THE RIEMANN PROGRAM

Hyperspatializing the Absolute, 1854-

1905 James E. Beichler

©1985, 2013

All rights reserved

Preface

This book was originally written, submitted and approved for a Master’s thesis in the History of

Science at the University of Maryland, College park in 1985. My Master’s committed consisted of

Professors Stephen G. Brush (Physics and History of Science) who was also my advisor and

committee chair, Frank Haber (History of Ideas and Cultural History) and Robert Friedel (History

of Technology). The thesis represented an exploration into one of the most important eras in the

History of Physics, the final decades of the nineteenth century just prior to the onset of the Second

Scientific Revolution. From all the time, sweat and research that had gone into studying the Physics

of that time period, I came to the conclusion that specific shortcomings existed in all of the previous

histories of science that I studied in graduate school. Quite frankly, important issues that influenced

both late Newtonian science and the course of the Second Scientific Revolution were literally

missing from the history books. I was particularly interested in this topic because I came to the

History of Science from a background in Physics. In particular, why had no serious physical

applications of the new non-Euclidean geometries been conducted before Einstein in spite of the

fact that Riemann’s work was considered groundbreaking a half century earlier?

The few references that I found to this subject all seemed to either say or strongly imply that

scientists, philosophers and other concerned scholars of the time period in question separated the

mathematical possibilities of the new geometries from their physical implications, so no serious

attempts were ever made to find space curvature or the physical consequences of possible

hyperspaces. Yet I also knew that mathematics was considered the “queen” of the sciences during

the nineteenth century, which meant that mathematics, mathematical possibilities and scientific

realities were intimately tied together. These two notions seemed at complete odds with each

another, so I sought out to investigate what really occurred during the nineteenth century just prior

to Einstein’s development of general relativity in 1915.

What I found was astonishing. The history taught and repeated by scientists and historians was

at least sadly incomplete if not completely false. There actually exists a very rich history of attempts

to apply the non-Euclidean geometries and hyperspaces to explain physical phenomena during the

last three decades of the nineteenth century. I also found numerous references to attempts to

measure space curvature through astronomical observations, which were not supposed to have

existed during the nineteenth century. These historical events were not conducted in secret of by a

few scientists and known to only a select group within the scientific community. The concepts and

ideas involved were well known to scientists and non-scientists alike. They were even popular within

the lay population. Once I placed these new discoveries within the historical context of the era

whole new questions regarding the originality of Einstein’s general theory of relativity, its

interpretation and acceptance by the scientific community were raised. Answering these questions

became the goal of my work.

The central and defining event of this story was not hard to find. Riemann’s 1854 inaugural

lecture or Habilitationsschrift at the University of Göttingen, which set forth a new program for the

development and application of his new differential geometry of surfaces and defined the new non-

Euclidean geometries relative to one another, was the starting point of all the ensuing changes in

science. Even Einstein credited Riemann for his mathematical discoveries and referred directly to

Riemann’s 1854 lecture. Given these historical facts, it is hard to understand why the history of the

last few decades of nineteenth century physics is so wrong, at least relative to these events and

developments. Yet even recognizing and defining this problem was not enough because this study

also raised serious questions about the factors, causes and reasons for the Second Scientific

Revolution as a whole, whether we truly understand what the revolution was about and what really

changed in science during the revolution. Was the Second Scientific Revolution really what we

believe it was? Did if happen in the way that we have been taught?

These particular questions were answered directly in this study although some answers were

implied. The answers were not even attempted in this study because the possibilities simply stagger

the mind and a wider study would have been required to pay the questions any justice. So answers to

these questions have been left to a later and more comprehensive history of physics from ancient to

modern times to the present that will be published later under the title Phallacies in Fysics. However,

more details on this particular problem and the central role of William Kingdon Clifford in late

nineteenth century physics have been published under the title Twist and Shout: Clifford’s (not so) secret

program for “Solving the Universe”.

This work is analytical throughout. The writing style is argumentative in order to prove the

point being made. The story presented may therefore seem somewhat convoluted and prove

difficult to follow for some people interested in the history of science but untrained in science

and/or its history, but the read will be well worth the effort. I have edited the book to the best of

my ability, but I still seem to find typing errors here and there. I apologize for any errors that I have

missed. I am not the best editor for my own writing because I tend to find more things to say

whenever I try to edit what I have already written. I have also made specialized modifications to the

text for publication as an ebook, such as deleting and changing many characters (i.e. Greek letters

and foreign characters with specialized markings) that are not available for use by ebook readers and

tablets. I hope that these changes do not make the text less readable and understandable in places. I

invite everyone to my webpage at “www.neuorcosmology.net” for more information on the subjects

mentioned in this book.

James E. Beichler

July 2013

Introduction

According to the commonly held simplistic view in the history of science, Newton’s concept of

absolute space was accepted without change until replaced by Einstein’s special and general theories

of relativity. Mach had successfully challenged the concept of an absolute space in the 1870s, while

the concept of relative space slowly eroded the foundations of Newtonian absolute space during the

final decades of the nineteenth century. Speculations were made by Clifford and Riemann which

anticipated Einstein’s relativity, but neither amounted to much and had little or no influence on the

science of the day. However, Riemann and Clifford did not just anticipate Einsteinian general

relativity.

Riemann suggested a specific mathematical program for developing a new physical model of

absolute space that represented a great advance over Newton’s concept without detracting from the

rising influence of relative space in physics. He proposed that physical space had an infinitesimal

structure that was quite possibly different from that expected by the physical investigation of

macroscopic geometries and only experiment could verify which of several geometries was true. In

this respect, he suggested that absolute space had real physical properties. Clifford and a small number

of other investigators developed theories following and expanding upon Riemann’s suggested

program. Their ideas contended with other concepts of space, especially the aether theories, until

Einstein finally established the dominance of relative space in 1905 and thereafter.

In other words, the new science that developed during the second half of the nineteenth

century has been incorrectly portrayed in the commonly accepted history of science. No one can

doubt that great changes in science took place during the late nineteenth century and that these

changes preceded the ‘Revolution in Physics’ concerning the quantum and relativity. These changes

are evident in the rise of thermodynamics, Maxwell’s electromagnetism and the discovery of

radioactivity. Even as these changes took place in the scientific theories that described our world,

other important questions were raised about the scope and methods of science. These philosophical

questions can best be exemplified by the rise of positivism and the decline of Kantian philosophy.

Despite the importance of these and other changes to the subsequent founding of quantum

theory and relativity, concurrent but far more basic and subtle fundamental changes were working

themselves out in scientific thought and philosophical attitude toward the overall concept of space.

The Newtonian concept of space was being directly challenged and the ensuing subtle changes in

the scientific attitude toward space provided the background to the new concepts of physics that

emerged after the turn of the century. The challenge to Newtonian absolute space has been noted

and documented by historians, but their investigations have been limited to only one of the

important factors in the change: The rise of the Machian or pre-Einsteinian concept of relative

space. Another important factor, the change in the concept of absolute space, has gone relatively

unnoticed and unheralded.

Newtonian mechanics depended upon and perhaps even took for granted an absolute

homogeneous Euclidean space of three dimensions, devoid of any empirical or inherent physical

properties. Yet Newton recognized the importance of the relative nature of space as did Leibniz,

Berkeley, Euler, Kant and others (to varying degrees). Neither absolute nor relative space could

completely account for all physical phenomena, so Newton was forced to develop a rudimentary

concept of the aether as a medium for the propagation of gravity, thus protecting the physical and

philosophical integrity of his absolute space. Even though the Newtonian concept of absolute space

came to be universally accepted, this alternative space-like material called the aether became

necessary to fulfill the mechanical functions (of light and gravity propagation) and physical

properties of which absolute space was itself devoid.

