this objection ignores a clear passage in the Physics* 2 and 
another passage in De caelo.* 3 As Euclid's Elements show, the 
science of solid geometry was developed only to a small extent 
and mostly confined to the mensuration of solid bodies, which 
is at least one reason why even the termini technici of solid ge- 
ometry, compared with those of plane geometry, were so little 
standardized. The idea of coordinates in the plane seems to go 
back to pre-Greek sources, the ancient Egyptian hieroglyphic 
symbol for "district" (hesp) being a grid (plane rectangular 
coordinate system). It would therefore be only natural to expect 
some reference to spatial coordinates in Greek mathematics. But 
in the whole history of Greek mathematics no such reference is 
found. Longitude (mekos) and latitude (platos) as spherical 

"Ibid., 557. 

" Aristotle, Physics, 209 a 4-6. 

" Aristotle, De caelo, 268 a 7-10. 



coordinates on the celestial sphere or on the earth's surface were 
obviously used by Eratosthenes, Hipparchus, Marinus of Tyre, 
and Ptolemy, being the ideal two-dimensional system for concen- 
tric spheres in Aristotle's world of spherical symmetry. Simplicius 
mentions in his commentary on the first book of Aristotle's De 
caelo that Ptolemy composed an essay On Extension (Peri dia- 
staseon) in which he demonstrated that bodies can have three 
dimensions. Moritz Cantor 44 refers to this passage and says: "Bei 
der Unbestimmtheit dieser Angabe miissen wir allerdings dahin 
gestellt sein lassen, ob man glauben will, es seien in jener Schrift 
Gedanken enthalten gewesen, welche dem Begriffe von Raum- 
koordinaten nahe kommen." So our assertion of the absence of 
spatial coordinates in Greek mathematics may stand. The use of 
a three-dimensional coordinate system, and in particular of a 
rectangular spatial coordinate system, was not thought reasonable 
until the seventeenth century (Descartes, Frans van Schooten, 
Lahire, and Jean Bernoulli), when the concept of space had 
undergone a radical change. Undoubtedly, Greek mathematics 
dealt with three-dimensional objects; Euclid himself, as related 
by Proclus, 45 saw perhaps in the construction and investigation 
of the Platonic bodies the final aim of his Elements. Yet space, as 
adopted in mechanics or in astronomy, had never been geome- 
trized in Greek science. For how could Euclidean space, with its 
homogeneous and infinite lines and planes, possibly fit into the 
finite and anisotropic Aristotelian universe? 

"M. Cantor, Vorlesungen iiber Geschichte der Mathematik (Leipzig, 
1880), vol. 1, p. 357. 

"Procli Diadochi, In primum Euclidis elementorum librum commentarii 
(Leipzig, 1873), p. 64. 



Apart from metaphysics and physics proper, theology 
proved to be a most important factor in the formulation of 
physical theories of space from the time of Philo to the New- 
tonian era and even later. Because of the great effect of theo- 
logical considerations upon the development of mechanics, as 
shown in the case of d'Alembert 1 or of Maupertuis 2 who derived 
from his theoretical physics a proof of the existence of God, it 
will be worth while looking into the matter. It may be objected 
to such an inquiry that a religious physicist would quite naturally 
insist, without recourse to tradition, upon linking science and re- 

1 J. R. d'Alembert, Oeuvres philosophiques (Paris, 1805), vol. 2, Elemens 
de philosophie, chap. 6, p. 124. 

a P. L. M. de Maupertuis, Emi de cosmohgie (Lyons, 1756). 



ligion. Yet it should be bome in mind that the general climate of 
opinion is historically conditioned. 

Let us take for example Colin Maclaurin's Account of Sir 
Isaac Newton's philosophical discoveries, published in London 
in 1748 by Patrick Murdoch from the author's manuscript papers. 
In Book One, Chapter One, we read: "But natural philosophy 
is subservient to purposes of a higher kind, and is chiefly to be 
valued as it lays a sure foundation for natural religion and moral 
philosophy; by leading us, in a satisfactory manner, to the knowl- 
edge of the Author and Governor of the Universe. To study 
nature is to search into his workmanship: every new discovery 
opens to us a new part of his scheme." As this passage shows, 
the scientist's attitude toward the very function of science may 
effect his work, while his mental disposition is generally to a 
great extent determined by his place in history and by his en- 
vironment. As far as our problem is concerned, there is no doubt 
that a clearly recognizable and continuous religious tradition 
exerted a powerful influence on physical theories of space from 
the first to the eighteenth century. 

This influence culminated in the assertion that space is but 
an attribute of God, or even identical with God. To Newton, 
absolute space is the sensorium of God; to More, it is divine ex- 
tension. What are the sources of these doctrines and where did 
they originate? It is the purpose of this chapter to show that 
these sources can be traced back to Palestinian Judaism during 
the Alexandrian period. But this by itself is not enough; we have 
also to point out the possible channels through which this East- 
ern lore passed into Western thought. 

The earliest indication of a connection between space and 
God lies in the use of the term "place" (makom) as a name for 
God in Palestinian Judaism of the first century. "In Greek philos- 
ophy the use of the term 'place' as an appellation of God does 
not occur." 3 The only intimations of Greek influence in this usage 

' H. A. Wolfson, Philo: Foundations of religious philosophy in Judaism, 
Christianity, and Islam (Harvard University Press, Cambridge, 1947), vol. 
1, p. 247. 


are found in Sextus Empiricus and perhaps in Proclus. 4 In Sextus 
Empiricus we read: 

And so far as regards these statements of the Peripatetics, it seems 
likely that the First God is the place of all things. For according to 
Aristotle the First God is the limit of Heaven. Either, then, God is 
something other than the Heaven's limit, or God is just that limit. 
And if He is other than Heaven's limit, something else will exist out- 
side Heaven, and its limit will be the place of Heaven, and thus the 
Aristotelians will be granting that Heaven is contained in place; but 
this they will not tolerate, as they are opposed to both these notions, 

both that anything exists outside of Heaven and that Heaven is 

contained in place. And if God is identical with Heaven's limit, since 
Heaven's limit is the place of all things within Heaven, God — ac- 
cording to Aristotle — will be the place of all things; and this, too, 
is itself a thing contrary to sense. 5 

With reference to these words Fabricius adds the following 
interesting remarks: "Deum Hebraei non dubitant, quia a nullo 
continetur, ipse vero immensa virtute sua continet omnia, appel- 
lare "makom" sive locum, ut saepe fit in libello rituum Paschalium 
quern edidit Rittangelius." 6 

We have quoted at length the passage from Sextus Empiricus 
in order to emphasize the fact that for Greek thought the associa- 
tion of God with space is, if admissible at all, only a very remote 
abstract deduction of an almost paradoxical character; while in 
Jewish theology of this period, and probably even earlier, the 
substitution of "place" for the name of God is a common pro- 
cedure. It seems reasonable to assume that originally the term 
"place" was used only as an abbreviation for "holy place" 
(makom kadosh), the place of the "Shekinah." Incidentally, the 
Arabic term makam designates the place of a saint or of a holy 
tomb. Since theological conceptions in Judaism soon became 
more and more abstract and universal, the original connotation 
of the term "place" fell into oblivion and the word became an 

*E. Diehl, ed., Prodi Diadochi in Platonis Timaeum commentarium 
(Leipzig, 1903-06), 117 d. , 

8 Sextus Empiricus, Adversus mathematicos (Against the physicists), II, 
33, trans, by R. G. Bury (Loeb Classical Library; Harvard University Press, 
Cambridge, 1936), vol. 3, P- 227- . . , „„ 

'J. A. Fabricius, ed., Sexti Empirici opera (Leipzig, 1840-41), p. boi. 




appellation of God without the implication of any spatial limita- 
tion. For the notion of God's omnipresence very early became 
an important idea, as seen, for example, in the writings of the 
Hebrew Psalmist. Thus in Psalm 139 we read: 

Whither shall I go from they spirit? or whither shall I flee from 
thy presence? 

If I ascend up into heaven, thou art there: if I make my bed in 
hell, behold, thou art there. 

If I take the wings of the morning, and dwell in the uttermost 
parts of the sea; 

Even there shall thy hand lead me, and thy right hand shall 
hold me . . . 

My substance was not hid from thee, when I was made in secret, 
and curiously wrought in the lowest parts of the earth. 7 

In the Midrash Rabbah we find the following discussion: "R. 
Huna said in R. Ammi's name: Why is it that we give a changed 
name to the Holy One, blessed be He, and that we call Him 
'the place'? Because He is the place of the world. R. Jose b. 
Halafta said: We do not know whether God is the place of His 
world or whether His world is His place, but from the verse 
'Behold, there is a place with me' 8 it follows that the Lord is 
the place of His world, but His world is not His place. R. Isaac 
said: It is written "The eternal God is a dwelling-place'; 9 now 
we do not know whether the Holy One, blessed be He, is the 
dwelling-place of His world or whether His world is His dwell- 
ing-place. But from the text 'Lord, Thou hast been our dwelling- 
place' 10 it follows that the Lord is the dwelling-place of His 
world but His world is not His dwelling-place." n 

The Mishna could also be cited as illustrating the frequent use 
of the term "place" to denote God. 12 It has been maintained that 
this use of the term is of Persian origin, but according to Wolf- 

'Ps. 139: 7-10, 15. 
"Exod. 33:21. 
•Deut. 33:27. 
M Ps. 90:1. 

" Midrash Rabbah, Genesis II, LXVIII, 9 (trans, by H. Freedman; Son- 
cino Press, London, 1939), p. 620. 
a Aboth II, 9; Pessah X, 5; Middoth V, 4. 



son 13 it is undoubtedly of native Jewish origin. Elisaeus Landau 14 
referred to certain Pahlavi texts and to the much earlier Zend 
Avesta in which space is deified and reverenced, as mentioned 
also by Damascius. Landau traces the use of the term "place" or 
"space" as an appellation of God back to Simon b. Shetah who, 
according to Jerus. Berachoth 15 had frequent contact with Per- 
sians. But Marmorstein 16 has shown that the use of "space" in 
this sense dates back before Simon b. Shetah, at least to Simon 
the Just (c. 300 b.c), which disproves Landau's theory of Persian 
influence; and Geiger 17 disposes of a possible Alexandrian origin. 

The Jewish Palestinian metonymy is not to be taken simply as 
a metaphor, but is evidently the result of a long process of theo- 
logical thought which culminated in the concept of the Divine 
Omnipresence. And if this is so, it provides additional support 
for the assumption of Palestinian origin. For such an ideological 
development is congenial only to a monotheistic religion and is 
alien to any form of polytheism whatever. Thus Schechter notes 
the various steps in the gradual widening of the Divine abode 
in Jewish theology: "Say the Rabbis, 'Moses made Him fill all 
the space of the Universe, as it is said "The Lord he is God in the 
heaven above, and upon the earth beneath; there is none else," 18 
which means that even the empty space is full of God." 19 

In the Palestinian Talmud we find an interesting legend that 
admirably illustrates our thesis: R. Tanhuma narrates how a 
horrible storm threatened a boat on which a company of pagans 
and one Jewish boy were sailing; as their life seemed to be in 
danger, each passenger reached for his idol to worship it, but 
without success; finally the Jew yielded to the pagans' request 

"Wolf son, Phib, vol. 1, p. 247. 

14 E. Landau, Die dem Raum entnommenen Synonyma fiir Gott (Zurich, 

1888), p. 42- 
16 Palestinian Talmud, Berachoth, VII, 2, Nasir V, 5. 

16 A. Marmorstein, The old rabbinic doctrine of God (London, 1927), vol. 

1, p. 92. 

17 A. Geiger, Nachgelassene Schriften (Breslau, 1885), vol. 4, p. 424. 

18 Deut. 4:39- 

U S. Schechter, Some aspects of rabbinic theology (New York, 1910), p. 





and prayed to his God, whereupon the sea calmed down. When 
they came to the nearest port, all of them went ashore to buy 
provisions, except the Jew. When asked why he stayed aboard, 
he replied, "What can a needy stranger like me do?" Whereupon 
they answered, "You a poor stranger? We are the strangers. We 
are here, but our gods are in Babylon or Rome; and others 
amongst us who carry their gods with them derive not the least 
benefit from them. But you, wherever you go, your God is with 
you." 20 

This notion, that God was at one and the same time here and 
there, had no pantheistic consequences in Jewish theology, but 
led to the association of God and space as an expression of his 
ubiquity. This usage had spread to Alexandrian philosophy, 21 
was incorporated in the Septuaginta 22 and was adopted also in 
the pre-Mohammedan world of thought, as we see from the 
Divan of Lebid. 23 In later Jewish literature the term "space" or 
"place" as a name for God became so frequent that an explana- 
tion, if only post facto, appeared to be called for. Thus the gema- 
tria explains that both the name of God (the nomen ineffabile) 
and the word "place" lead to the same number: adding the 
squares of the numbers corresponding to the letters of the holy 
name one gets the sum of the numbers that correspond to the 
letters of the word "place": 

D 6 = 1 100 = p 40 = D, 5 = H 6 = 1 5 = n 10 = "•; 


40 + 6 + 100 + 40 = 186, 

5 2 + & + 5 2 + 10 2 = 186. 

The designation "place" for God and the mystical conception 

of God as the space of the universe are frequently encountered 

in the post-Talmudic-Midrashic literature. The Zohar vindicates 

this use by saying that God is called "space" because He is the 

space of Himself. 24 As is well known, the Zohar is a collection 

" Berachoth, IX, Halacha 1. 
21 See Philo, De somniis, I, 575. 
"Compare Exod. 24:10 with the Hebrew original. 
ffl Jusuf Dija-ad-Din al-Chalidi, ed., Der Diwan des Lebid (Vienna, 
1880), p. 12. 
24 Zohar, I, 147 b; II, 63 b and 207 a. 


of treatises, texts, and extracts belonging to different periods, 
but with one common purpose: to reveal the hidden truth in the 
Pentateuch. Tradition claims that Simon b. Yohai, a sage of the 
second century and pupil of Rabbi Akiba, was the author of this 
book of mystical interpretation. According to the legend, he had 
spent many years in complete solitude, when he received sacred 
revelations from the prophet Elijah. It is said that the Zohar has 
been hidden for more than a thousand years in a cave in Galilee 
until it was discovered by Moses of Leon at the end of the thir- 
teenth century. According to another version, Moses of Leon 
compiled the Zohar himself, using an Aramaic idiom to give it 
an air of antiquity. Whatever its source may be, it is a collection 
of ancient Jewish folklore and oral tradition that had a great 
influence, and not on Jewish thought alone. Italy especially, that 
shifting conglomeration of .republics, states, and cities in the 
Renaissance, became a fruitful soil for Jewish esoteric teachings 
and, in particular, for the spread of cabalistic ideas. In the second 
half of the fifteenth century, when Greek scholars streamed west- 
ward after the fall of Constantinople in 1453, there were among 
them Jewish scholars who found a refuge in Italy, as we know 
from the case of Elijah del Medigo. In 1480 Elijah was called 
to the University of Padua, where he made the acquaintance of 
Giovanni della Mirandola, who invited him to Florence. Pico 
della Mirandola is generally considered as the first to introduce 
the cabala into Christianity. By the time of Mirandola's death, 
in the year 1494, cabalistic ideas had already spread farther to 
the north. Henry Cornelius Agrippa of Nettesheim, who became 
a lifelong devotee of the cabala, delivered a lecture in 1509 at 
the University of D61e on Reuchlin's De verbo mirifico in which 
he preached the doctrine of the cabala and which led to a con- 
troversy with a Franciscan, who accused him of being a "Judaiz- 
ing heretic." The promulgation of the Zohar toward the end of 
the sixteenth century in Italy added new impetus to the spread- 
ing of cabalistic ideologies to the north, and the number of 
scholars interested in this rabbinical learning increased yearly. 



One of the most erudite scholars of his time, John Rainoldes 
(1549-1607), president of Corpus Christi College in Oxford, 
made an extensive study of rabbinical lore. The German alchem- 
ist Michael Maier (born in Ruidsburg, Holstein, 1568), who be- 
came court physician to the Emperor Rudolph, visited England 
in 1615, where he very likely exerted great influence on Robert 
Fludd. Fludd was one of the early English Platonists whose im- 
portance for English theories of space and time in the sixteenth 
century cannot be overlooked. He taught "the immediate pres- 
ence of God in all nature" and illustrated his ideas from Tris- 
megistus. "God is the center of everything, whose circumference 
is nowhere to be found." One of the major sources of Fludd's 
rabbinical knowledge was probably Rainoldes' Censura librorum 
apocryphorum Veteris Testamentis. 25 

But before we continue with the history of the English Chris- 
tian Cabalists and Platonists and their influence on theories of 
space and time, let us return to Italy, where Campanella, a lead- 
ing figure in the new Italian natural philosophy, was engaged in 
formulating a spiritualized conception of space. 

If we bear in mind that both Newton's and Locke's theories 
of space originated in part, at least, in Gassendi's natural philos- 
ophy and that Gassendi was in personal contact with Campa- 
nella, 26 we are in a position to see that Campanella's ideas about 
space are of no little importance for the history of later natural 
philosophy. As can be shown in detail by comparing the writings 
of Campanella, especially his astrological and metaphysical 
works, or his Medicinalium," 27 with the writings of Paracelsus 
and of Agrippa of Nettesheim, the influence on Campanella of 
German mystical thought of the middle of the sixteenth century 
was very great. But, as we know, Paracelsus, and Agrippa much 
more, were ardent students of the Jewish cabala. It is no wonder, 

" Oppenheim, 1611. 

*" They met, for example, in Aix. See Gassendi's biography of Peiresc. 
"Thomas Campanella, Medicinalium juxta propria principia libri septem 
(Lyons, 1635). 



therefore, that Campanella's ways of thinking show strong caba- 
listic tendencies. In his De occulta philosophic (liber I) Agrippa 
restates in the spirit of the cabala the doctrine of the realization 
of the Divine Thought through the creation of a hierarchy of 
worlds. With Campanella the hierarchy of five mutually envelop- 
ing and penetrating worlds comprises the mundus mathematicus 
seu spatium, the third realization after the mundus Archetypus 
and the mundus mentalis. In his Metaphysicarum rerum juxta 
propria dogmata 29 Campanella characterizes this mathematical 
world or space as the "omnium divinitas substentas, portansque 
omnia verbo virtutis suae . . ." In words analogous to those of 
R. Huna, 29 "The Lord is the dwelling-place of His world but 
His world is not His dwelling-place," Campanella states that 
space is in God, but God is not limited by*space, which is His 
"divina creatura." 30 The idea of the identification of space with 
at least an attribute of the Divine Being gains new impetus in 
the second book, 31 where we read: "Spatium, entia locans invenio 
primum immortale, quia nulli est contrarium." In Campanella's 
conception, space becomes an absolute, almost spiritual entity, 
characterized by divine attributes. Its reality guarantees a sound 
foundation for mathematical speculations, which, according to 
Campanella, ought to be based not on hypothetical artifacts but 
on reliable sense data. 

Together with this cabalistic-Platonic conception we find in 
Campanella's theory of space the strong influence of his teacher 
Bernardino Telesio. We shall have occasion in the next chapter 
to speak of Telesio since his doctrine represents a turning point 
in the history of physical thought of the sixteenth century owing 
to its anti-Aristotelian conception of space and time. But in the 
frame of the present chapter we are concerned only with the 

28 Paris, 1638. 

29 See p. 28. 

80 Campanella, De sensu rerum et magia, libri quatuor (Frankfurt, 1620), 
I, c. 12. 
"Ibid., c. 26. 



change brought about in Campanella's original conception of 
space by Telesio's critique. Campanella, too, came to the con- 
clusion that space was completely homogeneous and undifferen- 
tiated, immovable and incorporeal, penetrated by matter and 
penetrating matter, destined for the collocation of mobile en- 
tities. "Up" and "down," "right" and "left" were only pure crea- 
tions of the intellect, designed to facilitate practical orientation 
among the multitude of concrete bodies, but with no real direc- 
tional differentiations in space corresponding to them. Campa- 
nella is here following in the footsteps of his master, as he gen- 
erally did, and to such an extent that he was held to be the 
reincarnation of him: ". . . horum clarissimus erat Thomas Cam- 
panella Stylensis, cujus in corpus Telesii ingenium transmigrasse 
dicebatur." 32 

Another trend in the history of the theory of space, very similar 
in its mystical-theological character and its association of God 
and space, was the identification of space and light. From pre- 
historic times light was the symbol of supernatural forces. The 
oldest religions are of astral character, as Egypt's Re and ancient 
Persia's Ahura Mazda bear witness. In the Sutran a3 Brahman is 
personified as the Primordial Light. Atman is honored by the 
Gods "as immortal life, as the light of lights." S4 

Even the Bible, which prohibits all images of God, still uses 
the element of light as the medium in which God becomes visible 
to man: God appeared to Moses in a burning bush; 35 a column 
of fire showed the children of Israel the way out of Egypt. 36 We 
read in the Psalms: "Who coverest thyself with light as with a 
garment." 3T In the New Testament God is explicitly identified 

** Erythraeus, Pinacotheca, I, p. 41. 
" Sutran, I, 3- 22-23. 

" Brih ad-aranyakam, 4, 4, 16; cf . P. Deussen, The system of the Veddnta 
(Chicago, 1912), p. 130. 
* Exod. 3:4. 

"Exod. 13:21; Num. 14:14. 
"Ps. 104:2. 



with light: "Ego sum lux mundi." 38 Light plays an important 
role in the metaphysical systems of Philo and of Plotin. In the 
vocabulary of the Jewish Midrashim and of the cabala "light" 
is one of the most important terms signifying the most holy 
conception. The word Zohar, 39 the title of the most important 
book in Jewish mysticism, denotes "fight," "splendor," or "glim- 
mer." According to the cabala, the Infinite Holy One, whose light 
originally occupied the whole universe, withdrew his fight and 
concentrated it on his own substance, thereby creating empty 
space. 40 

This apotheosis of light became a fundamental characteristic 
of later Neoplatonism and medieval mysticism. Even the more 
sober natural philosophy of the Middle Ages, though still an- 
thropomorphic with its hierarchy of values in nature, accepted 
light as the most "noble" entity in the world. Plotinus set the 
example of ranking light as highest in existence. Through its 
various degrees and emanations the macrocosm formed one co- 
herent and organic unit. Light is the means by which universal 
order is maintained. In its purest actuality light is Deity. Accord- 
ing to Saint Bonaventure, God is "spiritualis lux in omnia-moda 
actualitate." 41 The theories that identify space with light under 
the influence of Neoplatonism and religious mysticism are there- 
fore essentially theological in character. In our treatment of them 
here we shall confine ourselves in the main to two representative 
examples, one in antiquity, the other in the thirteenth century. 
The first is the theory of Proclus; the second is that of Witelo. 

Fragment No. 6 of The fragments that remain of the lost writ- 
ings of Proclus 43 discusses space as being the interval between 
the boundaries of the surrounding body. This is similar to the 

"John 8:12. 

"The name is derived from Dan. 12:3. 

" The cosmogony of Genesis does not conceive space as a product of 
creation. But God's first words were: "Let there be light" (Genesis 1:3). 
Thus light was created before the stars or the sun existed. 

a Liber sententiarum, II, 13 a. Cf. Etienne Gilson, La philosophie de 
Saint Bonaventura (Vrin, Paris, 1943), p. 217. 

"English translation by Thomas Taylor (London, 1825), p. 113. 




Stoic definition. But Proclus draws the conclusion that space must 
therefore be commensurable with things that are corporeal and 
since it is able to contain other bodies it must be in some sense 
corporeal in itself. Yet space must be immaterial and immovable; 
for were it material, it would not be able to occupy the same 
place as another body, and were it movable, it would be moving 
in space, that is, in itself. To Proclus, space contains the whole 
material world, but is not contained by the world and becomes 
therefore coextensive with the domain of light. 43 It is regrettable 
that Proclus's work On space, mentioned in various texts, 44 is not 
extant; thus we know very little about the immediate source 
of this identification of space and light. Perhaps it was an old 
Pythagorean theory about the primitive light or the more poetical 
variation in Plato's Republic where the myth of the Pampylian 
Er, son of Armenios, is related: "For this light binds the sky to- 
gether, like the hawser that strengthens a trireme, and thus holds 
together the whole revolving universe." 45 

The conceptions of the Neoplatonic "light-metaphysics," propa- 
gated persistently also by Jewish philosophy and mysticism 
(Saadya of Fayum, Ibn Gabirol and many cabalists), exerted a 
strong influence on Robert Grosseteste. Assuming light (lux) as 
the first corporeal form and the first principle of motion, Grosse- 
teste reduced the creation of the universe in space to the "auto- 
diffusion" of light. Light, which in his view is propagated by 
itself instantaneously, as shown by its physical manifestation, 
the visible light, is conceived by him as the basis of extension 
in space. As A. C. Crombie rightly points out in his book on 
Grosseteste, 48 it was this fundamental assumption that made 
Grosseteste believe that the key to the understanding of the 
universe lies in the study of geometric optics. Thus, in the last 

** Simplicius, Physics, 612, 32. 
" For example, in the Fihrist. 
45 Plato, Republic, X, 616. 

"A. C. Crombie, Robert Grosseteste and the origins of experimental 
science (Clarendon, Oxford, 1953), p. 104. 



analysis, it was the Neo-Platonic conception of space that led 
to the great interest in optics and mathematics exhibited by the 
thirteenth century. 

Among the most conspicuous figures in this respect were the 
unknown author of the Liber de intelligentiis and the Silesian 
Witelo, both of whom were certainly influenced by Grosseteste. 47 
In fact, the writings of these two are so similar both in contents 
and in style that Clemens Baeumker erroneously attributed the 
Liber de intelligentiis to Witelo. 48 As with Proclus, the point of 
departure in the unknown author's theory is Aristotle's physics. 
We read at the very beginning of the Liber de intelligentiis that 
space is the "ultimum continentis immobilis," in contrast to Aris- 
totle's "ultimum immobile continentis." As Baeumker remarks, the 
author accepted the introduction of two additional celestial 
spheres, the second of which is immovable, the motive being to 
bring science into line with Scripture. 49 But when the author 
comes to the famous passage in Aristotle's Physics reading, 
"Moreover the trends of the physical elements (fire, earth, and 
the rest) show not only that locality or place is a reality but also 
that it exerts an active influence," 50 he becomes Platonic in his 
conception of "dynamis" (power, in the sense of exertion of 
force) and interprets it as the faculty of space. In his view, the 
nature of space is characterized by two functions: to inclose (the 
periechein of Aristotle) and to support "continere" and "con- 
servare." Light, the source of all existence, the all-pervading 
power, ranking highest in the hierarchy of Being, light alone 
fulfills these two conditions. Hence space and light are one. 

"A. Birkenmajer, "fitudes sin- Witelo," Bull, intern, acad. polon., Classe 
hist, philos. (Cracow), p. 4 (1918), p. 354 (1920), p. 6 (1922). 

48 C. Baeumker, "Zur Frage nach Abfassungszeit und Verfasser des 
irrtiimlich Witelo zugeschriebenen Liber de intelligentiis," in Miscellanea 
Francesco Ehrle (Studi e testi, 37-47) (Biblioteca Apostolica Vaticana, 
Vatican City, 1924). The Liber de intelligentiis was edited with a com- 
mentary by Baeumker (Minister, 1908). 

49 Cf . the "caelum Empyreum" of William of Auvergne. 
60 Aristotle, Physics, 208 b 10-11. 

:l : 





The assertion, "Unumquodque primum corporum est locus et 
forma inferiori sub ipso per naturam lucis," 51 is proved by a 
series of syllogisms. 62 

These writers were not the last proponents of the importance 
of light for a theory of space. Light played an important role in 
most cosmologies of the Italian natural philosophers of the 
Renaissance. Francesco Patrizi, the predecessor of Campanella, 
was also fascinated by the mysteries of light and made light an 
integral part in his speculations. He was faced, like most philos- 
ophers of nature, in the sixteenth century, with the formidable 
task of incorporating the inherited supernatural world of the 
Middle Ages in the newly discovered world of nature of the 
Renaissance. The problem was how to unite the corporeal con- 
crete world of nature with the incorporeal world of spirit. Space, 
light, and soul, and the Neoplatonic doctrine of emanations, to- 
gether constitute the means by which he tried to solve the prob- 
lem. Space, the entity that is neither corporeal nor immaterial, 
serves as the intermediary between the two worlds. Indeed, it 
was created by God for the fulfillment of this function. Space is 
infinite, for an infinite cause can give rise only to an infinite effect. 
The traditional scholastic principle, according to which the effect 
is always feebler than the cause, applies, according to Patrizi, 
only to finite entities. The first thing to fill this space is light, the 
all-pervading, all-preserving medium of three dimensions, whose 
importance is not confined to its physical function as a transmit- 
ter of heat, power, and other influences; it is also metaphysically 
the way to God. 

An outstanding example of a strong religious bias in the con- 

a Liber, de intelligentiis, VIII, 4. ( See reference 48. ) 

M "Cuius expositio est quod locus est ultimum continentis immobilis; 
illud autem ultimum caeli est ultimum per comparationem ad id ad quod 
determinatur locus unicuique inferiori sub ipso, sicut manifestum est de 
naui et palo fixo in aqua, mutat enim superflciem corporis continentis, 
scilicet aquae, non tamen mutat locum quia caeli non mutat partem, per 
comparationem ad quam determinabatur ei locus, unde caeli ultimum locus 
est. Hoc autem habet naturam lucis. Illud enim ultimum est continens et 
conseruans, cum sit locus . . ." 



ception of space is the theory of Henry More. 53 In order to sub- 
stantiate his vigorous religious beliefs, More thought it necessary 
to augment the science of Descartes with cabalistic and Platonic 
concepts. That the cabala, apart from Neo-Platonic philosophy, 
was a major factor in More's conception of space can be proved 
not only by an analysis of the conception itself, but on historical 
grounds as well. First of all, we know that More, together with 
Fludd, was regarded as one of the greatest rabbinical students of 
his time. He certainly studied Hebrew and read the Scripture 
in its original version, as may be seen from his wide use of that 
language in his various writings, especially in his Discourses on 
several texts of Scripture. 54 Having studied the writings of Mar- 
silius Ficinus, Plotinus, and Trismegistus, he was persuaded that 
cabalistic philosophy, as expounded, for instance, in the Liber 
drushim (Book of dissertations) of Isaac Luria, was of the great- 
est importance. As far as his theory of space is concerned, More 
himself refers to the cabalistic doctrine as explained by Cornelius 
Agrippa in his De occulta philosophia, 55 where space is specified 
as one of the attributes of God. 

More's metaphysical writings are in the main a somewhat 
desultory expansion of his fundamental views on the nature of 
incorporeal substances, whose existence he was convinced that 
he had proved on the basis of his cabalistic studies. During the 
last twenty years of his life he wrote numerous publications on 
mystical subjects, including a Cabalistic catechism. 56 Some of 
these writings were addressed to Baron Knorr von Rosenroth and 
published in the Kabbala denudata 5 '* a translation into Latin of 

"For his biography and intellectual development, see B.. Ward, The life 
of Dr. H. More (London, 1710). 

64 "By the late pious and learned Henry More, D. D." (London, 1692). 

55 III, 11. Cf. Friedrich Barth, Die Cabbala des H. C. Agrippa von Net- 
tersheim (Stuttgart, 1855). 

66 For a list of More's cabalistic writings, see Gerhard Scholem, Biblio- 
graphia kabbalistica (Berlin, 1933). p- no. 

"Kabbala denudata sen doctrina Hebraeorum transcendentalis et meta- 
physica (Sulzbach, 1677). 




cabalistic writings that has done an immense service to many 
occultists by furnishing material for their reveries. 58 

More exerted a great influence on Locke, Newton, and Clarke, 
and through them on eighteenth-century philosophy in general, 
so that his doctrine is worth analyzing in detail. His writings re- 
veal that the problem of space occupied his mind as early as 
1648 at least and continued to interest him till 1684; that is, his 
interest dates from his correspondence with Descartes to his 
correspondence with John Norris. In his letters to Descartes More 
places himself in strong opposition both to ancient Greek atomism 
and to the Cartesian identification of space and matter. Both 
seem to him to lead inevitably to materialism and atheism, and 
for the refutation of these he deems it necessary to demonstrate 
the existence of a spiritual Being, which permeates and acts in 
nature. To Descartes the attribute that distinguishes spirit from 
matter is thought, as manifested in contemplation and conscious- 
ness; to More it is spontaneous activity, the source of all change 
and motion. Since change and motion are present in all realms 
of nature, the question arises how this interaction can be per- 
formed on matter at all. The answer lies, according to More, in 
the nature of space, the clear understanding of which can alone 
save philosophy from an otherwise inevitable atheism. "Atque ita 
per earn ipsam januam per quam Philosophia Cartesiana Deum 
videtur velle e Mundo excludere, ego, e contra, eum introducere 
rursus enitor et contendo." 59 

The chief motive behind More's concern with the problem 
of space, as it is the motive of his whole philosophy, is to find 
a convincing demonstration of the indubitable reality of God, 
spirit, and soul. In accord with this purpose he rejects Descartes' 
absolute identification of matter and extension. In order to 
prove the reality of spirit, it suffices to show that extension is 

"On the relations between More, Knorr, and Van Helmont, see John 
Tulloch, Rational theology and christian philosophy in England in the 
17th century (Edinburgh and London, 1872), vol. 2, p. 345. 

68 Henry More, Enchiridion metaphysicum sive de rebus incorporeis (Lon- 
don, 1671), part I, chap. 8, 7. 



spiritual provided extension itself is real. On the basis of this 
reasoning More's treatment of the space problem may be divided 
into three parts: (1) extension is not the distinguishing attribute 
of matter; (2) space is real, having real attributes; (3) space is 
of divine character. We shall comment on each of these in turn. 

(1) In his correspondence with Descartes More points out 
that, apart from the primary qualities of matter, mentioned by 
Descartes, matter has also the property of impenetrability or 
"solidity," as impenetrability was called at that time. Impenetra- 
bility (and the related tangibility) is the criterium diferentionis 
between matter and extension. As is well known, Locke in his 
Essay concerning human understanding 60 takes account of these 
ideas. In order to understand the mutual interaction between 
the world of spirit and the world of matter it is necessary to find 
a common ground between them. This common ground is space. 
Hence it follows that extension characterizes the world of spirit 
no less than the world of matter. In a word, extension is not a 
distinguishing attribute of matter, but belongs to both spirit and 

(2) In order to prove that space is real, More used different 
arguments at different times. Already in his correspondence with 
Descartes he undertakes to refute the latter's plenum against 
the existence of space as such. To both thinkers empty space 
does not exist; but if space may be empty as far as matter is 
concerned, it is yet, according to More's view, always filled with 
spirit. Descartes contended that the walls of a vessel that is 
exhausted of air must necessarily collapse. "Si quaeratur, quid 
fiet, si Deus auferat omne corpus quod in aliquo vase continetur, 
et nullum aliud in ablati locum venire permittat? Respondendum 
est: Vasis latera sibi invicem hoc ipso fore contigua." 61 Descartes' 
prediction of this presumably too complicated experiment, which 
in his view only a God is able to perform, was based on his 
philosophy of space and matter. Descartes published his Principia 

00 Book II, c. 4. 

" Descartes, Principia philosophiae, II, 18. 



in 1644. Only a few years later, or perhaps even at the same 
time, 62 a simple burgomaster performed such an experiment. 
Sed vasis latera non fierunt contigua! When reading today Des- 
cartes' argumentation on this subject one has to bear in mind 
that the concept of "a sea of air," surrounding the earth, evolved 
only toward the middle of the seventeenth century, as the 
history of pneumatics shows, and that it was unknown to 
the French philosopher. 63 Mores reply to Descartes' argument 
must be understood with these same cautions in mind: It is 
not necessary that the walls of the vessel collapse, since all 
motion of matter in Descartes' own view is originated in God, 
so that God may impart to the walls of the vessel some counter-- 
acting motion and so prevent its collapse. 64 Yet even if God 
can provide for the existence of an empty space, it still would 
not be an absolute vacuum because of the "divine extension" 
which permeates all space. 

Still another proof for the reality of space, more scholastic 
in manner, is given in an appendix to More's Antidote against 
atheism. 65 The existence of space is guaranteed by its very 
measurability "par aunes ou par lieues." 66 In other words, space 
has indubitably the attribute of measurability, even when empty 
of all matter; as there are no accidents without substance, 
measurability as an accident demonstrates the substantiality of 
space. It is of course an incorporeal substance, since it includes 
"certain notions, such as immobility and penetrability, which are 
inconsistent with matter." 67 The penetrability of space affords 

"We do not know the exact date when Otto von Guericke began his 
famous line of experiments. It was between 1635 and 1654. In 1657 the 
first accounts of the air-pump experiments were published by Kaspar Schott 
in his Mechanica hydraulico-pneumatica, through which Robert Boyle be- 
came acquainted with Guericke's experiments. 

""See J. B. Conant, On understanding science (Mentor, New York, 1952), 
P- 54- 

"Oeuvres de Descartes (Paris, 1824-1826), vol. 10, p. 184. 

"First edition published 1653 (appendix 1655, London). 

w Oeuvres de Descartes, vol. 10, p. 214. 

"More, Enchiridion metaphysicum, VI-VIII; letter to Descartes, March 


in More's view another proof of its incorporeality, and hence 
of its total discriminability from matter. More differs from 
Descartes, whose doctrine in this respect he characterizes in the 
following words: "For though it is wittily supported by him for 
a ground of more certain and mathematical after-deductions in 
his philosophy; yet it is not at all proved, that matter and exten- 
sion are reciprocally the same, as well every extended thing 
matter as all matter extended. This is but an upstart conceit of 
the present age." 68 

In the book just quoted, another very interesting demon- 
stration of the reality of space is attempted, which, because of 
its originality and the probability of its direct influence upon 
Newton, calls for detailed treatment. Philotheus, the "zealous 
and sincere Lover of God and Christ, and of the whole Crea- 
tion," in his discourse with Hylobares, the "young, witty, and 
well-moralized Materialist," gives the following example: 

Philotheus: Shoot up an Arrow perpendicular from the Earth; the 
Arrow you know, will return to your foot again. 

Hylobares: If the wind hinder not. But what does this Arrow 
aim at? 

Philotheus: This Arrow has described only right Lines with its 
point, upwards and downwards in the Air; but yet, holding the motion 
of the Earth, it must also have described in some sense a circular or 
curvilinear line. 

Hylobares: It must be so. 

Philotheus: But if you be so impatient of the heat abroad, neither 
your body nor your phancy need step out of this cool Bowre. Consider 
the round Trencher that Glass stands upon; it is a kind of a short 
Cylinder, which you may easily imagine a foot longer, if you will. 

Hylobares: Very easily, Philotheus. 

Philotheus: And as easily phansy a Line drawn from the top of 
the Axis of that Cylinder to the Peripherie of the Basis. 

Hylobares: Every jot as easily. 

Philotheus: Now imagine this Cylinder turned round on its Axis. 
Does not the Line from the top of the Axis to the Peripherie of the 
Basis necessarily describe a Conicum in one Circumvolution? 

Hylobares: It does so, Philotheus. 

Philotheus: But it describes no such Figure in the wooden Cylinder 
itself: As the Arrow in the aiereal or material Aequinoctial Circle 

"More, Divine dialogues (London, 1668), I: XXIV. 



describes not any Line but a right one. In what therefore does the 
one describe, suppose, a circular Line, the other a Conicum? 

Hylobares: As I live, Philotheus, I am struck as it were with Light- 
ning from this surprizing consideration. 

Philotheus: I hope, Hylobares, you are pierced with some measure 
of Illumination. 

Hylobares: I am so. 

Philotheus: And that you are convinced, that whether you live or 
no, that there ever was, is, and ever will be an immovable Extension 
distinct from that of movable Matter. 69 

In these last words, reminiscent of the style of Jewish medieval 
Piutim, Philotheus concludes with the absolute reality of space. 
It is interesting to note that in the case of the arrow, the proof 
is based, in the last analysis, on the relative motion of the earth 
according to the Copernican theory, and in the case of the 
rotating cylinder the proof is based on the assumption that 
rotation is always relative to something, a question that occupied 
Newton, as we know from his famous experiment with the ro- 
tating pail. Whereas in the first case no observable phenomena 
result, the second, according to Newton, makes the existence of 
absolute space physically demonstrable. 

Space, then, is the medium in which the curved line or 
the cone is formed. But here the discussion is interrupted by 
Cuphophron, "a zealous, but Aiery-minded, Platonist and Carte- 
sian, or Mechanist," who contends, "that it may be reasonably 
suggested, that it is real Extension and Matter that are terms 
convertible; but that Extension wherein the Arrow-head de- 
scribes a curvilinear Line is only imaginary." This remark ushers 
in a line of arguments that for the modern reader suggests the 
Kantian conception of space. For Hylobares replies: "But it is 
so imaginary, that it cannot possibly be dis-imagined by humane 
understanding. Which methinks should be no small earnest that 
there is more than an imaginary Being there." 

