About the Work
Space and Geometry in the Light of Physiological, Psychological and Physical Inquiry is a philosophical and scientific treatise by Ernst Mach, translated from the original German by Thomas J. McCormack. The work investigates the nature of space and the foundations of geometry by examining them from three distinct but interrelated perspectives: the physiological (how space is perceived through the senses), the psychological (how spatial concepts develop in the mind), and the physical (how geometry relates to the behavior of material bodies). The book is structured as a sustained inquiry rather than a mere exposition, drawing on experiments, historical analysis, and theoretical reasoning to challenge conventional assumptions about the nature of space.
“The space of the Euclidean geometry is everywhere and in all directions constituted alike; it is unbounded and it is infinite in extent.”
The text is organized into three principal chapters. The first concerns physiological space as distinguished from geometrical space, exploring how vision, touch, and bodily movement generate spatial sensations that differ markedly from the idealized space of Euclidean geometry. The second chapter addresses the psychology and natural development of geometry, tracing how primitive experiences with physical bodies gradually gave rise to abstract geometrical concepts. The third chapter examines space and geometry from the standpoint of physical inquiry, considering how measurement, rigid bodies, and empirical observation ground geometrical knowledge, and how alternative geometries—such as those of Lobachevski and Riemann—relate to physical reality.
Mach’s approach throughout is comparative and historical. He draws on the work of numerous investigators—physiologists like Johannes Müller, E. H. Weber, and Hering; mathematicians and geometers from Euclid through Gauss, Lobachevski, Bolyai, and Riemann; philosophers such as Kant—to build a comprehensive account of how human beings come to know space. The book is notable for its insistence that space perception is a biological phenomenon shaped by the needs of organisms, and that geometry, far from being a purely a priori discipline, has its roots in physical experience with material objects.
Context
The work emerges from a period of intense intellectual ferment in the nineteenth century regarding the foundations of geometry and the nature of human perception. By the time Mach was writing, several developments had converged to make the question of space’s nature particularly urgent. The discovery of non-Euclidean geometries by Gauss, Lobachevski, and Bolyai had demonstrated that logically consistent geometrical systems could be constructed that denied Euclid’s parallel postulate, raising profound questions about which geometry, if any, truly described physical space. Riemann had extended these inquiries by proposing that space might be understood as a special case of a multiply-extended manifold, with its properties determined by experience rather than by pure reason.
Simultaneously, physiological and psychological research had revealed that human spatial perception is far more complex and less “geometrical” than had been assumed. Investigators such as E. H. Weber had demonstrated that tactile sensitivity varies dramatically across different parts of the body, while studies of visual perception had shown that visual space is neither homogeneous nor isotropic—it is bounded, distorted, and organized around the peculiarities of the human body. The phenomenon of binocular vision, the apparent oblateness of the celestial vault, and the distortions of perceived size and shape all suggested that the space of immediate experience differs fundamentally from the space of geometry.
Mach’s own position in this intellectual landscape was distinctive. He was a physicist and philosopher who rejected Kantian claims about the a priori nature of space, insisting instead that all knowledge, including geometrical knowledge, derives from experience. Yet he was equally critical of naive empiricism, recognizing that the sensations underlying spatial perception are themselves structured by the biological organization of the perceiving organism. His approach was to analyze space perception as a biological phenomenon, geometry as a physical science, and the relationship between them as a matter of ongoing investigation rather than settled doctrine.
The book also reflects the broader nineteenth-century interest in the history of science as a tool for understanding scientific method. Mach devotes considerable attention to the historical development of geometry—from primitive measurement practices through Euclid’s axiomatization to the non-Euclidean geometries of his own century—because he believes that understanding how a science actually developed illuminates its logical foundations. This historical approach was characteristic of Mach’s wider philosophical project, which sought to understand scientific concepts as products of human intellectual evolution rather than as timeless truths.
Structure of the Book
The book unfolds in three major movements, each corresponding to one of the perspectives announced in its title. The first chapter, “On Physiological, as Distinguished from Geometrical, Space,” establishes the fundamental distinction between the space of immediate perception and the space of geometry. Mach begins by examining visual space, showing through a series of examples and experiments that it is neither homogeneous, nor isotropic, nor unbounded. Objects in visual space cannot be freely moved without undergoing expansion or contraction; the heavens appear as a finite sphere; the celestial vault appears flattened vertically. He then turns to the space of touch, demonstrating through Weber’s experiments that tactile sensitivity varies enormously across the body surface, and that the skin’s spatial sense has little in common with metrical space. The chapter concludes by considering how these physiological spaces relate to biological function, arguing that space perception exists where it serves the organism’s needs and is absent where it does not.