By the beginning of the nineteenth century, the Newtonian concept of absolute space had

become a superfluous hypothesis even though it was not shown to be so until Einstein developed

special relativity in 1905. The idea was generally accepted since only relative space was necessary for

the explanation of all mechanical phenomena: Physicists such as Joseph-Louis Lagrange and Pierre-

Simone Laplace were able to complete the application of Newton’s mechanics of the world a

century earlier without any reference to the concept of absolute space. Yet the currents of history

never follow such simple paths and new factors that directly influenced the fundamental concepts of

science were developing at the same time. With an increasing knowledge of the world came new

ideas and the discovery of new phenomena in the study of electricity and magnetism while light was

shown to have a wave nature and interacted with matter in strange unsuspected ways. These

discoveries raised new questions about space which quickly became burning issues within the

scientific community. Just as the aether had been developed to transmit the forces of gravity across

the expanses of space, the expanding notions of these new phenomena increased the role and scope

of the aether, and thus absolute space, in so far as absolute space was associated with the aether.

Meanwhile, the advent of non-Euclidean geometry, especially with regard to the Riemannian

concept of space, heavily influenced the whole evolutionary process by which concept of absolute

space was progressing. Riemann not only developed a new non-Euclidean geometry that was

subsequently named after him, but he offered a complete philosophy of physical absolute space

which challenged the prevalent Kantian views of absolute space and suggested a new relationship

between physics and geometry: Riemann proposed a completely new program for physics based on

his new concepts of geometric space.

The completeness of Euclidean geometry had been questioned (by way of the parallel postulate)

since the time of Euclid, but no alternative or competing system of geometry emerged until the

development of hyperbolic geometry by Karl Friedrich Gauss, Nicholai Lobachevski and Janos

Bolyai between 1815 and 1835. Until this time there had been only one ‘geometry’, Euclid’s, and that

geometry alone was associated directly with absolute space by Newton. If other geometries were

mathematically possible, then the possibility existed, however small, that space could ultimately be

described in terms of these new geometries and that was a very novel and revolutionary idea.

However, this new non-Euclidean geometry still remained closely connected with a Newtonian

concept of absolute space with three dimensions. It was not until Riemann’s thesis (reinforced by

the independent developments of Helmholtz) that the concept of space in general and

dimensionality in particular came under closer philosophical and empirical scrutiny with results that

many times seemed quite unscientific. After Riemann, several attempts were made by well-known

and respected scientists to show the physical importance and relevance of hyperspaces based upon

the Riemannian model of absolute space. Riemann’s work and its physical extensions during the late

nineteenth century have been considered by scientists, philosophers and historians to be of little or

no consequence to later developments in physical science, but the historical records of the era show

something entirely different. They tell another story. The philosophical changes which resulted from

the debates over the new geometries were of vast importance in reformulating the attitudes of

scientists and scholars on the relationship between space and physics and these changes in attitude

in science took place before Einsteinian relativity, not after.

Other historical treatments of the evolution of space theory only contend that Mach provided

the first valid philosophical argument, which was later adopted by Einstein, against Newton’s

absolute space. They have not fully treated the context into which Mach’s ideas were born. Mach’s

criticism must be viewed within the historical context of the overall debate on non-Euclidean

geometries and spaces of higher dimensions. When other developments (i.e. the work of J.K.F.

Zöllner and Clifford) are mentioned at all, they are seen as isolated historical events which had no

specific relation to the mainstream of physical theory during the period. On the contrary, these

events were intimately related to one another other as well as the rest of physics demonstrating a

strict pattern in the development of physical theories of space from Riemann to Clifford, Karl

Pearson, J.J. Sylvester, Charles Sanders Peirce, W.W.R. Ball, Simon Newcomb and Charles Howard

Hinton, among others. They were an integral part of the more general philosophical debate on the

role of space in physics, in both its quantitative and qualitative aspects before Einstein. The new role

of space in physics revolved around three basic ideas; the concept of dimensionality, whether

physical space was ultimately Euclidean or non-Euclidean, and whether space itself could have

physical qualities or properties.

The Riemannian concept of space and the geometry associated with it offered a quantitative

method for showing that space could have physical properties without invoking the hypothetical

aether for the first time. Yet in many ways the concept of a hyperspace bore an important

resemblance to the aether which was employed with limited success by many scientists, especially

Newton, Euler, Kelvin and Maxwell. The debate over the nature of space itself and the role of

geometry in physics must then be looked at within a more general context which includes the

relation of hyperspace theories to aether theories. Those people who accepted the reality of absolute

space, devoid of physical properties, supported either a Euclidean concept of physical space under

any circumstances (such as Henri Poincare) or relegated physical space to a reality beyond absolute

knowledge due to the inaccuracies of human sensations (such as Ernst Mach). In this way they could

ignore the possibility that absolute space could exhibit real physical properties. By denying the

possible reality of absolute space with properties, Poincare, Mach and others were forced to accept

the alternative view of a relative space without properties and account for physical phenomena in

some other way. When faced with the problem of an expanding role for space in physics, Poincare

relied more heavily on the aether while Mach denied the aether and placed the scientific error in our

interpretation of external reality in the way by which we sense our world, leading to the rise of

philosophical positivism.

On the other hand, the possibility of testing space for inherent physical properties led, in some

cases, to a wild spree of scientific speculations which became entangled in the web of supernatural

beliefs rampant during the era. This evolutionary trend was embodied in the popular modern

spiritualism movement. The unfortunate association of physical hyperspaces with spiritualism

ultimately led to the ill-will of the scientific community toward hyperspace theories in general as well

as the direct applications of non-Euclidean geometry to physical theories. Any complete study of the

physics of the latter part of the nineteenth century must include the work done on hyper-

dimensional space theories despite the prejudice against such theories due to their spiritual

connections.

Riemannian geometry was not successfully employed in physics until Einstein enunciated his

general theory of relativity, although the implications and suggestions made by Riemann with regard

to the foundations of geometry were employed unsuccessfully before Einstein. Having found it

impossible to measure space curvature in the large-scale structure of the known universe, a few

scientists instead tried to follow Riemann’s suggestions and look at the infinitesimal structure of

space where curvature might directly influence physical phenomena. In other words, scientists

applied Riemannian geometry and the concept of curvature to explain the electrical properties of

hypothetical atoms and tried to change the fundamental basis of matter theory from gravity to

electromagnetism. The fact that these theories of space were unsuccessful does not detract from

their overall significance of these theories to science. They offered necessary alternatives to Mach’s

and Poincare’s views of space and they raised new questions on the scope and role of space theory

and geometry in physics (i.e., could space have real physical properties which went unanswered until

Einstein).

In those instances where these historical events have been studied by historians or other

scholars, they have been mistakenly viewed as anticipations of Einstein’s relativity. In actuality, there

are no known direct causal connections between the earlier attempts to use Riemannian geometry in

physics and Einstein’s later work. If anything, Einstein’s work restored the gravity theory of matter

to its dominant position in physics, reversing the more recent trends to base matter theory on non-

Euclidean (Riemannian) interpretations of electromagnetism. The earlier scientists were trying to

employ Riemann’s suggestions in ways totally different from the way in which Einstein applied

Riemannian geometry. In the first place, Clifford and the others who followed Riemann sought to

develop theories of electricity and magnetism rather than gravitation. Secondly, they worked

specifically with higher-dimensional absolute spaces so their theoretical work did not affect or

influence the relative nature of our common three-dimensional space in any way. In most cases they

used quaternions rather than tensors to mathematize space. The tensor calculus that Einstein later

used was not fully developed until 1901. And finally, they sought to explain the infinitesimal reaches

of physical and material reality, not the seemingly infinite expanses of the universe at large as

Einstein did.

Einstein’s theory dealt primarily with gravitation and had nothing to do with absolute space. It

employed tensors and was not even applicable in the infinitesimal regions of space. Quite clearly, the

theories of Clifford and the others within the Riemann program could not have anticipated Einstein’s

concepts since they were questioning completely different aspects of nature. This, however, has not

stopped scholars from viewing Riemann, Clifford and others as merely anticipating or foreshadowing

Einstein with wild untenable speculations. The only way in which Clifford and others could have

anticipated Einstein was in the application of physical properties to space, albeit an absolute space,

but historians have yet to recognize this fact in their histories.