Collateral confirmation of the reality of space is adduced by 
calling upon the authority of the ancient atomists, of Aristotle, 
and of the Pythagoreans with reference to the famous analogy 

"Ibid. (London, ed. 2, 1713), p. 52. 



that the "Vacuum were that to the Universe which the Air is 
to particular Animals." Finally, More reverts to his old argument, 
which we characterized as scholastic, when he makes Hylobares 
say: "And lastly, O Cuphophron, unless you will flinch from the 
Dictates of your so highly-admired DesCartes, forasmuch as 
this Vacuum is extended, and measurable, and the like, it must 
be a Reality; because Non entis nulla est Affectio, according 
to the Reasonings of your beloved Master. From whence it seems 
evident that there is an extended Substance far more subtile 
than Body, that pervades the whole Matter of the Universe." 

(3) It is contended with regard to this "subtile" substance, 
called further on the "Divine Amplitude," that it exists neces- 
sarily, and would exist even if all matter were annihilated. The 
necessary existence of space, even without matter, leads More 
to the final identification of space with God. For it might be 
argued, as Cuphophron argues toward the end of the discussion 
on the nature of space, that, if God as well as matter were anni- 
hilated from the world, extension would seem necessarily to 
remain. The spokesman for More's reasoning here is Bathynous, 
"the deeply-thoughtful and profoundly-thinking Man," who re- 
plies that God's essence implies existence (the ontological proof). 
In other words, to assume the annihilation of God is a contra- 
dictio in adjecto. God and space have both the property of 
necessary existence; they are therefore one and the same. 

This conclusion that God and space are one is drawn also in 
the appendix to the Antidote against atheism: 

If after the removal of corporeal matter out of the world, there 
will be still space and distance, in which this very matter, while it 
was there, was also conceived to lie, and this distant space cannot but 
be conceived to be something, and yet not corporeal, because neither 
impenetrable nor tangible, it must of necessity be a substance in- 
corporeal, necessarily and eternally existent of itself; which the clearer 
idea of a being absolutely perfect will more fully and punctually 
inform us to be the self-subsisting God. 70 

TO A collection of several philosophical writings of Dr. Henry More ( Lon- 
don, ed. 2, 1655), appendix, p. 338. Cf. F. I. MacKinnon, ed., Philosophical 
writings of Henry More (London and New York, 1925). 





The attributes of space are attributes of God. A list of these 
attributes is given in Mores Enchiridion metaphysicum: 

Neque enim Reale duntaxat, (quod ultimo loco notabimus) sed 
Divinum quiddam videbitur hoc Extensum infinitum ac immobile, 
<W tarn certo in rerum natura deprehenditur) postquam Divma 
ilia Nomina vel Titulos qui examussim ipsi congruunt enumeravimus, 
qui & ulteriorem fidem facient illud non posse esse nihil, utpote cm 
tot tamque praeclara Attributa competunt. Cujusmodi sunt quae 
sequuntur, quaeque Metaphysici Primo Enti speciatim attribunt Ut 
Unum, Simplex" Immobile, Aeternum, Completum, Independens, 
A se existens. Per se subsistens, Incorruptible, Necessarium, Immen- 
sum, Increatum, ^circumscriptum, Incomprehensible, Ommpraesen^, 
Incorporeum, Omnia permeans & complectans, Ens per Essentiam, 
STStu, Purus Actus. Non pauciores quam viginti Tituli sunt quibus 
insigniri solet Divinum Numen, qui infinite huic Loco interno, quern 
in rerum natura esse demonstravimus aptissime conveniunt: ut 
omittam ipsam Divinum Numen apud CabbaW appellan makom, 
id est, Locum. 71 

This list of attributes and names, a recurrent theme in the 
cabalistic writings, is quoted in full to show the extent to which 
More was influenced by Jewish mysticism. In his Divine dia- 
logues as well More mentions the cabalists in connection with 
the divine nature of God. The discussion that we cited from the 
Dialogues ends with Psalm 90: 

Lord, thou hast been our dwelling-place in all generations. 

Befoe the mountains were brought forth, or ever thou ^ hadst 
formed the Earth or the World, even from everlasting to everlasting, 
thou art God. 

No doubt, the general tenor of the cabala may easily have 
provoked such spiritual ideas about space as were harbored by 
More. Anyone who has read the Book of formation (Sepher 
Yezkah), which deals with cosmogonical problems of the uni- 
verse, or who has read on Luria's cabalistic notion of the "Zim- 
zum" 72 the divine self-concentration, creating space by self- 
restriction, will certainly accept the thesis that a somehow 

71 More, Enchiridion metaphysicum, part I, chap. 8. 
™"Deus creaturus mundos contraxit praesentiam suam, 
data (Sulzbach, 1677), part II, p. 150. 

Kabbala denu- 



pantheistic interpretation of the cabala must necessarily lead to 
More's conception of space. 

Indeed, a similar intellectual process was most probably in- 
fluential on the philosophy of Spinoza. With reference to his 
fundamental dictum; "Quidquid est, in Deo est et nihil sine Deo 
esse neque concipi potest," 73 Spinoza admits in a letter to Olden- 
burg: ". . . omnia, inquam, in Deo esse et in Deo moveri cum 
Paulo affirmo . . . et auderem etiam dicere, cum antiquis omni- 
bus Hebraeis, quantum ex quibusdam traditionibus, tametsi 
multis modis adulteratis conjicere licet." 74 As A. Franck in La 
cabbale'' 5 and much earlier (1699) Johann Wachter in Der 
Spinozismus in Jiidenthumb 76 have shown, Spinoza's remarks 
can refer only to cabalistic writings. 

In contrast to More, Spinoza includes not only extension but 
also matter as an attribute of God, thus changing the conception 
of God into that of an absolutely impersonal, almost mechanical 
God, as is shown in his Ethics. Leibniz, who seems to have read 
Wachter's De recondita Hebraeorum phttosophia,' 1 '' in which the 
author repeats his thesis concerning Spinoza's dependence on 
the cabala, wrote in this context in a letter to Bourget, "Verissi- 
rrmm est, Spinozam Cabbala Hebraeorum esse abusum." 78 

This digression on Spinoza's philosophy and its possible 
cabalistic sources has been inserted only to show that elements 
of Jewish esoteric writings, perhaps owing to the rise of Neo- 
platonic ideas, could have been easily integrated into the 
philosophy of the seventeenth century. We reserve judgment 

78 "Ethica more geometrico demonstrata," I, prop. 15, in Spinoza, Opera 
(ed. C. Gebhart; Heidelberg, 1925). P- 56. 

™H. Ginsberg, ed., Der Brief wechsel des Spinoza im Urtext (Leipzig, 
1876), p. 53, Epistola XXI; also Epistola LXXXIII. 

"Adolphe Franck, La cabbale ou la philosophie des HSbreux (Paris, 

"J. G. Wachter, Der Spinozismus in Jiidenthumb oder die von dem 
heutigen Jiidenthumb und dessen geheimen kabbala vergotterte Welt (Am- 
sterdam, 1699). 

"Published in 1706. 

™ Letter of the year 1707. 



on the question how far in detail Spinoza, More, or any other 
thinker of that time has actually been influenced by the cabala, 
but we claim — as definitely demonstrated in the case of the 
concept of space — that certain general ideas of cabalistic origin 
have been absorbed into the intellectual climate of that period. 

Our investigation into the influence of Judeo-Christian reli- 
gious speculations on the conception of absolute space in the 
seventeenth century has not been presented as an uninterrupted 
chain of incontestable conclusions. Owing to the evasive charac- 
ter of the rather obscure and mystical ideas involved, the con- 
tention is rather based on the tradition of a certain "climate of 
opinion" or attitude of mind, and not on a direct communica- 
tion of definite statements. In the case under discussion it was at 
least possible to expose all the major stages in this transmission. 

More problematic, and still more conjectural, is the theory of 
a religious background in the rival conception of space, that is, 
Leibniz's relational point of view. As we shall see in Chapter IV. 
Leibniz rejected Newton's theory of an absolute space on the 
ground that space is nothing but a network of relations among 
coexisting things. In his correspondence with Clarke, Leibniz 
likens space to a system of genealogical lines, a "tree of gene- 
alogy." or pedigree, in which a place is assigned to every person. 
The assumption of an absolute space, according to Leibniz's 
view, is wholly analogous to a hypostatization of such a system 
of genealogical relations. 

Now it is important to recall that a similar theory had already 
been propounded by the eleventh-century Muslim philosopher 
Al-Ghazali, or possibly by one of his predecessors. Here again 
it becomes apparent that theological and metaphysical specula- 
tions were influential on the formulation of a theory of space. In 
fact, the whole issue in question is based on the ideological 
conflict between Aristotelian cosmology and the Koranic dogma 
of divine creation. For a full understanding of the situation it is 
perhaps most instructive to discuss both the temporal and the 


spatial aspect of the problem. Time was defined by Aristotle 79 
as the "number of motion" (for example, of the revolutions of 
the celestial spheres). Since without natural bodies there cannot 
be motion, Aristotle concluded that outside the finite heaven 
there is no time. Place, according to the Stagirite, presupposes 
the possibility of the presence of bodies. 80 Since outside the finite 
heaven no body can exist, as proved in his writings previously, 
Aristotle deduced that outside the finite heaven there is no 
place. So far Muslim philosophy complies with Aristotelian cos- 
mology. But now the conflict becomes apparent: whereas Aristo- 
telian cosmology assumes the eternity of substance, the Koranic 
dogma affirms divine creation. Thus for Muslim philosophy a new 
problem is raised which did not exist for Aristotelian cosmology. 
It is the question whether there were space and time prior to 
the act of creation. Obviously, the answer must be negative, since 
space and time have no existence apart from matter per defini- 
tionem; they are mere relations among bodies. Space and time 
are consequently also products of creation. Muslim philosophy 
even went so far that it rejected the logical validity of the state- 
ment, "God was before the world was" 81 ("Kdna allahu walA 
'dlama"), if kdna is to be understood in a temporal sense. Even 
in the most fundamental proposition, "God created the world" 
("halaka allahu al 'dlama"), the verb "created" {halaka) has to 
be understood in a causal and not in a temporal sense. The rela- 
tion between God and the work of his hands is essentially a 
causal relation and not a relation in space and time. The ques- 
tion, "Where was God before the creation?" is meaningless, since 
space is a "pure relation" (iddfat mahda) among created bodies. 
Theological polemics, as we see, led to a conception of space 
as a network of relations a conception that shows a striking 
resemblance to Leibniz's idea about space. To be sure, it is al- 

™ Aristotle, De caelo, I, 9, 279 a. 


81 Algazel, Tahafot Al-Falasifat (ed. by Maurice Bouyges, S. J.; Beyrouth, 

1927), P- 53- 

■1 f 




ways very difficult to assess any influence where thought proc- 
esses are involved. Certainly, allowance has to be made for a 
complete independence between similar conceptions, especially 
if they are separated so much by time, space, and language. In 
the case under discussion it is certainly wise to defer judgment 
until a comparative study of the relevant texts establishes the 
fact of an indubitable intellectual dependence or until historic- 
biographic research proves the indebtedness without doubt. In 
our case it is rather tempting to suspect such a dependence if it 
is also noted that Leibniz's monadology shows a striking re- 
semblance to the atomistic theory and occasionalism of the 
Kalam, a famous Muslim school of thought, called also the "Muta- 
kallimun," or "Loquentes," as mentioned by Saint Thomas Aqui- 
nas. Details about their theory of space will be explained in 
Chapter III. As far as our question of theological influence on 
ideas about space is concerned, we have to stress the following 
fact: It has been established that the theory of atoms in Islam 
and the corresponding conception of space were originally of 
a purely profane character and became adapted to an extreme 
theistic dogma only during later stages of their development. 
From a strictly historical point of view it must be admitted, 
therefore, that the Kalamic theory of space did not originate on 
the background of religious speculations. However, it was this 
background from which it drew, at the climax of its vitality, its 
emotional strength of conviction. Our exposition of the Kalamic 
conceptions of space, in Chapter III, refers to this later stage, 
the "orthodox" Kalam, which once was defined as "the science 
of the foundations of the faith and the intellectual proofs in 
support of the theological verities." 82 

M Sir Thomas Arnold and Alfred Guillaume, The legacy of Islam (Oxford 
University Press, London, 1949), p. 265. 




In Aristotle space is identified with place and defined 
as the adjacent boundary of the containing body. This definition, 
it is clear, is in line with Aristotle's fundamental assumption 
of the impossibility of a vacuum. Theophrastus' critique of his 
master's doctrine had no immediate import for the development 
of physical thought. In this respect the teachings of Strata of 
Lampsacus, the "physikos," the successor of Theophrastus, ex- 
erted a greater influence. Strata's realistic attitude, probably a 
result of the Alexandrian climate of opinion, led him to divest 
the Aristotelian system of its transcendental elements and to seek 
for a compromise with everyday experience. He came to the 





conclusion that a vacuum is not an absolute impossibility, but 
may exist, and in fact does, in matter itself, forming minute 
interstices between the material particles. That this deviation 
from Aristotle's teaching induced Strato to revise the Aristo- 
telian conception of space seems to be highly likely, if we re- 
member that Strato wrote a book on the vacuum, which is now 
lost, but which was still known to Simplicius. Regrettably little, 
except some experimental features of this work, are referred to 
in later scientific literature, as for instance in the writings of 
Heron, who construed the penetration of light rays assumed to be 
of corporeal nature, and of heat through water as a proof of the 
existence of small vacua inside matter, as Strato had contended. 
In Heron's Pneumatics we find the assertion that continuous 
vacua can be produced, though only by artificial means, whereas 
in nature only small, discontinuous vacua can exist. 

The first major contribution to the clarification of the concept 
of absolute space was made by Philoponus, or John the Gram- 
marian, as he is often called (fl. c. a.d. 575). Philoponus is well 
known as the forerunner of the so-called "impetus theory" in 
mechanics, which was the subject of profound investigation 
during the fourteenth century and which became in its later 
development the main point of departure for Galilei's formula- 
tion of the basis of modern dynamics. We shall have occasion 
to see how Philoponus' revision of the Aristotelian conception of 
space is intrinsically connected with his impetus theory. He 
begins by pointing out an inner inconsistency in Aristotle's theory 
of space. To Aristotle, place is the adjacent boundary of the 
containing body, provided this containing body itself is not in 
motion. If, for example, we hold a stone in a current of water, 
the constantly changing envelope of water clearly is not the 
"place" of the stone; otherwise the motionless stone would change 
its place continuously, which is self-contradictory. The stone's 
place must therefore be the inner surface of the first immobile 
containing body, as, for instance, the river bed. 

Philoponus now asks what actually is the place of the sub- 



lunary world of matter, subjected to generation and decay. Ac- 
cording to Aristotle it is the concave surface of the first celestial 
sphere, the orbit of the moon. But, says Philoponus, this surface 
itself is constantly rotating and therefore not immobile; on 
the contrary, a certain part of this surface successively touches 
other parts of the contained matter, even if these parts them- 
selves happen not to be moving. To ascribe the place of our 
changing world to one of the higher spheres is of no avail, since 
all of them are in rotational motion. Philoponus rejects the argu- 
ment that rotation about a fixed axis or a fixed point is not 
local motion, since the sphere as a whole always occupies, so 
to say, the same place. 1 Philoponus concentrates upon a fixed 
part of the rotating sphere and shows how this part occupies 
different places in the course of time. Hence he concludes that 
Aristotle's definition of "place" leads to a cul-de-sac and must 
be rejected. The definition not only makes it impossible to deter- 
mine the place of the sublunar world, but provides no answer 
to the question of the place or the space in which the outermost 
sphere is moving, for moving it certainly is. 

Philoponus was not the first to notice this difficulty in ex- 
plaining the motion of the last sphere consistently with Aris- 
totle's principles. In fact, Aristotle's statement, "It is clear that 
there is neither place nor void nor time beyond the heaven," 2 
began to be a subject of serious doubt. As adopted by most of 
the commentators, the usual solution to this question was to 
point out that the place of each individual part of the rotating 
sphere was determined by the parts of the same sphere that 
were contiguous to it. But, contends Philoponus, if this is the 
case, what parts of the rotating sphere actually "change their 
places" during the rotation? 

A solution of this problem was offered, for instance, by 
Themistius, whose Paraphrasis in libros quatuor Aristotelis de 

1 In modern words, being a surface of constant curvature it is transfer- 
able into itself. 
•Aristotle, De caelo, 279 a 12. 


caelo is extant in a Hebrew translation and also in a later Latin 
translation of this Hebrew text by Moses Alatino. 3 In his attempt 
to overcome the difficulty, Themistius lands himself in a vicious 
circle, for what he says boils down to the statement that the place 
of the outermost sphere is the convex surface of the sphere of 
Saturn, as much as the place of Saturn is the concave inner 
surface of the last sphere. 4 

Inconsistencies of this kind proved to Philoponus that a new 
definition of "place" or space was necessary. According to him, 
the nature of space is to be sought in the tridimensional incor- 
poreal volume extended in length, width, and depth, different 
altogether from the material body that is immersed in it. "Space 
is not the limiting surface of the surrounding body ... it is a 
certain interval, measurable in three dimensions, incorporeal in 
its very nature and different from the body contained in it; it is 
pure dimensionality void of all corporeality; indeed, as far as 
matter is concerned, space and the void are identical." 5 

However this identification of space and void does not assume 
the existence of a void as such "in actu." The void, although a 
logical necessity, is always coexistent with matter. Void and body 
are two inseparable correlates, each of them requiring the exist- 
ence of the other. As soon as one body leaves a certain part of 
space, another body "replaces" the first. A certain region of space 
can receive different bodies in succession without taking part in 
the motion of the occupying matter. Philoponus' phoronomy is 
completely analogous, as Duhem 6 points out, to Aristotle's doc- 
trine of substance and form, where one form is succeeded by 
another continuously, so that substance is never void of form. 
Just as matter successively receives one form after another, so 
a section of space may be occupied by one body after another, 
space itself remaining immobile. 

"Moses Alatino, Themistii in libros Aristotelis De caelo paraphrasis, 
Hebraice et Latine (ed. S. Landauer; Berlin, 1902). 

* Cf . Simplicius, Physics, p. 589. 

B H. Vitelli, ed., Ioannis Philoponi in Aristotelis physicorum libros quinque 
posteriores commentaria (Berlin, 1888), p. 567. 

"P. Duhem, he systeme du monde (Paris, 1913), vol. 1, p. 381. 


It is clear that this rather abstract notion of space is incom- 
patible with Aristotle's dynamics, for Philoponus conceives space 
as pure dimensionality, lacking all qualitative differentiation. 
Space can no longer be conceived as the efficient cause of motion, 
compelling the body to move to its "natural place." "It is ridic- 
ulous to pretend that space, as such, possesses an inherent power. 
If every body tends to its natural place, it is not because it seeks 
to reach a certain surface; the reason is rather that it tends to 
the place which was assigned to it by the Demiurgus." 7 

Such concepts as "up" and "down," whose justification Philo- 
ponus does not deny, are no longer an intrinsic quality of space 
or place, but owe their validity to purely geometrical arrange- 
ment on the one hand, and to cosmological-theological predes- 
tination, as assigned by the Demiurgus, on the other. As to the 
first of these it should be pointed out that Philoponus, following 
Aristotle, accepts the fundamental tenet of the finiteness of the 
universe. Since matter is finite, its correlate, space, which is 
indissolubly connected with it, must be finite as well. Thus the 
universe possesses a final boundary, a last sphere, which deter- 
mines its "upper" regions. The center, which as decreed by the 
Demiurgus is occupied by the earth, is by definition the direction 
of the "down." A body falls "down," not because its new place 
exerts such a strain as to direct it to its "natural place," where 
this strain ceases to operate, but rather because it possesses 
an inherent tendency to reach the place assigned to it by the 
Demiurgus. It is this tendency, inherent in the moving body 
and not in the medium or in space, that corresponds to the 
"impetus" in the case of forced motion. Philoponus' explanation 
of the fall of heavy bodies shows a remarkable resemblance 
to the explanation of gravity suggested by Copernicus: "Equidem 
existimo gravitatem non aliud esse, quam appententiam quan- 
dam naturalem partibus inditam a divina providentia opificis 
universorum, ut in unitatem integritatemque suam sese conferant 
in forman globi coeuntes." 8 

7 Reference 5, p. 581. 

8 De revolutionibus orbium coelestium, liber I cap. IX. 



Iamblichus' theory of space, because of its influence on 
physical thought in antiquity, should be mentioned here. The 
theory stands in complete contrast to that of Philoponus. As 
Simplicius relates in detail, 9 Iamblichus defines place as a mate- 
rial force which sustains the body and holds it together, which 
raises what has fallen and brings together what has departed, 
fining their volume and surrounding them from all sides. Du- 
hem 10 thinks it probable that Iamblichus was influenced by the 
writings of Archytas. 

A profound inquiry into the nature of space is to be found 
in Damascius' treatise Peri arithmou kai topou kai chronou, of 
which Simplicius gives us a detailed account in the Corollarium 
de loco 11 of his Commentaria to Aristotle's Physics. The primary 
term in Damascius' inquiry is not place or space, but position or 
location, which is for him an inseparable attribute of the object. 
In fact, this notion has two meanings, one denoting the relative 
location of the different parts of the object and the other denoting 
the position of the whole in the universe. Speaking in modern 
terms, we find here, probably for the first time, the notion of 
the three degrees of freedom of a complex body as a whole as 
opposed to its internal degrees of freedom. Space is as different 
from position as time is different from motion. Just as time for 
Aristotle is the numerical measure of motion, so for Damascius 
space or place is the numerical measure of position. If we look 
upon position as a certain quality of the object, space makes it 
possible to determine this quality quantitatively. But the essence, 
the nature of this quality, is not accessible to geometric formula- 
tion. In the same way as every part of the universe has a "nat- 
ural" position which is best for that part, so the whole universe 
has a "natural disposition," attained when all its parts are in 
their corresponding "natural" positions. 

* Simplicius, Physics, p. 639. 

10 Le systeme du monde, vol. 1, p. 333. For a detailed discussion on 
Iamblichus' theory of space, see also Eduard Zeller, Die Philosophic der 
Griechen (Leipzig, 1881) vol. 3, 2, p. 706. 

11 Book IV, cap. IV. 




In contrast to the traditional conception of place in Greek 
thought, Damascius holds that position is inseparable from the 
object, even when the object is in motion. Place was usually 
supposed to be capable of receiving different bodies in succes- 
sion; but position, like any other attribute, was not directly 
transferable from one object to another. Just as a body, when 
changing its color from white to black, does not leave whiteness 
behind as an independent existent, so the position of a body in 
motion, although constantly changing, never becomes the posi- 
tion of another body, but ceases to exist when the moving body 
acquires a new position. 

Damascius' conception of space as the geometric measure of 
position led him to an important conclusion which again does 
not conform to the traditional Peripatetic doctrine. It is the 
famous question whether motion presupposes rest. Aristotle, or 
whoever wrote De motu animalium, lays it down that whenever 
a body is in motion, there must be something that is motion- 
less. 12 Duhem thinks that the writer of De motu animalium did 
not himself refer explicitly to this axiom as an argument for the 
immobility of the earth. Still, the majority of commentators 
undoubtedly saw in it a proof of the necessity of the earth's 
immobility. For example, Themistius in his Paraphrasis in libros 
Aristotelis de caelo says: "Sed conversio, immo omnis motus, 
super manente ac quiescente aliquo omnino celebratur. In iis 
autem, quae De Animalium Motu a nobis dicta sunt, monstra- 
tum est id, quod manet ac quiescit, illius autem partem esse non 
posse, quod super ipso movetur." 13 As a matter of fact, the au- 
thor of De motu animalium seems himself to have been aware 
of the cosmological implications of his biological conclusion, as 
when he writes: "And it is worthwhile to stop and consider this 
dictum; for the reflection which it involves applies not merely 
to animals, but also to the motion and progression of the uni- 

" Cf. Aristotle, Movement of animals, 698 b 10, trans, by A. L. Peck 
(Loeb Classical Library; Harvard University Press, Cambridge, 1937)- 
"Reference 3, p. 97- 




verse. For just as in the animal there must be something which 
is immovable if it is to have any motion, so a fortiori there must 
be something which is immovable outside the animal, supported 
on which that which is moved moves." 14 

The question whether motion presupposes the existence of 
something immobile or not divided Aristotle's followers during 
the centuries. This question was held to be a problem in dynam- 
ics and to be different from the purely kinematic phenomenon 
of apparent motion which pertains to sense perception. The 
latter question was dealt with adequately by Euclid, who stated 
that an object which appears to be at rest to an observer who 
regards himself as being moved, would appear to be retrogressing 
to the same observer if he regards himself as being at rest. 15 
Averroes, who expounds Aristotle's theory of space and motion in 
detail, supports the view that a concrete motionless body is a 
necessary condition for the existence of motion. The problem 
assumed a vital importance at the end of the thirteenth century 
when its implications for theology became apparent. In the 
council of the doctors of the Sorbonne, which took place under 
the presidency of Etienne Tempier in 1277, Averroes' interpreta- 
tion was declared to be heretical, since the recognition of an 
absolutely immobile body, immobile even for the Creator of the 
universe, was thought to be incompatible with the fundamental 
ideas of Christian theology. Their belief in God's omnipotence 
forced the theologians to the conclusion that God could move 
the whole universe — which of course was thought to be of 
finite extent — through space. We shall later have an opportunity 
to discuss the physical implications of this condemnation of 
Averroistic conceptions of motion and shall show how the at- 
tempt to reconcile it with Aristotelian physics led to a new 

" Aristotle, Movement of animals, 698 b 10. 

18 "Si aliquibus latis pluribus inaequali celeritate simul transportetur in 
easdem partes et oculus, quae quidem oculo aequali celeritate feruntur, 
videbuntur stare, tardiora vero in contrarium ferri, celeriora vero in prae- 
cedentia. — Si aliquibus latis appareat aliquid, quod non feratur, videbitur 
illud non latum retrorsum ferri. ' I. L. Heiberg and H. Menge, ed., Euclidis 
opera omnia (Leipzig, 1883-1916), vol. 7, Optica, p. 110. 



interest in the problem of space and motion. At present we are 
interested only in the fact that the theologians of Paris accepted 
the doctrine of Damascius as the only orthodox doctrine. 

For according to Damascius motion presupposes no immobile 
body. It is only our perception of motion which necessitates 
reference to something that is supposedly not in motion; we need 
the assistance of a fixed object if we are to distinguish motion 
from rest through the change of certain geometric measures. The 
absence of a motionless body does not preclude the possibility 
of local motion; it merely prevents our recognition of motion 
as such. A motion of the whole universe is therefore not im- 
possible. It is also possible, says Damascius, that the heavens 
would continue in their habitual diurnal revolutions even if 
neither east nor west nor south existed. 16 Although this appears 
to be the earliest affirmation of the merely relative value of 
geographical or astronomical directions in space, we must be 
careful not to exaggerate the point. For Damascius still holds to 
the traditional doctrine of natural places, which remain immobile 
and fixed, that is, independent of the real motion of the concrete 
parts of the universe. 

If we were to translate Damascius' conception into modern 
terms, we might say that his set of natural places is identical 
with an extended field whose coordinates stand in a one-to-one 
correspondence with the material parts of the universe. This 
field is invariant and independent of the actual motion of the 
universe, but, as determining the final cause of natural motion, 
it may be thought of as a regulative force, making for an in- 
creasing degree of perfection of the universe. In this sense 
natural place is characterized by Damascius as the "telesiurgus," 
or the driving force to perfection. 17 As the final system of refer- 
ence for the actual positions of all mobile bodies, "natural place" 
does duty in this sense for Aristotle's outermost immobile sphere, 
the final containing concave surface. However, as the things 

"Simplicius, Physics, p. 634. 
"Ibid., p. 601. 



of the universe have not reached their natural places, this sys- 
tem of reference remains an ideal abstraction, of no avail for 
the physical determination of the position of actual bodies. 

One question has still to be answered: Does Damascius en- 
dow natural place with efficient as well as final causes? In other 
words, does a natural place, in terms of his theory of space, 
exert a direct and directive force on the material body to which 
it corresponds? Duhem answers this question in the affirmative, 18 
basing himself on Simplicius' remarks that Damascius admired 
Iamblichus' doctrine and saw in him a predecessor of his own 

Turning from these commentators of the Stagirite to Muslim 
exponents of Aristotelianism and Greek philosophy in general, 
we note that once the Arabian world became familiar with the 
system of thought contained in the writings of the Greeks and 
Syrians, the authority of Aristotle became paramount. As far as 
our subject is concerned, only a few major deviations from the 
system of "al-failasuf ("the philosopher" par excellence) are 
encountered: the atomistic doctrine of space in the Kalam, and 
the theories of Al-Razi and of Abu'l Barakat. 19 The Kalam may 
be compared with scholastic philosophy of medieval Europe 
not only because of its dialectic method in theological specula- 
tion but also because of its object of supporting a dogma by 
discursive thought. Abu' l'Hasan al-Ash'ari of Bagdad and Abul- 
Mansur al-Maturidi of Samarquand, both of the tenth century, 
are usually named as the principal founders of the orthodox 
Kalam, although its origin most certainly dates back to the ninth 
century. 20 

In order to bring into special prominence the divine creative 
act, the Kalam attributes to matter (as well as to space, as we 

M Duhem, he syste'me du monde, p. 350. 

" Cf. S. Pines, Etudes sur Awhad al-Zaman Abu'l Barakat al-Baghdadi," 
Rev. etudes Juives 3, 5 (1938)- 

"According to Ibn-Khaldun it was Al-Baqilani (d. 1013) of Basra, the 
most remarkable disciple of Al-Ash'ari, who introduced the concept of 
atomism into the Kalam. See G. Sarton, Introduction to the history of 
science (Baltimore, 1931), vol. 1, p. 706. 



shall see immediately) only a transient existence of extremely 
short duration and range, requiring thereby a constant divine 
creative interference for the maintenance of coherence and 
continuity in the universe. Everything contained in this universe 
was conceived as being composed of atoms and accidents, and 
not of substance and properties as Aristotle taught. This atomistic 
doctrine seems at first sight to stand in opposition to the theo- 
logical tenet of the Kalam; but as soon as the principle of 
causality or the Democritian concept of "necessity" is abandoned 
and replaced by the adoption of a transcendental principle of 
divine interference, the contrast dwindles away. The refutation 
of a mutual interaction among the atoms is also in accordance 
with Aristotle's argumentations 21 with regard to a consequential 
atomism. Such a revised atomism proves to be a most suitable 
ground for the extreme theistic philosophy of the Kalam. The 
doctrine of atomism was regarded as the first fundamental propo- 
sition in the system, as we see in chapter 73 of Moses Maimon- 
ides' Guide for the perplexed. 22 This work, written with the 
purpose of reconciling Aristotle with Jewish theology, serves as 
an important source for our information on Muslim philosophy 
in general and the Kalam in particular, although it does not 
treat the latter in an impartial way. 

The atoms of the Kalam are indivisible particles, equal to 
each other and devoid of all extension. Spatial magnitude can 
be attributed only to a combination of atoms forming a body. 
Although a definite position (hayyiz) belongs to each individual 
atom, it does not occupy space (makdn). It is rather the set of 
these positions — one is almost tempted to say, the system of 
relations — that constitutes spatial extension. Thus, for instance, 
according to Mu'ammar (or Ma'mar), one of the older advocates 
of the Kalamic theory, two atoms, if connected, related, or 
attached (indamma) with each other, constitute length; four 

21 Aristotle, De generatione et corruptione, I, 9, 326 a. 

22 For an analysis of this chapter, see D. B. MacDonald, "Continuous re- 
creation and atomic time," Isis 9, 342 (1927). 



atoms constitute length and breadth, that is, a two-dimensional 
spatial extension; a three-dimensional body is composed of a 
pile (tabaka) of two-dimensional extensions and contains con- 
sequently at least eight atoms. 23 As Isaac Israeli's Liber de 
elementis clearly shows, 24 the problem of reconciling spatial 
extension of bodies with the supposedly unextended nature of 
atoms was a much discussed topic already in early Muslim- Jewish 
natural philosophy. 

In the Kalam, these rather complicated and surprisingly ab- 
stract ideas were deemed necessary in order to meet Aristotle's 
objections 25 against atomism on the ground that a spatial con- 
tinuum cannot be constituted by, or resolved into, indivisibles 
nor can two points be continuous or contiguous one with another. 

In view of the surprising resemblance between the atomistic 
theory of the Kalam and the monadology of Leibniz, as well as 
between the corresponding conceptions of extension and space, 
we face the interesting problem whether this is a mere fortuitous 
coincidence. We know with certainty that Leibniz read Maimon- 
ides' exposition of the Kalam in Buxtorfius' Latin translation of 
the Guide. The copy used by Leibniz shows many marginal notes 
written by his own hand, demonstrating what great inspiration 
he drew from reading the book. Foucher de Careil, one of the 
editors of Leibniz's works, adduces additional information on 
this point in his book Leibnitz, la philosophie juive et la Cabbale 
(Paris, 1861). M. Guttmann draws our attention in this con- 
nection to the following phrase in Leibniz's Epistolae ad P. des 
Bosses: "Substantia nempe simplex etsi non habeat in se ex- 
tensionem, habet tamen positionem, quae est fundamentum 
extensionis." 26 

An important feature of the atomistic doctrine of the Kalam 

28 S. Pines, Beitrage zur Islamischen Atomenlehre (Berlin, 1936), p. 5. 

"Isaak b. Salomon Israeli, Sefer Hayesodoth (Frankfurt a. M., 1900), 
p. 43 of the Hebrew text. 

85 Aristotle, Physics, IV, 6; VI, 4, 6; 213 b, 234 b, 237 a. 

"Moritz Guttmann, Das philosophische System der MutakaUimun (Bres- 
lau, 1885), p. 20. 



is its affirmation of the existence of empty space. Empty space 
is not only a necessary presupposition for the possibility of 
motion — combination and separation among atoms explain the 
processes of generation and corruption — but the disjunctive 
character of empty space is also claimed as a necessary pre- 
requisite for the separateness and independence of the individual 
atom. Consequential thought led the Kalam to the conclusion 
that space, as well as matter (and time), is of atomistic struc- 
ture. Otherwise, that is, on the assumption of a spatial and 
temporal continuity, matter could be proved to be divisible ad 
infinitum, contrary to the first fundamental proposition. Dis- 
continuity of space and time leads to the peculiar, but logical, 
explanation of motion as a series or sequence of momentary 
leaps: the atom occupies in succession different individual space- 
elements. Physical motion becomes thus a discontinuous proc- 


To be exact, the discrete structure of space, according to the 
theory of Kalam, can be inferred from the two premises ( 1 ) of 
the discreteness of time (the third fundamental proposition 
of the Kalam, according to the enumeration of Maimonides); 
(2) of the Aristotelian inference from the continuity of space 
to that of motion, and from the continuity of motion to that of 
time. 27 Since the consequent, according to the first premise, is 
denied, the formal application of the modus tollens leads to the 
conclusion that space is not continuous. 

The atomistic theory of space gave rise to many complications. 
First of all, it became evident that differences in speed can no 
longer be attributed to the fact that the body which has moved 
through a larger distance had a greater velocity, but must be 
due to the circumstance that the "faster" body has been inter- 
rupted by fewer moments of rest. Fundamentally, only one 
common speed (frequency) lies at the basis of all physical 
processes. Had motion pictures or electric advertising with its 
stroboscopic illusions been an invention of the Middle Ages, the 

" Aristotle, Physics, IV, 1, 3. 


proponents of the Kalam would have faced no difficulties in 
finding suitable illustrations for their teachings. 

That this peculiar conception of motion led to considerable 
complications was soon realized. Thus Maimonides argued: 

"Have you observed a complete revolution of a millstone? Each 
point in the extreme circumference of the stone describes a large 
circle in the same time in which a point nearer the center describes 
a small circle; the velocity of the outer circle is therefore greater 
than that of the inner circle. You cannot say that the motion of the 
latter was interrupted by more moments of rest; for the whole moving 
body, i.e., the millstone, is one coherent body." They reply, "During 
the circular motion, the parts of the millstone separate from each 
other, and the moments of rest interrupting the motion of the por- 
tions nearer the center are more than those which interrupt the 
motion of the outer portions." We (Maimonides) ask again, "How is 
it that the millstone, which we perceive as one body, and which 
cannot be easily broken, even with a hammer, resolves into its atoms 
when it moves, and becomes again one coherent body, returning to 
its previous state as soon as it comes to rest, while no one is able to 
notice the breaking up of the stone?" Again their reply is based on 
the twelfth proposition, which is to the effect that the perception 
of the senses cannot be trusted, and thus only the evidence of the 
intellect is admissible. 28 

The argument of the revolving millstone, which was ad- 
vanced by Maimonides with the obvious purpose of showing 
the inner inconsistency of the Kalamic space theory, is but 
another version of the well-known problem which from the late 
fifteenth century on became renowned under the ambitious name 
of "rota Aristotelis." It was known, however, throughout the 
Middle Ages and led to many investigations into the structure 
of space and in some cases to a rejection of the traditional Aris- 
totelian doctrine of continuity. The problem essentially is this: 
two concentric circles with different radii, rigidly connected to 
each other, move in such a way that each of them, during one 
complete rotation, rolls along a straight line (Fig. 1); how can 
these two lines be equal in length, being produced by circum- 
ferences of different radii? 

28 Moses Maimonides, The guide for the perplexed, trans, by M. Friedlaen- 
der (Pardes Publishing House, New York, 1946), chap. 73, p. 122. 



As late as in the seventeenth century "interposed vacua" or 
infinitesimal "moments of rest" were postulated in order to solve 
the problem. It will be recalled that Galilei also discusses the 
problem in his Discorsi e dimostrazioni matematiche, intorno a 
due nuove scienze™ and that his treatment of the "infinite and 
indivisible" is reminiscent of the ancient teachings of the Kalam. 

By assuming the discontinuity of motion the Kalam protected 
itself against Aristotle's famous attacks 30 on atomism, later 
repeated in another form in Al-Ghazali's Makdsid-al-faldsifa, ac- 

Fig. 1. 

cording to which the conception of a continuous motion in an 
atomic universe leads necessarily to a division of the indivisible 
and is therefore incompatible with an atomic theory of space 
and time. 

Another important result of the theory of the Kalam was the 
rejection of a possible incommensurability between spatial data 
and a denial of the existence of irrational magnitudes (lines, 
etc. ) . If every spatial extension, say a line, is atomistic in struc- 
ture, that is, composed of an integral number of atoms, clearly 
no incommensurable lines can exist and no irrational measures 
can be conceived. The Pythagorean discovery of the irrational 
was in the view of the Kalam but an unhappy chimera. 

Very little is known about a possible influence of the Kalamic 
space theory on scholastic thought in medieval Europe. As a 

"Edizione Nazionale, p. 70 and 96; Dialogues concerning two new 
sciences trans, by H. Crew and A. de Salvio (Dover, New York, 1952), p. 
22, 51. 

*° Aristotle, Physics, VI, 232 a, 233 b. 



pure speculation, which ignores any scientific relevance of sense 
data and consequently is extremely averse to experiment and 
observation, it remained necessarily restricted in its physical 
contents to the discussion of motion. It is possible, therefore, to 
trace its influence only in scholastic investigations into the nature 
of space, motion and the continuum, a subject which was one 
of the major attractions for fourteenth-century schoolmen. But 
even here we have little evidence of any direct dependence and 
all our conclusions are only conjectural. 

To be sure, it is an established fact that the works of Al- 
Ghazali 31 and of Maimonides, with their references to the atom- 
istic space theories of the Kalam, were widely read by the 
scholastics. The Guide for the perplexed was most probably 
translated into Latin as early as in the middle of the thirteenth 
century at the court of the Emperor Frederic II. 32 Is it possible 
that the atomistic theory of space could have escaped the atten- 
tion of William of Auvergne, Francis of Sales, Vincenz of Beau- 
vais, Albertus Magnus, Thomas Aquinas, Duns Scotus, and many 
others who were well acquainted with Maimonides' Guide? 

On the other hand, the problem whether spatial magnitudes 

— lines, areas, and volumes — were constituted of indivisibles 
("compositio ex indivisibilibus") or of points ("compositio ex 
punctis") was widely discussed in the course of the fourteenth 
century. The majority of the schoolmen retained the Aristotelian 
doctrine according to which the continuum is characterized as 
being composed of parts which themselves can be divided and 
subdivided ad infinitum ("continuum est constitutum ex quanti- 
bus divisibilibus in alias quantitates"); it is not composed of 
indivisibles. Aristotle recognized that this regression of an end- 
less division introduces the concept of infinity. But this concep- 
tion — in contrast to the notion of an infinitely extended space 

— requires, in his view, only the idea of a potential infinite. 

* Al-Ghazalfs works were translated into Latin in the twelfth century by 
Dominic Gundisalvi. 

a See Graetz, Monatschrift (Jan.- June 1875). 