The second chapter, “On the Psychology and Natural Development of Geometry,” traces how geometrical concepts arise from physical experience. Mach argues that geometry originated in practical concerns—measuring fields, constructing buildings, comparing volumes of grain and liquids. He examines the role of measurement, showing how comparing bodies with bodies gradually led to the concept of congruence and the idea of rigid bodies. He considers how the straight line, the plane, and the circle came to be singled out for investigation, and how the angle-sum theorem of the triangle was discovered through physical manipulations such as rotating a ruler or folding paper. The chapter also addresses the relationship between physical and mental experiments, arguing that what we call “thought experiments” in geometry are really copies of physical experiments, and that visualization alone cannot establish geometrical truths.
The third chapter, “Space and Geometry from the Point of View of Physical Inquiry,” examines geometry as a physical science. Mach discusses the role of rigid bodies in establishing geometrical concepts, the idealization involved in treating physical objects as perfectly rigid or perfectly straight, and the sense in which geometrical propositions are empirical hypotheses. He provides an extended historical account of attempts to prove Euclid’s parallel postulate, from ancient geometers through Saccheri and Lambert to Gauss, Lobachevski, and Bolyai. He explains how Lobachevski’s geometry of negative curvature and Riemann’s geometry of positive curvature arose from questioning the parallel postulate, and he considers whether these alternative geometries might apply to physical space. The book concludes by discussing the mathematical extension of geometry to higher dimensions and multiple manifolds, suggesting that such extensions are valuable as intellectual exercises even when they have no immediate application to reality.
Inside the Book
The Book’s Design and the Problem of Two Spaces
The treatise opens with a clear statement of purpose, announced on its title page and contents: Ernst Mach’s Space and Geometry in the Light of Physiological, Psychological, and Physical Inquiry, translated by Thomas J. McCormack. The work is organized into three chapters, each addressing a distinct mode of inquiry: the physiological, the psychological, and the physical. The first chapter, “On Physiological, as Distinguished from Geometrical, Space,” immediately establishes the central tension of the book. Mach draws a sharp line between the space of our immediate, sensible experience and the idealized space of Euclidean geometry. The latter is homogeneous, infinite, and unbounded—everywhere alike in all directions. The former, the space we actually perceive, is none of these things. It is a space of feeling, where “up” and “down,” “near” and “far” are not neutral coordinates but distinct qualities of experience.
The Space of Vision
Mach begins his investigation with the sense of sight, which Johannes Müller and Ewald Hering had already identified as presenting a space fundamentally different from the geometric ideal. Visual space is not constituted alike everywhere. The feeling of looking upward is qualitatively different from looking downward, just as the sensation of an object receding into the distance is unlike one approaching. Even the distinction between right and left, while less pronounced due to the body’s physiological symmetry, carries its own specific feeling-tone. This anisotropy is not a mere curiosity but manifests in observable phenomena. Mach cites the example of stones at a tunnel entrance, which appear to swell dramatically as one approaches rapidly and to shrink upon emerging, demonstrating that objects in visual space cannot be moved without undergoing expansion or contraction. This suggests that visual space is closer to the curved, non-Euclidean spaces of the metageometricians than to Euclid’s flat plane. A simple experiment reinforces this: a cylindrical glass vessel tipped over the face, or a walking-stick laid endwise against the eyebrow, appears conical, revealing the inherent distortion of our visual field.
Furthermore, visual space is bounded, sometimes narrowly so. Plateau’s experiment shows that an after-image does not appreciably diminish in size when projected onto a surface beyond thirty meters, implying a limit to our spatial extension. The naive perception of the heavens as a finite sphere, shared by ancient astronomers, is a direct consequence of this boundedness. The oblateness of the celestial vault—the fact that the sky appears to us as a flattened dome rather than a perfect hemisphere—was known to Ptolemy and discussed by Euler, and it provides further proof that visual space is unequal in extent in different directions. Mach notes that Zoth found a physiological explanation for this phenomenon, interpreting it as a result of the elevation of the line of sight relative to the head, a line of research later completed by F. Hillebrand. The very possibility of panoramic painting, which seeks to envelop the viewer in a scene, depends on the narrow boundaries of our visual field. Crucially, Mach argues that visual space is not originally metrical. Localities and distances differ in quality, not in quantity; our ability to measure visually is a derived skill, built upon primitive physical and metrical experiences rather than being an innate property of sight.