To completely ignore the historical significance of the Riemann program in physics is to deny the

existence of one of the most important aspects of science in the late nineteenth century. There may

only have been a few scientists willing to openly support hyperspace theories and the physical

interpretations of non-Euclidean geometry and the theories developed in the program may have been

ultimately unsuccessful, but these facts cannot be allowed to detract from the significance of their

scientific work from the overall history of science. Whether they were successful or not, the

theoretical attempts to apply hyperspace and non-Euclidean geometries to explain physical space

directly influenced the scientific discourse of the day. These theories directly affected the attitudes of

the scientific and scholarly communities and thus laid the foundations for the introduction and

success of Einstein’s relativity theories. To note the existence of the Riemann program, but belittle the

ideas and theories associated with it as irrelevant, isolated, trivial, or mere anticipations of Einstein’s

later theories, is to propagate historical inaccuracies. The only answer to this dilemma is to view the

theories which used hyperspaces and non-Euclidean geometries, those theories that were developed

in accordance with the Riemann program, as legitimate and important contributions to the science of

the late nineteenth century.

Chapter 1: Defining the problem

Redeeming bad history

If asked, most scholars would inform the casual inquirer that no significant changes occurred in

the concept of physical space between the eighteenth century adaption of Newtonian doctrine and

the Einsteinian revolution of the twentieth century.1 Max Jammer, in essence, made this claim by

noting that the only possible exceptions came with Ernst Mach’s criticism of the Newtonian concept

absolute space and William Kingdom Clifford’s speculation on the relations between motion, matter

and the changes in spatial curvature. But these instances were not significant since the Newtonian

absolute space was not overthrown until Einstein’s theory established the new paradigm of space-

time.

Jammer also wrote about “Riemann’s anticipation” of Einstein, stating that “Riemann’s

allusions were ignored by the majority of physicists. His investigations were deemed too speculative

and theoretical to bear any relevance to physical space, the space of experience.” 2 He further stated

that Clifford’s “speculations aroused great opposition among academic philosophers who still

adhered to the Kantian doctrine...” 3 while asserting that Mach made “the earliest proclamation of

the principle of general relativity.” 4 Yet Jammer, one of the few authors to write a history of the

concept of space during the crucial period when general relativity was beginning to become more

popular due to the space program, new developments in astronomy and cosmology as well as other

factors, did not offer supporting evidence for the claims.

On the other hand, he claimed that Mach developed the first successful “modification of the

traditional interpretation of Newton’s pail experiment” and objected “to accepting the experiment as

a proof of the existence of absolute space,” 5 implying that these were the only significant changes in

the concept of physical space which could have been made by Mach and other relativists. Unlike his

representation of Clifford and Riemann’s contributions to science, Jammer’s representation of

Mach’s advances and influence on the concept of physical space is closer to what actually happened.

In other words, Jammer downplayed the roles of Riemann, Clifford and their followers, implying

that the rise of non-Euclidean geometries in second half of the nineteenth century had no physical

consequences as did Mach’s work. He saw only those ideas on space which directly influenced

Einstein’s relativity as significant, a view which has been all too common in the history of science.

Although Jammer’s book, Concepts of Space, is somewhat dated (it was originally published in

1954), it is still one of the more important works in the small but growing collection of studies on

the theories of space and remains a well accepted, authoritative work on the concept of space. As

such, Jammer’s view on the development of concepts of space can be considered fairly common

relative to the accepted views of scientists, scholars and academics. However, it is not necessarily

true that no changes in the concept of physical space occurred during the closing decades of the

nineteenth century. In a later publication, Alfred Bork (1964) 6 wrote that some evidence exists that

physicists were interested in the new concepts of multi-dimensional physical space, but only to the

extent that they anticipated the later space-time of Hermann Minkowski (1907).7 Bork considered

those ideas which he deemed precursors to Minkowski’s space-time notions as isolated and trivial.

... we want to show that the physicist too was interested in these ideas. Most of the

examples are isolated, and some taken individually are rather trivial. But considered as a

group, they furnish evidence that the Minkowskian view of the universe had a prehistory in

nineteenth century physics.8

These changes were neither trivial nor isolated as Bork claimed.

Certainly neither trivial nor isolated

Riemann’s program

Chapter 2

Newton’s absolute space

To begin at the beginning

From Newton’s day, space was considered either absolute or relative. Relative space was used

successfully in the application of Newtonian mechanics as the normal space of the real physical

world, but the consistent underlying notion of an absolute space was also necessary. Without a

concept of absolute space, Newtonian mechanics would have lost a good deal of its philosophical

foundation. Absolute space, which was originally implied by specific phenomena such as Newton’s

example of the spinning bucket,1 was quite beyond any hope of experimental testability. Absolute

space in the Newtonian sense was a three-dimensional Euclidean space, closely associated with the

properties of immutability, penetrability, indivisibility, homogeneity, continuity, lack of resistance,

immutability, eternality and infinity. All of these properties placed it beyond the range of

experimental verification. Although absolute space was considered an objective thing, the qualities of

absolute space were idealizations that were well beyond the mechanical world of material existence.

Newtonian relative space alone was more than sufficient to describe all mechanical and related

phenomena, while absolute space was necessary to philosophically account for inertia and centrifugal

forces.

Early criticisms that Newton’s Principia was atheistic2 were made by the German philosopher

and mathematician Gottfried Leibniz as well as the English theologian and philosopher Bishop

Berkeley. Newton added a ‘General Scholium’ to the second edition of the Principia in 1713 in

answer to those criticisms. Newton associated his absolute space with the infinity and eternity of God

in the “General Scholium”, describing absolute space as God’s Sensorium.3 As such, Newton merged

the scientific objective purity of absolute space with his subjective religious views.

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 endures and is present. He endures forever, and

is everywhere present; and, by existing always and everywhere, he constitutes duration and

space. Since every particle of space is always, and every indivisible moment of duration is

everywhere, certainly the Maker and Lord of all things cannot be never and nowhere. ... Every

man, so far as he is a thing that has perception, is one and the same man during his whole

life, in all and each of his organs of sense. God is the same. God is always and everywhere.

He is omnipresent not virtually only, but also substantially; for virtue cannot subsist without

substance. In him all things are contained and moved; yet neither affects the other: God

suffers nothing from the motion of bodies; bodies find no resistance from the

omnipresence of God.4

This stratagem perpetuated a much earlier tradition of associating space with the virtues of God5 that

dated back centuries.

In all, several different types of physical space plus another purely mental concept of space were

popular to varying degrees during the period following Newton’s publication of the Principia.

However, each was ultimately based on the more general Newtonian concept of space and later

interpretations of Newton’s absolute space. The first two types were simple absolute and relative

space, both of which sprung directly from Newton’s hand. Neither could be characterized by

intrinsic physical properties. Absolute space had no physical characteristics by definition and relative

space could have no physical characteristics since it was merely an abstract relationship between

material and thus real objects’ positions. In other words, relative space was not a thing in the same

manner as absolute space, but still had some minimal reality as a ‘thing’. Leibniz’ concept of a

‘relational’ or relative space also enjoyed some popularity.

A century earlier, Leibniz argued that space exists only as a relation between different material

objects, so space could have no existence apart from the existence of those objects. Therefore,

motion could only exist as a relation between those objects. Leibniz therefore rejected the

Newtonian concept of an absolute space since the Newtonian concept of space provided an

absolute frame of reference against which objects moved. For Newton, absolute space was (in a

sense) both a container and a frame of reference that existed independently of the material objects it

contained. Absolute space was thus a real thing based on its own merits. Objects did not move in

relation to one another, but instead in relation to space itself in a relative manner.

The notion of space as a purely mental construct, as represented by Immanuel Kant’s concept

of schema, arose from Kant’s attempt to find a compromise between the Leibnizian conception of

relativity and the Newtonian concept of absolute.6 As a product or direct result of the mind rather

than a product of the physical world, absolute space in the Kantian sense could exhibit no physical

properties. Thus the Newtonian absolute and relative spaces, as well as the Kantian schema,

constituted those concepts of space which were completely divorced from physically testable

properties. However the last form of absolute space was altogether different because it was based

upon a hypothetical aether that either coexisted with absolute space or, in some extreme cases,

almost came to be equivalent and inseparable from absolute space.