Against those, like Nicolaus of Autrecourt or Henry of Harclay, 
who contended that spatial extension is composed of dimension- 
less and indivisible points, the following principle was quoted: 
"Indivisibile indivisibili additum non facit maius," a principle 
that expressively contradicts the Kalam conception of space. 
Nicolaus of Autrecourt's reasoning, incidentally, rests on his ob- 
jection to the Aristotelian conception of space as a plenum. If 
space were a plenum, he contends, one of three "inconveniences" 
would necessarily follow: either rectilinear motion would be 
impossible, or two bodies would be at the same place at the same 
time, or the motion of one body would imply the motion of all 
the other bodies in the universe. 33 Reason and experience de- 
mand, therefore, in his view, the postulation of the existence of 
vacuities and the rejection of the Peripatetic theory of space. 

The greatest resemblance to the Kalam theory of space is 
exhibited by the teachings of Duns Scotus' disciple Nicolaus 
Boneti, 34 who advanced an extreme atomistic theory of spatial 
extension. Unfortunately, very little has been published about 
his teachings. 

After this digression to questions relating to the continuity 
of space, let us resume the main line of our story and explain 
how an intrinsic critique of the traditional Peripatetic concep- 
tion of space led gradually to far-reaching consequences, cul- 
minating finally in the emancipation of the concept of space 
from the doctrine of substance and accident. 

So far our discussion has been confined mostly to the major 
theories of space in Antiquity and their recurrences in scholastic 
thought. They may be classified under three headings: the atom- 
istic view (with its emphasis on the physical character of space), 
the Platonic view (with its emphasis on mathematics), and 
finally the Aristotelian view (with its ontology). With regard 
to our special problem, as well as generally, the early periods of 

88 Julius Rudolph Weinberg, Nicolaus of Autrecourt (Princeton University 
Press, Princeton, 1948). 

"See Anneliese Maier, Die Vorlaufer Galileis im 14. Jahrhundert (Storia 
e Letteratura, Rome, 1949), p. 177. 

! V 




medieval thought exhibit a strong inclination toward Platonism 
which gives way in the late scholastics to Aristotelianism, until 
with the dawn of modern science war is declared on Peripatetic 
thought and Neoplatonism becomes the main ingredient in 
Italian natural philosophy. The early Middle Ages contribute 
little to the development of the concept of physical space. On 
the other hand, and far more important, is the physical thought 
of the later period in which Aristotle is of supreme influence. 

It will be recalled that place is the concave surface of the 
containing body and is by its nature immobile. In the light of 
this definition let us refer to the following passage in Aris- 
totle's Physics: "Of things which are in motion some are moved 
by the actualizing of their own inherent potentialities, and others 
only by being involved in the movement of something else in 
which they inhere." S5 For example, a nail in the side of a ship 
does not move by itself (hath' auto), but is moved per accidens 
(symbebekos), without changing its place. Here we come upon 
the first conceptual difficulty in Aristotle's doctrine of space, a 
difficulty that became one of the major problems of medieval 
physics. It is this: If space is the concave surface of the contain- 
ing body, and motion is change of space, how can the concept 
of "motion per accidens" be reconciled with these definitions? 
Looking at the problem in modern terms, it is clear that Aristotle 
was fully aware that motion can be inferred only with reference 
to a second body, that is, by the choice of an immediately sur- 
rounding body as a reference system. Thus Aristotle raised a 
difficulty that has baffled many thinkers throughout the ages. 
Sextus Empiricus in his Against the physicists was already 
struggling with this obvious contradiction. He writes: 

These motions, then, are omitted from their description; but there 
is also another more surprising kind of transitional motion, in which 
the moving object is conceived as not going out from the place wherein 
it is either as a whole or part by part; and this too is omitted from 
their definition, as is obvious at once. And the peculiar character of 
this motion will be more evident when we have explained it by an 

"Aristotle, Physics, IV, 4, 211 a 18. 



example. For if we should suppose that, when a ship is running be- 
fore the wind, a man is carrying an upright rod from the prow to 
the stern and moving at the same speed as the ship, so that in the 
time in which the latter completes the distance of a cubit in a forward 
direction, in an equal time the man who is moving in the ship passes 
over the distance of a cubit in a backward direction, then, in the case 
thus supposed there will certainly be transitional motion, but the 
moving object will not go out from the place wherein it is either 
wholly or in part; for the man who is moving in the ship remains in 
the same perpendicular both of air and of water owing to the fact 
that he is borne just as far forward as he seems to proceed backward. 
It is, then, possible for a thing which does not quit the place wherein 
it is wholly or in part to move transitionally. 36 

In these words Sextus Empiricus attacks Aristotle's definitions 
of place and motion, but he fails to see the possibility of adopt- 
ing as a system of reference some distant body to which 
the body under consideration is brought into spatial relation. 
This would not only have eliminated the difficulty with which 
he is concerned, but would also have it made possible to correlate 
the motions of various bodies under one common aspect. But 
such a step would have obliged him to reject Aristotle's defini- 
tion of place as the surface of the adjacent body. The authority 
of Aristotle was too great for such a radical change. Even Wil- 
liam of Occam, the revolutionary nominalist of the fourteenth 
century, considered it necessary to adhere to Aristotle's definition 
of space: 

Various explanations are suggested by various people in order to 
maintain the immobility of place. Thus some say that place has two 
aspects, namely, that which is material in place, viz., the surface of 
the containing body; secondly, that which is formal in place, viz., its 
order with regard to the universe (ordo ad universum). This order in 
relation to the universe, however, is always immobile. For place, with 
regard to its formal aspect, cannot be moved either for itself or per 
accidens . . . 3T 

By way of illustration Occam refers to the classical example 
of the ship lying at anchor. 

86 Sextus Empiricus, Against the physicists, II, 55. 

"William of Occam, Summulae in libros physicorum (Bologna, 1494). 



Although new masses of water are constantly coming up around 
the ship and although the ship does not always occupy the same 
order in relation to the parts of the river, as these are constantly 
moving, still, in regard to the river as a whole, the ship rests in the 
same place as long as it is at anchor ... If you are at rest, and 
even if all the air around you, or any body which surrounds you, is 
moving, you are always at the same place; for you are always at the 
same distance from the center and the poles of the universe. With 
regard to these the place is therefore called immobile. 38 

We appear to have here for the first time the introduction of 
distance for the identification of place. Thus immobility of a 
given place was reduced to the constancy of distance from a 
given reference body, or from a set of such bodies. As Occam's 
words "the same distance from the center and the poles of the 
universe" indicate, this reference body was usually the outermost 
sphere of Aristotelian-Ptolemaic cosmology. 

So we come to the second problem that arises within the 
framework or Aristotelian physics. The outermost sphere in the 
Aristotelian universe was regarded as moving with constant 
angular velocity, but was itself without place, 39 being uncon- 
tained in any further sphere. We have already seen that this 
difficulty was the occasion for much subtle discussion. 40 

All attempts to reconcile the obvious contradiction between 
the notion of absence of place (or space) for the last sphere 41 
and the assumption that it moves (and therefore changes its 
place, according to Aristotle's definition of motion) were doomed 
to failure. It remained a major problem in scholastic philosophy 
until Copernicus finally came to the conclusion that the two 
ideas were irreconcilable, and that at least one of them would 
have to be rejected. Either the definition of "place" had to be 
revised, or the dogma of the motion of the outermost celestial 


"Aristotle, Physics, IV, 5, 212 b 10, for instance. 
10 See p. 37 and also p. 53. 

41 Dante, Paradise-, XXVII, 109, with poetic license, suggests a theological 
solution of the problem: 

E questo cielo non ha altro dove 
Che la mente divina. 



sphere had to be repudiated. As we know, Copernicus preferred 
the second alternative. That this problem was really one of the 
major incitements to Copernicus' drastic revision of the accepted 
cosmological conception may be seen from various remarks in his 
De revolutionibus orbium caelestium ( 1543). In chapter V of the 
first book he says: "Cumque caelum sit, quod continet et caelat 
omnia, communis universorum locus, non statim apparet, cur non 
magis contento quam continentj, locato quam locantj motus attri- 
buatur." 42 Copernicus contends that it would be much simpler 
to attribute motion to the contained body than to the container, 
for this evidently would solve the problem. In chapter VIII of the 
same book 43 he goes so far as to call it "absurd" to attribute mo- 
tion to the containing last body. He writes: "Addo etiam, quod 
satis absurdum videretur, continenti sive locanti motum adscribi, et 
non potius contento et locato, quod est terra." And when, in 
chapter X of the first book, Copernicus gives a preview of his 
new cosmology, he feels justified in saying: "Prima et suprema 
omnium est stellarum fixarum sphaera, se ipsam et omnia con- 
tinens, ideoque immobilis; nempe universi locus, ad quem motus 
et positio caeterorum omnium syderum conferatur." 44 It seems to 
us clear that the word ideoque (therefore) indicates that to con- 
tain both itself and all other bodies implies the absence of mo- 
tion. It is not usually noted that the Copernican revolution was 
the outcome, in part, of the solution of a difficulty involved in 
the Aristotelian definition of place or space. We are well aware 
that the problem of the "placeless" motion of the outermost 
sphere was not the only factor that led Copernicus to his new 
conceptions. Copernicus' way of solving this problem had been 
suggested already by Alexander Aphrodisiensis, who according to 
Narboni's Kawtvanot ha-Pilosophim conceived of an immobile 
outermost sphere that does not exist in place; also, as we have 
seen on page 37, additional spheres have been proposed to con- 

"F. and C. Zeller, ed., Nicolai Copernici Thorunensis De revolutionibus 
orbium caelestium libri sex (Oldenbourg, Munich, 1949), p. 14. 
"Ibid., p. 20. 
"■Ibid., p. 25. 





tain the sphere of the fixed stars; but in all these cases we do 
not know of any major change in the cosmological conceptions 

Concerning the first of the two alternatives mentioned above, 
namely, the rejection of Aristotle's definition of place, it was 
adopted more than 150 years before Copernicus. It was a step 
that led to drastic revisions of the whole of Aristotelian physics. 
As much a revolution as that of Copernicus, it failed to bear 
fruit owing to adverse conditions of a political and religious na- 
ture. We are speaking of Hasdai Crescas' critique of Aristotelian 
physics in his Or Adonai (c. 1400). If we are to see Crescas' 
contribution in true historical perspective, we must take up 
again the question of space outside the universe. 

Aristotle's doctrine provided a clear and accurate definition 
of place, while the rival doctrine, as expounded by the atomists, 
and later, leaving out of account ancient Pythagorean lore, by 
the Stoics and Philoponus, omitted to give a strict definition of 
space or place, taking space as a more or less primitive concept 
in the construction of the system. As a matter of fact, the intuitive 
conception of a vastly extended space, surrounding the material 
universe, seems to have been dormant all through the ages and 
may be found underlying even the most conservative doctrines of 
theological cosmology in the Middle Ages. So, for example, 
among the errors condemned in 1277 there is this: "Quod Deus 
non possit movere Coelum motu rectu. Et ratio est quia tunc 
relinqueret vacuum." 45 In terms of Aristotelian physics the very 
idea of motion of the universe as a whole is absurd and senseless. 
For motion presupposes place — a place in which the moving 
object is and a place to which it tends. But in Aristotelian physics 
place by definition is found only inside the universe. 

In order to illustrate the way in which fourteenth-century 
thought struggled with the problem of outer space, we quote in 

45 Denifle-Chatelain, Chartularium Universitatis Parisiensis (1889-97), 
vol. 1, p. 546. 

detail a passage from Richard of Middleton's Super quattuor 
libros sententiarum quaestiones subtilissimae. 

I answer that God could move the outermost sphere (whether by 
creating or not creating space outside it) in rectilinear motion, al- 
though it would be impossible for any power whatever to apply 
such motion to a body taken in itself and as a whole as far as there 
is no space outside it. From this (it may be inferred) that if only a 
single angel existed, God could not move him in such a motion, if 
it were not by creating space outside him or around him; yet God 
could move any body in rectilinear motion, even if there were no 
space outside it, on condition that the motion is partial and accidental. 
Likewise, if there were a hole in the empyrean sphere and if the 
littlest man had a lance the lower end of which he drove with a 
rectilinear motion towards the outermost surface of the empyrean 
heaven, he would cause a certain part of the lance, in the course of its 
motion, to pass through the last sphere, although outside this sphere 
no space exists. So I say that God, if He moved in proper rectilinear 
motion a part of the empyrean heaven towards the earth, while the 
dimension and quantity of the former remain unchanged, would 
cause another part of heaven to move in a rectilinear motion, albeit 
not in space. Thus too it is clear, that He can move the whole of 
heaven in rectilinear morion by the rectilinear motion of that part 
which He causes to move in space. 46 

46 Richard of Middleton, Super quattuor libros sententiarum quaestiones 
subtilissimae, p. 186. The Latin text is also quoted in Alexander Koyre, 
"Le vide et l'espace infini au XlVme siecle," Archives d'histoire doctrinale 
et UttSraire du Moyen Age (1949), p. 71: Respondeo quod Deus posset 
movere ultimum coelum ( sive creando spacium extra ipsum sive non creando ) 
motu recto, quamvis enim eandem rem per se, et secundum se totam im- 
possibile sit moveri motu locali recto, per quamquamque potentiam nisi 
extra ipsam sit aliquod spacium (unde si nulla creatura esset nisi unus 
angelus, Deus non posset ipsum angelum tali motu movere nisi in quantum 
posset creare aliquod spacium extra ipsum, vel circa), tamen per accidens 
vel secundum partem Deus posset movere corpus aliquod motu locali recto, 
quamvis extra ipsum nullum esset spacium, inde si esset aliquod foramen 
in coelo empyreo, et minimus homo habeat lanceam, impellendo per motum 
rectum, partem lanceae inferiorem versus ultimam superflciem coeli em- 
pyrei, faceret, quod lancea motu recto transcenderet quantum ad aliquam 
sui partem, ultimam superflciem coeli empyrei, quamvis extra ipsam nullum 
sit spacium, sic diso, quod Deus si moveret motu proprio recto unam par- 
tem coeli empyrei usque ad terram, flgura coeli et quantitate salvis manenti- 
bus, faceret quod alia pars coeli moveretur motu recto; quamvis non in 
aliquo spacio. Sic ergo patet, quod posset totum coelum movere motu 
recto per rectum motum illius partis quam moveret in spacio. 

I ' 



! U 




Aristotelian physics in general and his theory of space in par- 
ticular became subjected to penetrating and detailed criticism 
in the fourteenth century. Henri de Gand, Richard of Middleton, 
Walter Burleigh, and Thomas Bradwardine discussed the con- 
troversial problem of space and void extensively. At the same 
time, however, it must not be forgotten that revisions and criti- 
cisms were advanced mostly from the theological point of view. 
Nor were the revisions incorporated into a consistent system; 
often they were adopted merely as probable assumptions as for 
example the theory of the vacuum of Nicolaus of Autrecourt. 47 
In Hasdai Crescas we encounter an independent thinker who 
deals with his opponents only after having presented their case 
as objectively as possible. His critique of Aristotelian physics, 
though reared on the foundations of Jewish orthodox theology, 
does not confine itself to mere assertions or refutations, as was 
the case with the Kalam in Muslim philosophy. Crescas succeeds 
in clearly pointing out inherent contradictions and inconsistencies 
and only then proceeds to a new formulation and a revised con- 
ception. In proposition I, part II of his Or Adonai iS Crescas 
refutes Aristotle's definition of place as the adjacent surface of 
the containing body by pointing out the many absurdities to 
which it necessarily leads. First of all, Aristotle's definition can- 
not be consistently applied to the heavens. In the Physics, where 
Aristotle discusses the problem whether the outermost sphere 
possesses a place or not, he says: "But heaven, as has been said, 
is not anywhere as a whole nor in a certain place, since there 
is no body embracing it; but as far as it is moved, it constitutes 
places for its own parts, since one part embraces another." 49 
If "heaven" in this connection is understood, as Themistius under- 
stood it, as the outermost sphere, the meaning of the term "place" 
when applied to this outermost sphere is different from its mean- 

47 Cf. J. R. O'Donnell, "The Philosophy of Nicholas of Autrecourt," 
Medieval Studies 4, 97 (1942) ■ 

48 H. A. Wolfson, Crescas critique of Aristotle (Cambridge, 1929), P- 

" Aristotle, Physics, IV, 5, 212 b 8-13. 



ing when applied to the other spheres. If, on the other hand, we 
accept the interpretations of Avempace and Averroes, according 
to which the place of the celestial spheres is their center around 
which they rotate, we are landed in still another inconsistency. 
The celestial bodies would be adapted to abide in a place that 
is beneath them, for every body is naturally adapted to abide 
in its place; yet fire is not adapted to abide in a place beneath 
it. Again, a continuously extended body, for example, the atmos- 
phere, raises a further difficulty. The proper place of the air as 
a whole is the concave inner boundary of fire. What then is the 
proper place of a part of air, which is surrounded by other parts 
of air? Is its place identical with the place of the air as a whole? 

Fig. 2. 

In that case, what happens to Aristotle's requirement 50 that 
place be equal to the object occupying it? On the other hand, if 
its place is the other parts of the surrounding air, the place then 
would not be distinct from what occupies it, so that here too 
Aristotle's requirements 51 would go by the board. Furthermore, 
the place of the part would not be a part of the place of the 
whole. In other words, Crescas' analysis shows that even for 
sublunar elements the requirements of "separateness," "sur- 
rounding," and "equality," all of which Aristotle holds essential 
for the concept of place, are incompatible. Further yet, the ac- 
ceptance of Aristotle's definition of place leads to a paradox 
which is hinted at by Crescas and expounded in detail by his 

°°lbid., 211 a 27. 
a Ibid., 211 a. 





7 6 


pupil Joseph Albo. 52 It is this: the place of a part may be greater 
than the place of the whole. Let us consider a spherical body, as 
illustrated in Figure 2(a), and let us make a deep cleft in it, 
as in Figure z(b). It is evident that the new body, which obvi- 
ously is only a part of the sphere, still has a greater "place" than 
the whole sphere, a conclusion that contradicts common sense 
and Euclidean geometry. 

According to Crescas, any definition of place (or space) must 
meet the requirement that the place of a body taken as a whole 
be equal to the sum of the places of the parts into which it can 
be broken up. Yet on the basis of Aristotle's definition, which 
identifies place with the adjacent boundary, the "place" of a 
cube is clearly smaller than the sum of the places of the little 
blocks into which it can be divided. 

Expounding his master s ideas, Joseph Albo criticizes Aristotle's 
definition of "place" and writes in his Sefer-Ha-'Ikkarim: 

Furthermore it follows according to him that the place of the part 
is greater than the place of the whole. For if you remove part of the 
inside of a sphere, it will require a greater surface to bound it out- 
side and inside than when it is solid. Besides, it would follow accord- 
ing to him that one and the same body will have many places differing 
in magnitude. For if you divide a body into parts, each of the latter 
will require a greater place than before the division and the same is 
true if you divide the parts into other parts, and those again into parts. 
But this is contrary to the statement of Euclid in his book Concern- 
ing the heavy and the light, m where he says that equal bodies occupy 
equal places. But according to the Aristotelian hypothesis this is not 
true. For of two equal bodies the one that is divided will require a 
greater place than the other. All these difficulties ^follow from the 
opinion that place is an external bounding surface." 84 

Such are the main arguments that Crescas advances against 
Aristotle's definition of place. They lead him to elaborate his 

""Joseph Albo, Sefer Ha-'Ikkarim (Book of Principles) (ed. by Isaac 
Husik, Philadelphia, 1929), vol. 2, p. 105. Cf. Dogmas II, 17 (Sonzino, 

68 About this spurious work see Steinschneider's article "Euclid bei den 
Arabern," Zeitschrift fur Mathematik, hist. litt. Abt., Bd. 31 (1886). 

" Albo, Sefer Ha-'Ikkarim, p. 106. 



own view, according to which space is a great continuum of 
infinite dimensions, an immovable void, ready to receive matter. 
"The true place of a thing is the interval between the limits of 
that which surrounds." 55 He adopts Aristotle's tentative, but 
finally rejected, definition that place is "some kind of dimensional 
extension lying between the points of the containing surface." 56 
The grounds of Aristotle's rejection of this view is that it 
contradicts the requirements of separateness and immobility 
of place. By identifying this "dimensional extension" with the 
vacuum which becomes place when it contains a body, Crescas 
proceeds to show that Aristotle's arguments do not hold. His 
answer to Aristotle's argument that since all bodies move, if 
space were the interval of a body, space would be moving in 
space, is that there are no various spaces; space is one, infinite 
and immovable. By admixture of matter the infinite void becomes 
the extension of physical bodies. 

Crescas' definition of space not only places him in opposition 
to Aristotelian physics, but makes him the first proponent of the 
reality of the vacuum in Jewish philosophy. Just as atomism, even 
in its Muslim theological form, was never endorsed by Jewish 
philosophers, unless exception is made of Abraham ibn Ezra, so 
the possibility of a void was always discarded by Jewish philo- 
sophical thought until Crescas became its great advocate. One 
reason for this traditional Jewish attitude toward the problem of 
the void was Aristotle's immense influence on Jewish thought 
in the Middle Ages; another was the empirical point of view 
adopted by Jewish thinkers. 

Crescas challenged this attitude by refuting one after another of 
Aristotle's arguments against the existence of a void. In his refuta- 
tion of Aristotle's argument that the existence of a void would 
preclude any motion, Crescas intimates his view on the physical 
structure of space; and hence his refutation is of special interest 
for us. According to Aristotle, the medium is an indispensable 

55 Crescas, Or Adonai, proposition I, part II; see reference 48, p. 195. 
"Aristotle, Physics, IV, 4, 211 b 8. 





condition for motion in that it serves constantly as both the 
terminus a quo and the terminus ad quern for the moving body 
whose natural motion is toward its natural place. Crescas tries 
to show that motion is not dependent on the existence of a me- 
dium. The first step in his argument is the assertion that weight 
and lightness are intrinsic qualities of the bodies and independent 
of any medium. "All moving bodies have a certain amount of 
weight, differing only secundum minus et majus," 67 upward mo- 
tion being the result of pressure exerted upon bodies by others 
that are heavier. 

In dispensing with the notion of an inner tendency in the 
elements to reach their natural places, Crescas revives certain 
views, particularly those of the atomists, according to which dif- 
ferences of weight are the result of differences in the internal 
structure of bodies. But these ideas are brought forward by Cres- 
cas merely as hypotheses. He does not follow them through to 
their logical conclusion and rejection of natural places altogether. 
He mentions them merely to show that the medium may be dis- 
pensed with as a cause of motion. To him, even if a more con- 
servative view be taken and the motion of matter be regarded 
as an inner striving of the elements toward their natural places, 
the medium is under no circumstances the efficient cause of mo- 
tion. For the parts of a vacuum can exhibit no differentiation in 
their constituent nature; for the vacuum is one homogeneous con- 
tinuum. Yet, with respect to relative distances of these parts from 
the lunar sphere (the periphery) and the earth (the center), they 
do show a differentiation, but one merely of external relation. 
In this way Crescas comes astonishingly near to the idea of actio 
in distans. "Thus when fire moves from one part of the vacuum 
into another in upward motion, it is not because it tries to escape 
one part of a vacuum in order to be in another, but rather in its 
endeavor to get nearer to its proper place, which is the con- 
cavity of the lunar sphere, it naturally has to leave those remote 
"Reference 48, p. 185. 




parts of the vacuum and occupy the parts which are nearer to 
its proper place." 58 

The void or space, which according to Crescas becomes mate- 
rial extension when occupied by matter, is conceived by him as 
infinite in extension. Thus Crescas inaugurates a new front in his 
struggle with Peripatetic physics, according to which the universe 
is finite and limited. Aristotle's demonstration of the impossibility 
of infinity is set forth in De caelo, m in a classical passage which 
was plausible to the point of hypnotizing medieval thought. 

Upward and downward motions are contraries, and contrary mo- 
tions are motions to opposite places; and if one of a pair of opposites 
is determinate, the other must also be determinate. But the center 
is determined, for wherever the downward-moving body may come 
from, it cannot pass farther than the center. The center then being 
determined, the upper place must also be determined; and if their 
places are determined and limited, the bodies themselves must be 

Aristotle attempts also in the Physics and in the Metaphysics 
to prove the impossibility of both a corporeal and an incorporeal 
infinite extension. An account of Crescas' refutation of all these 
arguments would lead us too far afield, and we cannot do better 
than refer the interested reader to Wolfson's book. 

As in his previous arguments, here too in his refutation of 
Aristotle's view of infinity, Crescas undermines his opponent by 
sound arguments and does not confine himself merely to express- 
ing his disagreement. Accordingly, as the first proponent of 
infinite homogeneous space, Crescas made an outstanding con- 
tribution to the history of scientific thought. For he not only 
turned his back on Aristotelian conceptions, but through strict 
logical reasoning anticipated some fundamental ideas of six- 
teenth- and seventeenth-century physics. It was a great misfor- 
tune that he was never able to bring his ideas to their full flower. 
Political instability in Spain in the fifteenth century put an end 
to the intellectual activities of Catalonian Jews. 

"Ibid., p. 402. 

M Aristotle, De caelo, I, 273 a 10. 





Crescas' theory of space solved the problem of the outermost 
sphere: The infinite vacuum provides this sphere with space, so 
that its eternal rotation becomes a special kind of local motion 
and the sphere ceases to be the final limit and boundary of 

Crescas' solution of the problem was not the only one ad- 
vanced in the beginning of the fifteenth century. Nicholas of 
Cusa offered another. In his view, universal motion has no center, 
since in terms of his principle of the "coincidentia oppositorum" 
the absolute minimum must coincide with the absolute maximum. 
But God alone may be thought of as the absolute maximum of 
existence, so that Cusanus comes to the conclusion: "Qui igitur 
est Centrum mundus? scilicet est Deus benedictus, ille est 
Centrum terrae, et omnium sphaerarum." 60 However, from the 
purely physical point of view, the identification of the center of 
the universe with its circumference is an obvious absurdity. To 
Cusanus the world has neither a center nor a circumference. 
"Quia minimum cum maximo coincidere necesse est. Centrum 
igitur mundi coincideret cum circumferentia. Non habet igitur 
mundus circumferentiam." 61 So it is clear that the earth is not 
the center of the universe or of space. "Terra non est centrum 
mundi." 62 The manner in which Cusanus goes on to derive the 
motion of the earth, thereby anticipating certain ideas of the 
Copernican theory, is not part of our subject. But it is important 
for us to note that the absence of a body absolutely at rest (the 
earth) does away with the possibility of absolute motion and 
absolute space. It is this relative character of position and motion 
that brands Cusanus' theory of space as modern. Another modern 
feature is its rejection of the idea that a hierarchy of values rules 
different regions of space. Of Aristotelian origin, the idea is im- 

"° Nicholas of Cusa, De docta ignorantia, II, 11; see A. Petzelt, ed., 
Nicolaus von Cues, Texte seiner philosophischen Schriften, nach der 
Ausgabe von Paris 1514, sotoie nach der Drucklegung von Basel 1563 
(Kohlhammer, Stuttgart, 1949), vol. 1. 

n Ibid., 21. 




plied in the doctrine of physico-moral parallelism. As is well 
known, Aristotelian biology assigns to the upper parts of the 
human body a greater degree of nobility than to its lower parts. 
In consequence of this conception, as well as that of the parallel- 
ism between macrocosm and microcosm, the terms "high" and 
"low," though primarily purely geometric notions of spatial orien- 
tation, came in most languages to stand for distinctions of value. 63 
The conception of a spatial hierarchy of values found its most 
perfect expression in Dante's Divine comedy, which from this 
point of view is a spatial metaphor of the gradations of sin and 
blessedness. How far this anthropomorphic conception became 
an integral part of medieval natural philosophy can be illus- 
trated by the fact that Nicolaus of Autrecourt had to renounce 
his untimely thesis: "Quod non potest evidenter ostendi nobi- 
litas unius rei super aliam." 64 

Cusanus, objecting the spatial hierarchy of values, states ex- 
plicitly: "Neque dici debet, quod quia terra est minor sole et 
ab eo recipit influentiam, quod propterea sit vilior." 65 To Cu- 
sanus the earth is certainly not the smallest celestial body, the 
moon and Mercury being smaller; nor can any conclusion be 
drawn from the fact that the earth depends on the sun since the 
earth as a celestial body influences also in some degree the sun 
and its region. 

The rejection of a spatial hierarchy of values is the logical 
conclusion of a more general principle which Cusanus advances 
in his Docta ignorantia: wherever in the heavens anyone may 
be placed, it would seem to him as if he were the center of the 
universe. This statement is evidently a rudimentary expression 
of the so-called "cosmological principle" of modern science as 

*" The designations "right" and "left" ("dextra," "sinistra") have their 
origin in a somewhat opposite development: the "propitious" or "faithful" 
(Hebrew: "yamin") hand became the "right," the "sinister," malignant, 
the "left." A reference to the widespread belief that the left side is ill- 
omened is encountered in the Ebers Papyrus, the famous document on early 
Egyptian medicine, dating most probably from 3400 B.C. 

** Denifle-Chatelain, Chartularium Universitatis Parisiensis, II, 544. 

" De docta ignorantia; see reference 60, p. 106. 



far as the spherical symmetry of space is concerned. The general 
validity of the principle that the universe presents the same 
aspect from every point (and according to a modern school of 
cosmologists also at every time), except for local irregularities, 
is accepted in modern science as a necessary condition for the 
repeatability of experiments, since space and time are the only 
parameters which, at least in principle, are beyond the control 
of the experimenter and cannot be reproduced at his will. Since 
this postulate in modern cosmology — not only with respect to 
the purely geometric aspect of space, but also with regard to its 
kinematic and dynamic aspects — has gained so much importance 
recently, it is not without interest to note that in Cusanus' writ- 
ings we encounter, probably for the first time, an explicit enuncia- 
tion of its spatial implications. If there were any justification for 
regarding Nicholas of Cusa as marking the turning point in the 
history of astronomy, it would be rather because of this enuncia- 
tion than on account of the insufficient evidence of his astronomi- 
cal discoveries (the triple motion of the earth). 66 One has, how- 
ever, to keep in mind that Cusanus' principally mystic-speculative 
approach to his conclusions is fundamentally different from the 
scientific method of the Renaissance. 

The theories of both Crescas and Cusanus, nevertheless, were 
far in advance of their time. If the notion of space was to be 
emancipated from the Aristotelian tradition, it would have to be 
done, as history proved, more gradually. It was not done until 
the sixteenth century. Even in Cardan's De subtilitate, space is 
still conceived in accord with Aristotelian tradition as the con- 
cave surface of the limiting body. "Est igitur locus ultima cor- 
poris superficies, corpus contentum ambicus." 67 

In contrast to Cardan, Scaliger identifies space with the void, 
which is coextensive with the body occupying it. Under the in- 
fluence of atomistic thought, Scaliger presupposes the vacuum 

w See Lynn Thorndike, Science and thought in the fifteenth century (New 
York, 1929), p. 133. 

"Jerome Cardan, De subtilitate, lib. I. 




as a necessary condition of motion. "In natura vacuum dari ne- 
cesse est." 6 * Scaliger's vacuum, however, is not an infinite empty 
extension beyond all bodies, but merely the receptacle coexistent 
with matter and penetrated by matter. The terms "vacuum," 
"locus," and "spatium" are synonymous in Scaliger's doctrine. 
"Idemque esse vacuum et locum; neque differre, nisi nomine." «• 
Although Scaliger's theory represents an important step forward 
in the demolition of Aristotelian doctrine, it is still not the de- 
cisive step. For to Scaliger space, in its logical as well as meta- 
physical significance, is only secondary to matter. In a word, 
Scaliger's physics is still dominated by Aristotelian categories! 
As Ernst Cassirer points out, the real turning point is Bernardino 
Telesio's and Franciscus Patritius' theories of space. 70 

In his general philosophic outlook Telesio adopted certain 
materialistic and Stoic conceptions of Antiquity, which led him 
to ascribe to spiritual functions a certain degree of corporeality. 
This may account for his tendency to attribute independent real- 
ity to space and time, to place them on the same level with con- 
crete matter. Space ceases with Telesio to be a mere quality and 
assumes an independent existence, parallel to matter or "moles," 
moles being a concept that comes very near to the Newtonian 
notion of mass. Space is the great receptor of all being whatever. 
If a body leaves its place or is expelled from it, place itself does 
not leave, nor is it expelled, but remains the same, promptly 
becoming the receptacle of another body. 

Itaque locus entium quorumvis receptor fieri queat et in existenti- 
bus entibus recedentibus expulsisve nihil ipse recedat expellaturve 
sed idem perpetuo remaneat et succedentia entia promptissime susci- 
piat omnia, tantusque assidue ipse sit, quantaquae in ipso locantur sunt 
entia; perpetio iiimirum iis, quiae in ea locata sunt, aequalis, at eorum 
nuUi idem sit nee fiat unquam, sed penitus ab omnibus diversus sit. 71 

"J. C Scaliger, Exotericarum exercitationum UbeH ad Hieronumum 
Caraanutn (Lutet, 1557). " 


™E. Cassirer, Das Erkenntnisproblem in der PhUosophie und Wissenschaft 
der neueren Zeit (Berlin, 1911). ' 

"Telesio, De natura rerum juxta propria principia libri novem (Naples 
i5°o;» 1, 25. r ' 


Thus space, though equal to the things which occupy it, is not 
the same as any of these things. First of all, space is incorporeal, 
and, being pure aptitude to receive matter ("aptitudo ad corpora 
suscipienda"), it is free of all actions and operations. Space shows 
no qualitative differentiation; it is completely homogeneous in 
its structure, so that the existence of "natural places" is impos- 
sible. All parts of space show equal aptitude to receive any kind 
of matter. The motion of bodies in space is not caused by any 
qualitative differences inherent in space itself, but is the result 
of physical forces. Space as a whole is immobile ("universum 
perpetuo immobile permanet"). It is accessible to sense percep- 
tion ("ipso comprehensum est sensu"), as experiments with vacua 
clearly show. Basing himself on physical grounds, Telesio attacks 
Aristotle's argument against the possibility of empty space, while 
disdaining to deal with demonstrations of the nonexistence of 
things whose existence is yet patently observable. 

The considerations adduced by Telesio show clearly the new 
spirit of Italian natural philosophy of the sixteenth century. Noth- 
ing less than the formulation of a new physics is at issue. But 
another obstacle has still to be removed before these ideas could 
be assimilated and a new mechanics reared on their basis. The 
traditional substance-accident doctrine, the great bulwark of 
scholastic thought, had to be set aside. It was not enough to 
revise the physical foundations of the theory of space: it had to 
be provided with a new metaphysical foundation as well. 
Franciscus Patritius undertook this task. 

Quid ergo est? hypostasis, diastema, est, diastasis, ectasis est, ex- 
tensio est, intervallum est, capedo est, atque intercapedo. Ergo quan- 
titas? Ergo accidens? Ergo accidens ante substantiam? & ante corpus? 
Architas uterque, & senior Pythagorae auditor, & iunior Platonis ami- 
cus, & quicos secuti sunt scriptores categoriam, hoc spacium non 
cognovere. 72 

Is space a substance or an accident, is it corporeal or incorporeal, 
he asks in the chapter called "De spacio physico" of his com- 

ra Patritius, Nova de universis philosophia libris quinquaginta compre- 
hensa (Venice, 1593), foL 65. 




prehensive work. None of these concepts applies to space, since 
they are only ways of characterizing things in space. Space must 
be presupposed as a necessary condition of all that exists in it. 
"Id enim ante omnia necesse est esse, quo posito alia poni possunt 
omnia; quo ablato alia omnia tollantur." 73 Further, qualities 
themselves are still dependent on space. It is therefore clear that 
space does not fit into the substance-accident scheme. "Nulla 
ergo categoriarum spatium complectitur; ante eas est, extra eas 
omnes est . . ." Patritius thus achieves the important result of 
emancipating the concept of space from the Aristotelian doctrine 
of categories. But, he asks, has not space magnitude? And is it 
not therefore subjected to the category of quantity? And this is 
his answer: 

Itaque aliter de eo philosophandum est quam ex categoriis. Spatium 
ergo extensio est hypostatica per se substans, nulli inhaerens. Non est 
quantitas. Et si quantitas est, non est ilia categoriarum, sed ante earn 
ejusque tons et origo. 74 

This view of space as being ontologically and epistemologicaUy 
the primary basis of all existence led Patritius, as Cassirer points 
out, 7 * to reverse the relation between mathematics and physics. 
The study of space must come before the study of matter. To 
Patritius, since space conditions not only matter as such, but its 
qualities as well, the investigation of space is an indispensable 
prerequisite to all natural science. Space makes not only nature, 
but the knowledge of nature, possible. 

Before we go on to analyze Patritius' influence on the develop- 
ment of seventeenth-century physics, we may pause for a mo- 
ment to say something about Giordano Bruno's place in the his- 
tory of the development of the concept of space. As the exponent 
of the philosophy of infinity, Bruno is obliged to dispose of the 
idea of the world's finiteness, and he is thus confronted with the 
Peripatetic physics, in particular, with Aristotle's definition of 
place. "If the world is finite, and beyond the world there is 

n Ibid. 

™ Patritius, Pancosmia. De spatio physico, 65 f . 

75 Cassirer, Das Erkenntnisproblem, vol. 1, p. 232. 





nothing at all, where then is the world?" asks Bruno. Aristotle's 
answer that the world is in itself, although it follows logically 
from the definition of place, does not satisfy Bruno. So without 
attacking the validity of the logical conclusion Bruno confines 
himself to the premise itself. It is the definition itself that is 
wrong, and only a wrong conclusion could follow. To define 
place as the adjacent boundary of the containing body is to pre- 
clude the existence of space for the outermost sphere, and this 
renders meaningless any question as to what is outside the world. 
Before stating his own ideas, Bruno, in the manner of Crescas, 
mentions the arguments of Aristotle: "The convex surface of the 
primal heaven is universal space, which being the primal con- 
tainer is by naught contained. For position in space is no other 
than the surfaces and limit of the containing body, so that he who 
hath no containing body hath no position in space." 76 On the 
question "Where is the universe?' Aristotle, on the basis of these 
definitions, can only answer: "It is in itself." Here it is where 
Bruno's criticism begins. He says: "What then dost thou mean, 
O Aristotle, by this phrase, that 'space is within itself? What will 
be thy conclusion concerning that which is beyond the world? If 
thou sayest, there is nothing, then the heaven and the world will 
certainly not be anywhere." After the discussion on the impor- 
tance of the convex surface of the outermost sphere for spatial 
relations, Bruno (through the words of Philotheo) confesses: 
"Thus let this surface be what it will, I must always put the 
question, what is beyond?" 77 Bruno's restless temperament and 
his constantly searching disposition of mind did not let him find 
satisfaction with Peripatetic dialectic. Bejecting the finite cate- 
gories of Peripatetic thought, he forms an ecstatic vision of an 
infinite universe in his mind. 

Henceforth I spread confident wings to space; 

I fear no barrier of crystal or of glass; 

I cleave the heavens and soar to the infinite. 
™ Bruno, On the infinite universe and worlds, trans, by Dorothea Waley 
Singer in Giordano Bruno (Schuman, New York, 1950), p. 251- 
""Ibid., p. 254. 


And while I rise from my own globe to others 

And penetrate ever further through the eternal field, 

That which others saw from afar, I leave far behind me. 78 


It is therefore only natural that Bruno expresses a new concep- 
tion of infinite space on the ground that "Si non superficies sed 
spatium quoddam locus est, nullum corpus neque ulla corporis 
illocata erit sive maximum, sive minimum sive finitum sit ipsum, 
sive infinitum." 79 Bruno's definition of space is contained in Phil- 
otheo's answer on Albertino's theses in the fifth dialogue of On 
the infinite universe and worlds. Beplying to Albertino's fifth and 
sixth arguments Philotheo says: 

There is a single general space, a single vast immensity which we 
may freely call void; in it are innumerable (innumerabili et infiniti) 
globes like this one on which we five and grow. This space we declare 
to be infinite, since neither reason, convenience, possibility, sense- 
perception nor nature assign to it a limit ... It diffuseth throughout 
all, penetrateth all and it envelopeth, toucheth and is closely attached 
to all, leaving nowhere any vacant space; unless, indeed, like many 
others, thou preferest to give the name of void to this which is the 
site and position of all motion, the space in which all have their 



Although this definition, or description, of space is characteris- 
tic of the spirit of Italian natural philosophy of the sixteenth 
century, it is yet the case, as Wolfson points out, that a certain 
indebtedness of Bruno to Crescas is likely. Both Crescas and Bruno 
focus their critique of Aristotle's definition on the problem of the 
outermost sphere; both attempt to demonstrate the existence of 
a vacuum on similar grounds; both refute Aristotle's theory of 
lightness in much the same way. 

That two men separated by time and space and language, but 
studying the same problems with the intention of refuting Aristotle, 
should happen to hit upon the same arguments is not intrinsically im- 
possible, for all these arguments are based upon inherent weaknesses 
in the Aristotelian system. But knowing as we do that a countryman 

™Ibid., p. 249. 