The Space of Touch
Turning from the eye to the skin, Mach finds a similar story. The skin, a closed surface of complicated geometrical form, serves as an agency of spatial perception. We distinguish the quality of an irritation and, through a supplementary sensation, its locality. This supplementary sensation need only differ from place to place to serve biological needs. E. H. Weber’s investigations revealed the profound discrepancies between the skin’s space-sense and metrical space. Two divider points, for instance, are recognized as distinct at a distance fifty to sixty times smaller on the tip of the tongue than on the middle of the back. The skin’s space is not homogeneous; it is a distorted map of the body. Weber’s experiments with divider points show that points enclosing the upper and lower lips appear to shut when moved horizontally toward the side of the face, and points on adjacent finger-tips, when carried over the fingers, palm, and forearm, appear to close completely at the forearm. The skin can distinguish the forms of touching bodies, though it is inferior to the eye; the tongue tip can recognize the circular cross-section of a tube only two millimeters in diameter. Mach offers a personal case: objects placed in his paralyzed hand were unrecognized, but when another person closed his hand, he recognized all of them. The initial failure was not a failure of sensation but an erroneous inference about impaired sensibility. Skin space, Mach suggests, is analogous to a two-dimensional, finite, unbounded, closed Riemannian space. The movements of body members—arms, hands, fingers—superimpose something like a third dimension, allowing us to estimate the thickness of a plate grasped in the dark. Both haptic and visual space, he concludes, have as little in common with metric space as they do with each other; both are anisotropic and non-homogeneous, with cardinal directions that are not equivalent.
The Biological Function of Space-Sense
The sense of space is not a universal gift but is developed only where it has a biological function. We lack information about our internal organs, over which we have no control. Our space-sense does not extend far into the nostrils; we cannot tell whether scents from paired pipettes enter the right or left nostril. Tactual sensibility of the ear extends to the tympanum, enabling us to determine whether the louder of two sound-impressions comes from the right or left, but this is a rough locality information, inadequate for exact purposes. This biological dependence is the key to understanding the nature of physiological space.
Correspondence and the Origin of Dimensions
Despite their differences, physiological and geometric space share a fundamental correspondence. Both are threefold manifoldnesses. To every geometric point A, B, C, D corresponds a physiological point A′, B′, C′, D′. If C lies between B and D, then C′ lies between B′ and D′. Continuous motion in geometric space corresponds to continuous motion of a co-ordinate point in physiological space. Mach remarks, however, that this continuity is a convenient fiction, not an actual necessity. Every system of sensations, including space-sensations, is finite; an endless series of sensational qualities or intensities is psychologically inconceivable. The properties of visual space are biologically adapted, and biological needs would not be satisfied with pure geometric relations. The distinction between “rightness,” “leftness,” and “aboveness” is not a geometric luxury but a biological necessity.
The origin of our familiarity with the three dimensions of geometric space lies in the anatomy of our bodies. The system of space-sensations is broadly similar across animals that, like humans, have three cardinal directions distinctly marked on their bodies. Above/below, front/behind, and right/left are anatomically unlike in such animals. Though right and left appear externally symmetrical, this geometrical and mechanical symmetry, which serves rapid locomotion, masks a deeper anatomical and physiological asymmetry. The plaice, a flatfish, exemplifies how closely allied species to symmetrical animals can assume strikingly unsymmetrical forms, while the externally symmetric slug contrasts instructively with the unsymmetric shapes among its nearer relatives. This trinity of conspicuously marked cardinal directions may be regarded as the physiological basis for our familiarity with the three dimensions of geometric space.
The Primacy of Tactual Space and Its Correlation with Vision
While visual space is the clearest and broadest system of space-sensations, tactual space is biologically perhaps more important. Skin irritations are spatially registered from the outset, disengaging protective movements. These movements then induce sensations in the extended or contracted skin, joints, and muscles, which become associated with space-sensations. The first localizations in tactual space presumably occur on the body itself, as when the palm is carried over the thigh’s surface. However, Mach argues that deriving tactual space psychologically from such experiences, via the concept of time and assuming spaceless sensations, is “altogether futile.” Instead, he holds that touch-space and vision-space may be conceived in the same manner, by transferring Hering’s view of visual space to tactual space. A newly-hatched chick, noticing a small object, looks toward it and pecks immediately. A central-organ area is excited by the irritation, automatically disengaging eye and head and neck movements. This excitation, determined by the geometric locality of the physical irritation, is the basis of the space-sensation. The chick behaves like an infant spying a shining object and snatching at it. Acoustic, thermal, and gustatory irritations can also evoke prehension or defense movements, especially in blind people, corresponding to the same irritated central-organ parts and thus the same space-sensation. Blind people’s irritations are more restricted and less sharply localized, so their spatial-sensation system must initially be meager and obscure, as when a blind person defends against a buzzing wasp. Education can perfect this sense, as the achievements of the blind geometer Saunderson show, though his table, divided into quadratic spaces with pins and threads, indicates his orientation difficulty. He demonstrated the volume of a pyramid as one-third of a prism of the same base and height by dividing a cube into six congruent pyramids.