Aether as absolute space incarnate

Non-Euclidean geometries enter the picture

Chapter 3

Changes before Riemann

Debating Gauss

The basic mathematical notion of space with curvature was developed before Riemann’s lecture

on the Bases of Geometry.1 The first non-Euclidean geometry is commonly thought to have been

developed independently by Gauss (after 1815),2 Lobachevski (1826),3 and Janos Bolyai (1832),4

while important contributions were made at earlier dates by Giovanni Girolamo Saccheri (1733),5

Johann Lambert (1766),6 Adrien-Marie Legendre (1794),7 Ferdinand Karl Schweikart (1818),8 Franz

Adolph Taurinus (1825),9 and Wolfgang Bolyai (1804-1833).10 Gauss also developed a new metric-

differential geometry in 1827 in connection with two-dimensional surfaces curved in a third

dimension,11 but does not seem to have connected this with his work on non-Euclidean geometry as

Riemann did at a later date.

In other words the extent to which Gauss connected the concept of curved surfaces with

differentiating between different geometrical spaces in the same manner that Riemann later

developed is not known. Perhaps that is why Gauss was so happily and zealously supported

Riemann’s 1854 lecture. However, it only seems that Gauss never made the connection. Gauss’ own

later testimony indicates that he took the matter of space curvature far more seriously than his

published accounts could possibly indicate. Many historians and scientists believe, and rightly so,

that Gauss hypothesized and attempted to measure the actual curvature of space12 during a geodetic

survey of Hanover in the early 1820’s.13

This view has recently been refuted by Arthur I. Miller (1972)14 and Roberto Toretti (1978).15

Toretti argued that Gauss would never have made such measurements. He would have known as

early as 1819 that such measurements were useless since any curvature could only be detected on the

astronomical scale. So Toretti was not arguing that Gauss did not believe in a possible curvature of

space, but rather than he thought it impossible to measure. On the other hand, Miller went much

further and argued that modern writers projected their Einsteinian biases onto Gauss’ early work

while trying to find precursors for general relativity. In spite of these arguments to the contrary,

there exists ample evidence that Gauss at least applied his geodetic survey data to the problem of

curved space even though he did not attempt a specific experiment to measure space curvature.

Miller’s claims are symptomatic of a far more pervasive general attitude that anything that

happened before the scientific revolution could not have been similar to what happened after the

scientific revolution because the revolution, by definition, presented something completely new and,

well, revolutionary. This attitude implies that no one could have developed theoretical notions of

space curvature since space curvature was uniquely a product of the Second Scientific Revolution

and if anyone had done so, by chance or accident rather than design, the idea could not have been

seriously considered by the scientific community. Under these circumstances, Miller and others’

claims to the same effect can be easily refuted. The Gauss story was well known and publicized

throughout the late nineteenth century while other astronomers followed the example Gauss set and

attempted to determine the curvature of space through their parallax measurements since parallax

triangles represented the largest known example of triangles. However, refuting these

misconceptions of history raises several questions about what really happened in science prior to as

well as during the Second Scientific Revolution.

In any case, the question of whether the physical space of experience is curved or not took on

new meaning after Einstein’s general theory of relativity (1916), which utilized a Riemannian

geometry. After that time a greater historical emphasis came to bare on the question and a few

scientists and scholars may have slightly embellished the Gauss story by extrapolating back from

Einstein’s work to Gauss’ 1827 paper (without reading it, of course), keeping in mind that Gauss’

theorems were applied to the largest triangle in his geodetic survey (of Hanover). ...16 All that

historians can legitimately claim is that Gauss could have easily used his data from that survey to

look for space curvature sometime after the fact and probably did. No separate measurement or

experiment was ever necessary. Furthermore, a mistaken association with Lobachevski’s attempts to

measure curvature would be a more likely hypothesis for the myth of Gauss’ measurement of space

curvature, but it would still be historically inaccurate.

Gauss never published his views on the non-Euclidean geometry for fear of the “clamor of the

Boeotians.” 17 So it seems that there is no direct evidence that Gauss attempted to measure the

physical curvature of space, automatically rendering all available evidence circumstantial. Yet it was

known that astronomical measurements had been made by Lobachevski. It would be easy to assume

that Gauss had come to the same conclusions regarding a non-Euclidean application to physical

space as Lobachevski. Gauss, after all, was a first rank scientist as well as a mathematician.

Miller missed several other essential points in his criticism; (1) Since Gauss feared the clamor of

others that might have resulted from the public disclosure of his purely mathematical speculations

on geometry, he would have been far more careful concerning any possible physical consequences

of such ideas, i.e., the notion of a real physical curvature of space, and would never have published any

speculations or measurements of space curvature without disguising them, as he did in his essay on

metrical geometry; (2) During the period of time in question, mathematics was still very closely

associated with physics and any new mathematical concepts, no matter how radical or improbable,

would have automatically implied corresponding physical consequences, making it rather difficult for

Gauss to keep his concepts mathematically pure. So, once again, he would not have published his

speculations concerning the physical reality of curvature in so open a manner as his essay on the

differential-metric geometry; (3) There is ample and perhaps even overwhelming evidence that

Gauss was well aware of the concept of space curvature and its physical implications; (4) It would be

scientifically unthinkable to claim, as does Toretti with respect to mathematics, that Gauss would

assume that any curvature would be of astronomical proportions without verifying the fact by

looking for possible curvature along the largest earthly triangle possible; and (5) It is inaccurate to

assume that Gauss was not aware of Lobachevski’s later attempts to measure the physical curvature

of space even if such a physical application had not been his own original idea. Even if his notions

of a real physical curvature came several years after his geodetic survey of Hanover, he could always

refer back to the data from that survey at a later date to prove that earthly triangles did or did not

exhibit space curvature. How Gauss interpreted his data in the privacy of his own mind is still open

to speculation and debate by historians since they can never know for certainty unless an original

letter in Gauss’ own handwriting to that effect is discovered.

The best argument that can be made to refute Gauss’ possible attempt to measure space

curvature lies in the fact that there is no direct evidence he ever equated his metric geometry to his

non-Euclidean speculations. Therefore, he may not have associated a real curvature with his

geodetic survey of the earth which was made in connection with his metric geometry. But this still

does not prove that he did not use his measurements to prove or show space curvature to his own

satisfaction and independent of his results for the survey. He could have used the survey to disguise

his results on space curvature from the Boeotians, which seems more likely to have happened in light

of related historical data on the subject.

Although the question of whether or not Gauss actually measured the curvature of space while

surveying in Hanover is still open to debate, there can be little or no doubt that his conceptions of a

non-Euclidean geometry took into consideration the nature of a real physical space. In a letter to

Tarinus in 1824, Gauss wrote that:

The propositions of this geometry appear partly paradoxical and absurd to the

uninitiated, but on closer and calmer consideration it will be found that they contain in

them absolutely nothing that is impossible. Thus, the three angles of a triangle, however

great the sides may be taken, can never exceed a definite limit, nay, can never once reach it.

All my endeavors to discover contradictions or inconsistencies in this non-Euclidean

geometry have been in vain, and the only thing in it that conflicts with our reason is the

fact that if it were true there would necessarily exist in space a linear magnitude quite

determinate in itself, yet unknown to us. But I opine that, despite the empty word-wisdom of

the metaphysicians, in reality we know little or nothing of the true nature of space, so much

so that we are not at liberty to characterise as absolutely impossible things that strike us as

unnatural. If the non-Euclidean geometry were the true geometry, and the constant in a

certain ratio to such magnitudes as lie within the reach of our measurements on the earth

and in the heavens, it could be determined a posteriori. I have, therefore, in jest frequently

expressed the desire that the Euclidean geometry should not be the true geometry, because

in that event we should have an absolute measure a priori.18

Since Gauss stated that “we know little or nothing about the true nature of space,” it is clear that he

seriously considered the application of a non-Euclidean geometry to a real physical space: There can

be no doubt that he was aware of the physical implications of space curvature, but just how he

defined those physical implications is unknown.

It would be more reasonable to conclude that Gauss was well aware of the physical implications

and assumed that anyone reading his mathematical theory on non-Euclidean geometry would

automatically jump to the conclusion that Gauss was referring to a real curvature of space whether

or not he believed the curvature was real. Gauss would have left the reality of curvature to

experimental verification and would not have withdrawn his ideas on non-Euclidean geometry even

if they proved to be physically untrue. This argument would more appropriately explain why Gauss

feared the “clamor of the Boeotians.”19 Mathematics either could or could not represent the real world:

There was no guarantee that any given mathematical situation was analogous to real physical

phenomena. The Boeotians, or common people, would not understand this principle of

mathematics since mathematics was still directly associated with physical reality during Gauss’

lifetime.