TO Bruno, Acrotismus (Vitebergae, 1588), I, 1, p. 121. 
"Bruno, On the infinite universe and worlds; see reference 76, pp. 363, 



of Bruno, Giovanni Francesco Pico della Mirandola, similarly sep- 
arated from Crescas in time and space and language, obtained knowl- 
edge of Crescas through some unknown Jewish intermediary, the 
possibility of a similar intermediary in the case of Bruno is not to be 
excluded. 81 

Campanella develops Patritius' theory of space still further, 
maintaining that space is the immovable basis of all existence: 
"basin omnis creati, omniaque praecedere esse saltern origine et 
natura." 82 At another place he calls space "locus, basis existen- 
tiae, in quo pulcrum Opificium, hoc est mundus, sedet." 83 In 
Campanella's view space is homogeneous and undifferentiated, 
penetrated corporeally and penetrating incorporeally. Its homo- 
geneity excludes such differentiations as "down" or "up," which 
attach to the diversities of bodies, rather than to space. It goes 
without saying that the existence of "natural places" is em- 
phatically rejected. God created space as a "capacity," a recep- 
tacle for bodies. "Locum dico substantiam primam incorpoream, 
immobilem, aptam ad receptandum omne corpus." 8 * 

The works of Telesio, Patritius, and Campanella show that 
Italian natural philosophy must be credited with having emanci- 
pated the concept of space from the scholastic substance-accident 
scheme. In the physics of the early seventeenth century space 
becomes the necessary substratum of all physical processes. It is 
this emancipated concept, divested of all inherent differentiations 
or forces. Gilbert in his Philosophia nova expresses these ideas in 
a concise way: 

Sed non locus in natura quicquam potest: locus nihil est, non existit, 
vim non habet; potestas omnis in corporibus ipsis. Non enim Luna 
movetur, nee Mercurii, aut Veneris Stella, propter locum aliquem in 
mundo, nee stellae fixae quietae manent propter locum. 85 

81 Wolfson, Crescas' critique of Aristotle, p. 36. 

82 Thomas Campanella, De sensu rerum (1620), I, c. 12. 

83 Campanella, Metaphysicarum rerum juxta propria dogmata (1638), 
pars I, lib. 2, c. 13. 

84 Campanella, Physiologia (Paris, 1637), I, 2. 

"William Gilbert, De mundo nostro sublunari philosophia nova (Amster- 
dam 1651), lib. II, cap. 8, p. 144. 



Place does not affect the nature of things, it has no bearing on 
their being at rest or being in motion. 

Although these words are directed first of all against the theory 
of "natural places" and of the attractive forces exerted by them, 
they mean much more. In order to understand the full mean- 
ing of "vim non habet," in which Gilbert agrees with Telesio, 
Patritius, and Campanella, we have to refer to an instrument 
which generally has been used since Antiquity for the measure- 
ment of time, but which paradoxically had a most important 
effect on the formation of concepts of space: the clepsydra. Since 
the Italian natural philosophers mention this device in their writ- 
ings and since they draw important conclusions from the way it 
works, it is perhaps most fitting to explain the historical impor- 
tance of the clepsydra for our subject in this chapter. 

Aristotle, 86 referring to Anaxagoras, had stressed the impor- 
tance of experiments with the clepsydra for the investigation into 
the existence of empty space. He, as well as his followers, quoted 
the rising of water in exhausted tubes, for instance, as a demon- 
stration for the impossibility of a void. Philoponus, as we have 
seen, 87 denied the existence of a void "in actu," notwithstanding 
his criticism of his masters conception of space; and one of his 
arguments for his contention was just this kind of experiment. 88 
A similar attitude was adopted by the author, or the authors, of 
the Problemata, 89 as well as by most of the other Peripatetics. 

A different interpretation of the same phenomena is found in 
the writings of various Arabian authors. Biruni, 90 Avicenna, 91 
and in particular al-Razi 92 not only cite these experiments as a 
verification of the existence of the void, but ascribe to empty 

88 Aristotle, Physics, TV, 6, 213 a. 
"Page 54- 

88 See his commentaries to the Physics, 569. 

89 Problemata, XVI, 8 (trans, by E. S. Forster; Oxford, 1927). 

90 Biruni, Al-atar al-baqiya, p. 263; cf. C. E. Sachau, ed. and trans., The 
chronology of ancient nations. An English version of the Arabic text of the 
Athdr-ul-Bakiya of Albiruni (London, 1879). 

"Avicenna, Sufficientia (Venice, 1508), fol. 30 b. 
"According to Fahr al-DIn and Sirazl. 




space a force of attraction (gddiba). 93 Although al-Razi's con- 
ception of space is similar to that of Philoponus — with the im- 
portant difference, of course, that al-Razi affirms the existence of 
a void "in actu" — it may also be compared with the Stoic con- 
ception. For al-Razi assumes, as did the Stoa, two kinds of void, 
the intramundane vacuities (gawhar al-hala or the "substance 
of emptiness") and the void beyond the material universe (al- 
fada). Whether the force of attraction exerted by the void is 
caused by a tendency of the intramundane vacuities to con- 
glomerate and to unite among themselves, or whether it is to 
be explained by their tendency to reach the fada is hard to 
ascertain. Both alternatives are conceived under the influence of 
Platonism and both alternatives are instrumental for the explana- 
tion of the rising of light bodies, since lightness was explained 
as a preponderance — not in the literal sense of the word — of 
intramaterial vacua. 

This conception of an empty space endowed with forces had 
already been submitted to severe criticism by Roger Bacon 94 
and is now emphatically rejected by the Italian natural philos- 
ophers. 95 

Pierre Gassendi, who introduced his contemporaries to the 
oldest complete source of atomism, 96 was particularly obliged to 
face the problem of space and void. For as the proponent of a 
revised atomism he was under the necessity of defending the 
reality of the vacuum, which becomes in his view identical with 
space. Although space and time appear to be "nothing," if meas- 
ured by the scale of corporeal-concrete reality, they yet have real 

88 It is not impossible that the origin of these ideas lies in the following 
simple experiment, cited by Heron in his Pneumatics: If you exhaust the 
air from a small bottle with your mouth, the bottle remains attached to 
your lips as if the void produced attracted your flesh. See H. Diels, "Ueber 
das physikalische system des Straton," Sttzber. preuss. Akad. Wiss. Berlin 
(1893), p. 101. 

"P. Duhem, Roger Bacon et I'horreur du vide (Oxford, 1914). 

"Telesio, De natura rerum, I, 25; Patritius, Pancosmia, I; Campanella, 
De sensu rerum, I, 10. 

"Pierre Gassendi, Animadversiones in decimum librum Diogenis Laertii 
(Lyons, 1649). 



existence, as the very preconditions of kinematics or physics in 
general. Further, Gassendi accepts Patritius' thesis of the prior- 
ity of space over matter: "Ideo videntur Locus et Tempus non 
pendere a corporibus, corporeaque adeo accidentia non esse." 9T 
According to Gassendi, this priority is not only logical or onto- 
logical, but also temporal, for he says explicitly: "Unum est, 
spatia immensa fuisse, antequam Deus conderet mundum." Al- 
though the atoms were created by God, space was ever-existent, 
uncreated and independent. Gassendi was fully aware of the dif- 
ficulties involved in this statement, which as we know was at- 
tacked later by Leibniz as an assertion of God's limitation. But 
Gassendi stresses the fact that famous theologians adhered to it. 
To Gassendi space is a necessary, infinite, immobile, and in- 
corporeal datum of three dimensions. It is certainly no fiction, 
not even the mode of a substance. "Cum ex deductis constet 
posse quidem ea spatia dici nihil corporeum, seuquale substantia, 
aut accidens est, sed non nihil incorporeum ac specialis sui 
generis: constat quoque esse ea posse, etsi intellectus non cogitet, 
ac non quemadmodum chimaeram merum esse opus imagina- 
tionis." 98 Space is neither a mode nor an attribute; both of these 
exist in subordination to the object to which they belong, whereas 
space is independent of any substance. Bernier, the expounder of 
Gassendi's doctrine, in his Abrege de la philosophie de Gas- 
sendi 99 emphasizes the difference between Gassendi's space and 
the common notion of corporeal extension, and warns his readers 
not to confuse the two. Whereas space is infinite, corporeal ex- 
tension is finite. Space can be occupied by bodies, but corporeal 
extension is impenetrable, subjected to all the vicissitudes of 
matter, whereas space is unchangeable and immovable. 

It is certainly an important fact from our point of view that 
Gassendi and Campanella met personally. That Campanella's 
conception of a homogeneous and infinite space must have found 

"Gassendi, Syntagma philosophicum (Florence, 1727), part II, sec. 1, 
lib. II, cap. 1. 
"Ibid. I, 189. 
" Second edition, Lyons, 1684, vol. 2, p. 9, 


a ready support in Gassendi may also be seen in detail in Gas- 
sendi's Epistolae tres de motu impresso a motore translate. 100 In 
the atomism of Democritus and Epicurus he found the undif- 
ferentiated void through which the atoms move. From Gilbert 
and Kepler he adopted the idea that attraction and forces in 
general are not inherent to certain regions of space but have their 
causes in matter alone. The independence, autonomy, and pri- 
ority of space, all vigorously propounded by Gassendi, were a 
timely concession to the requirements of the new physics. Physi- 
cal phenomena could now be explained on the assumption of an 
infinite space that was partly filled and partly empty. Hence 
Gassendi's conception of space became the foundation, both of 
the atomistic theories of the seventeenth century with their dis- 
continuous matter filling continuous space, on the small scale, and 
of celestial mechanics on the large scale. It was Newton who 
incorporated Gassendi's theory of space into his great synthesis 
and placed it as the concept of absolute space in the front line 
of physics. 

100 Gassendi, Opera omnia (Florence, 1727), vol. 3. 




Newton's conceptual scheme, as expounded in his Philo- 
sophiae naturalis principia mathematica, became the basis of 
classical physics and as such the subject of much profound anal- 
ysis. We need mention only Neumann and Mach, who investi- 
gated its epistemological implications, and Wolff and Hegel, who 
explored its metaphysical foundations. So far as the purely physi- 
cal teachings of the Principia are concerned, they are susceptible 
of different epistemological and metaphysical interpretations; for 
that work, as the first comprehensive hypothetico-deductive sys- 
tem of mechanics, lends itself, as does every system of the kind, 
to a variety of philosophical constructions. And so questions arise 









that allow of no absolute answer. Newton himself appears to 
have understood the distinction between the purely theoretical- 
deductive part of a theory and its practical application. In the 
Scholium to Proposition LXIX in the first book he says: "In 
mathematics we are to investigate the quantities of forces with 
their proportions consequent upon any conditions supposed; then, 
when we enter upon physics, we compare those proportions with 
the phenomena of Nature, that we may know what conditions of 
those forces answer to the several kinds of attractive bodies." 1 
The comparison to which Newton here alludes (conferendae 
sunt) 2 seems to correspond to an "epistemic correlation" 3 in 
modern philosophy of science, except for Newton's quite different 
conception of the character of mathematics (mathesis) . For to 
Newton mathematics, particularly geometry, is not a purely hypo- 
thetical system of propositions, logically deducible from axioms 
and definitions; instead geometry is nothing but a special branch 
of mechanics. "Therefore geometry is founded in mechanical 
practice, and is nothing but that part of universal mechanics 
which accurately proposes and demonstrates the art of meas- 
unng. 4 

This view of the relation of geometry to mechanics, Newton 
believes, follows from the impossibility of abstract geometry. 

For the description of right lines and circles, upon which geometry 
is founded, belongs to mechanics. Geometry does not teach us to draw 
these lines, but requires them to be drawn, for it requires that the 

1 F. Cajori, ed., Sir Isaac Newton's Mathematical principles of natural 
philosophy and his System of the world. A revision of Mott's translation 
(University of California Press, Berkeley, 1934) [quoted as Principles], 
p. 192- 

"For the original Latin text, references are given from the Thomson- 
Blackburn edition of the Principia (Glasgow, 1871) [quoted as Principia]. 
On p. 188 we read: ". . . deinde, ubi in physicam descenditur, conferendae 
sunt hae rationes cum phaenomenis . . .' 

"Cf. F. S. C. Northrop, The logic of the sciences and humanities (New 
York, Macmillan, 1947), p. 119. 

'Newton, Principles, p. xvii; Principia, p. xiii, "Auctoris Praefatio ad 
Lectorem," reads: "Fundatur igitur geometria in praxi mechanica, & nihil 
aliud est quam mechanicae universalis pars ilia, quae artem mensurandi 
accurate proponit ac demonstrat." 

learner should first be taught to describe these accurately before he 
enters upon geometry, then it shows how by these operations problems 
may be solved. To describe right lines and circles are problems, but 
not geometrical problems. The solution of these problems is required 
from mechanics, and by geometry the use of them, when so solved 
is shown. 8 ' 

Newton's view of the unity of geometry and mechanics (cf. 
his conception of "fluxions" and his aversion to handling geo- 
metric problems algebraically) can be traced back to his teacher 
Isaac Barrow, for whom geometric curves had essentially a 
geometric character. In "De quadratura curvarum" Newton 
writes: "Quantitates mathematicas, non ut ex partibus quam 
minimis constantes, sed ut motu continuo descriptas, hie con- 
sidero . . . Hae geneses in rerum nature locum vere habent et 
in motu corporum quotidie cernuntur." 6 This realistic concep- 
tion of mathematics is of the first importance for Newton's no- 
tion of absolute space, as we shall soon see. At this point it 
interests us as being an important feature of Newton's meth- 
odology, showing, as it does, that the primary concepts under- 
lying the structure of Newton's system are not hypothetical and 
unreal, justified only by subsequent experimental verification. 

It should be borne in mind, too, that such a remark applies 
not only to the mathematical apparatus employed in the Prin- 
cipia, but to its fundamental laws, as for example the laws of 
motion. We can see today that these laws are assumptions in- 
accessible to experimental verification, but to Newton they were 
facts of immediate experience. For although Newton calls the 
laws of motion "axioms" (Axiomata sive leges motus), the term 
"axiom" as employed by Newton in this context certainly does 
not have the modern meaning of an arbitrary assumption; phrases 
like "lex tertia ... per theoriam comprobata est" 7 or "certa sit 
lex tertia motus" 8 show clearly that Newton by his use of the 
term axiom thought the relevant statement to be the point of 

5 Newton, Principles, p. xvii. 

'Opuscula Newtoni (Lausanne and Geneva, 1744), vol. 1, p. 203. 

'Newton, Principia, p. 25. 

8 Ibid., p. 27. 



departure for further investigation, and thus in conformity with 
his general plan, of which he writes: "To derive two or three 
general principles of motion from phaenomena, and afterwards 
to tell us how the properties and actions of all things follow from 
those manifest principles would be a very great step in philoso- 
phy." 9 It is in the light of these remarks that the historical treat- 
ment of Newton's theory of space must proceed. In other words, 
as historians we are bound to view Newton's system of mechanics 
not from the vantage point of a modern textbook on classical 
mechanics, but from that which Newton himself adopted. Ac- 
cordingly we shall not confine ourselves to the Principia alone, 
but will take into consideration other writings of his as well, 
for example, the Opticks, the correspondence, and especially the 
famous exchange of letters between Leibniz and Newton's dis- 
ciple, Samuel Clarke, who wrote under the guidance of the 

Although Newton cannot, as we have already remarked, be 
regarded as a positivist in the modern sense of the word, yet he 
drew a clear line of demarcation between science on the one 
hand and metaphysics on the other. The famous "Hypotheses 
non fingo," although originally expressed only with relation to 
an explanation of gravitation, became his motto for the exclusion 
of the occult, metaphysical, or transcendental religious entities. 
His aim was not to abolish metaphysics, but to keep it distinct 
from physical investigation. It is well known that Newton, himself 
a religious man, never denied the existence of beings and en- 
tities that transcend human experience; he contended only that 
their existence had no relevance to scientific explanation: In its 
mundus discorsi, science has no place for them. Intimately ac- 
quainted with the problems of religion and metaphysics, Newton 
managed to keep them in a separate compartment of his mind, 
but for one exception, namely, his theory of space. Space thus 
occupies a unique place in his teachings. 

In order fully to understand the Newtonian idea of space, it 

"Newton, Opticks (ed. 4, London, 1730; Dover, New York, 1952), p. 401. 




is necessary to bear in mind the general conceptual background 
of his physical system. Apart from space and time, force and 
mass are the fundamental concepts of the Newtonian physics. 
In Newton "force" is not the sophisticated notion of modern 
physics. It is not a mathematical abstraction, but an absolutely 
given entity, a real physical being. As for "mass," Newton, revert- 
ing to the view of Galileo, conceives of it as the most essential 
attribute of matter and thus places himself in diametrical oppo- 
sition to Descartes, who identified matter with extension and 
regarded extension as the chief characteristic of matter. The 
Newtonian concept of "mass-point," still used in present-day text- 
books, marks the chasm that separates Newton's concept of mass 
from Descartes' concept of spatial extension. A priori, it was per- 
haps a matter of predilection or preference which of the two, 
mass or extension, was to be given priority, since every real body 
has both and is inconceivable apart from either. Newton's ab- 
straction proved to be the more fruitful. 

Since mechanics deals with motion, space as the correlate of 
mass-point — just as the void was the correlate of the atom — has 
to be introduced at the very beginning of the system. It is there- 
fore no accident that almost at the very beginning of the Prin- 
cipia we find the famous Scholium dealing with the concept of 

I do not define time, space, place, and motion, as being well known 
to all. Only I must observe, that the common people conceive those 
quantities under no other notions but from the relation they bear to 
sensible objects. And thence arise certain prejudices, for the removing 
of which it will be convenient to distinguish them into absolute and 
relative, true and apparent, mathematical and common . . . 

Absolute space in its own nature, without relation to anything ex- 
ternal, remains always similar and immovable. Relative space is some 
movable dimension or measure of the absolute spaces; which our 
senses determine by its position to bodies; and which is commonly 
taken for immovable space; such is the dimension of a subterraneous, 
an aerial, or celestial space, determined by its position in respect to 
the earth. Absolute and relative space are the same in figure and 
magnitude; but they do not remain always numerically the same. For 
if the earth, for instance, moves, a space of our air, which relatively 



and in respect of the earth remains always the same, will at one time 
be one part of the absolute space into which the air passes; at another 
time it will be another part of the same, and so, absolutely understood, 
it will be continually changed. 10 

In believing that time, space, place, and motion are concepts 
well known to all, Newton, as we see, does not feel called upon 
to give a rigorous and precise definition of these terms. Yet, be- 
cause these notions arise only in connection with sensible objects, 
certain prejudices cling to them, and to overcome these Newton 
deemed it necessary to set up the distinctions of absolute and 
relative, true and apparent, mathematical and common. Since 
space is homogeneous and undifferentiated, its parts are imper- 
ceptible and indistinguishable to our senses, so that sensible 
measures have to be substituted for them. These coordinate 
systems, as they are called today, are Newton's relative spaces. 

But because the parts of space cannot be seen, or distinguished from 
one another by our senses, therefore in their stead we use sensible 
measures of them. For from the positions and distances of things from 
any body considered as immovable, we define all places; and then 
with respect to such places, we estimate all motions, considering 
bodies as transferred from some of those places into others. And so, 
instead of absolute places and motions, we use relative ones; and that 
without any inconvenience in common affairs. 11 

In modern physics, coordinate systems are nothing but a useful 
fiction. Not so for Newton. Given Newton's realistic conception 
of mathematical objects, it is easy to understand why these rela- 
tive spaces form "sensible measures." Not only is the reference 
body accessible to our senses, but likewise the "relative space" 
is dependent on it. But this accessibility to sense perception 
yields a notion that is of temporary validity only and lacking 
in generality. It is quite possible that there is no body at rest, 
to which the places and motions of other bodies may be re- 
ferred; in a word: all these relative spaces may be moving 
coordinate systems. But moving in what? In order to answer this 
question, Newton takes flight from the realm of experience 

"Newton, Principles, p. 6. 
a Ibid., p. 8. 



altogether, at least for the time being. In his famous words, "But 
in philosophical disquisitions, we ought to abstract from our 
senses," 12 Newton introduces absolute and immutable space, of 
which relative space is only a measure. The final degree of 
accuracy, the ultimate truth, can be achieved only with reference 
to this absolute space. And it is therefore rightly called "true 

What, it may be asked at this point, guarantees the final truth 
of absolute space, the very conception of which appears to con- 
tradict Newton's methodological rule: "We are to admit no more 
causes of natural things than such as are both true and sufficient 
to explain their appearances"? 13 In Newton's time this question 
became a highly controversial one and remained so until the 
beginning of the twentieth century. Is the concept of an absolute 
space a necessity for physics? Or can a consistent conceptual 
scheme be constructed that explains all physical phenomena 
without the use of such a concept? As every historian of physics 
knows, the problem reappeared in the nineteenth century as the 
problem of the ether and gave rise to an immense amount of 
discussion and experiment. 

To Newton, absolute space is a logical and ontological neces- 
sity. For one thing, it is a necessary prerequisite for the validity 
of the first law of motion: "Every body continues in its state of 
rest, or of uniform motion in a right line, unless it is compelled 
to change that state by forces impressed upon it." u Rectilinear 
uniform motion requires a reference system different from that 
of any arbitrary relative space. Further, the state of rest presup- 
poses such an absolute space. Newton explains: 

Absolute motion is the translation of a body from one absolute place 
into another; and relative motion, the translation from one relative 
place into another. Thus in a ship under sail, the relative place of a 
body is that part of the ship which the body possesses; or that part of 
the cavity which the body fills, and which therefore moves together 

"Ibid., p. 398. 
"Ibid., p. 13. 



with the ship: and relative rest is the continuance of the body in the 
same part of the ship, or of its cavity. But real, absolute rest, is the 
continuance of the body in the same part of that immovable space, in 
which the ship itself, its cavity, and all that it contains, is moved. 
Wherefore, if the earth is really at rest, the body, which relatively 
rests in the ship, will really and absolutely move with the same veloc- 
ity which the ship has on the earth. But if the earth also moves, the 
true and absolute motion of the body will arise, partly from the true 
motion of the earth, in immovable space, partly from the relative mo- 
tion of the ship on the earth; and if the body moves also relatively in 
the ship, its true motion will arise, partly from the true motion of the 
earth, in immovable space, and partly from the relative motions as 
well of the ship on the earth, as of the body in the ship; and from 
these relative motions will arise the relative motion of the body on 
the earth. 15 

Since the first law of motion, as we have seen, is for Newton 
a matter of immediate experience, and since the law depends for 
its validity upon an absolute reference system, absolute space 
becomes indispensable to Newtonian mechanics. The interesting 
point, however, is that for Newton the introduction of the con- 
cept of absolute space into his system of physics did not result 
from methodological necessity only. Newton was led by his 
mathematical realism to endow this concept, as yet merely a 
mathematical structure, with independent ontological existence. 
He realized that there was a great difficulty to be overcome: the 
"inertial system," or, in less modern words, the system in which 
the first law holds, is not uniquely determined. Newton's 
mechanics is invariant for a translational transformation with 
constant velocity, that is, a Galilean transformation. Newton rec- 
ognizes that a whole class of "spaces" or reference systems com- 
ply with this requirement. In Corollary V we read: "The motions 
of bodies included in a given space are the same among them- 
selves, whether that space is at rest, or moves uniformly forwards 
in a right line without any circular motion." 16 

If Newton had been a confirmed positivist he would have 
acknowledged all uniformly moving inertial systems as equivalent 

15 Ibid., p. 7- 
M Ibid., p. 30. 


to each other. As it was, only one absolute space existed for him. 
How is this space to be distinguished from among the multitude 
of inertial systems? For the solution of this problem Newton re- 
sorts to cosmology. In Hypothesis I of his The system of the 
world 17 he states: "That the centre of the system of the world 
is immovable. This is acknowledged by all, while some contend 
that the earth, others that the sun, is fixed in that centre." 

To Newton, now, the center of the world is the center of 
gravity of the system composed of the sun, the earth, and the 
planets; 18 this center either is at rest or moves uniformly forward 
in a straight line; the latter alternative, however, is eliminated 
by Hypothesis I. In this way Newton defines the unique absolute 
space among all possible inertial frames. It it interesting to note 
that in the last-mentioned Corollary Newton is concerned to find 
the astronomical location of this universal center of gravity, 
which is his reference point for the determination of absolute 
space. He maintains that the movable centers of the earth, sun, 
and planets cannot serve as such a center, since they all gravitate 
toward each other. However, if the body toward which other 
bodies gravitate most has to be placed in the center, then it is 
the sun that should be allowed this privilege. Yet, since the sun 
itself is moving, a fixed point has to be chosen from which the 
center of the sun recedes least, and from which, if its density 
and volume were greater, it would recede still less. 

All this points to the rather limited scope of Newton's cosmo- 
logical conceptions. It is also interesting to note that Newton did 
not take into account the fixed stars when trying to determine 
the center of gravity of the world. Had he done so, he might 
have come very near the conception of the body "Alpha," which 
was introduced by C. Neumann 19 at the end of the last century. 
The fact that Newton ignored the fixed stars in this respect is the 
more curious, since for him they were still really "fixed," that is, 

"Ibid., p. 419. 

w Ibid., p. 419, corollary to Proposition XII. 

19 C. Neumann, Ueber die Prinzipien der Galileir-Newton'schen Theorie 



not moving in space. For although Bruno had already imagined 
the sun to be in motion, and although Halley confirmed this 
anticipation in 1718, when he announced 20 that Sirius, Aldeba- 
ran, Betelgeuse, and Arcturus had unmistakably shifted their 
positions in the sky since Ptolemy assigned their places in his 
catalogue, it was only after the death of Newton that the proper 
motion of the stars became an accepted truth. 

Newton's cosmological assumption that the center of the world 
is at rest escapes all possibility of experimental or observational 
verification. The fact was clearly recognized by Berkeley, one 
of the great opponents of the theory of absolute space. In "De 
motu" he writes: "Uti vel ex eo patet quod, quum secundam 
illorum principia qui motum absolutum inducunt, nullo sympto- 
mate scire liceat, utrum integra rerum compages quiescat, an 
moveatur uniformiter in directum, perspicuum sit motum abso- 
lutum nullius corporis cognosci posse." 21 

As we shall see in what follows, Newton was convinced that 
dynamically, though not kinematically, absolute space can be 
determined through the existence of centrifugal forces in rota- 
tional motion. Although Newton does not explicitly draw the 
conclusion that centrifugal forces determine absolute motion 
which in its turn determines absolute space, it is clear that this 
was his intention and it was always recognized as such by his 
commentators. If space is a physical reality, as Newton un- 
doubtedly assumes, and if accelerated motion furnishes a cri- 
terion for its identification, it would appear to be a serious 
inconsequence to hold that uniform translational motion, since 
it fails to provide such a criterion, is different from all other 
kinds of motion; furthermore, space would seem to possess a 
dual structure, absolute for accelerated motion and relative for 
uniform translation. Newton's cosmological assumption protects 

"E. Halley, Phil. Trans. 30, 737 (1718). 

21 A. A. Luce and T. E. Jessop, ed., The works of George Berkeley (Nel- 
son, London, 1951), vol. 4, p. 28. 



him against such an objection, which incidentally was raised by 
Leibniz in his correspondence with Huygens. 

According to Newton, as we have seen, the first law of motion 
presumes the necessary existence of absolute space but provides 
no means by which it can be identified experimentally. Hence 
Newton's next step. Since absolute space and time "do by no 
means come under the observation of our senses," it becomes 
necessary to investigate the dynamics of motion. For motion, ac- 
celerated motion in particular, is the means and medium through 
which space can be explored. Inasmuch as they refer to relative 
or to absolute space, motions are either relative or absolute, so 
that if it were possible to identify absolute motion, the identifica- 
tion of absolute space would follow. Now absolute motion, ac- 
cording to Newton, can be distinguished from relative motion by 
its "properties, causes, and effects." 

The causes by which true and relative motions are distinguished 
one from the other, are the forces impressed upon bodies to generate 
motion. True motion is neither generated nor altered, but by some 
force impressed upon the body moved; but relative motion may he 
generated or altered without any force impressed upon the body. For 
it is sufficient only to impress some force on other bodies with which 
the former is compared, that by their giving way, that relation may 
be changed, in which the relative rest or motion of this other body did 
consist ... 

The effects which distinguish absolute from relative motion are, the 
forces of receding from the axis of circular motion. For there are no 
such forces in a circular motion purely relative, but in a true and abso- 
lute circular motion, they are greater or less, according to the quantity 
of the motion . . . 

It is indeed a matter of great difficulty to discover, and effectually 
to distinguish, the true motions of particular bodies from the apparent; 
because the parts of that immovable space, in which those motions 
are performed, do by no means come under the observation of our 
senses. Yet the thing is not altogether desperate; for we have some 
arguments to guide us, partly from the apparent motions, which are 
the differences of the true motions; partly from the forces, which are 
the causes and effects of the true motions. 22 

" Newton, Principles, pp. 10, 12. 



Thus Newton's first argument with regard to absolute motion 
is based on the idea that real force creates real motion. To 
Newton, at least in this context, forces are metaphysical entities 
conceived anthropomorphicaUy. However, if we leave out of ac- 
count the import of forces for the determination of absolute 
space, the notion of force in Newton's mechanics may be in- 
terpreted in the modern functional way, as in Heinrich Hertz's 
Die Prinzipien der Mechanik: "Was wir gewohnt sind als Kraft 
und als Energie zu bezeichnen ist dann fuer uns nichts weiter 
als eine Wirkung von Masse und Bewegung, nur braucht es 
nicht immer die Wirkung grobsinnlich nachweisbarer Masse und 
grobsinnlich nachweisbarer Bewegung zu sein." 23 

But undoubtedly there is no question of this functional con- 
ception of force in Newton's discussion of absolute space. It is 
foreign to the general character of his system. His argument 
"from causes" is based on the traditional metaphysics, the inclu- 
sion of which in the framework of physical explanation is strongly 
objected to by Newton himself. In order to see the vicious 
circle inherent in Newton's reasoning, we have only to think for 
a moment of a world of moving masses in which no living 
organism existed. For in such a world an absolute force could 
be determined, according to Newton, solely by the absolute 
motion of the body on which this force was exerted. 

The second argument for the existence of absolute motion 
proceeds from the effects that such motion produces, in partic- 
ular, the appearance of centrifugal forces ("vires recedendi ab 
axe motus circularis"). So we have Newton's famous pail experi- 
ment, which he describes as follows: 

If a vessel, hung by a long cord, is so often turned about that the 
cord is strongly twisted, then filled with water, and held at rest to- 
gether with the water; thereupon, by the sudden action of another 
force, it is whirled about the contrary way, and while the cord is 
untwisting itself, the vessel continues for some time in this motion; the 
surface of the water will at first be plain, as before the vessel began 
to move; but after that, the vessel, by gradually communicating its 

**H. Hertz, Die Prinzipien der Mechanik (Leipzig, 1894), p. 31. 



motion to the water, will make it begin sensibly to revolve, and recede 
by little and little from the middle, and ascend to the sides of the 
vessel, forming itself into a concave figure (as I have experienced), 
and the swifter the motion becomes, the higher will the water rise, till 
at last, performing its revolutions in the same times with the vessel, 
it becomes relatively at rest in it. This ascent of the water shows its 
endeavor to recede from the axis of its motion; and the true and abso- 
lute circular motion of the water, which is here directly contrary to 
the relative, becomes known, and may be measured by this endeavor. 
At first, when the relative motion of the water in the vessel was great- 
est, it produced no endeavor to recede from the axis; the water showed 
no tendency to the circumference, nor any ascent towards the sides 
of the vessel, but remained of a plain surface, and therefore its true 
circular motion had not yet begun. But afterwards, when the relative 
motion of the water had decreased, the ascent thereof towards the 
sides of the vessel proved its endeavor to recede from the axis; and this 
endeavor showed the real circular motion of the water continually 
increasing, till it had acquired its greatest quantity, when the water 
rested relatively in the vessel. And therefore this endeavor does not 
depend upon any translation of the water in respect of the ambient 
bodies, nor can true circular motion be defined by such translation. 24 

For a clear analysis of this experiment, let us consider also 
the final phase — not described by Newton — when the rota- 
tion of the pail is stopped while the water continues its circular 
motion (owing to conservation of angular momentum). During 
this final stage of the experiment, as long as friction can be ig- 
nored, the water contained in the vessel maintains its parabo- 
loidal surface. 

The gist of this experiment can be summarized in modern 
terms as follows: Both in the beginning of the experiment (when 
the pail spins alone) and at the end of the experiment (when the 
water spins alone) pail and water are moving relative to each 
other in the same manner. Rigorously considered, the directions 
of the relative rotations are reversed; but owing to the assumed 
isotropy of space this reversal can obviously have no effect on 
the dynamical result. If in the second case the time parameter 
had been reversed, as is permissible in a purely mechanical 
phenomenon, exacdy the same relative motion would have re- 

M Newton, Principles, p. 10. 




suited. Now, were all motion (rotation) purely relative, no 
physical difference should become apparent between the two 
states. However, since the surface of the water contained in the 
pail is level in the first case and paraboloidal in the second, 
rotation, thus concludes Newton, must be absolute. 

This experiment was the cause of much controversy in the 
history of modern physics and the situation was clarified only 
with the appearance of Einstein's principle of equivalence in his 
general theory of relativity. In Newton's interpretation of the 
pail experiment he obviously again transcends the realm of ex- 
perience. His simple assumption that the surface of water in the 
pail would be as curved, even if it were rotating in empty space, 
as when rotating in space filled with starry matter, is not sus- 
ceptible of physical verification. And the same inaccessibility to 
physical verification characterizes all the other attempts to in- 
force his argument, as, for example, his experiment with the two 
cord-connected spheres revolving around their common center 
of gravity, the tension in the cord being taken by him as an indi- 
cation of the absolute motion of the spheres. "And thus we might 
find both the quantity and the determination of this circular 
motion, even in an immense vacuum, where there was nothing 
external or sensible with which the globes could be compared." 25 
But such conditions can never be realized, any more than in the 
case of the astronomical effects of centrifugal forces, as for ex- 
ample the spheroidicity of the earth and of Jupiter, as Newton 
expounds the matter in the third book of his Principia. 26 

Berkeley rejects Newton's implicit assumption that the pail 
experiment, if performed in empty space, would yield the same 
result. As Berkeley explains in his "De motu," the real motion of 
the pail is far from being circular, if the diurnal rotation of 
the earth and its annual revolution are taken into account. For 

" Ibid., p. 12. 

"Proposition XVIII, Theorem XVI; also Proposition XIX. Problem III 
(Principles, p. 424); et alia. 



Berkeley the idea of an absolute motion and an absolute space 
is a mere fiction, lacking all experimental foundation. Relating 
all such motions as are exhibited in the pail experiment ulti- 
mately to the system of stars as frame of reference, Berkeley 
assumes the existence of such a system as necessary for the intel- 
ligibility of motion. He says: 

If we suppose the other bodies were annihilated and, for example, 
a globe were to exist alone, no motion could be conceived in it; so 
necessary is it that another body should be given by whose situation 
the motion should be understood to be determined. The truth of this 
opinion will be very clearly seen if we shall have carried out thor- 
oughly the supposed annihilation of all bodies, our own and that of 
others, except that solitary globe. 

Then let two globes be conceived to exist and nothing corporeal 
besides them. Let forces then be conceived to be applied in some way; 
whatever we may understand by the application of forces, a circular 
motion of the two globes round a common centre cannot be conceived 
by the imagination. Then let us suppose that the sky of the fixed stars 
is created; suddenly from the conception of the approach of the globes 
to different parts of that sky the motion will be conceived. 27 

Berkeley's statement obviously cannot be considered as being 
equivalent to what is called in modern cosmology "Mach's 
principle" (that is, that the inertia of any body is determined 
by the masses of the universe and their distribution), as Berkeley 
confines himself to the problem of the perception and compre- 
hensibility of motion and ignores in this context the dynamical 
aspect of motion. 

Newton's final argument, based upon the distinction between 
absolute and relative, or apparent, motion, is not further devel- 
oped in his works. His idea seems to be that a body which moves 
in relative motion may be moving in absolute motion or may be 
at rest relative to absolute space; there is no way of deciding 
between these two alternatives. However, if a second body moves 
relative to the first, it is clear that one of these two at least must 
be endowed with absolute motion. It is impossible that both of 

17 Luce and Jessop, ed., The works of George Berkeley, vol. 4, p. 47. 



them should be at rest with respect to absolute space. 28 The 
weakness of this argument is its indefensible assumption that an 
absolute reference system is an essential prerequisite for the 
description of the behavior of these bodies. 

So Newton takes over from Patritius, Campanella, and Gas- 
sendi the concept of an infinite space, which is homogeneous and 
isotropic and, in addition, succeeds in convincing himself that 
he has proved the reality of this concept by physical experiment. 
He thought he had demonstrated that space has an existence 
proper to itself and independent of the bodies that it contains. In 
his view, it makes sense, therefore, to state that any definite body 
occupies just this part of space and not another part of space, 
and the meaning of such a statement does not presuppose a rela- 
tion to any other bodies in the universe. He was not aware that 
his procedure violated the very principles of the methodology he 
professed. Since he was a younger contemporary of Henry More, 
whose personal acquaintance he made in his youth and whose 
teachings, via Isaac Barrow, exerted a great influence upon him, 
it is no wonder that Newton found support for his theory of 
space in the doctrine of that thinker. More's important works had 
been published about seven years before the appearance of the 
Principia. But it was the religious element, originating, as we 
saw, in Jewish cabalistic and Neoplatonic thought, that gained 
ascendency over Newton in his later years. So a comparison of 
the first and later editions of the Principia shows that the iden- 
tification of absolute space with God, or with one of his attributes, 
came into the foreground of Newton's thought only toward the 
end of his life, that is, at the beginning of the eighteenth century. 
However, his interest in Biblical and post-Biblical literature may 
be traced back to the influence of one of the teachers in Cam- 
bridge, Joseph Mede, a fellow of Christ's College. Mede, apart 
from his studies in apocryphical and other esoteric literature, 

28 This argument in defense of absolute motion reappeared later in Alois 
Hoefler's Studien zur gegenwartigen Philosophie der Mechanik (Leipzig, 
1900), p. 133. 



stimulated philological interest among his students in the Hebrew 
of the Bible by his etymological theory, quite popular at that 
time, that Hebrew was the mother of all languages. 

It also has been established that Durand Hotham's book on 
Jacob Bohme exerted a strong influence on young Newton. 
Bohme's Mysterium magnum, a commentary on Genesis, shows 
extraordinary parallels to the Zohar and to other sources of 
Jewish theosophy. The Hebrew Chokmah, a body of books 
ascribed to King Solomon, seemed to have passed over to the 
Gnostic Sophia and by another transition to the "Virgin Wisdom" 
of Bohme. We know also with certainty that Henry More 29 and 
Isaac Barrow exerted a very strong influence on Newton at that 
time. Henry More was the spiritual leader at Christ's College in 
Cambridge and the chief disseminator of Cabalistic and Neo- 
platonic ideas, as described in detail in Chapter II. Isaac Barrow, 
Newton's famous teacher, promulgated More's ideas in mathe- 
matized form in his Mathematical lectures. In Barrow's geometry, 
space is the expression of divine omnipresence, just as time is the 
expression of the eternity of God. Under the influence of these 
strong forces it seems most probable that Newton, even when 
writing on purely physical problems, had similar ideas in the 
back of his mind. In fact, that he had theological and religious 
ideas in his mind when writing the Principia is evident from his 
letter (December 10, 1692) to Bichard Bentley in which he 
confessed: "When I wrote my treatise about our system, I had 
an eye upon such principles as might work with considering 
men for the belief of a Deity; and nothing can rejoice me more 
than to find it useful for that purpose." It was, however, only in 
1713 that Newton prepared the General Scholium of Book III 
to be published in the second edition (1713). It is in this 
Scholium, in addition to Queries 19-31 of the Opticks (missing 
in the first edition), that we find explicit statements of Newton's 
ideas on the relation between his theory of absolute space and 

" For the facts of personal contact between More and Newton, see L. T. 
More, Isaac Newton (Scribner, New York, 1934), pp. 11, 31, 182. 



theology. Undoubtedly, Newton's increasing interest in theo- 
logical and spiritual questions during his later years was one 
of the motives for the preparation of the Scholium. Another 
reason was Cotes's request that he prevent any recurrence of 
criticisms which pronounced Newton's theory of space as lead- 
ing to atheism. In a letter (March 18, 1713) to Newton, the 
editor of the second edition of the Principia writes: "I think it 
will be proper to add something by which your Book may be 
cleared from some prejudices which have been industriously laid 
against it . . . That You may not think unnecessary to answer 
such Objections You may be pleased to consult a Weekly Paper 
called 'Memoires of Literature and sold by Ann Baldwin in 
Warwick-Lane." The article referred to is Leibniz's letter (Feb- 
ruary io, 1711) to the Dutch physician Hartsoeker, in which 
Leibniz attacks Newton's theory of gravitation. Of greater rele- 
vance for our subject, however, is Berkeley's attack on Newton's 
theory of space, which Cotes certainly had in mind, although he 
did not mention Berkeley by name. Berkeley published in 1710 
his Principles of human knowledge, in which he criticizes New- 
ton's concept of absolute space on theological grounds as being 
a pernicious and absurd notion. Space, according to Berkeley, 
has to be conceived as relative only, "Or else there is something 
beside God which is eternal, uncreated, infinite, indivisible, un- 
mutable." 30 

It is therefore not surprising that in the General Scholium 
Newton gives free reign to his religious enthusiasm: 

It is the dominion of a spiritual being which constitutes a God: a 
true, supreme, or imaginary dominion makes a true, supreme, or im- 
aginary God. And from true dominion it follows that the true God is 
a living, intelligent, and powerful Being; and, from his other per- 
fections, that he is supreme, or most perfect. He is eternal and infinite; 
omnipotent and omniscient; that is, his duration reaches from eternity 
to eternity; his presence from infinity to infinity; he governs all things, 
and knows all things that are or can be done. He is not eternity and 
infinity, but eternal and infinite; he is not duration or space, but he 

"Berkeley, Principles of human knowledge, in A new theory of vision 
and other writings (Dent, London, 1938), p. 173- 



endures and is present. He endures for ever, and is everywhere pres- 
ent; and by existing always and everywhere, he constitutes duration 
and space. 31 

Here, for the time, Newton identifies space and time with God's 
attributes. God is not eternity and infinity, but he is eternal and 
infinite. Eternal and omnipresent, God constitutes duration and 
space. A few lines further on we read, "In ipso continentur 
& moventur universa," "In him are all things contained and 
moved," a statement to which Newton adds the marginal remarks 
that this was the opinion of the Ancients: St. Paul (Acts 17:27, 
28), St. John (14:2), Moses (Deut. 4:39), David (Ps. 139:7-9), 
Solomon (I Kings 8:27), Job (22:12-14), Jeremiah (23:23, 24). 
Here we have unmistakably an echo of More's Enchiridion 
metaphysicum and his Divine dialogues, but with this difference, 
that Newton's expressions are more reserved and more carefully 
chosen. He seemed to be aware that he might easily be misunder- 
stood and counted among the pantheistic thinkers of his time, who 
in orthodox circles were identified with the atheists. 