Feelings of Space and the Stimulus to Motion
Tactual space exhibits the same anisotropy and dissimilarity in its three cardinal directions as visual space, differing from Euclid’s geometric space. Optical and tactual space-sensations accord at many points: stroking a stationary surface with tangible objects feels them at rest, just as visual objects appear at rest during voluntary eye movement, even though the retinal images move. Moving objects appear in motion whether the organ is at rest or following them. Physiological symmetry and similarity find the same expression in both domains, yet the two systems cannot be identical, because different central-organ portions are affected when an object excites looking versus grasping. Both systems readily coalesce by association and adapt to one another. A chick can look, pick, run, and turn toward or around an object; a creeping child eventually rises and runs. These continuously passing cases require a similar viewpoint: certain brain parts, simply irritated, give rise to space-feelings and produce automatic movements, sometimes complicated ones. Extensive locomotion and orientation stimuli proceed from optical excitations but also from chemical, thermal, acoustic, and galvanic excitations in blind animals. Constitutionally blind worms and retrogressively blind animals like moles and cave animals show extensive movements, so space-sensations may be conceived analogously in sighted and sightless animals. A centipede creeping uniformly suggests a uniform stimulation stream from some organ, answered by rhythmic automatic movements of successive segments, with phase differences producing a longitudinal wave through the legs. Analogous phenomena exist in higher animals: during active or passive rotation about the vertical axis, labyrinthine irritation disengages nystagmic eye movements. The organism adapts to regular excitation alterations, and on cessation, negative after-images occur, as with Plateau and Oppel’s expanding spiral appearing to shrink when stopped. Mach long ago assumed that the rate of alteration of a stimulus with time corresponds to a special, possibly felt process associated with a definite organ. The rate of motion is thus directly felt within adaptive limits, not merely abstractly conceived, but as a specific sensation furnishing the original impulse to idea-formation. A person feels a line’s direction and curvature, not just successive points.
The Central Motor Organ and Primary Sensations
If an organ exists that, on simple irritation, disengages complicated locomotion movements, then this simple irritation, if conscious, may be regarded as the will or attention appurtenant to that locomotion. The organism needs the locomotion’s effect felt in a correspondingly simple manner. Space perception proceeds from biological need. The retina’s single specific spot with greater distinctness and discrimination is an economic device, necessitating eye movement in response to attention changes while excluding the disturbing effects of willed eye movements on space-sensations from resting objects. Sensations of movement, the last-discussed species, are sharply distinguished from previously investigated ones by their uniformity and inexhaustibility. These movement sensations appear only in animals free to move about; animals confined to a single spot are restricted to the first-considered sensations, now designated primary sensations of space, distinguished from secondary sensations of movement. A fixed animal possesses necessarily a bounded space, whose symmetry depends on the symmetry of its own body. A vertebrate confined to one spot and restricted in orientation could construct only a bounded space, dissimilar above and below, before and behind, and, accurately speaking, right and left, presenting an analogy with the physical properties of a triclinic crystal. If the animal acquired free movement, it would obtain an infinite physiological space, as movement sensations can always be produced anew unless prevented by accidental external hindrances. Untrammeled orientation and the interchangeability of every orientation with every other invest physiological space with equality in all directions. However, this result is not obtained through physiological factors exclusively, because orientation with respect to the vertical, the direction of gravity’s acceleration, is not altogether optional for any animal. Primary space cannot be absolutely supplanted by secondary space, being phylogenetically and ontogenetically older and stronger. Physiological space is in no wise concerned with metrical relations.
A Biological Theory of Spatial Perception
Mach proposes that physiological space is an adaptive result of the interaction of elementary organs, constrained to live together and absolutely dependent on co-operation. Of cardinal importance to animals are the parts of their own body and their relations; outward bodies matter only as they relate to the animal’s parts. Geometric space, by contrast, embraces only the relations of physical bodies to one another, leaving the animal body out of account. Only one species of consciousness elements is known: sensations. Spatial perceptions depend on sensations, whose character and operating organs must be left undecided. The view is this: an elementary organ’s feeling when in action depends partly on the irritation quality, the sense-impression, and a second part is determined by the organ’s individuality, the same for every stimulus, varying only from organ to organ, inversely proportional to ontogenetic relationship. This second part is the space-sensation. It is produced only with irritation of elementary organs; every time the same organ or complexus is irritated, the same space-sensation is evoked. This opens a phylogenetic and ontogenetic understanding of spatial perception, and after these conditions are elucidated, a physical and physiological explanation seems possible. Mach is far from thinking this explanation absolutely adequate or exhaustive, but he is convinced he has made some approach to truth.
The A Priori Theory and Physiological Influences in Geometry
Mach then addresses Kant’s assertion that one could never picture space as not actually existing, though one might imagine no objects in it. Today, he notes, scarcely anyone doubts that sensations of objects and of space enter consciousness only in combination, and vice versa leave only in combination. If for Kant space is not a “concept” but a “pure intuition a priori,” modern inquirers incline to regard it as a concept derived from experience. We cannot intuite our system of space-sensations per se, but may neglect object-sensations as subsidiary; overlooking this, the notion arises of a pure intuition.