Lobachevski gets the credit

Chapter 4

Riemann’s Habilitationsschrift A unique contribution to science

Riemann’s inaugural lecture or Habilitationsschrift was unique for several reasons and therefore

stands alone as a pivotal event in the evolution of the concept of physical space. To begin with, (1)

Riemann sought no less than a complete change to the physical concept of space. His ideas were

highly empirical despite the ingenuity of his pure mathematics and the role that his ideas played in

influencing new studies in pure mathematics. A great deal of attention has been directed toward the

empirical aspect of Riemann’s lecture, but only to the extent that he offered a new physical

philosophy of space without considering the actual attempts to apply his philosophy to real physical

situations. In other words, although Riemann’s work was mathematical it offered a physical theory

without any specific references as to how that theory could be experimentally verified. He made no

specific physical predictions as would a scientist proposing a new theory of space.

Within this context, the philosophical roots of Riemann’s concept of space can be traced to

both Gauss and Johann Friedrich Herbart, under whom Riemann studied philosophy. Both of these

men espoused empirical philosophies which opposed the prevalent Kantian doctrine. Herbart

proposed a theory of sequence-forms. He considered our idea of a three-dimensional space as one

space within a sequence of spaces. This notion corresponded quite closely to Riemann’s concept of

space as an n-dimensional aggregate within an n+1-dimensional manifold.

Secondly, (2) Riemann’s geometric system was not only the culmination of the non-Euclidean

geometries which preceded his Habilitationsschrift, but heir to several different trends in mathematical

development. To name a few, these included the separation of the algebra of dimension from

geometry,1 the descriptive-analytical system of geometry as developed over the period of time from

Descartes’ initial work in analytic geometry to Gauss’ metric-differential geometry,2 and the concept

of n-dimensional analysis which itself had roots in these other trends.3 Riemann’s work could thus

be viewed as a focal point between those ideas which preceded his own and the later mathematical

extensions of his work rather than just one more contribution in a simple linear progression of ideas.

Thirdly, (3) the roots of the later developments in tensor calculus by Riemann himself,4 G.H.

Christoffel (1869),5 Tullio Levi-Civita and Gregorio Ricci (1901)6 can be found in Riemann’s

Hypotheses (see Appendix 1). This stream of development led, a half century after Riemann’s death, to

the first successful application of Riemann’s geometry to the physical world with Einstein’s general

theory of relativity (1916). Riemann’s geometry, following on the heels of Gauss’ early developments

in metric-differential geometry, was also a major step in the formation of modern topology. Since

Riemann’s geometry dealt with a stationary element of space, independent of the mobility or motion

in space with which Helmholtz dealt,7 it was more easily applied to the development of analysis situs

or topology (as it is now called). In topology there is no need to define the continuity of space in

terms of an intuitive sense of motion, as in either mechanics or the earlier concept of fluxions in

calculus. The continuity of space could instead be defined analytically and arithmetically.8 The seeds

of topology found in Riemann’s and Gauss’ work did not bear fruit until several decades later when

topology reached a level of sophistication whereby it could be called a mature field of pure

mathematics.9

Still other factors distinguish Riemann’s work as pivotal. His Habilitationsschrift lecture

alternatively brought a far greater amount of attention to the study of non-Euclidean geometries and

formed a proper basis by which the different geometries could be unified into a single mathematical

system. The unification of the various geometries was necessary for the advancement of geometrical

theory as a whole; and finally, (4) Riemann suggested a program for physical speculation and further

theoretical work on the nature of space which had been lacking in the Newtonian concept of

absolute space. Contrary to popular belief since the advent of Einsteinian physics, Riemann’s main

emphasis was not toward the infinite extensions of space with which curvature dealt, although he

did detail the relation of an overall curvature to space quite explicitly.10

In the extension of space-construction to the infinitely great, we must distinguish

between unboundedness and infinite extent, the former belongs to the extent relations, the latter

to the measure-relations. That space is an unbounded three-fold ‘manifoldness, is an

assumption which is developed by every conception of the outer world; according to which

every instant the region of real perception is completed constructed, and which by these

applications is forever confirming itself.11

In general, philosophers and scientists have regarded Riemann’s suggestion that infinite extension does

not necessitate an unbounded space as Riemann’s most radical and constructive contribution to

physical science. This suggestion has become the basis for most cosmological speculations made

after Einstein’s enunciation of the general theory of relativity. However, this does not mean that the

relation of unbounded to infinite spaces went unrecognized before Einstein. Riemann’s concept of an

unbounded and infinite space was popularized and became part of the common knowledge of

cosmological speculation well before Einstein. But far more attention was paid to the infinitesimal

aspects of Riemann’s theory of space before Einstein dealt with the infinite features of Riemann’s

geometry, which better fit the predisposition of science before 1900. Science was far more interested

in the microworld of physical reality than the macroworld of cosmology during the nineteenth

century.

The true problem of physics

The new emphasis on empiricism

Chapter 5

Popularization of the new geometries

The early history

Euclidean geometry was still the universally accepted form of geometry for representing space

in the 1860s. The non-Euclidean geometries had found only a small but tenacious audience of

adherents because the fundamental publications regarding the non-Euclidean were relatively

unknown except to a few specialists. Between 1865 and 1870, there seems to have been a general

confluence of ideas epitomized by the first publication of Riemann’s 1854 lecture, which brought

together the various strands of non-Euclidean geometry and related areas of mathematics.

Riemann’s work was not published until 1866;1 Lobachevski’s and Bolyai’s work was not widely

known or circulated until translated into French by Hoüel in 1866 and 1868;2 And, of course, Gauss’

work on non-Euclidean geometry, largely kept secret except for a few friends and colleagues during

his lifetime, was not published until after his death when a great deal of interest was shown for

finding the historical foundations of non-Euclidean geometry.

The first public knowledge of Gauss’ work emerged when a few of Gauss’ personal letters to

Schumacher were published in 1862 and 1863.3 Hermann Grassmann’s work in n-dimensional

analysis was also relatively unknown4 and Saccheri’s early work in geometry was not rediscovered, or

rather recovered from obscurity, until 1889.5 The Italian mathematician, Eugenio Beltrami’s analysis

of Lobachevski’s work using methods analogous to Gauss’ 1827 metric geometry was first published

in 1868. Only after he learned of Riemann’s work, in the last months of 1868, did Beltrami expand

his own analysis to include the more general n-dimensional analysis.6 Whether by historical chance or

design the table seems to have been set for a major change in geometrical attitudes by the end of the

decade of the 1860s.

An alternate path

Different trends in later development

Chapter 6

The program takes form

Clifford takes the high road

W.K. Clifford’s physical extensions of Riemann’s ideas were first made public in 1869 by J.J.

Sylvester.1 Clifford publicly summarized his ideas before the Cambridge Philosophical Society on 21

February 1870 in an abstract entitled “On the Space-Theory of Matter.” He specifically proposed

that the motion of matter amounted to no more than small ripples or variations in the curvature of

space. The distortions of space were continually being passed from one portion of space to another

in the manner of a wave.2 In his own words, Clifford stated that

Riemann has shewn that as there are different kinds of lines and surfaces, so there are

different kinds of space of three dimensions; and that we can only find out by experience

to which of these kinds the space in which we live belongs. In particular, the axioms of

plane geometry are true within the limits of experiment on the surface of a sheet of paper,

and yet we know that the sheet is really covered with a number of small ridges and furrows,

upon which (the total curvature not being zero) these axioms are not true. Similarly, he says

although the axioms of solid geometry are true within the limits of experiment for finite

portions of our space, yet we have no reason to conclude that they are true for very small

portions; and if any help can be got thereby for the explanation of physical phenomena, we

may have reason to conclude that they are not true for very small portions of space.3

If there could be any doubt upon whom Clifford had relied far his concept of space,

that doubt would clearly have been dispelled by his direct reference to Riemann. Once

Riemann’s primary ideas had been summarized, Clifford continued to state his own ideas.