Since every particle of space is always in existence, and every 
indivisible moment of duration is everywhere, "certainly the 
Maker and Lord of all things cannot be never and nowhere." 32 
Elsewhere Newton speaks of 

the Wisdom and Skill of a powerful ever-living Agent; who, being in 
all Places, is more able by his Will to move the Bodies . . . within 
his boundless uniform Sensorium, and thereby to form and reform the 
Parts of the Universe, than we are by our Will to move the Parts of 
our own Bodies. 33 

This identification of the omnipresence of space with the omni- 
presence of God leads to a serious difficulty, and Leibniz with his 
sharp intellect exploited it remarkably in his controversy with 
Clarke. For according to Newton's conception, the divisibility of 
space — relative spaces are parts of the absolute space — would 

41 Newton, Principles, p. 544. 

* Newton, Principia, p. 528: "Certe rerum omnium fabricator ac dominus 
non erit nunquam, nusquam." 
88 Newton, Opticks (Dover ed.), p. 403. 




appear to involve the divisibility of God. Clarke's response to 
Leibniz's argument may be summarized as follows: Absolute 
space is one; it is infinite and essentially indivisible. The assump- 
tion that it can be divided leads to a contradiction, since any 
partition — according to Clarke — would require an intermediary 
space. Hence divine infinity and omnipresence imply no divisi- 
bility of the substance of God. Clarke concludes that it is only 
because a pictorial and unjustified meaning is attached to the 
notion of divisibility that the difficulty arose. 

Another point of interest in this controversy is the term 
"sensorium," occurring in the above quotation, and earlier in 
Query 28: 

. . . does it not appear from Phaenomena that there is a Being 
incorporeal, living, intelligent, omnipresent, who in infinite Space, as 
it were in his Sensory, sees the things themselves intimately, and 
throughly perceives them, and comprehends them wholly by their 
immediate presence to himself. 34 

In the letter opening the controversy, which was to end only 
with Leibniz's death in 1716, Leibniz says: 

Sir Isaac Newton says, that Space is an Organ, which God makes 
use of to perceive Things by. But if God stands in need of any Organ 
to perceive Things by, it will follow, that they do not depend alto- 
gether upon him, nor were produced by him. 35 

But did Newton really identify space with an organ of God? Or 
was this expression only an unfortunate lapsus calmi? Clarke's 
response to Leibniz gives the answer to this question: 

Sir Isaac Newton doth not say, that Space is the Organ which God 
makes use of to perceive Things by; nor that he has need of any 
Medium at all, whereby to perceive Things; But on the contrary, that 
he, being Omnipresent, perceives all Things by his immediate Presence 
to them, in all Space whereever they are, without the Intervention 
or Assistance of any Organ or Medium whatsoever. In order to make 
this more intelligible, he illustrates it by a Similitude: That as the 
Mind of Man, by its immediate Presence to the Pictures of Things, 

"■Ibid., p. 370. 

83 A Collection of Tapers which passed between the late learned Mr. 
Leibnitz and Dr. Clarke (London, 1717), p. 3. 



form'd in the Brain by the means of the Organ of Sensation, sees 
those Pictures as if they were the Things themselves; so God sees all 
Things, by his immediate Presence to them: he being actually present 
to the Things themselves, to all Things in the Universe; as the Mind 
of Man is present to all the Pictures of Things formed in his Brain 
. . . And this Similitude is all that he means, when he supposes In- 
finite Space to be (as it were) the Sensorium of the Omnipresent 
Being. 36 r 

Accordingly, it seems to be clear that Newton used the term 
"Sensorium" merely as a comparison and did not identify space 
with an organ of God. 

With Newton's conception of space now before us we may 
turn to the question why he thought it needful and appropriate 
to introduce theological considerations into the very body of his 
scientific writings. Apart from the reasons dictated by polemic, 
as we have seen, there are certainly other motives; John Tull 
Baker, in his monograph entitled An historical and critical exami- 
nation of English space and time theories, discusses some of them. 
He writes that, in the first place, absolute space and time find 
a place in the Principia because as attributes of God they ren- 
dered to the Principia a completeness, as a cosmology, which 
it might have lacked otherwise. Furthermore, their inclusion in 
the very beginning of the Newtonian system gives to the founda- 
tions of mechanics and mathematical physics a theological justi- 
fication, an idea congenial to Newton: 

In the second place, the postulations of absolute time and absolute 
space suggest the construction of mathematical entities which might 
be approached as limits of perfection on the description of physical 
facts. Just as relative time always more nearly approaches absolute 
time as we refine our measurements and relative motion approximates 
absolute motion as we examine forces more carefully, so the scheme 
of things as a whole may be more clearly understood as we progress 
in more detailed experiment and analysis. 37 

According to this interpretation, the use of absolutes by New- 
ton may be understood as an ideal of perfection, an ideal attain- 

"Ibid., p. ii. 

OT J. T. Baker, An historical and critical examination of English space and 
time theories (Sarah Lawrence College, Bronxville, New York, 1930), p. 30. 



able in matters of space only. In addition it may be rightly 
claimed that absolute space and absolute time have always had 
a strong appeal to human emotion. Through their presence clarity 
and rigor, certainty and definiteness seem to be guaranteed. 

One thing is certain: Newton's mechanics, as expounded in the 
Principia, is one great vindication of his theory of absolute space 
and absolute motion. At the end of the Scholium in the first 
book he says: "How we are to obtain the true motions from 
their causes, effects, and apparent differences, and the converse, 
shall be explained more at large in the following treatise. For 
to this end it was that I composed it." 88 "Hunc enim in finem 
tractatum sequentem composui." 39 To demonstrate the existence 
of true motion and absolute space — such is the program of the 
Principia. All Newton's achievements and discoveries in the realm 
of physics are in his view subordinate to the philosophical con- 
ception of absolute space. The outstanding success of Newtonian 
mechanics in the physics and astronomy of the last two centuries 
seemed an indubitable guarantee of the soundness of its philo- 
sophical implications. It is not surprising, therefore, that the 
criticisms leveled by Leibniz and Huygens against the theory 
of absolute space found no echo in this long period. Today we 
are in a position to understand the force of these criticisms, which 
is not to say that the Principia ceases to be a landmark in the his- 
tory of human intellectual achievements. It is this not because of 
its philosophical conclusions but because of the wealth of its 
purely physical contents, backed by experimentation and hence 
verifiable, and further because of the wonderful systematization 
of this wealth of material. 

It is not the purpose of this chapter to provide a compre- 
hensive account of Leibniz's theory of space. In any case it is 
a task immensely complicated by the fact that Leibniz's theory 
in the course of its development passed through three differ- 

" Newton, Principles, p. 12. 
*" Newton, Principia, p. 12. 



ent stages at least. We shall confine our discussion here to his 
critique of Newton's conception, for the understanding of which 
it is necessary to bear in mind that in his view space is nothing 
but a system of relations, devoid of metaphysical or ontological 
existence. In his fifth letter to Clarke, Leibniz summarizes his 
conception of space as follows: 

I will here show, how Men come to form to themselves the Notion 
of Space. They consider that many things exist at once, and they 
observe in them a certain Order of Co-existence, according to which 
the relation of one thing to another is more or less simple. This Order 
is their Situation or Distance. When it happens that one of those 
Co-existent Things changes its Relation to a Multitude of others, which 
do not change their Relation among themselves; and that another 
Thing, newly come, acquires the same Relation to the others, as the 
former had; we then say it is come into the Place of the former; and 
this Change we call Motion in That Body, wherein it is the immediate 
Cause of the Change. And though Many, or even All the Co-existing 
Things, should change according to certain known Rules of Direction 
and Swiftness; yet one may always determine the Relation of Situa- 
tion, which every Co-existent acquires with respect to every other 
Co-existent; and even That Relation, which any other Co-existent 
would have to this, or which this would have to any other, if it had 
not changed or if it had changed any otherwise. And supposing, or 
feigning, that among those Co-existents, there is a sufficient Number 
of them, which have undergone no Change; then we may say, that 
Those which have such a Relation to those fixed Existents, as Others 
had to them before, have now the same Place which those others 
had. And That which comprehends all those Places, is called Space. 40 

Leibniz goes on to explain that the relation of situation is 
a wholly sufficient condition for the idea of space. No absolute 
reality need be invoked. He makes his point clear by an excellent 
illustration from genealogy: 

In like manner, as the Mind can fancy to itself an Order made up 
of Genealogical Lines, whose Bigness would consist only in the Num- 
ber of Generations, wherein every Person would have his Place: and 
if to this one should add the Fiction of a Metempsychosis, and bring 
in the same Human Souls again; the Persons in those Lines might 
change Place; he who was a Father, or a Grand-father, might become 

40 A Collection of Papers . . . , p. 195. 



a Son, or a Grand-son &c. And yet those Genealogical Places, Lines 
and Spaces, though they should express real Truths, would only be 
Ideal Things. 41 ' 

The illustration of a tree of genealogy, which shows the mutual 
relations of kinship between certain persons by attributing to 
them definite positions within the scheme, serves Leibniz very 
well. For nobody would hypostatize this system of relations and 
endow it with ontological existence. Newton's absolute space, in 
Leibniz's view, is nothing but a similar unjustified hypostati- 

Having thus outlined his concept of space, Leibniz realizes 
that what he has done is only to define the expression "having 
the same place," this being enough for the foundation of the 
concept of physical space. He then proceeds with great ardor to 
attack More and through him Newton. For the context the 
following words of Leibniz are worth quoting: 

If the Space (which the Author fancies) void of all Bodies, is not 
altogether empty; what is it then full of? Is it full of extended Spirits 
perhaps, or immaterial Substances, capable of extending and contract- 
ing themselves; which move therein, and penetrate each other with- 
out any Inconveniency, as the Shadows of two Bodies penetrate one 
another upon the Surface of a Wall? Methinks I see the revival of the 
odd Imaginations of Dr. Henry More (otherwise a Learned and well- 
meaning Man), and of some Others, who fancied that those Spirits 
can make themselves impenetrable whenever they please. Nay some 
have fancied, that Man in the State of Innocency, had also the Gift 
of Penetration; and that he became Solid, Opaque, and Impenetrable 
by his Fall. Is it not overthrowing our Notions of Things, to make 
God have Parts, to make Spirits have Extension? 42 

Leibniz's clear conception of space 43 as a system of relations 
and his well-known "principium identitatis indiscernibilium" 
are the two solid foundations from which he launches his 
criticism of Newton's absolute space and absolute motion. On 

a Ibid., p. 201. 

° Ibid., p. 205. 

"£ or a .luetic history of Leibniz's philosophy of space and time, see 
W. Gent, Leibnizens Philosophie der Zeit und des Raumes," Kantstudien 
31, 61 (1926). 



kinematic grounds there can be no doubt that Leibniz is the 
victor in this dispute. Clarke's refutations of Leibniz's arguments 
are often not to the point and show a great deal of misun- 
derstanding. However, as soon as Clarke leaves the subject of 
kinematics and brings forth — no doubt under the briefings 
of Newton himself — the dynamical arguments in favor of the 
existence of absolute space and motion, Leibniz faces an insuper- 
able difficulty. With regard to Clarke's reference to the Scholium 
and Newton's demonstrations therein of the existence of absolute 
space and absolute motion by means of centrifugal forces, Leib- 
niz feels obliged to admit: 

However, I grant there is a difference between an absolute true 
motion of a Body, and a mere relative Change of its Situation with 
respect to another Body. For when the immediate Cause of the 
Change is in the Body, That Body is truly in Motion; and then the 
Situation of other Bodies, with respect to it, will be changed con- 
sequently, though the Cause of that Change be not in Them. 44 

Having thus bowed to the idea of an "absolute true motion," 
Leibniz is placed in a dilemma from which he finally sees only 
one way out: namely, to allow for a double meaning of the con- 
cept of motion. On the one hand, it may denote the purely spa- 
tial change of situation, which saves his view of the conceptual 
structure of space; on the other hand, it may signify a dynamical 
process which is completely unrelated to space as such. But Leib- 
niz is aware that such a stratagem exposes him to the danger 
of having to fall back on doubtful scholastic concepts like 
quality, form, substance. It is especially clear from Leibniz's 
correspondence with Huygens that he tried desperately for years 
without success to find a dynamical argument for the relativity 
of motion. Yet it is a curious fact for us today to note that actually 
he came very near to Mach's solution of the problem. In his 
"De Causa Gravitatis, et Defensio Sententiae Autoris de veris 
Naturae Legibus contra Cartesianos" 45 Leibniz tried to demon- 
strate that gravity is not explicable as a force acting at a distance, 

" A Collection of Papers . . . , p. 213, 
16 Acta Eruditorum (1690). 



but is reducible to the contiguous action of the surrounding 
ether. In other words, he tried to reduce gravity to a centrifugal 
force, saying: "Etsi valde dudum inclinaverim ipse ad gravitatem 
a vi centrifuga materiae aethereae circulantis repetendam, sunt 
tamen aliqua quae dubitationes gravissimas injecere." 46 Directly 
opposite to this was Mach's daring description of centrifugal 
forces as a disguised gravitational action. So Leibniz, having 
failed to find the key to dynamical relativity, saw no need to 
revise what he had written some twenty years before, when he 
summarized his remarks on Cartesian physics in his "Animad- 
versions on Descartes' Principles of Philosophy":" 

On Art. 25. If motion is nothing but change of contact or imme- 
diate vicinity, it follows that it can never be determined which thing 
is moved. For as in astronomy the same phenomena are presented 
in different hypotheses, so it is always permissible to ascribe real 
motion to either one or other of those bodies which change among 
themselves vicinity or situation; so that one of these bodies being 
arbitrarily chosen as if at rest, or for a given reason moving in a given 
line, it may be geometrically determined what motion or rest must 
be ascribed to the others so that the given phenomena may appear. 
Hence if there is nothing in motion but this respective change, it 
follows that no reason is given in nature why motion must be ascribed 
to one rather than to others. The consequence of this will be that 
there is no real motion. Therefore in order that a thing can be said to 
be moved, we require not only its situation in respect to others, but 
also that the cause of change, the force or action, be in itself. 48 

It is to these lines that Huygens refers in his letter of May 29, 
1694 to Leibniz. He objects to the assertion "that it would be 
absurd, if there exists no real, but only relative motion" ("abso- 
num esse nullum dari motum realem sed tantum relativum"). If 
Huygens' quotation from Leibniz is verbally inaccurate, it is 
not so essentially. Huygens declares his intention to stick to his 
theory — perhaps by way of contrasting his own firmness with 
Leibniz's wavering — and says that he will not let himself be 

"G. I. Gerhardt, Leibnizens mathematische Schriften (Halle, i860), 
part 2, vol. 6, p. 197. 

"Published in 1692. 

48 G. M. Duncan, The philosophical works of Leibnitz (New Haven, 
1890), p. 60. 



influenced by the experiments in the Principia, convinced as he 
is that Newton is wrong. At the same time he hopes that Newton 
will retract his theory in the forthcoming second edition of the 
Principia, which he thought would be edited by David Gregory. 
Huygens' instinct toward his own theory was sound, although he 
was mistaken about the second edition of the Principia, which 
in fact was prepared by Roger Cotes, as he was mistaken about 
its possible revision by Newton. 

The subject occurs in Huygens' first letter to Leibniz, which 

Je vous diray seulement, que dans vos notes sur des Cartes j'ay 
remarque que vous croiez absonum esse nullum dari motum realem, 
sed tantum relativum. Ce que pourtant je tiens pour tres constant, sans 
m'arrester au raisonnement et experiences de Newton dans ses Prin- 
cipes de Philosophic, que je scay estre dans l'erreur, et j'ay envie de 
voir s'il ne se retractera pas dans la nouvelle edition de ce livre, que 
doit procurer David Gregorius. 49 

Leibniz's reply to this letter (June 22, 1694) is extremely 

Quant a la difference entre le mouuement absolu et relatif, je croy 
que si le mouuement ou plus tost la force mouuante des corps est 
quelque chose de reel comme il semble qu'on doit reconnoistre, il 
faudra bien quelle ait un subjectum. Car a et b allant l'un contre 
l'autre, j'avoue que tous les phenomenes arriveront tout le meme, quel 
que soit celuy dans le quel on posera le mouuement ou le repos; et 
quand il y auroit 1000 corps, je demeure d'accord que les phenomenes 
ne nous scauroient fournir (ny meme aux anges) une raison in- 
fallible pour determiner le sujet du mouuement ou de son degre; et 
que chacun pourroit estre concu a part comme estant en repos, et c'est 
aussi tout ce que je crois que vous demandes; mais vous ne nieres pas 
je crois que veritablement chacun a un certain degre de mouuement 
on, si vous voules de la force; non-obstant l'equivalence des Hypoth- 
eses. II est vray que j'en tire cette consequence qu'il y a dans la nature 
quelque autre chose que ce que la Geometrie y peut determiner. Et 

Sarmy plusieurs raisons dont je me sers pour prouuer qu'outre l'eten- 
ue et ses variations, qui sont des choses purement Geometriques, il 
faut reconnoistre quelque chose de superieur, qui est la force; celle-cy 
n'est pas des moindres. Monsieur Newton reconnoist l'equivalence des 

"Oeuvres completes de Christiaan Huygens (The Hague, 1905), vol. 10 
(correspondence, 1691-1695), p. 609. 



Hypothese en cas des mouuements rectilineaires; mais a l'egard des 
Circulaires, il croit que l'effort que font les corps circulans de s'eloig- 
ner du centre ou de l'axe de la circulation fait connoistre leur mouue- 
ment absolu. Mais fay des raisons qui me font croire que rien ne 
rompt la loy generate de l'Equivalence. II me semble cependant que 
vous meme, Monsieur, esties autres fois du sentiment de M. Neuton 
a l'egard du mouuement circulaire. 50 

As this letter shows, Leibniz finds himself in a precarious situ- 
ation, embracing the logical principle of kinematical relativity 
on the one hand, and the phenomenon of circular motion which 
demands the existence of absolute space, on the other. His "true 
motion," which differs from pure geometrical motion concep- 
tually, is obviously an attempt at a compromise. 

But Huygens is opposed to any compromise. Thus he writes in 
a letter dated August 24, 1694: 

Pour ce qui est du mouvement absolu et relatif, j'ay admire vostre 
memoire, de ce que vous vous estes souvenu, qu'autrefois j'estois du 
sentiment de Mr. Newton, en ci qui regard le mouvement circulaire. 
Ce qui est vray, et il n'y a que 2 ou 3 que j'ay trouve celuy qui est 
plus veritable, duquel il semble que vous n'estes pas eloigne non plus 
maintenant, si non ence que vous voulez, que lorsque plusieurs corps 
ont entre eux du mouvement relatif, ils aient chacun un certain degre 
de mouvement veritable, ou de force, enquoy je ne suis point de vostre 



Leibniz's reply of September 14, 1694, which brings this 
highly interesting exchange of ideas to an end, Huygens dying in 
1695, shows his great interest in Huygens' solution of the problem 
of circular motion. He agrees that no special privilege attaches 
to circular motion as compared with uniform translational motion 
and that all reference systems should be treated as equivalent. In 
Leibniz's opinion it is merely the principle of simplicity that leads 
to the exclusive ascription of certain motions to certain bodies. 
No doubt this was borrowed by Leibniz from the realm of 
astronomy, where for many years it played an important role in 
the controversy between the Copernicans and their opponents. 
Leibniz not only realized the inherent similarity, or near identity, 

"Ibid., p. 639. 

a Huygens, Oeuvres completes, vol. 10, p. 609. 



of the problem under discussion with the problem whether the 
Ptolemaic or the Copernican system is preferable, but he even 
composed a treatise, Tentamen de motuum coelestium causis, 62 
whose intention is to show how the arguments with regard to 
the mechanical relativity of motion suggest the equivalence of the 
two rival cosmological systems. It seems that he originally in- 
tended to publish this work in Rome during his visit to the Holy 
City. But caution prevailed and he submitted only a Promemo- 
ria, 53 whose theoretical part begins with the statement: "Ut vero 
res intelligatur exactius, sciendum est Motum ita sumi, ut involvat 
aliquid respectivum et non posse dari phaenomena ex quibus 
absolute determinetur motus aut quies; constitit enim motus in 
mutatione situs seu loci." 

We have mentioned Leibniz's last letter to Huygens which 
deals with the problem of absolute space. The letter is as follows: 

. . . Comme je vous disois un jour a Paris qu'on avoit de la peine 
a connoistre le veritable sujet du Mouuement vous me repondites que 
cela se pouuoit par le moyen du mouuement circulaire, cela m'arresta; 
et je m en souuins en lisant a peu pres la meme chose dans le liure 
de Mons. Newton; mais ce fut lorsque je croyois deja voir que le 
mouuement circulaire n'a point de privilege en cela. Et je voy que 
vous estes dans le meme sentiment. Je tiens done que toutes les hy- 
potheses sont equivalentes et lorsque j'assigne certains mouuements a 
certains corps, je n'en ay ny puis avoir d'autre raison, que la simplicity 
de l'Hypotheses croyant qu'on peut tenir la plus simple (tout con- 
sidere) pour la veritable. Ainsi n'en ayant point d'autre marque, je 
crois que la difference entre nous, n'est que dans la maniere de parler, 
que je tache d'accomoder a 1'usage commun, autant que je puis, salva 
veritate. Je ne suis pas meme fort elogne de la vostre, et dans un petit 
papier que je communiquay a Mr. Viviani, et qui me paroissoit propre 
a persuader Messieurs de Rome a permettre l'opinion de Copernic, je 
m en accommodois. Cependant si vous estes dans ces sentimens sur la 
realite du mouuement, je m'imagine que vous deuries en avoir sur la 
nature du corps de differens de ceux qu'on a coustume d'avoir. J'en ay 
d'assez singuliers et qui me paroissent demonstres. 84 

What is this singular conception of the nature of bodies on 

the basis of which Leibniz here claims to have found the solu- 

M Gerhardt, Leibnizens maihematischen Schriften, vol. 6, p. 144. 

"Ibid., p. 146. 

61 Huygens, Oeuvres completes, vol. 10, p. 681. 



tion of the problem of circular motion? We do not know. Leibniz 
does not explain bis solution, either here or elsewhere as far as 
is known. 

We are in a more fortunate position with regard to Huygens' 
solution of the same problem. How could Huygens maintain, in 
the light of certain dynamical effects such as the rise of centrif- 
ugal forces in circular motion, the kinematical principle of rel- 
ative motion, and at the same time dispense with the existence 
of absolute space and motion? 

In 1886 L. Lange drew attention to the possibility of finding 
Huygens' solution among his posthumous papers in the archives 
of Leyden. It was, however, only in 1920 that D. J. Korteweg 
and J. A. Schouten, having found in the Leyden archives four 
loose sheets written by Huygens, and all dealing with circular 
motion, published the solution. The fourth paper, in which 
Huygens summarized his solution, is quoted in part: 

Diu putavi in circulari motu haberi veri motus "criterion" ex vi 
centrifuga. Etenim ad ceteras quidem apparentias idem fit sive orbis 
aut rota quaepiam me juxta adstante circumrotetur, sive stante orbe 
illo ego per ambitum ejus circumferar, sed si lapis ad circumferentiam 
ponatur projicietur circumeunte orbe, ex quo vere tunc et nulla ad 
aliud relatione eum moveri et circumgyrari judicari existimabam. Sed 
is effectus hoc tantummodo declarat impressione in circumferentiam 
facta partes rotae motu relativo ad se invicem in partes diversas im- 
pulsas fuisse, ut motus circularis sit relativus partium in partes con- 
trarias concitatarum sed cohibitus propter vinculum aut connexum, an 
autem corpora duo inter se relative moveri possunt quorum eadem 
manet distantia? 

Ita sane dum distantiae incrementum inhibetur, contrarius vero 
motus relativus per circumferentiam viget. 

Plerique verum corporis motum statuunt cum ex loco certo ac fixo 
in spatio mundano transfertur, male nam cum infinite spatium undique 
extensum sit quae potest esse definitio aut immobilitas loci? 

Stellas aflixas, in Copernicano systemate, forsan revera quiescentes 
dicent. Sint sane inter se immotae sed omnes simul sumtae alterius 
corporis respectu quiescere dicentur, vel qua in re different a celerrime 
motis in partem aliquam ? nee quiescere igitur corpus nee moveri in 
infinito spatio dici potest, ideoque quies et motus tantum relativa 
sunt. 55 

80 D. J. Korteweg and J. A. Schouten, Jahresbericht der Deutschen Mathe- 
matiker-Vereinigung 29, 136 (1920). 


This may be translated: 


For a long time I had thought that rotational motion by means of 
centrifugal forces contains a criterion for true motion. Indeed, with 
regard to other phenomena it is the same whether a circular disk or 
a wheel rotates near me, or whether I circle round the stationary disk. 
However, if a stone is put on the circumference this will be projected 
only if the disk rotates, and therefore I formerly thought that circular 
motion is not relative to any other body. Still, this phenomenon 
showed only that the parts of the wheel, owing to the pressure acting 
on the circumference, are driven in relative motion among themselves 
in different directions. Rotational motion is therefore only a relative 
motion of the parts, which are driven to different sides, but held 
together by a rope or other connection. 

Now, is it possible to move two bodies relatively without changing 
their distance? This is indeed possible if an increase in their distance 
is prevented. An opposite relative motion exists on the circumference. 
Most people suppose that the true motion of a body consists in its 
being transferred from a certain fixed place in the universe. This is 
wrong; for if space is unlimited in all directions, what then is the 
definition of the immobility of a place? It will perhaps be said that 
the fixed stars in the system of Copernicus are really at rest; well, they 
may indeed be mutually immobile with respect to each other; but 
taken together, relative to what other body are they said to be at rest 
or in what respect are they to be distinguished from bodies moving 
very fast in a certain direction? It is therefore impossible to state that 
a body is at rest in infinite space, or that it moves therein; rest and 
motion are therefore only relative. 

Thus Huygens thought he had discovered that the dynamical 
effect of the appearance of centrifugal forces is merely an indica- 
tion of the relative motion of the different parts of the disk. Yet 
the relative motion of these parts can be transformed away by 
taking as a reference system just that system which has the same 
angular velocity (and the same origin) as the rotating disk. In 
this rotating coordinate system the parts of the disk are at rest. 
The dynamic effect, however, referred to this system, does not 
vanish: the "pressure" exerted by centrifugal forces has not been 
transformed away, as it should be were the centrifugal force but 
a dynamical effect of the relative motion of the particles. Huy- 
gens' explanation, therefore, certainly does not pass the test of 
modern scientific criticism. Nevertheless, it is a historical fact 




that Huygens, inspired by his sound scientific insight, was the 
first physicist who believed in the exclusive validity of a principle 
of kinematic as well as dynamic relativity, two hundred years 
before the rise of modern relativity. 

chapter 5 



Nothing that Leibniz and Huygens had to say in criticism 
of Newton's concept of absolute space could prevent its accept- 
ance. The letters that passed between Leibniz and Clarke, though 
widely read, were studied and discussed chiefly for their theo- 
logical implications. With the gradual acceptance of the Newto- 
nian system, and as the rival Cartesian theories fell out of grace, 
Newton's concept of absolute space became a fundamental pre- 
requisite of physical investigation. 

In this respect the words of John Keill, one of the early 
advocates of Newtonian physics in the University of Oxford, are 



typical. In his second lecture, which he delivered at Oxford in 
1700, he said: 

We conceive Space to be that, wherein all Bodies are placed, or, 
to speak with the Schools, have their Ubi; that it is altogether pene- 
trable, receiving all Bodies into itself, and refusing Ingress to nothing 
whatsoever; that it is immovably fixed, capable of no Action, Form 
or Quality; whose Parts it is impossible to separate from each other, 
by any Force however great; but the Space itself remaining immov- 
able, receives the Successions of things in motion, determines the 
Velocities of their Motions, and measures the Distances of the things 
themselves. 1 

Not only this sober, factual and scientific aspect of Newton's 
conception of absolute space gained ground; the divinization of 
space was acclaimed with no less enthusiasm by the early eight- 
eenth century, as it conformed so very well with the general 
outlook of the time for which science has become identical with 
the study of the works of God. "Nature was rescued from Satan 
and restored to God." 2 What wonder, therefore, that Joseph 
Addison praised Newton's religious interpretation with the words: 
"The noblest and most exalted way of considering this infinite 
space is that of Sir Isaac Newton, who calls it the sensorium 
of the Godhead." 3 The words of the Psalmist, "Coeli enarrant 
gloriam Dei," could now be interpreted in a new sense. 

The spacious firmament on high 
With all the blue aethereal sky 
And spangled heavens, a shining frame, 
Their great Original proclaim. 

The spread of such ideas was not confined to Europe. In the new 
world Jonathan Edwards, metaphysician and divine, who still 
one hundred years after his death was called "the greatest meta- 
physician America has yet produced," 4 expressed ideas about 

1 John Keill, An introduction to natural philosophy (London, 1745), p. 15. 

'Basil Willey, The eighteenth century background (Chatto and Windus, 
London, 1949), p. 4. 

'The Spectator, No. 565 (1714). 

* Georges Lyon, L'Iddalisme en Angleterre au XVIIIe siecle (Paris, 1888), 
p. 406. 



space which in their theological implications were most congenial 
to those expressed by Isaac Newton. 

In England, in particular, Newton's inclusion of religious ideas 
in his system of physics was hailed as an outstanding achieve- 
ment in natural philosophy. By analyzing the fundamental con- 
cepts of science, it was hoped, new material could be brought to 
light of avail in proving the existence of God. These proofs, based 
on the infinity and absoluteness of space, were to replace the 
traditional scholastic demonstrations which began to be consid- 
ered as logically deficient. Such ideas were expressed by nu- 
merous authors during the first half of the eighteenth century, 
of whom we mention only Jacob Raphson, 5 John Jackson, 6 
Joseph Clarke, 7 and Isaac Watts. 8 

The notion of absolute space triumphed on all fronts. Not 

only that, but, during the eighteenth century, attempts were 

made to demonstrate the logical necessity of the concept. Indeed, 

no less a man then Leonhard Euler grappled with the problem 

for more than thirty years. In his Mechanica sive motus scientiae 

analytice exposita Euler develops his mechanics on Newtonian 

lines and introduces the concept of absolute space and absolute 

motion in the spirit of the Principia. So his second definition 

reads: "Locus est pars spatii immensi seu infiniti in quo universus 

mundus consistit. Vocari hoc sensu acceptus locus solet absolutus, 

ut distinguatur a loco relativo, cuius mox fiet mentio." 9 To Euler, 

however, in his early work the question of the real existence of 

absolute space is a matter of indifference. Whether absolute 

space exists or not, it is only necessary to imagine such a space for 

the determination of absolute motion or absolute rest. But Euler 

" Jacob Raphson, De spatio reali seu ente infinito conamen maihematico- 
metaphysicum (London, 1702). 

• John Jackson, The existence and unity of god, proved from his nature 
and attributes (London, 1734). 

''Examination of Dr. Clarke's notion of space (Cambridge, 1734). 

'Isaac Watts, Philosophical essays on various subjects (London, ed. 2, 
1736), "Essay I: A fair enquiry and debate concerning space whether it be 
something or nothing, God or a creature." 

* L. Euler, Mechanica sive motus scientiae analytice exposita ( St. Peters- 
burg, 1736), p. 2. 



changed his mind and in his "Reflexions sur l'espace et le 
. temps" 10 emphasized the necessary existence of absolute space; 
for he had come to the conclusion that some real existing sub- 
stratum is indispensable to the determination of motion. Since 
this substratum appears not to exist in the casual surrounding 
material, it must be space itself that exists in this capacity. "On 
en devroit plut6t conclure, que tant l'espace absolu, que le 
temps, tels que les Mathematiciens se les figurent, etoient des 
choses replies, qui subsistent memes hors de notre imagination." u 
Euler's demonstration of the reality of absolute space on the 
basis of the law of inertia appears finally in his Theoria motus 
corporum solidorum seu rigidorum, 12 though the Mechanica al- 
ready insists that the laws of motion presuppose the existence of 
absolute space. "Si hac significatione expositae voces accipiantur, 
vocari solent motus absolutus, quiesque absoluta. Atque hae sunt 
verae et genuinae istarum vocum definitiones, sunt enim accomo- 
datae ad leges motus, quae in sequentibus explicabuntur." 1S Of 
course, Euler was not the first among Newton's successors who 
emphasized the intrinsic importance of the concept of absolute 
space for the formulation of the law of inertia. Maclaurin, in his 
Account of Sir Isaac Newton's philosophical discoveries, had 
stated explicidy: "This perseverence of a body in a state of rest 
or uniform motion, can only take place with relation to absolute 
space, and can only be intelligible by admitting it." " Euler did 
not confine himself to merely stating such an implication. If it 
were possible to demonstrate the logical necessity of the law of 
inertia, then, according to Euler, the logical necessity of absolute 
space would follow from it by implication. Thus, after he for- 
mulates the law of inertia, Euler attempts to give an a priori 
vindication of its necessity. The law is formulated in Axioma 2 

"Histoire de I'Academie Royale des Sciences et des Belles-lettres, 1748 
(Berlin, 1750), vol. 4, p. 324. 

u Reference 10. 

12 Rostock and Greifswald, 1765. 

M Euler, Mechanica, p. 2. 

" Colin Maclaurin, Account of Sir Isaac Newton's philosophical discoveries 
(London, 1748), book 2, chap. 1, sec. 9. 



as follows: "Corpus, quod absolute quiescit, si nulli externae 
actioni fuerit subjectum, perpetuo in quiete perseverabit." 15 In 
the "Explicatio" immediately following we read, "cum enim in eo 
(i.e. elemento corporis) nulla insit ratio, cur in unam potius 
directionem moveri incipiat, quam in omnes alias, atque extrin- 
secus omnis causa motus adimatur, secundum nullam directionem 
motum concipere poterit. Nititur igitur quidem haec Veritas 
principio sufficientis rationis." 

It was a fairly common assumption of the time that by means 
of the principle of sufficient reason the law of inertia, and hence 
indirectly the existence of absolute space, could be demonstrated. 
Much the same reasoning occurs in D'Alembert's Traite" de dy- 
namique and also especially in Kant's Metaphysical foundations 
of natural science. But before this Kant had entertained a dif- 
ferent opinion. In his youth Kant showed great interest in the 
natural sciences; in fact, from the time of his first work in 1747 16 
the problem of space and motion occupied him constantly. In 
his Principiorum primorum cognitionis metaphysicae nova diluci- 
datio of the year 1755 Kant attempted to reconcile Newton with 
Leibniz. Agreeing with Leibniz's relational point of view, Kant 
sees in spatial relations not reflections of simply qualitative data 
given within the order of coexisting matter but rather mutual 
effects and interactions among bodies; since causal interdepend- 
ence is not given with matter as such, but has been added and 
imparted by divine creation, space is an independent existent of 
absolute reality in the Newtonian sense. A similar compromise 
between Leibniz's metaphysics and Newton's physics is aimed at 
in the Monadologia physica of 1756. The metaphysician, accord- 
ing to Kant, asserts that all substantial reality is constituted of 
fundamental indivisible units or monads; the mathematician, 
on the other hand, claims that space is infinitely divisible; and 
the physicist finally applies mathematical space to metaphysical 
15 Euler, Theoria motus, p. 32. 

"Kant, "Thoughts on the true estimation of living forces," in Tohn 
Handyside, trans., Kant's inaugural dissertation and early writings on svace 
(Chicago, 1929). b r 



matter. This state of affairs is consistent only if space is not a 
substance but a phenomenon of relations among substances and 
if substance is but a center of action effecting other substances, 
and effected by them, through the mutual operation of forces. 
A simple substance "occupies" a larger or smaller space, not 
because it fills it up with a larger or smaller number of material 
parts, but because it exerts stronger or weaker forces of repulsion 
to prevent the approach of adjacent monads. Spatial magnitude 
is therefore only a measure of the intensity of acting forces 
exerted by the substance. In his Neuer Lehrbegriff der Bewegung 
und Ruhe Kant stresses again the pure relative character of space 
and writes: 

Now I begin to see that I lack something in the expression of motion 
and rest. I should never say, a body is at rest, without adding with 
regard to what it is at rest, and never say that it moves without at the 
same time naming the objects with regard to which it changes its rela- 
tion. If I wish to imagine also a mathematical space free from all 
creatures as a receptacle of bodies, this would still not help me. For 
by what should I distinguish the parts of the same and the different 
places, which are occupied by nothing corporeal? 1T 

Yet five years later, 18 apparently under the influence of Euler, 19 

Kant abandons this point of view and declares himself in favor 

of the Newtonian concepts of absolute space and absolute time. 

In his essay "On the first grounds of the Distinction of Regions in 

Space" Kant formulates his program as follows: "My aim in this 

treatise is to investigate whether there is not to be found in 

the judgments of extension, such as are contained in geometry, 

an evident proof that space has a reality of its own, independent 

of the existence of all matter, and indeed as the first ground of 

the possibility of the compositeness of matter." M Kant thought 

"Kant, Gesammelte Werke (Akademie Ausgabe; Berlin, 1905), vol. 2, 

P- 13- 

"Kant, "Versuch den Begriff der negativen Grosse in die Weltweisheit 
einzufiihren" (1763) in Gesammelte Werke (Akademie Ausgabe), vol. 2, 
p. 165. 

w On Euler's influence on Kant, see H. E. Timerding, "Kant und Gauss," 
Kantstudien 28 (1923). 

"Kant, "Von dem ersten Grande des Unterschiedes der Gegenden im 
Raume" (1769) in Gesammelte Werke (Akademie Ausgabe), vol. 2, p. 375. 



that here he had found an incontestable proof of the existence 
and reality of absolute space, independent of the existence of 
matter. As the first ground for the possibility of the composite- 
ness or disposition of matter, space is endowed with a reality of 
its own. Kant bases his proof on the distinction between left and 
right. It is observed, he says, that the intrinsic relations among 
the individual parts of our left hand with regard to each other are 
the same as in our right hand; and yet evidently a fundamental 
distinction makes it impossible to substitute one hand for the 
other. Now, if this fundamental difference cannot be explained as 
being merely the appearance of different relation in the order or 
disposition of the parts with respect to each other, it can be 
accounted for only by the assumption of a different disposition 
with regard to absolute space. Thus, absolute space must be 
introduced as a fundamental metaphysical notion necessary for 
the explanation of this phenomenon. 