Thibaut’s Deduction and the Draughtsman’s Method
The discussion of the triangle’s angle-sum theorem continues with a specific deductive proof attributed to Thibaut, a contemporary of Gauss, from his 1809 Grundriss der reinen Mathematik. The argument proceeds by considering the complete revolution of four right angles. A triangle possesses six right angles when its interior and exterior angles are combined. By subtracting the three exterior angles traced during a complete revolution—labeled a, b, and c—from this total of six right angles, one is left with two right angles, which is the sum of the triangle’s interior angles. A footnote directs the reader to Hankel’s history of mathematics for further context and notes that objections to this method will be considered later. The translator adds that Playfair independently gave the same proof in 1813.
A related, more practical demonstration arises from the work of a draughtsman. If a ruler is turned successively around the interior angles of a triangle, always in the same direction, its edge will be found to have swept out the entire interior angle sum. Initially lying outside the triangle, the ruler’s edge, upon returning to the first side, will now lie inside it, having performed half a revolution. This observation was noted during the act of drawing itself. The author connects this to a remark by Tylor, who suggested that the manipulation of cloth or paper-folding might have led to similar geometric insights. Indeed, folding a triangular piece of paper in a specific manner produces a double rectangle equal to half the triangle’s area, demonstrating that the three angles coincide at a single point and sum to two right angles. While acknowledging that paper-folding can yield astonishing results, as shown in Sundara Row’s work on the subject, the author expresses doubt that such processes were historically productive for the development of geometry. The material has too limited an application, and artisans lack the incentive for the exact observation required to derive general theorems from such practices.
Experimental Knowledge of Geometry
The angle-sum theorem, like the law of the lever or the gas law of Boyle and Mariotte, is ultimately reached through experience. Neither the unaided eye nor delicate instruments can absolutely demonstrate that the sum of a triangle’s angles is exactly two right angles; the same holds true for the lever law and Boyle’s law. All such principles are idealized, schematized experiences, and real measurements will always show slight deviations from the theoretical ideal. While the gas law proved to be approximate and required modification, the lever law and the angle-sum theorem remain in exact accord with facts within the limits of experimental error, as do all their consequences.
Parallels and Paving
The practice of paving with equal, similar triangles placed with their bases on a single straight line led to important geometric knowledge. When a triangle is displaced along a straight line without rotation, all its points describe equal paths. The same bounding line in two positions gives two straight lines that are equally distant at all points, which vouches for the equality of the angles made by the displacement line. Consequently, the sum of the interior angles on the same side equals two right angles, which is Euclid’s theorem of parallels. The possibility of extending the paving indefinitely increased the obviousness of this fact. Sliding a triangle along a ruler remains the simplest method of drawing parallel lines. The parallel theorem and the angle-sum theorem are thus inseparably connected, representing different aspects of the same fundamental experience.
Hexagons, Circles, and Inscribed Angles
Stone masons would naturally discover that regular hexagons can be composed of equilateral triangles, leading to the division of circles into six parts by the radius, and into three parts, among other divisions. Carpenters know instinctively that a rectangular, symmetric beam can be cut from a cylindrical tree-trunk in an infinite number of ways; the edges of such a beam lie in the cylindrical surface, and the diagonals of its section pass through the center. According to Hankel and Tylor, this practical knowledge likely led to the discovery that all angles inscribed in a semicircle are right angles.
Role of Physical Experiences
A stretched thread furnishes a visualization of the straight line, which is characterized by its physiological simplicity: all its parts induce the same direction sensation, every point evokes the mean of the space-sensations of its neighboring points, and every part resembles every other. However, the geometer requires more than this physiological characterization, which nonetheless influenced Euclid’s Definition 3. Physical experience with corporeal objects enriches the visual image. For instance, a string fastened at one point and passed through a ring at another, when pulled, approaches straightness. Fewer identical parts compose the straight line joining the two points than a curved line.
Straight Line as Shortest
It is erroneous to believe that visualization alone shows the straight line is the shortest distance between two points. We can reproduce in imagination the simultaneous change of form and length, but this is merely a reviviscence of prior experience, an experiment in thought. Passive contemplation of space never yields this knowledge. Measurement involves physical reaction and superposition, and visualized lines cannot be applied to one another forthwith; this must be experienced with material objects that are accounted unalterable. Animals lack instinctive knowledge of the straight line as the shortest path. If a stimulus excites an animal’s attention and it turns so its plane of symmetry passes through the stimulus, the straight line is the uniquely determined path, as shown in Loeb’s investigations on animal tropisms.