I wish here to indicate a manner in which these speculations may be applied to the

investigation of physical phenomena. 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

ethereal. (4) That in the physical world nothing else takes place but this variation, subject

(possibly) to the law of continuity. I am endeavouring in a general way to explain the laws

of double refraction on this hypothesis, but have not yet arrived at any results sufficiently

decisive to be communicated.4

This abstract was described as a space-theory in Clifford’s title, but he also referred to his ideas as

speculations while regarding them as factual. So what did Clifford have in mind? Was he just being

over enthusiastic about the new ideas in physics or was he really on to something?

Historians have berated Clifford based on his “Space-Theory” abstract alone and claimed that

he never had a theory, but he has been treated unjustly by historians and scientists alike who have

simply failed to understand both Clifford’s work and the scientific temper of the times. Clifford did

have a theory, but it was not a gravitational theory as later historians would have hoped for and have

looked for in his work. He may have even anticipated Einstein as they have claimed, but his theory

was an electromagnetic theory first and only later would it have been used to explain gravity. His

ideas quite definitely referred to a real physical space with a definite structure, based upon both a

general curvature in the large which approximated Euclidean flatness over astronomically measured

distances and a more specific curvature in the smallest portions of space that directly influenced

physical phenomena. Clifford thought that material particles themselves were electrically rather than

gravitationally constituted and gravity was a secondary force or rather effect of matter. This

prescription followed Riemann’s suggestions in his ‘Space-Theory” abstract quite closely.

Had Clifford’s abstract been his only contribution to the general concept of space, modern

historians would be justified in regarding his ideas as mere speculations out of context with his

times, but Clifford was not that far out of step with his peers regarding both the physics and

mathematics which were currently at the vanguard of research. According to Sylvester:

I know there are many, who, like my honoured and deeply lamented friend the late

eminent Prof. Donkin, regard the alleged notion of generalised space as only a disguised

form of algebraic formulisation; but the same might be said with equal truth of our notion

of infinity in algebra, or of impossible lines, or lines making a zero angle in geometry, the

utility of dealing with which as positive substantiated notions no one will be found to

dispute. Dr. Salman, in his extension of Chasles’ theory of characteristics of surfaces, Mr.

Clifford, in a question of probability (published in the Educational Times), and myself in my

theory of partitions, have all felt and given evidence of the practical utility of handling

space of four dimensions, as if it were conceivable space.5

Sylvester, at least, was convinced of the practical utility of using spaces of higher dimensions and

different curvatures. He continued and stated his belief, and the belief of other prominent

mathematicians, in the reality of such transcendental spaces.

If Gauss, Cayley, Riemann, Schlafli, Salmon, Clifford, Krönecker, have an inner

assurance of the reality of transcendental space, I strive to bring my faculties of mental

vision into accordance with theirs. The positive evidence in such cases is more worthy than

the negative, and actuality is not cancelled or balanced by privation, as matter plus space is

none the less matter. I acknowledge two separate sources of authority - the collective sense

of mankind, and the illumination of privileged intellects.6

It would seem that some of those mathematicians who worked most closely with the concepts of

hyper-dimensional or transcendental spaces, as they had become known to some people, had

become convinced that those spaces had some physical reality, or so Sylvester would have his

readers believe, and Clifford was included in that group which put him in the company of many

prominent mathematicians.

Clifford’s (not so) secret theory

Post Clifford

Equating curvature to the aether

Ball rolls on

Hyperspace theories in America

Tracing new trends

Halsted the popularizer

By the end of the 1870’s, notions of hyperspace and non-Euclidean geometries had spread to

North America and another line of theoretical development began. The advent of non-Euclidean

geometries in North America corresponded quite closely to J.J. Sylvester’s appointment as a

professor at Johns Hopkins University in Baltimore, where they were quickly popularized by G.B.

Halsted, Sylvester’s very first student.1 Throughout his career, Halsted published more than eighty

papers related to non-Euclidean geometries, their histories and foundations. He also claimed to have

taught the first course in the History of Mathematics in the United States (at Princeton in 1881)2 and

sought to clarify the issues in the continuing debates raised by the acceptance of non-Euclidean

geometries and their physical interpretations.3

Halsted offered a much less biased exposition of the new geometries and their relationship to

physical reality than any of his European contemporaries. He was physically and ideologically

separated from the subjective debates carried on in Europe and his primary interests in the subject

were historical rather than philosophical. Halsted represented a trend to historicize the new

foundations of geometry. His contributions are indicative of the overall interest in scholarly history

born of the same era, but they also represent the interest shown by many scholars in finding the

historical basis of the new and radical ideas of geometry that were sweeping the scholarly

establishment. The new geometries were so radical that any reference to a historically well known

and respected personality could only help to justify the geometries.4 In the absence of experimental

verification of curvature or any of the other physical aspects of a hyperspace, a reliance on the

authority of past scientific heroes was essential.

Newcomb the theoretician

Intrinsic versus extrinsic

Newcomb the relativist

Peirce challenges the norm

Hinton the outsider

An unfortunate situation develops

Chapter 8

Spirits, dimension and space

Prior comparisons and associations

Attempts by Hinton and others to equate the world of the spirits with hyperspaces were not

without important historical precedents, nor were they devoid of important historical consequences

for legitimate hyperspace theories. When scholars and commoners equated hyperspaces to a realm

of spirits in the late nineteenth century they were merely filling an intellectual and philosophical void

that the successes of Newtonian science had created, although that notion is something that scholars

and academics would never easily or willingly admit. Newtonian science had become so successful

that it was replacing some aspects of established religious beliefs which left an opening that was

filled by creating a new individual and secular search for religious meaning and purpose. This

situation was further stressed by the recent development of evolution theory, which was, after its

own manner, a Newtonian mechanism for biological advance and progress that ended with humans

and the human mind at the top of the evolutionary scale of being.

An extensive amount of philosophical and historical literature to which reference could be

made regarding the possibility of uniting the spiritual world with the physical world via either the

fourth dimension or absolute space already existed. Spirit and science (in so much as early pre-

Newtonian natural philosophy could be equated to science) coexisted before they were torn asunder

by Descartes’ separation of body and mind in the seventeenth century. This separation seemed to

have been necessary to foster the mechanical view of nature that developed during the seventeenth

century as well as the proper laws of motion and gravity. But a long history of scientists and scholars

who criticized the completely non-spiritual and non-subjective mechanistic worldview still stretched

without breaks from the present back to the seventeenth century and earlier. In other words, the

Cartesian line of demarcation between mind (the realm of God, religion and subjective reality) and

matter (the realm of science and objective reality) was in a state of flux due to recent advances and

developments in science and mathematics, including the open challenge to Newton’s concept of

absolute space, the “sensorium of God”, by the existence of the non-Euclidean geometries as well as

other recent successes of Newtonianism.

Contrary to the strictly mechanical view of nature, many scholars still sought to find some

connection whereby spirit and/or mind would have their own place within the mechanical world.

Newton found a niche for spirit within his concept of absolute space, a view that was preceded by

the earlier writings of Francesco Patrizi,1 Pierre Gassendi2 and others who had influenced Newton’s

concept of physical space.3 On the other hand, Leibniz reduced matter to the point where the

smallest possible bit of matter and spirit were intimately connected and called this his monad.4

Although he only spoke of it briefly, Riemann’s view of an atomic aether was quite similar to the

Leibnizian monad, except that it lacked a direct link to spirit. Henry More, whose religious influence

was also quite evident in the Newtonian concept of absolute space,5 went further and offered the

first association of the fourth dimension with God and spirit in his concept of the essential spissitude.6

If, during the Middle Ages, one was to associate God with a space which was infinitely

extended, then one was further forced to also admit to the non-dimensionality of God.7 This

particular doctrine was abandoned by Henry More in the middle of the seventeenth century,

breaking the philosophical deadlock between physicality and spirituality and helping to usher the way

for Newton’s concept of absolute space. In his Enchiridion Metaphysicum (1671), More offered the

definitive statement of his “incredibly bold and unheard-of step”8 of assuming “the tridimensionality

of God himself, ...” and accepted “the inevitable consequence that God must be a three-dimensional

being.”9 More’s views of God’s relationship to the physical world was thus a major influence on

Isaac Newton, especially regarding Newton’s religious as well as his physical views of absolute

space.10

The publication of the Enchiridion Metaphysicum marked the culmination of More’s ideas on

religion. In this book he clearly related a fourth dimension of space to spirits, an act which was made

quite explicit to anyone who merely read the heading to his section on the subject.