H. Weyl, who shows that mathematically the root of this 
distinction is of a purely combinatorial character ( a permutation 
of given linearly independent vectors determines the "sense" of 
rotation, as for example in left- or right-handed coordinated sys- 
tems), says of Kant's argumentation: 

Kant finds the clue to the riddle of left and right in transcendental 
idealism. The mathematician sees behind it the combinatorial fact of 
the distinction of even and odd permutations. The clash between the 
philosopher's and the mathematician's quest for the roots of the 
phenomena which the world presents to us can hardly be illustrated 
more strikingly. 21 

The left side of a straight line can be interchanged with its 
right side by rotating the line in a plane, the clockwise direction 
on a surface can be interchanged with an anticlockwise direc- 
tion by moving the surface in three-dimensional space (turning 
over), a left-hand screw with a right-hand screw — or the left 
hand with the right hand — by "moving" the object in four- 
dimensional space. It is clear, therefore, that mathematically no 

"H. Weyl, Philosophy of mathematics and natural science (Princeton 
University Press, Princeton, 1949), p. 84. 



essential mark distinguishes one sense from the other. Likewise, 
all natural laws are undoubtedly invariant with respect to an 
interchange of right and left, although certain asymmetries, in 
some cases still of an unexplained nature, hold in chemical and 
biological phenomena. Weyl mentions the following example: 

That homo sapiens contains a screw turning the same way in all 
individuals is proved in a rather horrid fashion by the fact that man 
contracts a metabolic disease called phenylketonuria leading to amen- 
tia when a certain quantity of levo-phenylalanine is added to his food, 
whilst the dextro form has no such disastrous effect. 22 

Phenomena of this kind have certainly no deeper significance 
and nobody today would use them as a proof for the existence 
of absolute space. 

But back to Kant. It is in his view only immediate intuition 
that distinguishes between left and right, a difference that cannot 
be formulated conceptually. Furthermore, it is immediate intui- 
tion that forms our general conceptions in geometry and makes 
their statements evident. In this intuition rests the proof for the 
reality of absolute space. "In den anschauenden Urteilen, derglei- 
chen die Messkunst enthaelt, ist der Beweis zu finden, dass der 
absolute Raum unabhangig von dem Dasein aller Materie und 
selbst als der erste Grund ihrer Zusammensetzung eine eigene 
Realitat habe." M The idea that intuition lies at the basis of our 
geometric cognition brings about a radical change in Kant's atti- 
tude toward these questions. The problem of space now appears 
to Kant in a new light. It ceases to be a problem of physics and 
becomes an integral part of transcendental philosophy. To Kant 
from now on space is a condition of the very possibility of experi- 
ence. In the inaugural dissertation "De mundi sensibilis atque 
intelligibilis forma et principiis," 2i the concepts of absolute space 
and absolute time are considered to be merely conceptual fic- 

82 Ibid., p. 208; see also H. Weyl, Symmetry (Princeton University Press, 
Princeton, 1952), pp. 16-38. 

" Kant, Gesammette Werke (Akademie Ausgabe), vol. 16. 

"Kant, "Dissertation on the form and principles of the sensible and 
intelligible world" (1770), in Handyside, reference 16, p. 33. 



tions, a mental scheme of constructed relations of coexistence and 
sequence among sense particulars. Not itself arising out of sensa- 
tions, the concept of space is a pure intuition, neither objective 
nor real, but subjective and ideal. 

Kant's critical theory of space, like his philosophy in general, 
was greatly influenced by the English empiricists Locke, Berke- 
ley, and Hume, and their analytical investigations into the 
formation of ideas. For Locke, extension, figure, size, and motion, 
in contrast to color, sound, and taste, were primary qualities, 
inherent in the object and independent of the perceiving subject. 
Berkeley, in an attempt to explain our conception of space, in 
his New theory of vision 25 reduced visual space to visual signs of 
tangible space, arguing that visual ideas are within the mind, 
whereas tangible space, in his view, needs no explanation. Dis- 
tances, sizes, or figures, in consequence, are not "seen" or per- 
ceived, but inferred by the mind, experience having shown that 
certain visual sensations are related to certain tactile relations. 
In his other works, however, Berkeley asserts that tangible ideas 
(objects) are also within the mind, but he does not reexamine 
his exposition of the perception of visual space in the light of his 
new position. Notwithstanding this inconsequence, extension, 
size, and figure appear in Berkeley's philosophy as secondary 

In his treatise Concerning the principles of human knowledge 
Berkeley describes how according to his empiristic view the con- 
cept of space is formed by the perception of extension, the notion 
of space being but an abstract idea of extension. For like other 
general ideas it is formed in the human mind by abstraction from 
sense perceptions relating to bodies. Newton's notion of absolute 
space, which contains all bodies and is retained if all bodies are 
thought away, is in Berkeley's view a false hypostatization of an 
abstraction. Only particular spaces, corresponding to extensions 
perceived by our senses through their colors, figures, and tactile 

25 Berkeley, A new theory of vision and other writings (Dent, London, 
1938), p. 37- 




qualities, are to be admitted. The notion of empty space, how- 
ever, is a mere verbal expression of a state of empirical facts. 

When I excite a motion in some part of my body, if it be free or 
without resistance, I say there is space: but if I find a resistance, then 
I say there is body: and in proportion as the resistance to motion is 
lesser or greater, I say the space is more or less pure. So that when I 
speak of pure or empty space, it is not to be supposed, that the word 
space stands for an idea distinct from, or conceivable without body 
and motion. Though indeed we are apt to think every noun substan- 
tive stands for a distinct idea, that may be separated from all others: 
which hath occasioned infinite mistakes. 29 

The adoption of such an extreme subjective idealism is the 
final conclusion of an empiristic approach, according to which all 
our knowledge, furnished by experience, goes back to elementary 
sense data subjected to the reflection of the mind. 

Kant's position in this context is best characterized by his own 
words: "Although all knowledge begins with experience, it does 
not necessarily all spring from experience." The object of sensa- 
tion is not identical with the object of thought. This applies in 
particular to the conception of space, which, according to Kant, 
is a form of intuition, instrumental in the process of cognition as 
an ideal organizer of the contents of sensations. 

Kant's claim of the transcendental ideality of space (and 
time) is expounded both in his Prolegomena and in the Critique 
of jmre reason, where its discussion, under the name of Transcen- 
dental Aesthetic, plays a fundamental role in his epistemology. 
It is well known how Kant tries to demonstrate that the imme- 
diate object of perception is rooted partly in external things and 
partly in the apparatus of our own perception. The first compo- 
nent, due to the "thing-in-itself," is called the "sensation," and 
the second component, the "form" of the phenomenon. It is this 
second component that brings order into the amorphous manifold 
of our sensations; it is an a priori element of our perception, 
antecedent to all experience. It is universal, since it does not 

"Berkeley, Principles of human knowledge in A new theory of vision, 
p. 173. 



depend on the particular data of our sensation. As a pure form 
of sensibility Kant calls this component "pure intuition" (reine 
Anschauung) . There are two pure intuitions: space and time. 

Four metaphysical arguments based on the nature of space 
and time, and one transcendental argument derived from the 
special character of Euclidean geometry, are set forth to prove 
the contention. 

The metaphysical exposition begins with the classical words: 
"Space is not an empirical notion which has been derived from 
external experience," 27 because, according to Kant, any possible 
sensation referred to something external presupposes the percep- 
tion of space. The second argument stresses the fact that space 
is a necessary perception a priori, underlying all external per- 
ception, because we cannot imagine that there is no space, 
although we can well imagine that there are no objects found 
in space. Thirdly, space is no discursive or general notion (as 
"animal" or "table"), for, speaking of spaces, we mean merely 
parts of one and the same space. Finally, space is conceived as 
an infinite magnitude. "Now a notion must be conceived, indeed, 
as common to an infinite number of different possible individuals," 
that is to say, the notion has to be contained in an infinite number 
of particulars subsumed by it. This relation does not hold in the 
case of the concept of space. Consequently, space is an intuition, 
a priori, and not a notion. 

The transcendental argument is based on Kant's characteriza- 
tion of geometry as synthetic and yet a priori; geometric proposi- 
tions are, in his view, apodictic, that is, they bring with them 
their own necessity, but they are not judgments of experience. 
Geometry can be known a priori without being a mere tautology, 
only because it lies at the basis of our perception. 

Kant's metaphysical exposition tries to demonstrate that space 
and time are conditions under which sense perception operates. 
The a priori ideas of space and time are not images which cor- 

27 Kant, The critique of pure reason (trans, by J. H. Stirling; Edinburgh, 
1881), p. 141. 



respond to external objects. In fact, there is no object in the ex- 
ternal world called space. It is not an object of perception, it 
is a mode of perceiving objects. Nevertheless, Kant explicitly 
opposes the relational theory of space and endows the form of 
perception with existence independent of the particular bodies 
contained in it. In the Critique of pure reason he says: 

Space does not represent any property of things in themselves, nor 
does it represent them in their relation to one another. That is to say, 
space does not represent any determination that attaches to the objects 
themselves, and which remains even when abstraction has been made 
of all the subjective conditions of intuition. 28 

It is easy to understand that Kant's doctrine of the transcen- 
dental ideality of space, hailed in its days as one of the greatest 
achievements in contemporary philosophy, was greatly influential 
on the course of idealistic philosophy and had its repercussions 
also on psychology. Indeed, Lotze's theory of "local signs," Bren- 
tano's psychological researches, and Stumpfs investigation into 
the origin of space perception show clearly how Kant's epistemo- 
logical inquiries were bound to be diverted into the field of the 
psychology of senses. 

Since a discussion of the psychological origin of our space 
conception is outside the scope of our topic, we need not enter 
here into the details of A. Bain's, J. Mueller's, or Helmholtz's 
important contributions toward a clarification of this complicated 
problem, nor are we here concerned with the history of the 
nativistic or of the empiristic theories on the formation of space 
perception. It is, however, important for us to note that Helm- 
holtz, one of the chief proponents of the empiristic school, has 
shown that the two parts of Kant's doctrine, the metaphysical 
and the transcendental expositions, are not so closely connected 
as was originally assumed. Although Helmholtz 29 does not reject 

" N. K. Smith, trans., Immanuel Kant's Critique of pure reason ( Mac- 
millan, London, 1950), p. 71. 

29 H. von Helmholtz, "Ueber die Tatsachen welche der Geometrie zu- 
grunde liegen," Gottinger gelehrte Nachrichten (1868), pp. 193-221. See 
also his "Ueber den Ursprung und die Bedeutung der geometrischen 
Axiome," Vortriige und Reaen (Brunswick, ed. 5, 1903), vol. 2, pp. 1-31. 



in principle the metaphysical argumentation, he strongly objects 
to the assumption of the a priori nature of Euclidean geometry. 
Space, as a pure form of intuition, leads, according to Helmholtz, 
to one single conclusion: that all objects of the external world 
must necessarily be endowed with spatial extension. The geo- 
metric character of this extension, however, is in his view purely 
a matter of experience. Helmholtz's opinion, imbued with the 
recognition of the validity of non-Euclidean geometry, may, on 
the whole, still be taken today as representative of the attitude 
of physicists toward Kant's doctrine of space. 30 Modern psychol- 
ogy denies the allegation that a determinate metric space must 
be assumed to exist as an organizing element of cognition. It 
should also be recalled that modern logic no longer respects 
Kant's double dichotomy of judgments into a priori and a pos- 
teriori, analytic and synthetic, as a fundamental classification for 
epistemological research. It is certainly a very conservative state- 
ment to say that the line between a priori and a posteriori has 
been drawn at different places at different times. 

It is interesting to note how little the actual progress of the 
science of mechanics was affected by general considerations con- 
cerning the nature of absolute space. Among the great French 
writers on mechanics, Lagrange, Laplace, and Poisson, none of 
them was much interested in the problem of absolute space. They 
all accepted the idea as a working hypothesis without worrying 
about its theoretical justification. In reading the introductions to 
their works, one discovers that they felt that science could very 
well dispense with general considerations about absolute space. 
It is interesting to note that the Encyclopedie of Diderot and 
d'Alembert expresses much the same view. In the fifth volume, 
under the heading "Espace," we read: 

Cet article est tire des papiers de M. Formey, qui l'a compose en 
partie sur le recueil des Lettre de Clarke, Leibnitz, Newton, Amsters. 
1740. & sur les inst. de Physique de madame du Chatelet. Nous ne 
prendrons point de parti sur la question de l'espace; on peut voir, 

w Cf. Viktor Henry, "Das erkenntnistheoretische Raumproblem," Kant- 
studien, Erganzungsheft No. 34 (1915). 



partout ce qui a ete dit au mot Elemens des Sciences, combien cette 
question obscure est inutile a la Geometrie fit a la Physique. 31 

It may even be claimed that this absence, so far from being 
a hindrance to mechanics in the eighteenth and early nineteenth 
century, in a certain degree facilitated the development of this 
science. In England, by the middle of the nineteenth century, it 
became clear that the concept of absolute space was useless in 
physical practice. In that country the great success of Newtonian 
physics led to the paradoxical situation of the adherence to the 
concepts of absolute time and absolute space, on the one hand, 
and their absence from practical physics, on the other. Clerk 
Maxwell's remarks on absolute space in his Matter and motion 
are characteristic: 

Absolute space is conceived as remaining always similar to itself and 
immovable. The arrangement of the parts of space can no more be 
altered than the order of the portions of time. To conceive them to 
move from their places is to conceive a place to move from itself. But 
as there is nothing to distinguish one portion of time from another 
except the different events which occur in them, so there is nothing to 
distinguish one part of space from another except its relation to the 
place of material bodies. We cannot describe the time of an event 
except by reference to some other event, or the place of a body except 
by reference to some other body. All our knowledge, both of time and 
space, is essentially relative. 32 

In 1885 an important attempt to find a way out of the para- 
doxical situation (that is, the adherence to the concept of abso- 
lute space on the one hand and its absence from practical physics 
on the other) was made by Ludwig Lange. 33 Lange thought that 
he had found the way to eliminate the concept of absolute space 
from the conceptual foundation of physics. In his view, the essen- 
tial (today he would say the operational) content of the law of 
inertia, and with it of the whole of mechanics, retains its full 

" Diderot and d'Alembert, Encyclope'die, ou dictionnaire raisonnS des 
sciences, des arts et des mitiers, vol. 5 (1755), p. 949. 

" J. C. Maxwell, Matter and motion, reprinted, with notes and appendi- 
ces by Sir Joseph Larmor (Dover, New York, n.d.), p. 12. 

38 L. Lange, ' Ueber die wissenschaftliche Fassung des Galileischen Behar- 
rungsgesetzes," Ber. kgl. Ges. Wiss., Math.-phys. Kl. (1885), pp. 333-35 1 - 



physical meaning if the somewhat "ghostly" idea of an absolute 
space is replaced by the concept of an "inertial system." Let 
there be given a mass-point A whose motion is arbitrary (even 
curvilinear). Then it is always possible to move a coordinate 
system S in such a manner that A moves relative to S along a 
straight line a. If in addition a second point B and a third point 
C are given with arbitrary motions, the coordinate system can 
still be moved in such a way that all three mass-points, relative 
to S, move along straight lines a, b, c. Now, three is the maximum 
number of mass-points for which in general the construction of 
a coordinate system S is possible in which the points move along 
straight lines. If we now assume that the three points A, B, C, 
being projected from the same origin, are left to themselves (that 
is, are not subjected to any forces), the corresponding coordinate 
system S; relative to which the three points describe three dif- 
ferent straight lines, is defined as an "inertial system." The physi- 
cal contents of the law of inertia, according to Lange, is equiv- 
alent to the contention that any fourth mass-point, left to itself, 
also moves along a straight line relative to S. In short, an "inertial 
system" is a coordinate system in respect to which Newton's law 
of inertia holds. Lange's suggestion of eliminating the idea of 
absolute space by introducing the concept of an "inertial system" 
was hailed by his contemporaries as an outstanding contribution 
to the foundations of physics. Seeliger 34 thought that it was pos- 
sible to compare Lange's inertial system with the empirical co- 
ordinate system used in astronomy and stated that the relative 
motion between these two systems is less than 2 seconds of arc 
within the span of a century. 

Toward the end of the nineteenth century it became obvious 
that absolute space evaded all means of experimental detection. 
Mach showed that the assumption of absolute space for the 
explanation of centrifugal forces in rotational motion was un- 
necessary. In the Science of mechanics he writes: 

M H. Seeliger, "Ueber die sogenannte absolute Bewegung," Sitzher. 
Miinchener Akad. Wiss. (1906), p. 85. 



Newton's experiment with the rotating vessel of water simply in- 
forms us, that the relative rotation of the water with respect to the 
sides of the vessel produces no noticeable centrifugal forces, but that 
such forces are produced by its relative rotation with respect to the 
mass of the earth and the other celestial bodies. No one is competent 
to say how the experiment would turn out if the sides of the vessel 
increased in thickness and mass till they were ultimately several 
leagues thick. 35 

Mach's modification of the traditional interpretation of New- 
ton's pail experiment and his objection to accepting the experi- 
ment as a proof of the existence of absolute space are the result 
of his conviction that all metaphysical concepts have to be elimi- 
nated from science. In the foreword to the first edition of his Die 
Mechanik in ihrer Entwicklung 3 * he writes: "Vorliegende Schrift 
ist kein Lehrbuch zur Einiibung der Satze der Mechanik. Ihre 
Tendenz ist vielmehr eine aufklarende oder, urn es noch deut- 
licher zu sagen, eine antimetaphysische." The very idea of an 
absolute space, that is, of an agent that acts itself but cannot 
be acted upon, is, in his view, contrary to scientific reasoning. 
Space as an active cause, both for translational inertia in rec- 
tilinear motion and for centrifugal forces in rotational motion, has 
to be eliminated from the system of mechanics. 

With reference to Newton's theory of space, Mach is ready to 
accept only the idea of relative spaces (see p. 98), which approxi- 
mate inertial systems. In the concluding passage of his foreword 
to the seventh edition (1912) of his Mechanik Mach says: 
"Beziiglich der Begriffsungetiime des absoluten Raumes und der 
absoluten Zeit konnte ich nichts zuriicknehmen. Ich habe hier 
nur deutlicher als vorher gezeigt, dass Newton zwar manches 
uber diese Dinge redet, aber durchaus keine emste Anwendung 
von denselben gemacht hat. Sein Coroll. V. (Principia, 1687, p. 
19) enthalt das einzig praktisch brauchbare (wahrscheinlich 
angenahrte) Inertialsystem." ST 

" E. Mach, The science of mechanics (trans, by T. J. McCormack; Chi- 
cago, 1902), p. 232. 
"E. Mach, Die Mechanik in ihrer Entwicklung (Leipzig, 1883). 
"Ibid. (Leipzig, ed. 8, corresponding with ed. 7, 1921), p. x. 



The elimination of what Mach calls "the conceptual mon- 
strosity of absolute space" (das Begriffsungetum des absoluten 
Raumes) is achieved, in his view, by relating the unaccelerated 
motion of a mass-particle not to space as such, but to the center 
of all masses in the universe. The assumption of an intrinsic func- 
tional dependence between inertia and a large-scale distribution 
of matter closes for him the series of mechanical interactions 
without resorting to a metaphysical agent. "Ueber den absoluten 
Raum und die absolute Bewegung kann niemand etwas aussagen, 
sie sind blosse Gedankendinge, die in der Erfahrung nicht auf- 
gezeigt werden konnen." The very fact that absolute space and 
absolute motion are physically imperceptible, even if their ob- 
jective existence may be admitted philosophically, characterizes 
them — in Mach's terminology — as "metaphysical" and demands 
their elimination from exact science. In the fourth edition of Die 
Mechanik Mach summarizes his ideas concerning space in a very 
clear statement (which curiously is omitted in the later editions) : 

Fur mich gibt es iiberhaupt nur eine relative Bewegung und ich 
kann darin einen Unterschied zwischen Rotation und Translation nicht 
machen. Dreht sich ein Korper relativ gegen den Fixsternhimmel, so 
treten Fliehkrafte auf, dreht er sich relativ gegen einen anderen 
Korper, nicht aber gegen den Fixsternhimmel, so fehlen die Flieh- 
krafte. Ich habe nichts dagegen, wenn man die erstere Rotation eine 
absolute nennt, wenn man nur nicht vergisst, dass dies nichts anderes 
heisst, als eine relative Drehung gegen den Fixsternhimmel. Konnen 
wir vielleicht das Wasserglas Newtons festhalten, den Fixsternhimmel 
dagegen rotieren, und das Fehlen der Fliehkrafte nun nachweisen? 
Der Versuch ist nicht ausfiihrbar, der Gedanke iiberhaupt sinnlos, da 
beide Falle sinnlich voneinander nicht so unterscheiden sind. Ich halte 
demnach beide Falle fur denselben Fall und die Newtonsche Unter- 
scheidung fur eine Illusion. 

It is obvious that these words may be considered as the earliest 
proclamation of the principle of general relativity and, in fact, 
they have been interpreted as such. 38 

Mechanics, it seemed, had to give up the notion of absolute 
space. Under the stress of these circumstances it was suggested 

88 For example, by W. Wien, Die Relativitatstheorie (Leipzig, 1921), 
P- 31. 



by Drude and Abraham, to mention only these names, that the 
ether, the carrier of electromagnetic waves, should be identified 
with absolute space. If the ether as an absolute reference system 
could be demonstrated, the notion of absolute space could be 
saved. Indeed, one of the most important experiments to this end, 
the Michelson-Morley experiment, was in 1904 interpreted by 
Lorentz in this sense. His interpretation fulfilled all physical re- 
quirements. As is well known, according to Lorentz every body 
moving with reference to the motionless ether or absolute space 
undergoes a certain contraction in the dimension parallel to the 
motion. However, the Michelson-Morley experiment served as the 
starting point for the development of the theory of relativity and 
was interpreted by Einstein on entirely different lines, adverse to 
the acceptance of absolute space. It was understood that both 
interpretations give a complete explanation of all observations 
known at the beginning of the twentieth century. An experimen- 
tum crucis could not decide between these two theories. As Laue 
explains the situation in 1911: 

A really experimental decision between the theory of Lorentz and 
the theory of relativity is indeed not to be gained, and that the former, 
in spite of this, has receded into the background, is chiefly due to the 
fact, that close as it comes to the theory of relativity, it still lacks 
the great simple universal principle, the possession of which lends the 
theory of relativity ... an imposing appearance. 39 

Epistemologically, Lorentz's theory shows its unsatisfactory 
character by the fact that it ascribes to the ether or absolute 
space certain definite effects which by their very assumed exist- 
ence preclude any possible observation of the ether. Similarly, 
all other experiments to identify the ether as a privileged system 
of reference had to go by the board. Physics, and not only me- 
chanics, was ready to abandon the concept of absolute space 
altogether. Poincare's words, "Whoever speaks of absolute space 
uses a word devoid of meaning," 40 became an accepted truth. 

" Quoted from E. Cassirer, Einstein's theory of relativity considered from 
the epistemological standpoint (Chicago, 1923), p. 376. 

"H. Poincare, Science and method (trans, by F. Maitland; London, 
1914), p. 93. 



Yet space, no longer an absolute entity, retained one property 
in common with such an entity: it was Euclidean in nature. 
Even in the theory of special relativity, the space-time continuum 
by which every observer identifies the events in his physical 
world was held to be Euclidean, or pseudo-Euclidean, if the 
Minkowski representation is adopted. The question whether the 
space of experience was Euclidean or not was already a subject 
of discussion before the rise of general relativity. To Newton 
and his immediate successors, with no alternative before them, 
absolute space was naturally thought to be Euclidean. The dis- 
covery of non-Euclidean geometry led to the elimination of this 
last traditional characteristic of space, and modern physics came 
finally to base its conception of space upon the Riemann notion 
of an n-dimensional manifold. 

It is an exciting story, and begins with Euclid and his fifth 
postulate, later simply called the "parallel axiom," which reads 
as follows: 

That, 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 are the 
angles less than the two right angles. 41 

Or, in an equivalent formulation, it states that in a given plane 

through a given point not more than one parallel to a given line 

exists. 42 That this postulate is not needed for the demonstration 

of the first z8 theorems of the Elements was noted early. In 

antiquity it was thought possible to prove the postulate on the 

basis of the other postulates. From Ptolemy and Proclus to Na- 

siraddin-at-Tusi, the Persian editor of the Elements, and John 

Wallis, down to Lambert and Legendre, all attempts at such a 

proof failed. Of all the agelong attempts to solve the problem, 

the most remarkable is that of Girolomo Saccheri. In his Euclides 

ab omne naevo vindicatus 43 Saccheri tries to show that a con- 

a Euclid, The elements (trans, by Sir Thomas Heath; St. John's College 
Press, Annapolis, 1947), vol. 1, p. 202. 

"The existence of at least one parallel can be proved by the other 

"Milan, 1733- 

i 4 4 


tradiction results if the postulate in question is replaced by 
another. Today we know that no contradiction with the other 
postulates would have arisen if Saccheri had not used unawares 
an assumption that is in fact equivalent to the fifth postulate. 
His book had a great influence on subsequent investigations into 
the nature of the postulate. The problem attracted many first- 
class mathematicians. One of the greatest of them, Carl Friedrich 
Gauss, seems to have recognized the logical possibility of a non- 
Euclidean geometry even before Lobachevski and Bolyai came 
out with their sensational discoveries. By the end of the first half 
of the nineteenth century it was clear that the fifth postulate 
could not be deduced from the others, since its negation led to 
no contradictions with the other postulates. Further, Klein suc- 
ceeded in showing that by using a Euclidean model of non- 
Euclidean geometry, that is, by systematically interpreting non- 
Euclidean geometric terms by Euclidean terms, non-Euclidean 
geometry is certainly as consistent as Euclidean geometry. So 
Euclidean geometry stood as one system among others with no 
privileged position, at least from the point of view of logic. 

Our interest here is not in non-Euclidean geometry as such, 
but in the remarkable effect it had on the concept of space in 
modern physics. Not only did it lead to a fuller understanding 
of the hypothetical nature of pure axiomatic geometry, and so 
to an understanding of the nature of mathematics in general, but 
it led also — and this is no less important — to the clarification 
of the concept of physical space as opposed to the concept of 
mathematical space. With the discovery of non-Euclidean geome- 
try it became clear that there were no a priori means of deciding 
from the logical and mathematical side which type of geometry 
does in fact represent the spatial relations among physical bodies. 
It was natural, therefore, to appeal to experiment and to find 
out whether the question of the true geometry could be settled 
a posteriori. 

Once the validity of non-Euclidean geometry was recognized, 
the question arose whether the space of physics was Euclidean 




or not. In the vanguard of the attack on this problem was F. K. 
Schweikart, professor of law at the University of Marburg, who, 
according to the historians of non-Euclidean geometry, must be 
reckoned among the first independent discoverers of this science. 
Schweikart published his geometric system under the title Astral- 
geometrie, intending to indicate thereby that only by experiments 
or observations on an astronomical scale could the difference 
between Euclidean geometry and his own geometry be detected. 
Better known is Gauss's attempt to find out whether the space of 
experience is Euclidean or not. He tried to measure directly by 
an ordinary triangulation with surveying equipment whether the 
sum of the angles of a large triangle amounts to two right angles 
or not. Accordingly, he surveyed a triangle formed by three 
mountains, the Brocken, the Hoher Hagen, and the Inselberg 
with sides measuring 69, 85, and 107 km. Needless to say, he 
did not detect any deviation from 180 within the margin of 
error and thus concluded that the structure of actual space is 
Euclidean as far as experience can show. 

This was the first accurate survey of a geodetic triangle on a 
very large scale, and it required no little work. Its outcome must 
have been somehow disappointing for Gauss. The negative re- 
sult, the fact that no deviation from Euclidean geometry has been 
ascertained, was not conclusive, that is, it could not serve either 
to prove or to disprove decisively his ideas about space, which 
had been in his mind already for a couple of years. For as early 
as 1817 he wrote to H. W. M. Olbers: 

I become more and more convinced that the necessity of our geome- 
try cannot be demonstrated, at least neither by, nor for, the human 
intellect. In some future life, perhaps, we may have other ideas about 
the nature of space which, at present, are inaccessible to us. Geometry, 
therefore, has to be ranked until such time not with arithmetic, which 
is of a purely aprioristic nature, but with mechanics. 44 

Gauss's experimental investigation of the geometric structure 

of space was based, as we see, on his conviction, suggested 

"K. F. Gauss, Werke (Konigliche Gesellschaft der Wissenschaften zu 
Gottingen; Leipzig, 1863-1903), vol. 8, p. 177. 


by the acknowledged validity of non-Euclidean geometry, that 
geometry is essentially different from arithmetic and analysis. 
Whereas the last two branches of mathematics are based on 
the idea of pure number, and remain, therefore, purely rational 
knowledge, geometry becomes an empirical science inasmuch as 
it requires experimental investigation. In a letter to Bessel 45 
Gauss wrote that we have to admit that number is a product of 
the mind but space has a reality outside the mind whose laws 
we cannot prescribe a priori. Gauss seemed to have realized 
that his conception of space has far-reaching epistemological 
consequences. It was perhaps the anticipation of an impending 
conflict with orthodox philosophy that made him guard his secret 
most carefully for a number of years, fearing "the clamor and cry 
of the blockheads." Only in 1844 he wrote to his friend Schu- 
macher, the director of the observatory in Kiel and editor of the 
Astronomische Nachrichten, the pungent remarks: 

You see the same sort of things [mathematical incompetence] in 
the contemporary philosophers Schelling, Hegel, Nees von Essenbeck, 
and their followers; don't they make your hair stand on end with their 
definitions? Read in the history of ancient philosophy what the big 
men of that day — Plato and others (I except Aristotle) — gave in 
the way of explanations. But even with Kant himself it is often not 
much better; in my opinion his distinction between analytic and syn- 
thetic propositions is one of those things that either run out in a 
triviality or are false. 46 

To Gauss, non-Euclidean geometry, or, as he called it, "anti- 
Euclidean" geometry, was logically impeccable but experiment 
seemed to preclude its application to physical space. Lobachev- 
ski, who independently shared this view, wrote in his New 
Anfangsgrunde der Geometrie: 

The futility of all these efforts of the last two thousand years since 
the time of Euclid made me suspect that in geometry the concepts 
themselves do not imply the truth whose proof we sought and whose 

"Ibid., p. 201. 

"Quoted from E. T. Bell, Men of mathematics (Simon and Schuster, 
New York, 1937), p. 240. 



vindication, like the vindication of other natural laws, can be achieved 
only through experience, as for example by astronomical observation. 47 

If physical space were different from Euclidean space, he 
thought, the difference could be established only by means of 
large-scale observations. 

For the benefit of the reader who has a rudimentary knowledge 
of non-Euclidean geometry, an astronomical method for deter- 
mining the space constant k, on the assumption of hyperbolic 
geometry, will be explained. The numerical value of k depends, 
of course, on the arbitrary unit of length employed, but k may 
be used itself as a natural unit of length. 

Let A and B (Fig. 3) be two opposite positions of the earth on 
its annual orbit around the sun S. Let F be a star, whose parallax 

Fig. 3. 

is defined to be the angle AFS subtended by the radius r of the 
earth's orbit. For a direct measurement of the parallax AFS the 
transit circle can be employed. On the assumption of Euclidean 
geometry the parallax is then given by the angle V2.1t — SAF. A 
second method of measuring the parallax consists in comparing 
the position of F with that of another star G whose distance is 
taken to be very much greater than that of F. By measuring the 
angle GAF, the parallax, on the basis of Euclidean geometry, is 
found immediately, since it is equal to GAF. However, if space 
is assumed to be of hyperbolic structure, the two methods ex- 
plained will yield different results, since the sum of the two 
angles GAF and FAS is different from &r. In fact, this sum is just 
the angle of parallelism n(r), corresponding to the radius r of 

"N. I. Lobachevski, "Neue Anfangsgriinde der Geometrie mit einer 
vollstandigen Theorie der Parallellinien," Kasaner Gelehrten Schriften 
(1835-1838), p. 67. 




the earth's orbit. Let 8 be defined by the equation 28 = &r — 
n(r). Then according to a fundamental theorem of hyperbolic 
geometry we have 

1 — tan 8 

1 + tan 8 

e~ rlk = tan $H(r) = tan \{\v - 25) = tan (J* - 8) 
Taking natural logarithms we obtain 

t , 1 + tan 5 

7 = In-; -1 

k 1 — tan 

and finally that r/k is approximately equal to 2 tan 8. A calcula- 
tion of 8, on the basis of the two different methods of parallax 
measurement, in combination with the knowledge of the radius 
r of the earth's orbit, determines the space constant of hyperbolic 
space. A finite (real) value of k could be interpreted in favor of 
the hypothesis of hyperbolic space structure. 

Lobachevski used a triangle whose base was the diameter of 
the earth's orbit and whose apex was the star Sirius, whose 
parallax he assumed to be 1.24" according to a determination 
made by Comte d'Assa-Montdardier. 48 Lobachevski's data were 
wrong and it was only in 1838 that Bessel succeeded in measur- 
ing the first parallax of a star (61 Cygni; 0.45"). The true value 
of the parallax of Sirius is less than 0.40", in other words, less 
than a third of the value accepted by Lobachevski. 

Like Gauss's, Lobachevski's attempt to prove the non-Eu- 
clidean structure of space empirically came to nothing. So he 
concluded that Euclidean geometry alone was of importance for 
all practical purposes. Thus he writes: 

However it be, the new geometry whose foundations are laid in this 
work, though without application to nature, can nevertheless be the 
object of our imagination; though not used in real measurements, it 
opens a new field for the application of geometry to analysis and 
vice versa. 49 

48 D'Assa-Montdardier, Memoire sur la determination de la parallaxe des 
itoiles (Paris, 1828). 

*" Lobachevski, "Neue Anf angsgriinde der Geometrie," p. 24. 




The problem of the relevance of non-Euclidean geometry to 
physical space became a subject of controversy among scientists 
and philosophers of science, especially after the posthumous 
publication of Riemann's great treatise "On the hypotheses which 
lie at the basis of geometry." 50 In this treatise Riemann pre- 
sented an analytic approach to non-Euclidean geometry, in con- 
trast to the axiomatic approach of his predecessors. Analyzing 
the mathematical properties of a manifold of undefined objects, 
called points, which are determined by a set of coordinates, Rie- 
mann stressed for the first time in the history of mathematics the 
important distinction between the unlimited and the infinite. 
Since the time of Gassendi space, as a homogeneous continuum, 
was always thought of as unlimited, a boundary being obviously 
a singularity, mathematically speaking. Riemann showed that 
homogeneity and finiteness are compatible. His generalization of 
Gauss's theory of surfaces, culminating in the concept of "curved 
space," made it clear that the space of Euclidean geometry and 
the space of the geometry of Lobachevski and Bolyai were only 
special cases of the generalized space, that is, spaces of constant 
zero curvature or constant negative curvature. By introducing 
an appropriate metric Riemann was able to show also that a space 
of constant positive curvature, a so-called "spherical" space, 51 is 

This mathematical analysis of the structure of space, initiated 
by Gauss and Riemann, is of such paramount importance for the 
formation of modern space conceptions, both in mathematics and 
in physics, that a more detailed account of these investigations 
has to be given. Riemann's paper "On the hypotheses which lie 
at the basis of geometry," written at the age of only 28 years, 
80 G F B. Riemann, "Ueber die Hypothesen, welche der Geometrie zu 
Grande liegen" ( Habilitationsschrift, 1854), Abhandl. kgl. Ges. Wiss. zu 
Gbttineen 13 (1868); see also H. Weber, ed., Collected worh, of Bemhard 
Riemann (Gesammelte mathematische Werke), (Dover, New York, ed. 2, 

n The surface of a sphere is a two-dimensional "space" of constant posi- 
tive curvature. 



became the foundation of a general theory of space. Furthermore, 
it gave a new impetus to the important development of modern 
tensor analysis, which, originally confined in its applications to 
the treatment of problems in elasticity, became through the 
works of Ricci, Beltrami, Christoffel, Lipschitz, Bianchi, WeyL 
and Einstein an indispensable tool for higher mathematics as 
well as for theoretical physics. 

Riemann successfully generalized Gauss's theory of the in- 
trinsic geometry on a surface. Gauss's interest in geodesy, car- 
tography, and allied branches of applied mathematics drew his 
attention to the problem of how far geometric properties on 
curved surfaces can be expressed without resorting to the geome- 
try of the embedding three-dimensional space. Such properties 
on curved surfaces, called intrinsic, have to be unaffected by a 
deformation without stress of the surface in the embedding space. 
Already in 1816 Gauss had engaged in geodetic problems, as 
Staeckel has shown in his article "Gauss als Geometer." 52 But 
his interest was particularly focused on this subject when he was 
asked by the Hanoverian government to serve as scientific ad- 
viser in an extensive geodetic survey of Hanover. (Gottingen at 
that time was under the government of Hanover.) As a result 
of his mathematical investigations in connection with this survey, 
which was performed under his direction until 1825, ne published 
two important papers. 53 These papers, and especially his "Dis- 
quisitiones circa superficies curvas," 64 published in 1827, broke 
new ground and became through the work of Riemann the foun- 
dation of modern mathematical investigations into the structure 
of space. Once again we see that, historically viewed, abstract 
theories of space owe their existence to the practice of geodetic 
work, just as ancient geometry originated in the practical need 
of land surveying. 

" Gauss, Werke, vol. 10, 2 Abh. IV. 

" "Bestimmung des Breitenunterschiedes zwischen den Sternwarten von 
Gottingen und Altona" (1828; Werke, vol. 9); "Untersuchungen iiber 
Gegenstfinde der hoheren Geodasie" (1843, Werke, vol. 4). 

~ Gauss, Werke, vol. 4. 



The theory of surfaces as such was not a new subject. Euler, 
Lagrange, and Monge already had investigated geometric prop- 
erties on certain types of curved surfaces. But it was left for 
Gauss to study the problem in its generality and to lay thereby 
the foundations for differential geometry. In his "Disquisitiones" 
(already mentioned above), his first systematic exposition of 
quadratic differential forms, he investigated the possibility of 
an intrinsic determination of the curvature of a surface (today 
called the "Gaussian curvature"). Gauss's great contribution to 
differential geometry rests in his proof that this curvature, which 
is determined as the reciprocal product of the two principal radii, 
can be expressed in terms of intrinsic properties of the surface. 

For this purpose Gauss assumed two families of curves to be 
drawn on the surface. Along each curve of the first family (the 
Xi-curves) x 2 is constant and along each of the other curves 
(the x 2 -curves) Xi is constant, much as in the ordinary Cartesian 
coordinate system along the ordinate y the value of the abscissa 
x is constant and vice versa. These curves are to cover the whole 
surface for varying values of the constants and any %i curve is 
to intersect any x 2 curve only in one single point. Any point P 
on the surface is consequently determined by the values of Xi and 
x 2 of the two curves that intersect in it; x% and x 2 are called today 
the "Gaussian coordinates" of the point P on the surface. A 
familiar example is the system of longitude and colatitude co- 
ordinates on a sphere; here the x 2 -curve with constant x x (longi- 
tude) is a meridian and the JCx-curve with constant x 2 (colati- 
tude) is a latitude circle. 

Now, if ds is the element of arc of a curve on the surface, it 
can be shown, using the Pythagorean theorem in Cartesian co- 
ordinates, that 

ds 2 = giid*! 2 + 2g 12 dx 1 dx 2 + gsndxi*, 

or, according to the familiar summation convention, 

ds? = g mn dx m dx n , 

m and n to be summed over 1 and 2. 



In this expression, as usual, the dx m are infinitesimal increments 
of the Gaussian coordinates and the g m „ are magnitudes which 
in general depend on the Gaussian coordinates of the point in 
whose immediate vicinity the element of arc is to be computed. 

In a continuous manifold of n dimensions its continuity and 
dimensionality do not yet allow us to infer any metrical proper- 
ties, that is, properties to be ascertained by measurement. All 
that is known is that every point of the manifold is characterized 
by n numbers and that to closely adjacent points closely adjacent 
numbers correspond. But how can the distance between two 
given points be computed if only their coordinates are known? 
In axiomatic geometry the notion of congruency lies at the basis 
of measurements of distance or length. In practical geometry, 
however, with which physics is concerned, distance must be re- 
ferred to the physical properties of a rigid body that can be 
transported from one place to another without change. To be 
sure, the rigid scale can be of arbitrary smallness. These ideas 
induced Riemann to see in ds, as used by Gauss in his surface 
theory, the appropriate mathematical expression for an infinitesi- 
mal element of length. Riemann thus assumed for a fine element 
in a manifold of n dimensions with the general coordinates x u 
X2,...,x„ the formula 

ds 2 = g^ dx„. dx„ 

/x and v to be summed over 1, 2, . . . , n, and investigated the 
problem how to explore the geometry of this space on the basis 
of this expression. 

This differential expression representing ds is usually called 
today the metric form or fundamental form of the space under 
consideration and the g/*v, owing to the invariance of ds, are 
the components of a covariant tensor of the second rank, the 
so-called fundamental tensor. A continuous n-dimensional mani- 
fold is called a Riemannian space, if there is given in it a funda- 
mental tensor. 

For the sake of historical accurateness we have to note that 



Riemann apparently assumed that the concept of distance is 
intrinsic in space. Modern mathematics has shown that a logically 
consistent theory of a non-Riemannian space (that is, nonmetrical 
space) can be advanced in which the notion of distance is never 
encountered. The space of physical experience lends itself to 
measurements of length or distance, but it should be borne in 
mind that the concept of length or distance is foreign to the 
amorphous continuous manifold and has to be put in or "im- 
pressed" from without. For 'length" and "distance" are opera- 
tional concepts which find their mathematical counterpart 
through epistemic correlations. As will be explained on page 
165, the element of length ds, as a mathematical invariant in 
Riemannian space, will be made to correspond to an infinitesimal 
"stretch" on a "practically rigid body." 