Triangle Inequality
Visualization alone does not prove that two sides of a triangle exceed the third. One can lay two sides upon the base by rotating them around the basal-angle vertices, and the imagination shows the free ends moving in arcs that overlap and fill the base. But this is only convincing because the procedure was previously witnessed with corporeal objects. Euclid, in Book I, Proposition 20, deduces this fact circuitously from the proposition that the greater side is opposite the greater angle. The true source of this knowledge is the experience of the physical motion of triangles, a fact laboriously concealed by the form of deduction.
Straight Line Properties
A wire of arbitrary shape laid on a board between two upright nails, when slid while contacting them, constantly changes its form and position between the nails; a straighter wire alters less. A straight wire slides in itself, and when rotated round two fixed points, a crooked wire changes its position, but a straight wire rotates within itself. Defining a straight line as completely determined by two points is an idealization of empirical notions derived from physical experience.
Dependence of Measures and the Problem of Geometry
The discovery that the area of a figure depends not only on the lengths of its sides but also on its angles introduced the real problem of geometry. A rectangle is larger in area than an oblique parallelogram with sides of the same length. A rectangle made of strips of wood parallel to the base can be displaced into any parallelogram of the same height and base without altering its area. Quadrilaterals with given sides are undetermined in their angles, a fact well known to carpenters, who add diagonals to convert quadrilaterals into rigid triangles, which are unalterable in their angles once their sides are given. The perception that measures depend on one another introduced the real problem of geometry, a problem addressed by Steiner in his principal work on the systematic development of the dependence of geometrical figures on one another, and by Snell in his treatise on elementary geometry.
Physical and Mental Experiments
A plane physical triangle can be constructed of wires. Rotating one side around a vertex to increase the interior angle changes the position of the moved side and makes the opposite side grow larger, requiring a new wire to complete it. Such experiments can be repeated in thought, but the mental experiment is only a copy of the physical one. The mental experiment would be impossible if physical experience had not led to the knowledge of spatially unalterable physical bodies, which is the concept of measure. A footnote indicates that Euclidean geometry shows traces of this foundation, as does the “geometric characteristic” of Leibnitz.
Geometry of the Triangle
From such experiences, we learn that of the six metrical magnitudes of a triangle—three sides and three angles—three, including at least one side, suffice to determine it. If only one angle is given, it must be the included angle or the angle opposite the greater side for unique determination. Since a triangle is determined by its three sides and its form is independent of position, an equilateral triangle has all three angles equal, and an isosceles triangle has the two angles opposite the equal sides equal. This is logically certain, but its empirical foundation is no more superfluous than in physics.
The Pythagorean Theorem
The dependence of sides and angles is first recognized in special instances. Computing the areas of rectangles and triangles formed by their diagonals revealed that a rectangle with sides 3 and 4 units gives a right-angled triangle with sides 3, 4, and 5. Rectangularity was thus connected to a rational ratio between sides, and this was used to stake off right angles with ropes of 3, 4, and 5 units. The equation 3² + 4² = 5², generalized as a² + b² = c², riveted attention. Neither Greek geometrical nor Hindu arithmetical deductions of the Pythagorean theorem could avoid considering areas. An essential point is that if a triangle is slid along a short distance in its own plane, the space it leaves behind is compensated by the new space it enters; the area swept out by two sides equals the area swept out by the third side. This assumes the conservation of the triangle’s area. Considering a surface as a body of minute but unvarying thickness leads to the conservation of volume as the fundamental assumption. This applies to translating a tetrahedron but yields no new viewpoints. Conservation of volume is common to rigid and liquid bodies and was idealized as impenetrability; rigid bodies also preserve distances between all their parts, while liquids have rigid properties only for the smallest elements of time and space.
Deriving the Pythagorean Theorem
If an oblique-angled triangle with sides a, b, and c is displaced in the direction of side b, only sides a and c describe equivalent parallelograms. If a makes a right angle with b, and the triangle is displaced at right angles to c, side c describes the square c², while the other two sides describe parallelograms whose combined areas equal the square’s area. These parallelograms are equivalent to a² and b², yielding the Pythagorean theorem. Alternatively, sliding the triangle a distance a at right angles to a, and then a distance b at right angles to b, gives a² + b² equal to the sum of the surfaces swept out by c, which is c². For an oblique-angled triangle, this gives the more general proposition c² = a² + b² − 2ab cos γ. The dependence of the third side on the other two is thus determined by the area of the enclosed triangle, or by a condition involving volume.
Limits of Visualization
If two lines of lengths a and b meet at a point, the length c joining their free extremities is between the limits c < a + b and c > a − b. Visualization alone cannot inform us of this; it requires experimenting in thought, which reposes on and reproduces physical experience. A straight line is primarily a unique concrete image with physiological properties, obtained from a physical body like a string or wire of constant thickness that interposes a minimum of volume between its extremities. In abstract space, differences of direction do not exist; a straight line through a point is determined only by assigning a second physical point. Defining a straight line as constant in direction, or an angle as a difference of directions, is a physiological definition.