That besides those THREE Dimensions which belong to all extended things, a FOURTH

also is to be admitted, which belongs properly to SPIRITS.11

In the following text, More further stated and explained his hypothesis of a fourth dimension.

And that I may not dissemble or conceal anything, altho’ all ‘Material’ things, consider’d in

themselves, have ‘three’ Dimensions only; yet there must be admitted in Nature a ‘Fourth,’

which fitly enough, I think, may be called ‘Essential Spissitude;’ which tho’ it most properly

appertains to those spirits which can contract their Extension into a less ‘Ubi,’ yet by a less

Analogie it may be referred also to Spirits penetrating as well the ‘Matter’ as mutually ‘one

another,’ so that wherever there are more Essences than one, or more of the same Essence

in the same ‘Ubi’ than is adequate to the Amplitude therrof, there this ‘Fourth’ Dimension

is to be acknowledged, which we call the ‘Essential Spissitude.12

For having made these statements, More has credited as the first person to seriously consider a real

fourth dimension of physical space.

Originally in Latin, the Enchiridion Metaphysicum was translated to English in 1691 by Joseph

Glanvil in the Saducismus Triumphatus under the title The Easie True, and Genuine Notion and Consistent

Explication of the Nature of a Spirit.13 Within the context of this work, the word dimension was

specifically used by More, while in the context of his earlier work, The Immortality of the Soul (l662),14

More had spoken of a fourth mode rather than a fourth dimension.

And as what was lost in ‘Longitude,’ was gotten in ‘Latitude’ or ‘Profundity’ before, so

what is lost here in all or any two of the dimensions, is kept safe in ‘Essential Spissitude:’

For so I will call this ‘Mode’ or ‘Property of a Substance,’ that is able to receive one part of

itself into another. Which ‘fourth Mode’ is as easy and familiar to my understanding, as that

of ‘Three dimensions’ to my Sense or phansy. For I mean nothing else by ‘Spissitude,’ but

the redoubling or contracting of Substance into less space than it does sometimes occupy.

And Analogous to this is the lying of two Substances of several kinds in the same place at

once.15

The purpose of this fourth mode, or dimension as he later called it, was to allow spirit some interaction

with the three normal dimensions of material space.

More’s work came many years before the final decades of the nineteenth century, but it was still

well enough known during that period to draw critical responses. A German critic, R. Zimmerman

(1881)16 argued that More’s notion of the fourth dimension was untenable and should have no place

in any history of the physical nature of a fourth dimension of space. On the other hand Florian

Cajori, an American historian who wrote on the history of the fourth dimension, greatly differed

with Zimmerman’s position and enumerated his reasons for doing so.

The most compelling reasons given by Cajori dealt with More’s concept of redoubling or

contracting which characterized a spirit.

More says that “when one part of an Extended Substance runs into another, something

both of Longitude, Latitude, and Profundity, may be lost, yet what is lost here in all (three

dimensions), or any two dimensions, is kept safe in Essential Spissitude”; (and) More’s

“redoubling” constituted a mental process by which the passage of a substance from a

space of lower dimensions to one of higher, including the fourth dimension, could be

effected.17

This very process of redoubling would seem to be analogous to a physical situation whereby an object,

such as a cube, is rotated through an angle perpendicular to a line of three-dimensional vision. That

side of the object originally in view of an observer would be diminished, while another side,

originally not visible but laying in a direction perpendicular to the line of vision, would be brought

into view proportionally to the amount that the original side had been diminished. In essence, More

very nearly described the geometric rotation of a three-dimensional object that occurs as a result of

its motion through a fourth dimension of space such as described in the 1870s by Newcomb.

Cajori interpreted the same phrase using the example of a string being folded upon itself into a

higher dimension. This phrase is strikingly similar to the notion that three-dimensional knots would

be simply extended strings relative to a fourth dimension, a mathematical characteristic of higher

spaces that was investigated in detail by Peter G. Tait in his theory of knots. In either case, this is

strictly a geometrical problem and its use in More’s writing could only mean that he was

commenting on a real geometrical fourth dimension of space, negating Zimmerman’s argument that

More was talking of something other than a physical fourth dimension of space.

Whether or not scientists and philosophers of his age took More seriously and understood his

ambiguous statements, they published no comments on his work. Only one seventeenth century

scientist, John Keill of Oxford, seems to have made any reference to More’s concept of the fourth

dimension. In the introduction to the Examination of Dr. Burnet’s Figure of the Earth (1698),18 Keill cited

More, among others, to prove that philosophers maintained “opinions more absurd than can be

found in any of the most fabulous Poets.”19 It would seem that Zimmerman shared Keill’s much

earlier view.

The extent to which Zimmerman’s commentary was accepted during his own time is not

relevant to any questions regarding Riemann’s program, whereas the fact that Zimmerman recognized

any controversy in More’s use of the term dimension would constitute ample proof that lesser

known works such as More’s were still read and referred to during the last decades of the nineteenth

century, which is relevant to the Riemann program for several reasons the least of which is More’s

early use of a hyperspace. The fact that More related hyperspaces to spirits has far more relevance to

the Riemann program and the biases that emerged against its goals of applying non-Euclidean

geometries and hyperspaces in physics to explain material reality in the late eighteenth century.

More’s and works like More’s publications were still available as possible philosophical evidence or

fodder for those who sought authoritarian support for a physically real fourth dimension. More’s

writings, to a greater extent than others, set a lasting precedent for associating spirit with a higher

dimension of space. It also demonstrated the simple truth that no new sophisticated scientific model

of space, like the modern hyperspace theories, was needed for developing an analogy between spirit

and space. Anyone who had only the most rudimentary knowledge about the possibility of a

hyperspace was capable of drawing the conclusion that spirits might inhabit a possible hyperspace,

automatically corrupting any possible scientific value in the concepts of hyperspace.

Chapter 9

Modern spiritualism meets hyperspace

The Unseen Universe

Continuity

Late in the nineteenth century, as in the past, one of the foremost philosophical foundations of

science was the Principle of Continuity.1 Any new contribution to science had to conform to

continuity. In the past, continuity had been used to link spirit and the natural world.2 This same

principle was evident in Newton’s connection of relative to absolute space, Leibniz’ notion of the

monad and was also used quite widely in mathematics.3 With regard to any relation between the world

of the living and the dead, the Principle of Continuity was evoked to establish a connection between

that aspect of a person’s material existence, the soul or spirit, which continued past physical death.

In their popular book, The Unseen Universe, Balfour Stewart and P.G. Tait used the Principle of

Continuity and the newly formulated Conservation of Energy to hypothesize the connection of our

phenomenal world with an aethereal world which corresponded to a continuation of the human soul

or spirit after physical death.4

Although their book was based on what were then considered sound scientific principles, it still

represented the far fringes of scientific speculation. Yet many scientists and scholars applauded their

work, which was so popular that it went into several editions of publication. The authors were

anonymous when the first edition was published, but the popularity of the book did not allow them

to remain anonymous and they were identified in all later editions. There was obviously a need, even

in the scientific community, for some level-headed explanations of subjects that were only

peripherally scientific and even then still highly questionable.

The book and thoughts of Tait and Stewart were in keeping with those cultural attitudes of the

late nineteenth century which had been known a half century earlier as Romanticism. Part of the

essence of Romanticism lay in the fact that quantitative and qualitative reduction in science would

only lead to a dead end whereby science could proceed no further in its quest to explain nature.

Romanticism represented a philosophy or worldview whereby the whole was greater than the sum

of its parts, implying that there was more to the physical world than science could discover through

its normal methods of reduction. Romanticism tried to put the subjective back into objective

science. In the rapidly changing cultural and philosophical atmosphere of the late eighteenth century,

some form of scientific spiritualism would not have seemed an unwanted or unwelcomed

development in science for those thinkers who were already predisposed towards such thoughts. However, the

majority of scientists and scholars stuck by their reductionist objective methods and pilloried and

even ostracized anyone who thought differently.