In postulating the above formula Riemann showed that it pro- 
vides sufficient, although not necessary, specifications for a line 
element satisfying the fundamental requirements of a distance 
function. The position of a point P is determined by n numbers 
xi, x 2 , . . . , x„. If xi + dxu x 2 + dx 2 , . . . , x„ + dx» denote the 
values of the coordinates of an adjacent (neighboring) point F, 
the length ds of the line element PP' must be expressed as a 
certain function of the increments dx u dx 2 , ..., dx*. If these in- 
crements are all increased in the same ratio, ds must also be 
increased in this ratio. If all the increments are changed in sign, 
the value of ds must remain unaltered. Assuming a simple alge- 
braic relation between ds and the increments, these conditions 
suggest that ds must be an even root, the square root, fourth 

root, ... of a positive homogeneous function of the dx\, dx^ 

dx n of the second, fourth, . . . degree. Riemann selected the 
simplest hypothesis, namely, that ds is the square root of a 
homogeneous function of the increments of the second degree. 
He was fully aware of the arbitrariness in his determination of 
the length of the line element and emphasized the possibility of 
other expressions, as, for instance, the fourth power of ds as a 
biquadratic form of the coordinate differentials. The problem is 


of course connected with the question of the validity of the 
Pythagorean theorem in the vicinity of a point. Helmholtz, 55 
Sophus Lie, and Weyl 66 attempted to show the necessity of 
assuming a quadratic form for the square of the line element. 
H. P. Robertson's investigations are also relevant to this problem. 
Hermann von Helmholtz began his research on the structure 
of physical space because of his interest in the physiological 
problem of the localization of objects in the field of vision. In 
order to solve the problem of the dependence of ds on the incre- 
ments dx„, Helmholtz advances the principle of free mobility 
of rigid bodies and the principle of monodromy, that is, the 
assumption that a body being rotated about any arbitrary axis 
returns unchanged to its original position. It can readily be shown 
that the notion of congruence which lies at the basis of Helm- 
holtz's principles imposes severe limitations on the a priori 
determination of the mathematical dependence of ds on the in- 
crements. Let there be given five points, A, B, C, D, E in three- 
dimensional space with the respective coordinates xa 1 , x a 2 , x a s , 
Xb 1 , . . • , x B a . The distance between any two points out of these 
five is given by a certain distance function, the variables of which 
are the corresponding coordinates of the two points involved. 
Let us now try to construct a congruent figure composed of five 
points A', B', C, D', E', in which the distance between any pair 
of points equals the distance between the corresponding pair of 
points in the original figure. Clearly, A' can be chosen arbitrarily 
in space; B', however, is then restricted to a certain surface, since 
its coordinates have to satisfy one equation; C has to lie on a 
curve, its coordinates being submitted to two conditions; D' and 
E' are completely determined since their distances from A', B', 

"Helmholtz, "Ueber die Tatsachen, die der Geometrie zu Grande 
liegen." Cf. F. Lenzen, "Helmholtz's theory of knowledge," in Studies and 
essays in the history of science and learning, offered in homage to George 
Sarton (Schuman, New York, 1946), p. 309. 

"H, Weyl, "Die Einzigkeit der Pythagoraischen Massbestimmung," 
Math. Zeit. 12, 114 (1922). 



and C" are given. The assumption that D'E' equals DE imposes 
a restriction on the mathematical formulation of the distance- 

In general, n points in three-dimensional space have 371 co- 
ordinates and n(n— i)/2 mutual distances. We thus have 
n(n — i)/2 equations involving Qn coordinates, whereas the set 
of these n points, if considered as a rigid body, is determined by 
6 parameters (degrees of freedom). The 311 coordinates may now 
be eliminated from the n(n— i)/z equations and %n(n — 1) — 
3n-\-6 = %(n — 3) (n — 4) conditions will result. 

Helmholtz's investigations found their strict mathematical elab- 
oration in the works of Marius Sophus Lie. 57 Lie replaced the 
concept of mobility in space by the mathematical notion of a 
transformation between two systems of coordinates and reduced 
the geometric concept of congruence to the requirement of a 
certain invariance under such transformations. The displacement 
of a rigid body becomes equivalent to a one-to-one transforma- 
tion of all space into itself, two successive displacements being 
replaceable by a third single transformation. Lie's theory of con- 
tinuous groups, apart from its importance for axiomatic geometry 
(by showing that "congruence" is capable of a definition in terms 
of other fundamental geometric notions), has demonstrated that 
metric geometry is but the theory of the properties of certain 
particular congruence groups. Without having recourse to Helm- 
holtz's assumption of monodromy, Lie comes to the conclusion 
that the only possible types of metric geometry are Euclidean, 
hyperbolic, and elliptic, a result which again imposes severe 
restrictions on the expression for ds. 

Before concluding this digression on geometric investigations 
of the structure of space and resuming the subject of the further 
development of Riemann's contribution to the problem of space, 
a question will be asked that certainly deserves our attention, 

W M. S. Lie, Theorie der Tranformationsgruppen (Leipzig, 1888-1893). 
See also Lie's Ueber die Grundlagen der Geometrie (Leipzig, 1890). 



for it logically precedes all inquiries concerning the form of ds: 
How is it possible to define coordinates of points in space at all, 
as long as the concept of congruence is not yet determined? Von 
Staudt, whom Klein calls "einen der am tiefsten eindringenden 
Geometer, die je gelebt haben," thought he had found the solu- 
tion in his Geometry of position 58 by using a repeated applica- 
tion of the quadrilateral construction for a harmonic range in the 
establishment of a system of projective coordinates independent 
of distance. 

A similar attempt was made by Arthur Cayley by means of his 
projective equivalent of metric distance, employing the concept 
of cross ratio and leading finally to a vicious circle. As a matter 
of fact, the nature of these highly technical problems was fully 
understood only within the last five decades and interfered little 
with the early investigations concerning the structure of Rieman- 
nian space. Riemann assumed the validity of the Pythagorean 
theorem in the infinitely small. His theory of space, therefore, 
rests on geometric assumptions about infinitesimally small magni- 
tudes. Being essentially a geometry of infinitely near points, it 
conforms to the Leibnizian idea of the continuity principle, ac- 
cording to which all laws are to be formulated as field laws and 
not by actions at a distance. 

Riemann's geometry, in this respect, contrasted with the finite 
geometry of Euclid, can be compared with Faraday's field inter- 
pretation of electrical phenomena that formerly had been ex- 
plained by actions at a distance. Weyl characterizes this situation 
as follows: "The principle of gaining knowledge of the external 
world from the behaviour of its infinitesimal parts is the main- 
spring of the theory of knowledge in infinitesimal physics as in 
Riemann's geometry." B9 

58 Georg Karl Christian von Staudt, Geometrie der Lage (Niirnberg, 1847). 
See also von Staudt's Beitrage zur Geometrie der Lage (Niirnberg, 1856- 

W H. Weyl, Space-time-matter (trans, by H. L. Brose; London, 1922), 
P- 92. 



"Straight lines" of Euclidean geometry are generalized in Rie- 
mannian space to "geodesic lines," 60 or simply "geodesies," to 
wit, lines of extreme distances between their terminal points. 
These geodesies, whose equations contain the components of the 
covariant fundamental tensor and their derivatives in certain 
definite combinations ( Christoff el's symbols of the second kind), 
form a natural network throughout the n-dimensional manifold 
and can be used as a basis for the determination of its curvature. 
At a given point in the manifold let two infinitesimal vectors be 
given and the pencil of vectors linearly dependent upon them. 
With these vectors as initial elements, geodesies can be drawn, 
originating at the given point and generating a two-dimensional 
"geodesic surface" o* v with its normal N. Riemann now defined 
the general curvature K N of the n-dimensional manifold at the 
given point with respect to the normal N as the Gaussian curva- 
ture of this geodesic surface. It is obvious that the Riemannian 
curvature K N depends on the orientation N of the geodesic sur- 
face and varies also from point to point. In other words, it is a 
measure both for the anisotropy and for the heterogeneity of 

Now let the geodesic surface be represented by the oriented 
surface element <r* lv , an antisymmetric tensor of the second rank. 
It can be proved that the Riemannian curvature is then given 
by the expression 

(afl, yS) o* o* 
(Say 9f» - Qai guv) a"* a yi ' 

in which (a/?,y8) is the Riemann four-index symbol. It is related 
to the Riemann-Christoffel tensor by the equation 

(«/3, y$) = gtJi'aly, 

60 As Klein remarks in his Vorlesungen tiber die Entwicklung der Mathe- 
matik im 19. Jahrhundert (Chelsea Publishing Company, New York, 1950), 
vol. 2, p. 148, with reference to Staeckel's Zur Geschichte der geodatischen 
Linien, the name "geodesic" as a technical term became common usage 
paradoxically only with Liouville (1850), that is, at a time when theoret- 
ical geometers were already not interested in practical geodesy. 

Kk = 





with the usual designation for the components of the Riemann- 
Christoffel tensor. 61 Consequently, 


Kn = 

(afi, yS) = Raw 

(g a (tg»i - gasgth) ^^ 
This equation shows clearly that the Riemannian curvature K s 
vanishes everywhere if the Riemann-Christoffel tensor R a p y s is a 
constant zero tensor. Since the vanishing of the last-mentioned 
tensor is merely an analytic expression for the Euclidean struc- 
ture of space, we infer that in Euclidean space the Riemannian 
curvature is everywhere equal to zero. 

If the Riemannian curvature is independent of the orientation 
N of the geodesic surface element 0+" at every point in space, 
which certainly holds if space is isotropic throughout, then it 
is easy to show that 

K(g ay gfis — g a ig»y) = R a »ii- 
Covariant differentiation of this equation and the use of Bianchi's 
identity leads to the result 


dx m 

= 0. 

In other words, the Riemannian curvature is a constant. Accord- 
ing to this theorem, which was proved for the first time by F. 
Schur in 1886, 62 isotropy in every point of a Riemannian space 
implies its homogeneity. 

Applying this result to physical space, which by the end of the 
nineteenth century was conceived to be isotropic as a matter of 
course, the following result was obtained: either (1) the Rieman- 
nian curvature is everywhere zero and space is Euclidean, or (2) 
it is a positive constant and space is spherical, or (3) it is 

™ See, for example, A. Einstein, The meaning of relativity (Princeton 
University Press, Princeton, 1953). 

ra F. Schur, "Raume konstanten Kriimmungsmasses, II," Math. Ann. 27 


a negative constant and space is hyperbolic. To sum up, only 
these three types of geometry are compatible with the isotropy 
of space. Riemann's complicated calculations seemed to have 
brought to light essentially nothing new. 

Space is a form of phenomena, and, by being so, is necessarily homo- 
geneous. It would appear from this that out of the rich abundance of 
possible geometries included in Riemann's conception only the three 
special cases mentioned come into consideration from the outset, and 
that all the others must be rejected without further examination as 
being of no account: parturiunt monies, nascetur ridictdus tnus! 63 

Riemann, however, thought differently. He felt that labor was 
not lost. The assumption of a homogeneous space, in his view, 
does not take account of the existence of matter. Just as a strictly 
homogeneous magnetic or electrostatic field is never encountered 
in reality, so a homogeneous metrical field of space is only an 
idealization. Just as the physical structure of the magnetic or 
electrostatic field depends on the distribution of magnetic poles 
or electric charges, so the metrical structure of space is deter- 
mined by the distribution of matter. With prophetic vision Rie- 
mann wrote: "The basis of metrical determination must be sought 
outside the manifold in the binding forces which act on it." 6 * 

These words were clearly an anticipation of some central ideas 
in Einstein's theory of gravitation, according to which the metri- 
cal structure, determined by the Einstein tensor R/tv, is related 
at every point of the space-time continuum to the mass-energy 
tensor Tfiv by the field equations 

R u 

2 <7juj<Xb — Kl pp. 

The left-hand member of these famous equations involves the 
gfiv and their derivatives, the right-hand member is an expression 
for the distribution of matter and energy, and k is a constant re- 
lated to the Newtonian constant of gravitation. The integration 

68 Weyl, Space-time-matter, p. 96. 

** "Es muss also entweder das dem Raume zu Grande liegende Wirkliche 
eine discrete Mannigfaltigkeit bilden, oder der Grund der Massverhaltnisse 
ausserhalb, in darauf wirkenden bindenden Kraften, gesucht werden." 
Riemann, Collected works, p. 286. 



of these field equations, in general no easy task, leads to the 
determination of the g/tv as functions of the mass-energy dis- 

Riemann's anticipation of such a dependence of the metric 
on physical data seems to have been the logical solution of a 
dilemma to which the assumption of a variable curvature of 
space would have led. Since such curvature is an intrinsic prop- 
erty of space, that is, can be determined by geometric measure- 
ments within space itself, its very existence as a function of 
position would make it possible to designate position in space 
without resorting to a material coordinate system. Marks or labels 
could be assigned to points in space in accordance with the vary- 
ing curvature, a process which could serve for their identification 
or distinction. In other words, absolute space would have been 
reinstalled. This difficulty is overcome by relating the local in- 
homogeneity to the material content of space. 

Riemann's allusions were ignored by the majority of contem- 
porary mathematicians and physicists. His investigations were 
deemed too speculative and theoretical to bear any relevance to 
physical space, the space of experience. The only one who allied 
himself firmly to Riemann was the translator of his works into 
English, William Kingdon Clifford. Moreover, already in 1870 
Clifford saw in Riemann's conception of space the possibility for 
a fusion of geometry with physics. For Riemann, matter was the 
causa efficiens of spatial structure. Ry identifying cause and 
effect, a methodological procedure often encountered in the his- 
tory of science, Clifford conceived matter and its motion as a 
manifestation of the varying curvature. He assumed that the 
Riemannian curvature as a function of time may give rise to 
changes in the metric of the field after the manner of a wave, 
thus causing ripples that may be interpreted phenomenally as 
motion of matter. 

We may conceive our space to have everywhere a nearly uniform 
curvature, but that slight variations of the curvature may occur from 
point to point, and themselves vary with the time. These variations of 



the curvature with the time may produce effects which we not un- 
naturally attribute to physical causes independent of the geometry of 
our space. We might even go so far as to assign to this variation of 
the curvature of space "what really happens in that phenomenon 
which we term the motion of matter. ' 65 

Clifford's work from which this quotation has been taken was 
published posthumously by Karl Pearson, who states in the pref- 
ace that the chapters on space and motion were dictated by 
Clifford himself in 1875. In 1876 Clifford published a paper "On 
the Space-Theory of Matter" in which he expressed similar ideas. 
He wrote: 

I hold in fact 

(1) That small portions of space are in fact of a nature analogous 
to little hills on a surface which is on the average flat; namely, that 
the ordinary laws of geometry are not valid in them. 

(2) That this property of being curved or distorted is continually 
being passed on from one portion of space to another after the manner 
of a wave. 

(3) That this variation of the curvature of space is what really 
happens in that phenomenon which we call the motion of matter, 
whether ponderable or etherial. 

(4) That in the physical world nothing else takes place but this 
variation, subject (possibly) to the law of continuity. 66 

Clifford's suggestion, today no longer deemed as fantastic as 
in his own day, is in a certain sense the climax of a long develop- 
ment. For Aristotle, space was an accident of substance; for Clif- 
ford, so to speak, substance is an accident of space. The concept 
of space, after its emancipation during the Renaissance, seized 
totalitarian power in a triumphant victory over the other con- 
cepts in theoretical physics. 

These speculations aroused great opposition among academic 
philosophers who still adhered to the Kantian doctrine according 
to which the axioms of Euclidean geometry were a priori judg- 
ments transcending reason and experience. In addition to such 
philosophical considerations, the advocates of Euclidean geome- 

" W. K. Clifford, The common sense of the exact sciences ( ed. by J. R. 
Newman; Knopf, New York, 1946), p. 202. 
" Proceedings of the Cambridge Philosophical Society ( 1876). 




try used the important argument that Euclidean geometry, in 
opposition to elliptic and hyperbolic geometry, was independent 
of any absolute length. This idea was strongly emphasized in 
particular by A. Gerstel, E. Konig, J. Cohn, K. Geissler, and 
H. Cornelius. 67 But marching along with truth is error. Thus it 
was maintained by many of these propounders of the impossi- 
bility of a non-Euclidean structure of physical space that only 
Euclidean space was homogeneous, 68 or it was erroneously as- 
serted that non-Euclidean geometry presupposes Euclidean 
geometry. 69 

As late as 1900 the possibility of exploring by observation 
whether space is "Euclidean" ("flat") or "curved" attracted the 
attention of distinguished scientists. Thus K. Schwarzschild 70 
published at the turn of the century a paper "On the admissible 
curvature of space" in which he tried to find an upper limit of 
the curvature of space (or its absolute value) without committing 
himself on the question whether physical space, if curved, is 
elliptic (curvature > o) or hyperbolic (curvature < o). On 
the basis of parallax statistics, taking into account the possible 

"Adolf Gerstel, "Ueber die Axiome der Geometrie," Beilage zum 16. 
Jahresbericht der philosophischen Gesellschaft, Wien (1903), pp. 97-1"- 
Edmund Konig, "Kant und die Naturwissenschaften," Die Wissenschaft, 
part 22 ( 1907). Jonas Cohn, "Voraussetzungen und Ziele der Erkenntnis," 
Untersuchungen iiber die Grundfragen der Logik (Leipzig, 1908). Kurt 
Geissler, Moaerne Verirrungen auf philosophisch-mathematischen Gebieten. 
Kritische und selbstgehende Untersuchungen (1909); cf. K. Geissler, Phi- 
losophie der Mathematik (Interlaken, 1933)- Hans Cornelius, Grundlagen 
der Erkenntnistheorie. Transzendentale Systematik (Munich, 1916; ed. 2, 

M Cf. R. H. Lotze, Grundztige der Metaphysik (Leipzig, 1884). Paul 
Natorp, Die logischen Grundlagen der exakten Wissenschaften (Leipzig, 
1910). Hans Driesch, Ordnungslehre, ein System des nicht-metaphysischen 
Teiles der Philosophie (Jena, 1912). 

**Cf. Alois Riehl, Der philosophische Kritizismus und seine Bedeutung 
fiir die positive Wissenschaft (Leipzig, 1879); Christoph von Sigwart, 
Logik (Freiburg, 1893), vol. 2, "Methodenlehre"; Logic (trans, by Helen 
Dendy (London, 1895), vol. 2; also H. Cornelius, Transzendentale 
Systematik, and H. Driesch, Ordnungslehre. For further literature on these 
erroneous assumptions, see the bibliography of R. Carnap, "Der Raum," 
Kantstudien, Erganzungsheft No. 56 (1922). 

™K. Schwarzschild, 'Ueber das zulassige Kriimrnungsmass des Raumes," 
Vierteljahrschrift der astronomischen Gesellschaft, vol. 35 (1900), p. 337. 



errors of observation, he came to the conclusion that, if space 
is hyperbolic, its radius of curvature is at least 64 light-years; on 
the assumption of an elliptic structure, the radius of curvature 
turned out to be at least 1600 fight-years. 

It was only toward the turn of the century that Poincare 
demonstrated once for all the futility of this controversy and 
the fallacy of any attempt to discover by experiment which of 
the mutually exclusive geometries applies to real space. Measure- 
ment, he insists, is never of space itself, but always of empirically 
given physical objects in space, whether rigid rods or fight rays. 
Regarding the structure of space as such, experiment can tell us 
nothing; it can tell us only of the relations that hold among 
material objects. Suppose, Poincare" says, a deviation from two 
right angles had occurred in the triangulation carried out by 
Gauss, would this necessarily have constituted a refutation of 
Euclidean geometry? For there would be nothing to prevent us 
from continuing to use Euclidean geometry on the assumption that 
light rays are curved. Nothing could disprove such an assump- 
tion. So the highly important conclusion emerges that experi- 
ence can neither confirm nor refute a geometry, whichever ge- 
ometry it be. What geometry one chooses is, for Poincare, merely 
a matter of convenience, a convention. We select that system of 
geometry which enables us to formulate the laws of nature in 
the simplest way. 

Poincar^ was convinced, on the basis of this conclusion, that 
Euclidean geometry, the familiar abstraction from common ex- 
perience with solid bodies and light rays, would always remain 
the favored system. He was wrong, as the development of general 
relativity has shown. 71 The classical example of gravitation will 
suffice to make this point clear. 

Gravitation, as understood by the theory of general relativity, 

71 He was wrong also as far as the logical simplicity of Euclidean geometry 
is concerned. As modern research has shown, Euclidean geometry Tacks the 
distinction of logical simplicity and "hyperbolic geometry is the only one 
which can be developed from a few simple assumptions concerning join- 
ing, intersecting, and continuity alone" (K. Menger). 



is to be comprehended in the geometric structure of space-time. 
This fusion not only made physical theory logically more unified 
and simpler, but led to the great triumph of the new theory over 
classical physics (in the famous observable effects: advance of 
the perihelion of Mercury, deflection of light rays in a gravita- 
tional field, etc.). Let the geometry in a coordinate system 
(x 1 , x 2 , x 3 , x 4 ) be determined by the field equations and have 
the line element 

ds 2 = g^ dx* dx'. 
If an observer, following Poincare's suggestion, adheres to the 
Euclidean ( or, in a four-dimensional continuum, to the so-called 
Galilean) metric and selects as his line element 

cfe 2 = -(dx 1 ) 2 - (dx 2 ) 2 - (dx 3 ) 2 + (dx*) 2 , 

he will soon realize that his ds cannot be made compatible with 
his observational results. Thus, a freely moving particle will not 
follow the path described by the condition 

dfds = 0. 

If our observer is not willing to revise his geometry and to change 
its metric, he will be led to the conclusion that the particle, al- 
though apparently undisturbed, deviates from the geodesic line 
of his geometry, that is, from uniform motion in a straight line. 
This clearly contradicts the Galilean principle of inertia. In order 
to remove this contradiction he will suppose the existence of a 
"field of force" (for example, gravitation) and inquire into its 
physical properties, without realizing that this "field of force" 
is but a fiction, invoked by the discrepancy between the appro- 
priate "natural" geometry — as required by the field equations — 
and the Euclidean geometry to which he adheres. Our observer's 
predilection for his familiar geometry has led him to an enor- 
mous complication in his physical theory. 

A system of pure axiomatic geometry does not suffice, if 
geometry is to be applied to the space of physics. What is needed 
is a correlation between the geometric concepts of the abstract 


system with physical objects or physical processes. As Einstein 
pointed out in a lecture 72 at the Berlin Academy of Sciences in 
1921, the most natural and simple assumption would relate the 
physical behavior of rigid bodies to the geometric properties of 
solid bodies in Euclidean geometry. This, however, does not 
necessarily imply that the space of physics is Euclidean. 

How such a correlation can be established is explained also 
in Einstein's "Physik and Realitat." 73 The conceptual construc- 
tion of the notion of space in modern physics is based on the 
empirical fact, already noted by Poincar^, that there exist two 
kinds of alteration of physical objects, changes of state and 
changes of position. In contrast to the former, it is the latter type 
of change that can be reversed by the arbitrary motions of our 
bodies. "That there are bodily objects to which we have to 
ascribe within a certain sphere of perception no alteration of 
state, but only alterations of position, is a fact of fundamental 
importance for the formation of the concept of space (in a cer- 
tain degree even for the justification of the notion of the bodily 
object itself)." Such a bodily object is called by Einstein "prac- 
tically rigid." The position of either of two given practically rigid 
bodies can be changed without changing the position of the pan- 
as such. So we get the concept of "relative position," a special 
case of which is "contact" of two bodies at a point. Any two 
points on a practically rigid body define a "stretch" (as Lenzen 
calls it in his article on "Einstein's theory of knowledge" 74 ). 
Two stretches on two practically rigid bodies, one on each, are 
defined as equal if the points of one are in contact ("coincide") 
with the corresponding points of the other. We have now to 

ra Later published under the title Geometrie und Erfahrung ( Erweiterte 
Fassung des Festvortrages, gehalten an der Preussischen Akademie, Berlin, 
1921). Cf. Sidelights on Relativity (trans, by G. B. Jefferey and W. Perrett; 
London, 1922). 

™ Published in the Journal of the Franklin Institute 2.2.x, 313-347 ( 1936); 
English translation by J. Piccard, pp. 349-382. The following outline is 
based primarily on this article. 

"In P. A. Schilpp, ed., Albert Einstein, Philosopher-scientist (Tudor, 
New York, 1950), p. 355. 



postulate that two stretches, once determined to be equal, are 
always and everywhere equal. This relation of equality, a sym- 
metric, reflexive, transitive relation, independent of position and 
time, can now be correlated with the abstract notion of congru- 
ence in Euclidean geometry. 

To be sure, experience can show equality between two 
stretches only if they are adjacent; in addition, it can show that 
two stretches found to be equal remain so when separated and 
then brought together again. But experience can tell us nothing 
about equality between two nonadjacent stretches. The relation 
of equality "at a distance," so to speak, is a postulating generaliza- 
tion of the original definition of equality. The postulate is essen- 
tially only a positive expression for the rejection of the concept 
of absolute space. Were absolute space assumed to exist, it would 
serve as a gage by which the validity of the "postulate" could 
be tested. 

The postulate also deprives the famous space fantasies, pop- 
ularized by Delboeuf 75 and especially Poincar^, 78 of all physical 
meaning. Poincare" imagines the case of a uniform expansion of 
the universe. All its dimensions increase over night a thousand 
times. What was formerly one meter, will now measure one 
kilometer. Clearly, such an expansion is beyond all physical veri- 
fication, for whatever the measuring instrument employed, it too 
will have increased in the same ratio. Even if the imagined expan- 
sion were not uniform, but anisotropic, say ten times as much in 
one direction as in another direction perpendicular to the first, 
it too would go unnoticed by any observer. The fact that our 
postulate precludes such an expansion of the universe is equiv- 
alent to the statement that such an expansion can be formulated 
only on the assumption of the existence of absolute space. 

75 J. R. L. Delboeuf, ProUgomdnes philosophiques de la gSometrie 
(Paris, i860), containing the substance of his lectures before the Royal 
Belgian Academy of Sciences, "Nains et eeants" and "Megamicros ou les 
effets sensibles d'une reduction proportionelle des dimensions de l'universe." 
See also his articles "L'Ancienne et les nouvelles geometries," Revue 
philosophique 36, 449 (1893). 

"Poincare, Science and method (Dover, New York, 1952), p. 94. 



From the historical point of view it is interesting to note that 
Delboeuf and Poincare were not the first who stated that a 
uniform expansion or contraction of all magnitudes in the uni- 
verse would be unobservable. Laplace" had pointed out in 1808 
that on the basis of Newtonian physics an expansion of all dis- 
tances, accelerations, and masses in the same ration would have 
no effect on the order of physical events: the behavior of the 
physical universe is independent of the absolute magnitude of 
the scale. Laplace did not mention these considerations because 
of their relevance to the problem of space, but rather to show 
the unique importance of Newton's law of gravitation, which for 
the end of the eighteenth century was the archetype of all physical 
laws. Only the inverse-square law of force seemed to be com- 
patible with the complete independence of the absolute mag- 
nitude of the scale. Let us give a simple example. The attraction 
of the sun (a sphere of radius R) on the earth at a distance D is 
proportional to R 3 /D 2 . Under the influence of this attraction the 
earth moves in unit time a distance A in a centripetal direction. 
Now assume that R, D, and A all increase in the ratio n:i. 
The force of attraction is now proportional to (nR) 3 /(nD) 2 - 
n(R 3 /D 2 ), that is, it is increased n times, and it stands to the 
increased centripetal distance nA in the same relation as before. 
Laplace's explications, one is tempted to say, may serve as 
an a priori demonstration of the validity of Newton's law of 
gravitation, once the independence of an absolute scale for all 
physical dimensions is granted. Incidentally, Delboeuf tried to 
prove, along similar lines, the validity of Euclid's axiom of paral- 
lels on the assumption of the homogeneity of space. 78 

So it comes about that the physical concept of equality is now 
made to correspond to the mathematical notion of congruence, 
just as the physical behavior of practically rigid bodies is made 

17 P. S. Laplace, Exposition du systime du monde (Paris, 1808), book 4, 

° ^L.Vouturat, "Note sur la geometrie non-euclidienne et la relativite de 
l'espace," Revue de mStaphysique et de morale 1, 302 (i»93)- 



to correspond to the mathematical properties of solid bodies in 
Euclidean geometry. 

It will be noticed that this operational or epistemic correlation 
between the physical behavior of practically rigid bodies and 
the purely abstract notion of congruency in deductive geometry 
holds whether axiomatic geometry adopts for its systematic con- 
struction the notion of movement through space or not. Most 
modern treatments of axiomatic geometry (for example, Hilbert's 
Foundations of geometry) eliminate this notion by a skillful adop- 
tion of a congruence axiom. 79 This procedure, most probably, 
has its historical origin in the predominant influence of Kantian 
critical philosophy on European, and in particular German, 
thought in the nineteenth century. For Kant's transcendental 
aesthetic comprises only the elements of space and time, whereas 
the concept of motion, not known a priori but only by experience, 
belongs to the realm of sensibility. Its introduction as a primary 
notion into deductive geometry seemed accordingly to violate the 
a priori synthetic character of the science of space. 

These considerations are only of historical importance for the 
changed attitude of the twentieth century. Nevertheless, it is 
satisfying to realize that Einstein's procedure is unaffected by 
this problem, although an inclusion of movement among the 
primary notions of geometry could perhaps lead to a certain 
simplification. More important, however, is E. A. Milne's criticism 
concerning the postulated invariance of the length of rigid bodies 
under transport. Milne argued that such a statement as "a rigid 
measuring rod is unaltered under transport" is void of any 
operational specification, since no standard of length is available 
at the new position of the rod. In his manuscript for the Edward 
Cadbury Lectures, which he was to deliver at the University of 
Birmingham in 1950, Milne wrote: 

It is part of the debt we owe to Einstein to recognize that only 
"operational" definitions are of any significance in science: we must 
be in a position to state a test by which we can tell whether a given 

" See Felix Klein, Elementary mathematics from an advanced standpoint 
(Dover, New York, 1939), vol. 2, "Geometry," p. 175. 



entity may be identified with one mentioned in the definition. A defini- 
tion, in other words, must be couched in terms of "observables." Ein- 
stein carried out his own procedure completely when he analysed the 
previously undefined concept of simultaneity, replacing it by tests 
using the measurements which have actually to be employed to rec- 
ognize whether two distant events are or are not simultaneous. But 
he abandoned his own procedure when he retained the indefinable 
concept of the length of a "rigid" body, i.e., a length unaltered under 
transport. The two indefinable concepts of the transportable rigid 
body and of the simultaneity are on exactly the same footing; they 
are fog-centres, inhibiting further vision, until analysed and shown to 
be equivalent to conventions. 80 

In his attempt to find an operational meaning for "permanence 
in length under transport" Milne adopted a method of reducing 
length to time determination. His procedure can be compared 
in principle with the well-known radar technique for measuring 
distances of remote objects. With clocks having suitable gradua- 
tions, 81 coordinates can be assigned to various observers or ob- 
jects. In particular, the distance r of observer B from observer A 
can be defined by taking an appropriate combination of clock 
readings t\ and t s — corresponding to the epochs of transmission 
and reception of light signals — yielding r = %c(t 3 — £1), where c 
is a universal constant that later turns out to be the velocity of 
light. In general, "distances" and "lengths" (differences of dis- 
tance coordinates) become definite for an observer equipped 
with a graduated clock. Now, if a measuring rod, originally at 
rest relative to such an observer is transported to a new position 
of rest relative to the same observer, and if its "lengths" (as 
measured by the radar technique) in the two positions are 
equal, then "the rod is said to have undergone a rigid body dis- 
placement by this clock." Thus, according to Milne, a meaning 
of "permanence of length under transport" depends on the provi- 
sion of graduated clocks. In contrast to Einstein, and perhaps 

80 Milne's Cadbury Lectures were published posthumously under the tide 
Modern cosmology and the Christian idea of God ( Clarendon Press, Oxford, 
1952); the quotation is from p. 35. 

81 For details see E. A. Milne, Kinematic relativity (Clarendon Press, Ox- 
ford, 1948). 



even more in contrast to Eddington, Milne regarded the measure- 
ment of time as much more fundamental than the measurement 
of length. By reducing the determination of length to the meas- 
urement of time Milne tried to dispense with rigid measuring 

The reason why it is more fundamental to use clocks alone rather 
than both clocks and scales or than scales alone is that the concept 
of the clock is more elementary than the concept of the scale. The 
concept of the clock is connected with the concept of "two times at 
the same place," whilst the concept of the scale is connected with the 
concept or "two places at the same time." But the concept of "two 
places at the same time" involves a convention of simultaneity, namely, 
simultaneous events at the two places, but the concept of "two times at 
the same place" involves no convention; it only involves the existence 
of an ego. 82 

Milne's analysis of the "permanence of length under transport" 
seems to be justified from the purely logical point of view. As 
far as its scientific fruitfulness is concerned, it certainly cannot 
be compared with Einstein's analysis of the concept of "simul- 
taneity." Einstein's critique of the traditional notion of simultaneity 
led to a radical revision of classical conceptions, whereas Milne's 
important contributions to theoretical physics are certainly not 
primarily the result of his critique of the "permanence of length 
under transport." 

It was Einstein who made it clear that geometry, when applied 
in this way to the exploration of physical space, ceases to be an 
axiomatic deductive science and becomes one of the natural 
sciences, indeed, the oldest of all. Poincare was only partly right: 
it is a matter of convention which geometry we adopt, but only 
as long as no assumptions are made concerning the behavior of 
physical bodies as implied in the measurements. Once these 
assumptions are laid down, the choice of the geometric system is 
determined. As Einstein explains, it is the sum total of the as- 
sumptions of correlation and of the system of abstract geometry 
that has to conform to experience. Once the principle that relates 

m Milne, Modern cosmology and the Christian idea of God, p. 46. 



rigid bodies to Euclidean solids is accepted, it is experience that 
conditions the choice of geometry. For example, in a reference 
system that rotates relatively to an inertial system, the laws of 
placing 83 rigid bodies no longer correspond, owing to the Lorentz 
transformation, to the rules of Euclidean geometry. In accord- 
ance with our fundamental postulate, only one choice is possible 
and Euclidean geometry must give way to Riemannian geometry. 
Hence it is clear that the structure of the space of physics is not, 
in the last analysis, anything given in nature or independent of 
human thought. It is a function of our conceptual scheme. 

Space as conceived by Newton proved to be an illusion, al- 
though for practical purposes a very fruitful illusion — indeed, so 
fruitful that the concepts of absolute space and absolute time will 
ever remain the background of our daily experience. Acknowl- 
edging this fact, Einstein writes in his "Autobiographical notes": 

Newton forgive me; you found the only way which, in your age, 
was just about possible for a man of highest thought and creative 
power. The concepts which you created, are even today still guiding 
our thinking in physics, although we now know that they will have to 
be replaced by others farther removed from the sphere of immediate 
experience, if we aim at a profounder understanding of relationships. 84 

Thus Newton's conception of absolute space and its equivalent, 
Lorentz's ether, were shorn of their ability to define a reference 
system for the measurement of velocities. This was accomplished 
by the special theory of relativity. Within the framework of this 
theory, however, space as such was still a basic concept. To be 
sure, as part of the four-dimensional Minkowski space-time con- 
tinuum it certainly had lost any individual distinction; an infinite 
number of coordinate systems were physically equivalent. It had, 
however, its own representation as an inertial system. Owing to 
the relativization of simultaneity, furthermore, the notion of ac- 
tion at a distance had to be discarded and the adoption of the 
field concept as the basic element of the theory had been sug- 
gested. This program was carried through by the general theory 

88 For example, along a circle whose center lies on the axis of rotation. 
"In P. A. Schilpp, ed., Albert Einstein, Philosopher-scientist, p. 31. 




of relativity, whereby the inertial system was replaced by the dis- 
placement field, a component part of the total field, 

this total field being the only means of description of the real world. 
The space aspect of real things is then completely represented by a 
field, which depends on four coordinate-parameters; it is a quality of 
this field. If we think of the field as being removed, there is no "space" 
which remains, since space does not have an independent existence. 85 

In the conceptual construction of space according to modern 
science, the three-dimensionality of space or the four-dimen- 
sionality of the space-time continuum appears as an accidental 
feature, justified only by experience. For three numbers or co- 
ordinates suffice to locate a "point-object" in space and four 
coordinates determine a "point-event" in space-time unambig- 
uously. That the three-dimensionality of space has to be accepted 
as accidental was considered already in antiquity as a serious 
flaw and an essential deficiency in a deductive theory of space. 
Aristotle considered this problem worthy of a detailed discussion 
in the opening chapter 88 of his De caelo and thought he could 
solve it in the spirit of Pythagorean mythical notions of perfec- 
tion. Aristotle's arguments are recapitulated in the discussion 
between Salviatus and Simplicius in Galilei's Dialogue on the 
great world systems. Says Simplicius with reference to Aristotle: 

Do you not have there the proof that there are no more than three 
dimensions, because those three are all things and are everywhere? 
And is this not confirmed by the doctrine and authority of the Pythag- 
oreans, who say that all things are determined by three, beginning, 
middle, and end, which is the number of All? And where do you 
leave that argument, namely, that, as it were by the law of Nature, this 
number is used in the sacrifices of the gods? And why, being dictated 
by Nature, do we attribute to those things that are three, and not to 
less, the title of all? . . . Moreover, in the fourth text, does he not, 
after some other doctrines, prove it by another demonstration, viz. 
that no transition is made but according to some defect (and so 
there is a transition or passing from the line to the surface, because 
the line is defective in breadth) and that it is impossible for the per- 

85 Albert Einstein, Generalization of gravitation theory, a reprint of 
Appendix II from the fourth edition of The meaning of relativity (Prince- 
ton University Press, Princeton, 1953), p. 163. 

88 Aristotle, De caelo, 268 a. 



feet to want anything, it being every way so; therefore, there is no 
transition from the Solid or Body to any other magnitude. Do you 
not think that by all these places he has sufficiently proved how there 
is no going beyond the three dimensions, Length, Breadth, and Thick- 
ness, and that therefore the body or solid, which has them all is 
perfect? 87 

That such a demonstration does not satisfy the requirements 
of scientific rigor is Galilei's firm conviction. In fact, Salviatus 

To tell you true, I do not think myself bound by all these reasons 
to grant any more than this — that that which has beginning, middle, 
and end may be, and possibly ought to be, called perfect. But I cannot 
grant that, because beginning, middle, and end are three, the number 
three is a perfect number and has a faculty of conferring perfection 
on those things that have it. Neither do I understand nor believe that, 
for example, of feet, the number three is more perfect than four or 
two; nor do I conceive their number of four to be any imperfection in 
the elements; nor that they would be more perfect if they were three. 
Better therefore if he had left these subtieties to the rhetoricians and 
had proved his intent by necessary demonstration; for so it behooves 
to do in demonstrative sciences. 88 

As a matter of fact, Simplicius, the great commentator of Aris- 
totle, referred to the insufficient demonstration in De caelo and 
compared it with Aristotle's theory of place as expounded in the 
Physics, where place (or space), in Simplicius's view, is essentially 
conceived as a two-dimensional extension. 88 Simplicius contrasts 
this latter treatment with the teachings of Strata, the Platonists, 
and the Stoics who stressed the three-dimensionality of space. 

Apart from these scanty remarks, the three-dimensionality of 
space as a problem was scarcely discussed in antiquity or in 
the Middle Ages. Euclid's definition I in Book XI of the Elements: 
"A solid is that which has length, breadth, and depth," with the 
implicit identification of solids and bodies, was accepted without 

"Galileo Galilei, Dialogue on the great world systems (trans, by T. 
Salusbury, ed. by Giorio de Santillana; University of Chicago Press, Chi- 
cago, 1953), p. 13. For the section of the proof omitted in this edition, see 
Emil Strauss's translation, Dialog tiber die beiden hauptsachlichen Welt- 
sy steme (Leipzig, 1891), p. 12. 

88 Ibid. 

88 Simplicius, Physics, 601. 



further questions. This seemed to be only natural since the 
notions of surface, line, and point came later to be defined by 
the process of abstraction from the concept of the solid. More- 
over, the problem was dismissed by identifying three-dimension- 
ality with body, as had been done already by Isaac Judaeus. 90 
Even Leibniz, who, as we have seen, submitted the concept of 
space to a most critical analysis, took little notice of the problem 
of the dimensionality of space. Recognizing that space has three 
dimensions, he bases this statement on purely geometric con- 
siderations: "Le nombre ternaire est determine ... par une ne- 
cessity geometrique: c'est parce que les Geometres ont pu de- 
montrer qu'il n'y a que trois lignes droites perpendiculaires entre 
elles, qui se puissent coupre dans un meme point." 91 

With the rise of non-Euclidian geometry and other generaliza- 
tions of classical geometry it became evident that pure mathe- 
matics, not logically confined to three dimensions, could operate 
consistently with concepts of space that possesses any arbitrary 
number of dimensions. The question why ordinary space pos- 
sesses just three dimensions was considered from now on as a 
problem in physics or in logic applied to real existence. Never- 
theless, it was not always understood that a discussion which 
proceeds wholly within the boundaries of geometric notions 
necessarily cannot be decisive. 92 

One of the first for whom the three-dimensionality of space 
became a problem of physics was Kant. Already in his Gedanken 
von der wahren Schdtzung der lebendigen Kraft he considers the 
possibility of spaces having different dimensionalities. 