Measurement of the Angle
An angle can be determined by the distance between two fixed points on its sides. Choosing points at a fixed distance from the vertex is inconvenient because equimultiples of an angle would not be measured by equimultiples of the distance. A simpler measure uses the aliquot part of the circumference or area of a circle.
Sources of Geometrical Knowledge
Geometrical knowledge derives from multiple sources: physiological acquaintance with spatial forms through direct visual and tactual contact; physical (metrical) experiences involving the comparison of space-sensations evoked by different bodies under identical circumstances; and these experiences themselves express relations between sensations. These diverse orders are so interwoven that only thorough analysis can separate them, which explains the divergent views of those who base geometry on pure visualization and those who base it on physical experience.
Imagined Spatial Framework
An unbiased person picturing Cartesian coordinates would describe a system of rigid, transparent, penetrable, contiguous cubes with bounding surfaces marked only by nebulous visual and tactual percepts—phantom cubes through which real bodies or their phantom counterparts move. Gauss’s investigation of curved surfaces concerns the application of infinitely thin, flexible laminate bodies to one another.
Fundamental Facts
All varied experiences reduce to a minimum: movable rigid bodies exist. From any point, three lines not all in one plane can be drawn; by three movements along these lines, any point can be reached from any other. Three dimensions, physiologically and metrically the simplest, suffice for all spatial determinations.
Idealization
Physical metrical experiences are conceptualized and idealized. Absolutely rigid bodies, perfect straight lines, and perfect planes no more exist than perfect gases or liquids. Theoretical geometry assumes objects that fulfill its requirements absolutely; practical geometry must consider deviations. Geometry has an advantage over physics in that detected deviations can be removed, whereas physics cannot construct more perfect gases than exist, being concerned with relations like pressure, volume, and temperature that are independent of our will.
Mental Experiment
Fundamental truths derive from physical experience since space sensations are inaccessible to measurement. Yet once relations between spatial visualizations and metrical experiences become familiar, geometrical facts can be reproduced by purely mental experiment. Sensuous spatial imaginings are qualitative, not quantitative. The example of one coin rolling around another of equal size shows that imagination alone cannot determine the angle described in a full revolution; metrical concepts fix idealized experiences while direction is retained in sensuous imagination.
Kant’s Theory
Kant stated that “thoughts without contents are empty, intuitions without concepts are blind.” Mach suggests instead that “concepts without intuitions are blind, intuitions without concepts are lame.” Regarding Kant’s claim about mathematics in natural knowledge, Mach asserts that all sciences are sciences only insofar as they operate with concepts whose contents we have determined.
Present Form of Geometry
Rigid movable bodies suffice for understanding any geometrical fact, but geometry answers recurring questions economically by selecting a few simple theorems and formulating general propositions. Propositions about triangles receive emphasis for their wide applicability.
Universal Validity
Kroman asks why a demonstration with a special figure holds universally, answering that rapid variations impart all possible forms to the figure in thought. Mach agrees in essentials but notes that individuals may not achieve this “with the rapidity of lightning”; errors prove that some rested on conjecture. Generations collectively verify geometry. Mach recounts a teacher who required demonstrations with incorrect figures—useful for showing the role of logical operations, but habitually employed, it misses that abstract concepts draw their power from sensuous sources.
The Geometry of Spaces and the Limits of Rigid Bodies
The discussion turns to a fundamental constraint: rigid bodies can move freely only in spaces of constant curvature, a point Helmholtz proved at length in 1868. Riemann extended the concept of finite but unbounded lines—like great circles on a sphere—to a three-dimensional space of positive curvature, conceiving analogues of the straight line and plane. A difficulty arises when considering a measure of curvature for four-dimensional space, which would allow an easy transition to the three-dimensional case, but moving from the special to the general case involves a degree of arbitrariness, with different inquirers like Riemann and Kronecker taking different courses. The text questions whether a measure of curvature exists for one-dimensional space, concluding it does not as an interior measure, and warns of a possible illusion where we operate with symbols corresponding to nothing real or representable to the senses.
The Early Discoveries in Geometry
The narrative then turns to the “preliminary history of two thousand years” behind Gauss’s conviction of the empirical foundations of geometry. Early geometers, working with rule in hand, studied simple figures like the straight line, plane, and circle. They observed that fixing one, then two, then three points of a body progressively restricts its mobility, distinguishing pure rotation, pure displacement, and compound motion. This early geometry conceded to physiological factors of sense, explaining two fundamental measures: straight length and circular measure. The straight line was a rigid measuring-rod, and the angle was the rotation of one line relative to another. A proof is sketched where rotating a line b about its intersection with c through angle a, then about its intersection with a through angle β, yields an exterior angle μ = a + β, and since μ + γ = 2R, it follows that a + β + γ = 2R. Displacing a rigid system of intersecting lines shows the sum of interior angles of the resulting triangle is 2R, revealing properties of parallel lines. Doubts about successive rotation or pure displacement—justified on curved surfaces—never troubled these early discoverers. The study of rigid body movement, which Euclid avoids, remains best for elementary instruction.