The hypothetical fourth dimension of space fulfilled the scientific needs of anyone who sought

to find more in the world than reductive science could normally account.5 However, the close

association of the fourth dimension with spirits and the spiritual movement also raised the specter of

an incursion of superstition and the supernatural into science at a time when scientists thought that

all superstition had been all but erased from science if not completely erased in some instances by

objective reductionism. Hostile and bitter debates resulted, directly challenging the foundations of

the fledgling movement in scientific spiritualism, rightly or wrongly, almost as much as it did

modern spiritualism, and indirectly influencing the possibility of legitimate questions regarding the

role, scope and methods of science in general. It is highly likely that the general prejudice of many

academics and scientists to anything that even hinted at superstition or the supernatural whether

warranted or not, coupled with the silence of many other scientists when confronted with the radical

possibility of a real fourth dimension, damaged the possibility of any gains that could have been

made by the adoption of a physically real fourth dimension of space.

Modern spiritualism advances

Zöllner’s fall from grace

Tait versus Zöllner

The hidden history of science

An unseen universe for the exploring

Larger more fundamental questions are implied

Chapter 10

An age of contradictions

Effect leads to cause

Of all the various forms and types of physical of space, the spiritualistic aspects raised the

greatest and bitterest of debates. Only the question of evolution raised greater debates within the

scholarly community as a whole and those debates were tied to the debates over modern spiritualism

as well as the hyperspaces. But the debates over evolution were considered part of the history of its

development as well as the history of science in general while modern spiritualism and the

hyperspaces are not. The possibility of a fourth dimension of space and the subsequent models of

reality that supported the hypothesis were radical enough in their own right to foster bitter debates,

but they did not. However, the further association of the fourth dimension with a world of the spirit

so radicalized many scholars and scientists that it adversely affected any possibility of accepting a

purely physical concept of a fourth spatial dimension.

Hermann Schubert, a German mathematician of the period, noted and confirmed this appraisal

of the historical situation.

The knowledge, however, that mathematicians can employ the notion of four-

dimensioned space with good results in their researches, would never have been sufficient

to procure it its present popularity; for every man of intelligence has now heard of it, and,

in jest or in earnest, often speaks of it. The knowledge of a four-dimensioned space did not

reach the ears of cultured non-mathematicians until the consequences which the

spiritualists fancied it was permissible to draw from this mathematical notion were publicly

known. But it is a tremendous step from the four-dimensioned space of the

mathematicians to the space from which the spirit friends of the spiritualistic mediums

entertain us with rappings, knockings and bad English.1

The fact that the legitimate mathematical systems of hyperspace became widely known through their

association with spiritualism led to hyperspaces as a topic of conversation, whether the

conversations were held in jest or earnest. Any knowledge of legitimate hyperspace theories could

then, quite easily, be regarded in jest and dismissed in spite of the possibility of any real scientific

significance and how earnestly they had been developed. This attitude did not bode well for the

hyperspace and non-Euclidean theories and placed the whole Riemann program in jeopardy.

This attitude was also reflected in a criticism of T. Proctor Hall’s method of visualizing the

fourth dimension by Edmund C. Sanford. Hall merely had to mention ghosts briefly as a possible

physical manifestation of the fourth dimension, “If ghosts are four-fold, the erratic nature of their

movements may become more comprehensible in the course of time,”2 to be criticized for that

digression by way of a short jibe by Sanford. It became quite easy for anyone who argued against

hyperspace theories to simply turn away or blunt the effect of any scientific argument for a real,

physical fourth dimension by turning the foundation of the hypothesis into a point for ridicule.

In regards to the benefit of a full knowledge of four-fold space, Dr. Hall should not

allow himself to hope too much. A really clever and elusive ghost would never stop at four

dimensions, but would surely lead him, Will-’o-the-wisp fashion, through all the series of n-

dimensions.3

By attacking Hall’s stated goals, Sanford clearly demonstrated contempt for Hall’s overall view of

space in spite of the scientific significance of the hypothesis that Hall explained. Sanford clearly

included in his statement one of the more popular arguments against the reality of hyperspaces. If a

fourth dimension were assumed, what would stop anyone from assuming ever higher and higher

numbers of dimensions, ad absurdum. Yet how could this reduction be considered unscientific

considering that Newtonian absolute space was infinite? How would it differ to claim that absolute

space had infinite dimensions? Sanford and those who made similar statements were really attacking

the whole notion of absolute space. Sanford simply implied that no real benefits, scientific or

otherwise, could ever be gained from an attempt to realize the fourth dimension or use it for

conducting theoretical research in physics. This opinion presents a good case of ‘throwing out the

bathwater with the baby’.

Mach speaks

The plot thickens

The historical microcosm of Mach

Chapter 11

Analysis and bias

Science just went too far too fast

Outrunning our sensations

Mach’s final solution to the problems and paradoxes facing physics as a whole during the late

nineteenth century resulted in his redefinition of the physical world of experience in terms of the

sensations which our minds receive from external physical reality. External physical reality literally

means outside of the mind that interprets the sensations. In so doing, he was reacting to the

influences of his time, including those philosophical and physical problems stemming from the

physical implications of non-Euclidean geometries, hyperspaces and absolute space, no less than

were the spiritualists, but he chose to interpret the synthesis of mind and matter in a different

manner than the modern spiritualists and other scientists. In a strict sense, he and other scientists

were trying to redefine the Cartesian limits between mind and matter that had been a fundamental

part of science dating from the mid-seventeenth century and the Scientific Revolution.

It is no coincidence that Mach’s principle of relativity is noted in the history of science as a

cornerstone in the defeat of the Newtonian concept of absolute space.1 For Mach there were no

absolutes and anything which could not be sensed, or perhaps in a more scientific sense measured,

had no reality beyond any one person’s imagination. So it seems natural for Mach to have challenged

the notion of absolute space. Mach was convinced that all mechanical systems were relative to one

another, a view which had earlier been supported by Leibniz and Christian Huygens against

Newton’s concept of absolute space. Although Mach’s motivation for refuting Newtonian

mechanism was different from the spiritualists, they shared common philosophical roots. Even

though Mach’s arguments against the Newtonian absolute space swayed the judgment of many

scientists and philosophers, not all were persuaded by the expediency of Mach’s concept of

relativity. C.S. Peirce actually argued against Mach’s ideas and became convinced of the possibility of

an absolute space with real physical properties that were characterized by hyperspaces and/or non-

Euclidean geometries.

The foundation of modern spiritualist anti-mechanism lay in the continuation of this

mechanical world into some unseen universe of the spirit. For Mach, mechanism attempted to explain

more than it was capable of explaining and for the spiritualists it did not explain enough. In other

words, science went too far in trying to explain the mechanism of the universe using unwarranted

hypotheses such as a mechanical (luminiferous) aether, hyperspaces, atoms, non-Euclidean

geometries and other such non-verifiable and non-observable ‘things’. This idea fit Mach’s anti-

spiritualist views every bit as well as his views against hyperspace theories.

The central point of Mach’s rejection of hyperspaces, whether they were equated to absolute

space or not, dealt directly with the limits of man’s ability to cope with and explain the world about

him. This question had been raised, in part, by the purely mathematical adoption of non-Euclidean

geometries2 and from that point of departure had been stretched beyond the endurance of many

thinkers as they attempted to apply hyperspaces to the real physical world. Mach and others, notably

Karl Pearson and J.B. Stallo, rejected the possibility of completely knowing the outer world through

our sensations.

Mach’s philosophical cohorts

The backlash grows

The air clears

Russell’s philosophical analysis

A different analytical strategy

Hyperspace gets relative

Modern physics

Endnotes

Appendix 1

Riemann’s inaugural Habilitationsschrift

On the Hypotheses which lie at the Bases of Geometry

Translated by William Kingdon Clifford

[From Nature, Vol. VIII. Nos. 183, 184, pp. 14--17, 36, 37.]

Appendix 2

MACH’S FOOTNOTE

Bibliography