Eine Wissenschaft von alien diesen moglichen Raumarten ware 
ohnfehlbar die hochste Geometrie, die ein endlicher Verstand unter- 
nehmen konnte . . . Wenn es moglich ist, dass es Ausdehnungen von 

"Isaaci operi omnia lot. (Leyden, 1515-16), "Liber de dementis." 

"J. E. Erdmann, ed., Leibnitii opera philosophica (Berlin, i860), p. 606. 

" This applies, for example, to Whewell's demonstration in his History of 

scientific ideas (London, 1858), vol. 1, p. 97, which is a variation of a 

geometric proof given already in the beginning of the eighteenth century 

by Leibniz in his Essais de tModicSe (Amsterdam, 1710), §§351 and 196. 



anderer Abmessung gebe, so ist es auch sehr wahrscheinlich, dass sie 
Gott wirklich irgendwo angebracht hat. 93 

However, he thought he had discovered the reason for the three- 
dimensionality of physical space of our experience in Newton's 
law of gravitation, according to which the intensity of the force 
decreases with the square of the distance. 94 

Gauss, in a letter to Gerling, 95 refers to a generalization of 
his considerations on symmetry and congruence for a geometry 
of more than three dimensions, "for which we human beings 
have no intuition, but which considered in abstracto is not incon- 
sistent." As Sartorius von Waltershausen reports, Gauss consid- 
ered the three-dimensionality of space not as an inherent quality 
of space, but as a specific peculiarity of the human soul. 96 Gauss 
seems to have understood that the question whether space has 
three dimensions or more stands on the same footing as the 
problem of the Euclidean or non-Euclidean character of space. 
Roth questions need for a decision an external criterion, foreign 
to pure mathematics. 

Certain ideas in Herbart's philosophy 97 seem to have had 
a great influence on Riemann and H. Grassman in their formula- 
tion of a manifold with an arbitrary number of dimensions. Grass- 
mann s ingenious Theory of extensions, 9 * published in 1844 and 

98 Kant, Gedanken von der wahren Schdtzung der lebendigen Kraft, §10; 
see reference 16. 

"For a detailed analysis of Kant's argument, see Ueberweg's similar 
demonstration on page 177. F. W. J. Schelling, in his System des transcen- 
dentalen Idealismus (Tubingen, 1800), pp. 176-185, also tries to explain 
the three-dimensionality of space, under the influence of Kant, by dynamical 
considerations. In addition to the Kantian-Newtonian forces of attraction 
and repulsion, Schelling postulates a third force (newly invented by him) 
and relates the dimensions of length, breadth, and depth to these three 
forces. According to Schelling, not only spatial three-dimensionality but 
also electricity and magnetism owe their origin to these forces. 

" Letter of April 8, 1844. 

M Sartorius von Waltershausen, "Gauss zum Gedenken," in Gauss, Werke, 
vol. 8, p. 268. 

"Cf. J. F. Herbart, Habilitationsthese (October 23, 1802); cf. Schriften 
zur Metaphysik (Leipzig, 1851), part 2, chap. 4, "Vom Korperlichen 
Raume," p. 203. 

98 Ausdehnungslehr e (1844; ed. 2, Leipzig, 1878). 



in a second revised and amplified edition in 1862, was completely 
ignored at that time. Only the twentieth century began to under- 
stand the importance of Grassman's generalized algebraic treat- 
ment of n-dimensional spaces. 

In mathematics proper, however, the nineteenth century was 
very successful in clarifying the concept of dimension, partic- 
ularly after the development of the affine and projective geom- 
etry. J. Pluecker, generalizing the basic idea of the principle of 
duality in projective geometry, showed that the dimensionality 
of a space depends not only on topological properties, but also 
on the choice of the elements out of which space is constructed. 
Thus, for example, a Euclidean plane is three-dimensional if con- 
sidered as a manifold of circles (two coordinates determine their 
centers and the third their radii). On the other hand, the plane 
appears as a five-dimensional manifold if conies are chosen as 
the basic space elements. In short, dimensionality in Pluecker's 
view is not an absolute attribute of space, but depends upon the 
basic elements that constitute the space. However, referring to 
Cartesian coordinates, it was shown by L. E. J. Brouwer in 1911 
that the dimensionality of space is a topological invariant, that 
is to say, it remains invariant under any continuous transforma- 
tion of the coordinates. 

This short digression into the history of mathematics has been 
inserted because the mathematical development stimulated new 
interest among philosophers and physicists in the corresponding 
problem of the dimensionality of physical space. Indeed, it almost 
seems as if these mathematical generalizations of spatial dimen- 
sionality were looked upon as a challenge for scientists to prove 
the three-dimensionality of ordinary space. So numerous attempts 
were made in the course of the nineteenth century to prove that 
space in physics has only three dimensions. Remarkable because 
of its method is Bolzano's attempt." He refers in his proof to 

" B. Bolzano, "Versuch einer objektiven Begriindung der Lehre von den 
drei Dimensionen des Raumes," Abhandlungen der bohmischen Gesellschaft 
der Wissenschaften (Prague, 1843). 



the temporal order of the immediately given contents in our 
consciousness and tries to show that the assumption of a three- 
dimensional space is indispensable for the construction of a cor- 
respondence between these contents and the external objective 
causal connectivity. 

A very popular proof, based on similar lines, was that of Ueber- 
weg. 100 Ueberweg deduced the reality of three-dimensional exten- 
sion from the reality of time or temporal sequence, founded on 
internal experience. In his view, the empirically given order of 
time, the succession of day and night, of winter and summer, is 
based on mafhematico-physical laws that presuppose three-dimen- 
sional space. The causal interconnection of physical processes de- 
mands three-dimensionality. Like Kant, Ueberweg mentions in 
this connection Newton's law of gravitation, according to which 
the intensity of gravitation between constant masses decreases 
with the square of the distance. This law, according to Ueberweg, 
presupposes a three-dimensional space. For in a space of only two 
dimensions the intensity would decrease with the distance itself 
(the circumference of the circle being proportional to the radius ) , 
in the space of three dimensions the intensity decreases with the 
square of the distance (the surface of the sphere being propor- 
tional to the square of the radius), and in a space with more 
dimensions the intensity would decrease with a higher power 
of the distance, in accordance with the condition that every point 
affected receives a proportional part of the total effect. In other 
words, since in an n-dimensional space the gravitational force 
must decrease according to the last-mentioned condition with the 
(n — i)th power of the distance, and since Newton's law estab- 
lished a decrease with the second power of the distance, n must 
be three. It is obvious that Ueberweg's proof fails if the a priori 
character of Newton's law is rejected. Moreover, even if the num- 
ber of dimensions of the objective world should disagree with 
that of our phenomenal world, some mathematical order among 
the phenomena could still be conceived as possible, even if it 
100 F. Ueberweg, System der Logfk (ed. 5, Bonn, 1882), p. 113. 


were a distorted order or a kind of projected order. F. A. Lange 
discusses this possibility and says: 

Astronomy is but a special case, for which, under other conditions, 
something else might be substituted. For the rest, we have no absolute 
standard as to what we might demand as regards the intelligibleness 
of the world, and for this reason alone Ueberweg's standpoint is really 
based upon a concealed petitio principii. 1 ® 1 

Natorp's attempt to find an upper limit for the number of 
dimensions is based on his imposed restriction to find the mini- 
mum number of dimensions necessary and sufficient to guarantee 
a unique, closed, homogeneous, and continuous connection among 
spatial directions. 102 Through a consequent application of the 
notion of continuous rotation he assumes that he has achieved 
his purpose. Progressing from the straight line to the plane, to 
complete the manifold of directions, and progressing from the 
plane to three-dimensional space to complete the manifold of 
rotations, his deduction comes to an end apparently because the 
conception of a motion of a three-dimensional space as a whole 
lacks all intuition based on experience. It would certainly be 
beyond the scope of this chapter to discuss the other so numerous 
"proofs" of the three-dimensionality of space, as, for example, 
that given by Hegel 103 or that proposed by Trendelenburg. 104 

It is, however, curious to note that the idea of a fourth dimen- 
sion was cordially welcomed in spiritual circles. Henry More had 
already applied this notion for his spiritualistic conception of 
what he called "spissitudo essentialis." In his Enchiridion meta- 
physicum he writes: "Ita ubicumque vel plures vel plus essentiae 

1(tt F. A. Lange, History of materialism (ed. 3, trans, by E. C. Thomas; 
Humanities Press, New York, 1950), book 2, "History of materialism since 
Kant," p. 226. For Ueberweg's reply to Lange's criticism, see Ueberweg, 
Geschichte der Philosophie (ed. 2, 1869), vol. 3, p. 303. 

1<B Natorp, Die logischen Grundhgen der exakten Wissenschaften, p. 306. 
For a similar proof, see Friedrich Pietzker, "Die dreifache Ausdehnung des 
Raumes," Unterr. — Bl. f. Mathem. u. Nat. 8, 39 (1902). 

103 G. W. F. Hegel, Encyklopadie der phibsophischen Wissensdhaften im 
Grundriss (Leipzig, 1905), part 2, sec. 255, p. 214. 

101 F. A. Trendelenburg, Logische Untersuchungen (Leipzig, 1870), p. 226. 



in aliquo ubi continetur quam quod amplitudinem huius adaequat, 
ibi cognoscatur quarta haec dimensio, quam apello spissitudinem 
essentialem." 105 Supernatural phenomena as provoked by spirit- 
ualists in their stances were accounted for on the assumption 
of a fourth dimension. Most famous in this respect are the ex- 
periments performed by the German professor of astronomy, 
J. K. F. Zollner of Leipzig, in which many of his distinguished 
colleagues served as witnesses. Experiments of topological charac- 
ter, such as the untying of knots tied in closed loops of string or 
the famous instances known as "apports," the sudden appearance 
and approach of an object from nowhere, were explained as 
motions or processes in the fourth dimension of space. In his 
comprehensive work Transcendental physics™ 6 Zollner tries to 
explain on the basis of this hypothetical dimension not only 
phenomena that occur in spiritualistic sittings but also religious 
miracles of all kinds. In his Wissenschaftliche Abhandlungen, of 
which the Transcendental physics forms the third volume, he 
refers to patristic theology ( Hieronymus, Augustine, Cassiodorus, 
Gregory the Great) 107 and even to the teachings of his contem- 
porary Mach, 108 as congenial to his theory of the fourth dimen- 
sion. In his search for modern theological support for his theory 
of a four-dimensional space Zollner refers to Friedrich Christoph 
Oetinger 109 and to Johann Ludwig Fricker, 110 Oetinger's friend. 
These two theologians used some vague unorthodox formulation, 

106 H. More, Enchiridion tnetaphysicum, I, 28, §7. See also Robert Zim- 
mermann, Henry More und die vierte Dimension des Raumes (Vienna, 

106 J. K. F. Zollner, Transcendental physics (trans, by C. C. Massey; Lon- 
don, 1880 ) . The original Transcendentate Physik, vol. 3 of Wissenschaftliche 
Abhandlungen (Leipzig, 1878), was dedicated to William Crookes. See 
also Zollner s article in Quarterly Journal of Science (April 1878) and G. C. 
Barnard, The supernormal (London, 1933), chap. 8. 

107 Zollner, Transcendentale Physik, p. 600. 

108 Ibid. p. lxxxvii. 

108 Des Wirttembergischen Pralaten Friedrich Christoph Oetinger samrnt- 
liche Schriften (ed. by K. C. E. Ehmann; Stuttgart, 1858). 

110 Johann Ludwig Fricker, ein Lebensbild aus der Kirchengeschichte des 
18. Jahrhunderts (ed. by K. C. E. Ehmann; Heilbronn, 1872). 





involving the notion of a fourth dimension, in their attempt to 
explain — in a quasi-geometric way — two Biblical passages: 

Canst thou by searching find out God? canst thou find out the 
Almighty unto perfection? 

It is as high as heaven; what canst thou do? deeper than hell; what 
canst thou know? 

The measure thereof is longer than the earth, and broader than the 
sea. 111 

That Christ may dwell in your hearts by faith; that ye, being rooted 
and grounded in love, 

May be able to comprehend with all saints what is the breadth, and 
length, and depth, and height; 

And to know the love of Christ, which passeth knowledge, that ye 
might be filled with all the fulness of God. 112 

With regard to Christian teratology Zollner says: "Das sacrifi- 
cium intellectus welches die christlichen Wunder vom Verstande 
bisher verlangten, ist durch die Entdeckung jenes neuen Ge- 
bietes der Physik — der Transcendentalphysik — zum ungerriib- 
ten Genusse des Neuen Testamentes nicht mehr erforderlich." 113 
That literature of this kind was popular not only at the end of 
the last century is shown today by the wide circulation of C. H. 
Hinton's books, A new era of thought 114 and The fourth dimen- 
sion, 115 and in particular by P. D. Ouspensky's Tertium organum, 
A key to the enigmas of the world. 116 

Poincare" attempted to demonstrate the three-dimensionality of 
the space of experience by the following simple topological con- 
sideration. 117 Space cannot be separated into parts by isolated 
points (as in the case of a one-dimensional extension) nor by 

m Job 11:7-9. 

112 Ephesians 3:17-19. 

us Zollner, Wissenschaftliche Abhandlungen, vol. 2, part 2, p. 1187. For 
a contemporary critique of Zollner's theory, see Gutberlet, Die neue Raum- 
theorie (Mainz, 1882). 

111 New York, ed. 2, 1923. See also P. D. Ouspensky, A new model of the 
universe (Knopf, New York, 4th ptg., 1944), chap. 2, "The fourth dimen- 
sion," pp. 61-100; G. B. Burch, "The philosophy of P. D. Ouspensky," 
Review of Metaphysics 5, 247 (1951). 

115 London, 1888. 

^ Allen and Unwin, London, 1934. 

m H. Poincare, DemHres pensSes (Paris, 1917), p. 61. 

curves (as in the case of two-dimensional extensions). Since, 
however, a closed surface separates space into disjunctive parts, 
Poincare thought that he had found the fundamental qualitative 
ground for ascribing three-dimensionality to ordinary space. It 
is, however, clear that his proof demonstrates at best only the 
existence of a lower limit to the number of dimensions. 

In an article written in the last year of his life Poincare devel- 
ops these ideas in detail. He says: 

The most important of all theorems of analysis situs is the state- 
ment that space has three dimensions . . . What do we mean when 
we say that space has three dimensions? ... To separate space into 
parts, cuts are necessary which we call surfaces; to disconnect surfaces, 
cuts are necessary which we call fines; to divide fines, cuts are neces- 
sary which we call points. But we cannot go further, since a point, not 
being a continuum, cannot be divided. Therefore, lines that can be 
disconnected by cuts which themselves are not continua are continua 
of one dimension; surfaces that can be separated into parts by one- 
dimensional continua are continua of two dimensions; and finally 
space, which can be separated by two-dimensional continua, is a con- 
tinuum of three dimensions. 118 

Poincare was primarily interested in the physical and philo- 
sophical implications of the meaning of the concept of dimension, 
and yet this essay can be counted as the beginning of modern 
topological research concerning the mathematical problem of 
dimensionality. Although it was not his intent to formulate a 
strict mathematical definition of the concept of dimension, he 
anticipated the two essential elements of the modern definition 
of this term: the use of disconnecting subspaces and the induc- 
tive character of the definition. In fact, Brouwer's well-known 
topological invariant definition of dimension, 119 which for locally 
connected separable metric spaces is at present in common use, 
is based on Poincare's considerations. Mathematicians, who until 
the beginning of the present century used the concept of dimen- 
sion in a rather vague sense, became interested in a precise 

US H. Poincar6, Revue de mitaphysique et de morale 20, 486 (1912). 
119 L. E. J. Brouwer, "Ueber den naturlichen DimensionsbegrifF," /. reine 
u. angew. Math. 142, 146-152 (1913). 




definition with the rise of the modern theory of sets. Cantors 
famous one-to-one correspondence between the points of a line 
and the points of a plane, and Peano's continuous mapping of an 
interval on the whole of space, showed the deficiencies of the 
traditional definition of dimensionality as the smallest number of 
continuous real parameters sufficient to determine the position 
of a point. It was only in 1911 that Brouwer 120 established 
the proof that Euclidean spaces of different dimensionality are 
nonhomeomorphic, that is, cannot be mapped on each other by a 
continuous one-to-one correspondence. Important contributions 
by H. L. Lebesgue, K. Menger, P. Urysohn, and W. Hurewicz 
led to a further clarification of the mathematical concept of 

To solve the problem of the dimensionality of space is also the 
ambition of modern physics. Among various attempts to this end 
the most noteworthy are probably those of Sir Arthur Eddington 
and H. Weyl. In his Fundamental theory Eddington succeeded 
by means of a complicated system of notions in reducing the 
problem to a reality investigation (in the mathematical sense of 
the word) of the so-called E-frame, a purely mathematical con- 
struct that is brought into physics to be identified with space- 
time. Eddington says: 

The three-dimensionality of space and the time-like character of the 
fourth dimension are thus deduced directly from the properties of 
the E-frame. To what extent this amounts to an a-priori proof that 
the space-time of physical experience must be of this kind, depends 
on our inquiry into the ultimate origin of the E-frame in Chapter 

Unfortunately, Eddington died without having completed Chap- 
ter XIII. A note, probably written on the last day of his working 
life, indicates that the proposed chapter was to have been based 

120 L. E. J. Brouwer, "Beweis der Invarianz der Dimensionenzahl," Math. 
Ann. 70, 161-165 (i9ii)- 

121 A. S. Eddington, Fundamental theory (Cambridge University Press, 
Cambridge, 1946), p. 124. 



on his article on "The evaluation of the cosmical number." 122 Not 
enough material was published by him to furnish a decisive 
answer to the problem. 

Weyl, in order to explain the three-dimensionality of space, 
refers to his generalization of Riemannian space to non-Rieman- 
nian gage-invariant geometry. He shows that only in a world 
conceived as a (3 + 1) -dimensional gage-invariant manifold 
(three spatial dimensions and one temporal dimension) does a 
most simple integral invariant exist in the form of action on 
which Maxwell's theory is founded. 123 The electromagnetic-field 
tensor is identified with "distance curvature" and Maxwell's 
equations appear as an intrinsic law. 

And since it is impossible to construct an integral invariant at all of 
such simple structure in manifolds of more or less than four dimen- 
sions the new point of view does not only lead to a deeper un- 
derstanding of Maxwell's theory but the fact that the world is four- 
dimensional, which has hitherto always been accepted as merely 
"accidental," becomes intelligible through it. 124 

The proof would be complete if it could be shown that all 
laws of gravitation as well as of electromagnetism are derivable 
from a variational principle that has to comply with the require- 
ments of this invariance. It must, however, be admitted that 
Weyl's approach, ingenious as it is, is still open to serious objec- 
tions. 125 In another attempt to include electromagnetic potentials 
in the metric of space, T. Kaluza 126 retained, in opposition to 
Weyl, the Riemannian character of space, but assumed an ad- 
ditional fifth dimension of the substratum underlying physical 
phenomena, thereby increasing the number of the components 

110 A. S. Eddington, Proc. Camb. Phil. Soc. 40, 37 (1944). See also 
Eddington, Relativity theory of protons and electrons ( Cambridge University 
Press, Cambridge, 1936), chap. 6, "Reality conditions," and p. 325. 

138 H. Weyl, Sitzber. preuss. Akad. Wiss. (1918), p. 465; Ann. Physik 59, 
101 (1919). 

"* H. Weyl, Space-time-matter (London, 1922), p. 284. 

136 See P. G. Bergmann, Introduction to the theory of relativity (Prentice- 
Hall, New York, 1950), p. 253. 

""T. Kaluza, Sitzber. preuss. Akad. Wiss. (1921), p. 966. 


of the metrical tensor. Similar procedures were employed in the 
so-called projective field theories of Veblen, 127 Hoffmann, 128 and 
Pauli, 129 and can be compared to the well-known representation 
in geometry of an n-dimensional space by (n + 1) homogeneous 
coordinates. The use of five-dimensional tensors gained much 
popularity in other generalizations of general relativity, as for 
example in the modification introduced by Einstein and Mayer 130 
and similar developments, 131 in particular in connection with the 
ambitious intention to account relativisticalry for the results of 
quantum mechanics. In conclusion of our treatment of the prob- 
lem of dimensionality we may state that up to date no satisfying 
solution has been given. H. Grassmann's words, announced in 
1844, have not yet been disproved: 

The concept of space can in no way be produced by thought, but 
always stands over against it as a given thing. He who tries to main- 
tain the opposite must undertake the task of deducing the necessity 
of the three dimensions of space from the pure laws of thought, a 
task whose solution presents itself as impossible. 132 

Heisenberg, in his attempt 133 to achieve a simplified general 
representation of quantum mechanics, tried recently to abandon 
the principle of continuity in Riemannian or Euclidean geometry 
and introduced the suggestion of a "smallest length" to meet 
certain difficulties in quantum electrodynamics. This introduc- 
tion of a discrete space with a quantum of length — Margenau 134 
calls it a "hodon" from the Greek hodos, path, in analogy with 
the term "chronon" — would lead to a drastic revolution in the 
whole of theoretical physics. All differential equations would 

127 O. Veblen, Projektive Relativitatstheorie (Berlin, 1933). 

128 O. Veblen and B. Hoffmann, "Projective relativity," Phys. Rev. 36, 810- 
822 (1933)- 

139 W. Pauli, Ann. Physik. 18, 337 (1933)- 

"°A. Einstein and W. Mayer, Sitzber. preuss. Akad. Wiss. (1931), P- 
541; (1932), p- 130. 

lm Cf . the concluding chapters in Bergmann's Introduction to the theory 
of relativity. 

"~ 1 Grassmann, Die Ausdehnungslehre (ed. 2, Leipzig, 1878), p. xxiii. 
'Earliest reference, W. Heisenberg, Z. Physik 110, 251 (1938). 
'Henry Margenau, The nature of physical reality (McGraw-Hill, New 
York, 195°). P- 155- 


188 1 
184 1 


have to be recast into difference equations for the solution of 
which mathematicians would have to face almost insurmountable 
difficulties, although the subject of a finite geometry of discrete 
space structure has already been investigated, in particular by 
O. Veblen and W. H. Bussey. 135 From the historical point of 
view it is interesting to note that this possibility had also been 
envisaged already by Riemann. In the remarkable passage quoted 
on page 159 he said: 

If in a case of a discrete manifold the basis for its metrical deter- 
mination is contained in the very idea of this manifold, then for a 
continuous one it should come from without. The reality which lies at 
the basis of space, therefore, either constitutes a discrete manifold, 
or the basis of metrical determination must be sought outside the 
manifold in the binding forces which act on it. 

For the introduction of a "system of linkages" impressed upon 
a discrete manifold as a substitute for the fundamental tensor, the 
reader is referred to L. Silberstein s The theory of relativity. 1315 

The concept of a smallest length, or rather a fundamental 
length, as characterizing the ultimate limit of resolution in phys- 
ical measurement of spatial extension, has recently gained some 
popularity amongst theoretical physicists. Apart from Heisenberg, 
as mentioned above, A. March in particular advocated the as- 
sumption of a universal smallest length l . 137 A physical theory 
of spatial extension, he claims, has to be built on concepts that 
can be specified by their operational contents. The traditional 
geometry of points and infinitesimal magnitudes, therefore, has 
to be discarded as far as its immediate application in atomic 
physics is concerned. For any measurement is based ultimately 
on the coincidence of a scale with the object to be measured; 
and for the physicist the elementary particle is the smallest scale 
(or unit) available. The application of a concept of a still smaller 
spatial extension, not to say a pointlike extension, must — accord- 
ing to this school of thought — lead inevitably to insurmountable 

186 O. Veblen and W. H. Bussey, Trans. Am. Math. Soc. 7, 241 (1906). 
""L. Silberstein, The theory of relativity (London, 1924), p. 362. 
m Arthur March, Natur und Erkenntnis (Springer, Vienna, 1948). 



difficulties. Indeed, the idea of a pointlike electron, for example, 
would imply the concentration of an infinite energy, while the 
conception of an extended rigid electron would contradict the 
principle of relativity. Since two particles whose distance apart 
is less than k cannot be distinguished by diffraction experiments, 
l becomes a universal length independent of the particular 
character of the particle in question. 

The traditional conception of point coincidences has to be 
replaced by the notion of coincidences of particles. The spatial 
extension of elementary particles becomes apparent from the 
fact that a coincidence of particles A and B on the one hand, 
and of particles B and C on the other, does not necessarily lead 
to the result that A and C also coincide. The distance between 
two particles is determined by the minimal number of particles 
necessary to form a "chain of coincidences" between the given 
particles. Distances are therefore always integral multiples of 
l . The repeated occurrence in atomic physics of a length of the 
order of io~ 13 cm, as the classical radius of the electron, the 
range of nuclear forces, or the critical energy of io 8 electron 
volts, corresponding to a wave length of io" 13 cm, leads to the 
assumption that this length may be identified with Jo- 
in view of the great mathematical difficulties involved in the 
construction of a geometry of discontinuous space, however, 
physics has still to resort to the traditional geometry of a con- 
tinuous space by a statistical treatment of the concept of length. 
Thus continuous space resumes its service, even for nuclear 
physics, but as a convenient fiction for the statistical mathematiza- 
tion of physical reality. 

A similar result seems to follow from even more general 
considerations. A profound epistemological analysis of certain 
quantum-mechanical principles seems to suggest that the tradi- 
tional conceptions of space and time are perhaps not the most 
suitable frame for the description of microphysical processes. 
Thus Heisenberg's uncertainty principle states that the uncer- 
tainty involved in the measurement of the coordinate x of a 



particle and the uncertainty involved in the simultaneous de- 
termination of the momentum p are governed by the relation 
Ax • Ap^h (h is Planck's constant). The impossibility of an 
exact localization in combination with the determination of the 
momentum, and the related dualistic wave-particle character of 
physical reality, can be interpreted as a challenge for a critical 
revision of the accepted space and time conceptions. In his dis- 
cussion of electron transitions between stationary states within 
the atom, Niels Bohr already called such processes "transcending 
the frame of space and time." The problem of an intelligent 
applicability of traditional space and time conceptions to atomic 
physics was the subject of a paper submitted by Louis de Broglie 
to the Tenth International Congress of Philosophy (Amsterdam, 
August 11-18, 1948). De Broglie admits frankly the difficulties 
involved in the use of our notions of space and time on a micro- 
physical scale, but also confesses in the conclusion of his article 
that up to the present no alternative conceptual categories are 
known that can be substituted. He says: 

Les donnees de nos perceptions nous conduisent a construire un 
cadre de l'espace et du temps ou toutes nos observations peuvent se 
localiser. Mais les progres de la Physique quantique nous amenent a 
penser que notre cadre de l'espace et du temps n'est pas adequat a 
la veritable description des realites de l'echelle microscopique. Cepen- 
dant, nous ne pouvons guere penser autrement qu'en termes d'espace 
et de temps et toutes les images que nous pouvons evoquer s'y ratta- 
chent. De plus, tous les resultats de nos observations, meme celles qui 
nous apportent le reflet des realites du monde microphysique, s'expri- 
ment necessairement dans le cadre de l'espace et du temps. C est 
pourquoi nous cherchons tant bien que mal a nous representor les 
realites microphysiques (corpuscules ou systeme de corpuscules) dans 
ce cadre qui ne leur est pas adapte. 138 

Not only the notion of continuity, but also the conception 

of "emptiness" have recently been subjected again to critical 

examination. P. W. Bridgman, the keen operational analyst of 

physical concepts, expounded the dilemma raised by submitting 

"L. de Broglie, "L'espace et le temps dans la physique quantique," 
-"■"*>— ** of the * — *'-•'—» *• — » -' ' -•'-- » '»* -• —* 

Proceedings of the tenth international congress of philosophy ( North-Holland 
Publishing Co., Amsterdam, 1949), vol. 1, p. 814. 




l8 9 

the concept of empty space to the operational point of view. 139 
Clearly no instrumental method can exist for testing such empti- 
ness. The mere introduction of an instrument for this purpose 
already invalidates the very conditions of the situation under 
test. Moreover, no theory that attempts to eliminate the per- 
turbations caused by the test body or test instrument (for ex- 
ample, a thermometer) can be applied to this situation, for such 
a theory has to be based on the variations in the reading of 
the instrument under changing conditions. Still "the intellectual 
compulsion remains to give some instrumental meaning to the 
purported emptiness of space. The simplest way of meeting this 
compulsion is simply to say that the space is empty if no instru- 
ment gives any reading when introduced into it." 140 But even 
this highly problematic specification seems to be untenable if 
confronted with the concept of an electrostatic field with fluc- 
tuating zero point, as advanced recently by quantum mechanics. 
If physics has to maintain the idea of empty space, it seems 
to be possible only by "ignoring part of the operational back- 
ground." m 

It will be recollected that it has been suggested by Riemann 
and Clifford, and later ingeniously corroborated by Einstein in 
his theory of general relativity, that the metric of space structure 
is a function of the distribution of matter and energy. In accord- 
ance with this principle the large-scale properties of space be- 
came the object of cosmological research during the last three 
or four decades. In fact, Einstein's paper on "Cosmological con- 
siderations in general relativity," 142 published in 1917, turned 
the investigations of macroscopic space structure (for example, 
the question whether space is finite or infinite) and cosmology 
in general from poetical and philosophical speculations into a 

138 P. W. Bridgman, The nature of some of our physical concepts (Philo- 
sophical Library, New York, 1952). 

140 Ibid., p. 19. 

141 Ibid. 

148 A. Einstein, "Kosmologische Betrachtungen zur allgemeinen Relativitats- 
theorie," Sitzber. preuss. Mad. Wiss. (1917), p. 142. 

respectable scientific discipline with solid foundations in physics, 
in spite of the fact that many of its most important issues are still 
under debate. The article has shown incontestably that the mathe- 
matical apparatus of general relativity, when applied to cosmo- 
logical problems, could bring science a decisive step nearer to 
a solution of the large-scale problems of space. 

It has been shown by H. P. Robertson 143 and independently 
by A. G. Walker 144 that essentially only the following metric is 
compatible with the assumption of a large-scale homogeneous 
isotropic space-time continuum: 

ds 2 = dt 2 - 

[R(t)] 2 [(da 1 ) 2 + (dx 2 ) 2 + (dx 3 ) 2 ] 
{i + Mfc[(x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 ]} 2 ' 

where k may have the values o, — 1, or 1. Confining our consider- 
ation to purely spatial extension [t = const, and R = R(t)], we 
see that if k = o, space is Euclidean: the surface of a sphere with 
radius r is 4^-r 2 . If k = -1, space is hyperbolic: the surface of a 
sphere with radius r is $4*R 2 sinh 2 (r/^R), that is, greater than 
4TT 2 , as can be seen at once from the expansion into a power 
series. Finally, if k= +1, space is spherical: the surface of a 
sphere with radius r is 647^ sin 2 (r/4R), that is, less than 4^. 
We see from the last formula that a sphere has a maximum sur- 
face if its radius is equal to 2nR; if its radius is greater than twice 
the "radius of the universe R," the surface of the sphere decreases 
until for r = 4^ it shrinks to a point. A null-geodesic forms a 
closed curve, that is, light rays return to their starting point. If 
the Robertson line element is transformed to polar coordinates, 
a simple integration shows that Euclidean and hyperbolic spaces 
are unlimited in volume, whereas spherical space, at any instant 
*, has a total finite volume z^R 3 . 

It is well known that Einstein's classical paper of 1917 char- 
acterized space as endowed with the last-mentioned of these 
properties. Much has been written about the astronomical im- 

143 H. P. Robertson, Proc. Nat. Acad. Sci. 15, 822 (1929). 

144 A. G. Walker, Proc. London Math. Soc. 42, 90 (1936). 




plications of this result. The curious history of the cosmological 
term, introduced by Einstein in connection with Mach's principle, 
the revisions performed by de Sitter for empty space, the as- 
tounding discovery of actual large-scale motion of stellar systems, 
and the following mathematical elaborations 145 by Friedmann, 
Lemaitre, and Robertson are too well known to be repeated here. 

In Milne's theory of kinematical relativity the four-dimensional 
space-time continuum had to be described as Euclidean and 
consequently the three-dimensional space as hyperbolic. 146 In 
Dirac's cosmology three-dimensional space must be Euclidean, 
for any curvature would imply a time-variant number of ele- 
mentary particles, in contradiction to the principle of conserva- 
tion of mass. 147 The steady-state theory, which rejects this prin- 
ciple, resumes de Sitter's metric. 148 And so forth. It certainly goes 
beyond the scope of our subject to give a detailed account of 
these conclusions or to describe the conceptions of space struc- 
ture according to still more controversial cosmological theories, 
such as those of Eddington 149 or Jordan. 180 

The purpose of these concluding remarks has been merely to 
show that the investigations of large-scale properties of space, 
although still highly debatable in their conclusions, have become 
finally the object of scientific research. It can only be hoped that 
progress in statistical studies of stellar systems together with ever- 
increasing depth of observational penetration into space will 
bring man to the solution of these questions. Like all science, 
the science of space must still be classed as unfinished business. 

145 Jean Becquerel recently deduced the non-Euclidean structure of space- 
time in a gravitational field from the fact that the energy of a photon is 
proportional to the frequency of the associated wave; see Compt. rend. 232, 
No. 18 (30 April 1951). 233, No. 11 (10 Sept. 1951)- 

146 E. A. Milne, Kinematic relativity (Oxford University Press, London, 

147 P. A. M. Dirac, Proc. Roy. Soc. (London) A 165, 199 (1938)- 
148 H. Bondi, Cosmology (Cambridge University Press, Cambridge, 1952). 
p. 146. 
14, A. S. Eddington, Fundamental theory. 
160 P. Jordan, Nature 164, 637 (1949). 


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fried (1776-1858), 146 

Neugebauer, Otto Eduard, 4 



Neumann, Carl (1832-1925), 93. 

Newton, Isaac (1642-1727), xiv, 

2, 26, 32, 40, 43, 92, 93, 125- 

129, 137, 139-143, 167, 171, 177 
Nicolaus of Autrecourt (c. 1308- 

c. 1360), 67, 74, 81 
Nicolaus Boneti (d. c. 1360), 67 
Nicolas of Cusa (Krebs, Cusanus) 

(1401-1464), 80-82 
Norris, John (1657-1711), 40 
Northrop, Filmer Stuart Cuckow 

(1893- ). 94 

O'Donnell, J. R., 74 

Oetinger, Friedrich Christoph 

(1702-1782), 179, 180 
Olbers, Heinrich Wilhelm Matthias 

(1758-1840), 145 
Oldenburg, Heinrich (1620-1677), 

Ouspensky, Peter Demianovich 

(1878-1947). 180 

Paracelsus, Theophrastus B. von 
Hohenheim (c. 1490-1541), 32^ 
Parmenides (b. c. 539 B.C.), 9 
Patritius, Fransciscus (Patrizi) 
(1529-1597), 38, 83-88, 90, 108 
Pauli, Wolfgang (1900- ), 184 
Peano, Giuseppe (1858-1932), 182 
Pearson, Karl (1857-1936), 161 
Peiresc, Nicolas Claude Fabri de 

(1580-1637), 32 
Philo Judaeus (c. 20 b.c.-c. a.d. 

40). 30, 35 
Philolaus (b. c. 480 B.C.), 7, 13 
Philoponus, Joannes (John the 

Grammarian) (c. 575), 52-56 
Pines, Salomon, 60, 62 
Pietzker, Friedrich, 178 
Planck, Max (1858-1947), 187 
Plato (429-348 b.c), 12-14, 36, 

Plotinus (c. 204-270), 35, 39 
Pluecker, Julius (1801-1868), 176 
Plutarch (c. 46-120), 12 
Poggio, Gian Franscesco Bracciolini 

(1380-1459), 11 
Poincar6, Henri (1854-1912), 142, 

163-167, 180, 181 


Poisson, Simeon Denis ( 1781— 

1840), 137 
Posidonius (c. 135-51 B.C.), 21 
Proclus (410-485), 24, 27, 36, 143 
Ptolemy, Claudius (c. 150), 24, 

102, 143 
Pythagoras of Samos (c. 530 b.c), 

7, 18, 172 

Rainoldes, John (Reynolds) (1549- 

1607), 32 
Raphson, Jacob (1648-1716), 127 
Reuchlin, Johann von (1455-1522), 

Ricci, Gregorio (1853-1925), 150 
Richard of Middleton (d. c. 1307), 

73, 74 
Riehl, Alois (1844-1924), 162 
Riemann, Bernhard (1826-1866), 

149-160, 171, 175, 183, 185, 188 
Rittangelius, Johann Stephanus (d. 

1652), 27 
Robertson, Howard Percy ( 1903- 

), 154, 189, 190 
Rosenroth, Baron Knorr von (b. 

1631), 39 

Saadya ben Joseph, of Fayum (c. 
892-942), 36 

Saccheri, Geronnimo (1667-1733), 
143, 144 

Sarton, George (1884- ), 60 

Scaliger, Julius Caesar (1484- 
1558), 82, 83 

Schechter, Salomon, 29 

Schelling, Friedrich Wilhelm Jo- 
seph (1775-1854). 146, 175 

Scholem, Gerhard, 39 

Schooten Franz van (d. 1659), 24 

Schott, Kaspar, 42 

Schouten, Jan Arnoldus ( 1883- 
), 122 

Schumacher, Heinrich Christian 
(1780—1850), 146 

Schur, Friedrich (b. 1856), 158 

Schwarzschild, Karl (1873-1916), 

Schweikart, Ferdinand Karl (1780- 

1859), 145 
Seeliger, Hugo (1849-1924), 139 

i 9 6 

Sextus Empiricus (end of second 

century), 12, 27, 68, 69 
Sigwart, Christoph von (1830- 

1904), 162 
Silberstein, Ludwik (1872- ), 

Simon ben Shetah (c. 100 B.C.), 29 
Simon ben Yohai (c. 90-c. 160), 31 
Simon the Just (c. 300 B.C.), 29 
Simplicius (c. 530), 7, 36, 52, 54, 

59. 60, 173 
Sitter, Willem de (1878-1934), 

189, 190 
Spinoza, Baruch (1632-1677), 47 
Staeckel, Paul Gustav (1862-1919), 

150, 157 
Staudt, Karl Georg Christian von 

(1798-1867), 156 
Steinschneider, Moritz, 76 
Strato of Lampsacus (c. 275 b.c), 

51, 173 
Stumpf, Carl (1848-1936), 136 

Tanhuma (c. 335-c. 370), 29 

Tannery, Paul, 7 

Telesio, Bernardino (1509-1588), 

32, 83, 84, 88, 90 
Tempier, Etienne (c. 1200-c. 

1261), 58 
Themistius (317-c. 387), 53, 57 
Theophrastus (b. c. 372 b.c), 21, 

Thomas Aquinas (c. 1225-1274), 

50, 66 
Thorndyke, Lynn (1882- ), 82 
Trendelenburg, Friedrich Adolf 

(1802-1872), 178 
Tulloch, John (1823-1886), 40 

Ueberweg, Friedrich (1826-1871), 
177. 178 

Urysohn, Paul (1898-1924), 182 

Veblen, Oswald (1880- ), 184 

Vincenz of Beauvais (1194-1264) 


Wachter, Johann (1663-1757), 47 
Walker, Arthur Geoffrey (1909- 

), 189 
Walks, John (1616-1703), 143 
Waltershausen, Wolfgang Sartorius 

von (1809-1876), 175 
Ward, Richard (c. 1657-1723), 39 
Watts, Isaac (1674-1748), 127 
Weinberg, Julius Rudolf, 67 
Weyl, Hermann (1885- ), 131, 
132, 150, 154, 156, 159, 182, 183 
Whewell, William (1794-1866), 

Whitehead, Alfred North (1861- 

1947), 1 
Wien, Wilhelm (1864-1928), 14! 
Willey, Basil, 126 
William of Auvergne (d. 1249), 37, 

William of Occam (Occham) (d. 

c- 1349), 69, 70 
Witelo (b. c. 1220), 35, 37 
Wolff, Christian (1679-1754), 93 
Wolfson, Harry Austryn, 26, 29, 74, 

Zeller, Eduard (1814-1908), 6, 12, 

13. 56 
Zeno of Elea (beginning of fifth 

century B.C.), 3, 16 
Zimmermann, Robert, 179 
Zoellner, Johann Karl Friedrich 

(1834-1882), 179-180