Deductive Geometry
This naive conception vanished with professional scholarly treatment. The goal became systematization, separating directly cognizable truths from deduced ones. Euclid’s system placed the simplest principles first, reducing them to a minimum and detaching geometry from its empirical soil. Derived truths were regarded as higher than perceived ones, leading to demands for proofs of obvious propositions. Euclid’s system arose, per tradition, to check the Sophists. This artificial method concealed research methods and lost the organic connection between principles, producing pedants rather than investigators. A footnote notes a German reaction from Herbart, Schopenhauer, and Trendelenburg favoring genetic methods. Scholasticism worsened matters, fostering an exaggerated respect for logical form. The entire period from Euclid to Gauss suffered under this influence.
Euclid’s Fifth Postulate
This postulate states that if a line meets two lines making interior angles on one side less than two right angles, the lines meet on that side. Euclid proves parallels make equal alternate angles only by this postulate. The converse—that only one parallel passes through a point—is equivalent, and the angle-sum theorem follows from it and vice versa. Since converging lines meet beyond observation, successors of Euclid futilely tried to prove the postulate from Euclid to Gauss—a “sublime spectacle” of men seeking hidden sources of an undoubted truth.
Saccheri’s Theory of Parallels
Saccheri and Lambert are notable figures in this pursuit. Rectangles cannot be demonstrated without the Fifth Postulate. Two congruent isosceles right triangles laid together form an equilateral quadrilateral whose angles at B and C are indeterminate by the first 27 propositions. Analogous theorems hold on spheres and surfaces of constant negative curvature, giving obtuse or acute angles. Saccheri’s achievement was his problem statement: if the postulate is involved in other assumptions, then in a quadrilateral with right angles at A and B and AC=BD, the angles at C and D must be right; assuming obtuse or acute leads to contradictions. He shows each hypothesis holds universally if it holds in one case. He adverts to physico-geometrical experiments supporting the right angle, rightly not assuming an equidistant line is straight. Other experimental proofs include the angle in a semicircle being right, the radius subtended thrice, and the existence of similar triangles. Saccheri refuted the obtuse hypothesis but struggled with the acute, reaching conclusions later rediscovered by Lobachevski and Bolyai. He rejected the acute hypothesis as incompatible with the straight line’s nature, showing bias.
Lambert’s Investigations
Lambert, in 1766, starts from a quadrilateral with three right angles, examining the fourth as right, obtuse, or acute. Similarity is incompatible with the latter two. The obtuse case, where the angle-sum exceeds 2R, is realized on spherical surfaces. He conjectured the acute case, where the angle-sum is less than 2R, might occur on a sphere of imaginary radius. The departure of the angle-sum from 2R is proportional to triangle area. Lambert nearly reached modern views.
Gauss’s View of Geometry
The passage discusses investigators who expressed radical views on geometry only briefly, through word of mouth or letters. According to Sartorius von Waltershausen, Gauss regarded geometry merely as a logically consistent system of constructs, with the theory of parallels placed at the pinnacle as an axiom. Gauss had reached the conviction that this proposition could not be proved, though it was known from experience—for example, from the angles of the triangle joining the Brocken, Hohenhagen, and Inselsberg—that it was approximately correct. If this axiom were not conceded, Gauss contended, there results from its non-acceptance a different and entirely independent geometry, which he had once investigated and called the Anti-Euclidean geometry.
Stolz’s Researches
O. Stolz, in a small but instructive pamphlet, sought to deduce the principal propositions of Euclidean geometry from purely observable facts of experience. Given one large triangle ABC having the angle-sum 2R, one draws the perpendicular AD on BC, completes the figure by congruent triangles, and adds a congruent figure to obtain a single rectangle. A rectangle can be divided into two congruent rectangles by a perpendicular erected at the middle point of one side, and by continuing this procedure the line of division may be brought to any point desired. From a given rectangle one can cut out a smaller one having sides bearing any proportion. The diagonal divides it into two congruent right-angled triangles, each having the angle-sum 2R. Every oblique-angled triangle can be decomposed into right-angled triangles, so 2R results for every triangle if it holds exactly of one. From these propositions, the two opposite sides of a rectangle are everywhere the same distance apart and never intersect, having the properties of Euclidean parallels. Two straight lines cut by a third making the sum of interior angles less than two right angles will meet on that side, and the straight line is infinite.
Geometry and Physics Compared
Geometry consists of the application of mathematics to experiences concerning space. Like mathematical physics, it becomes an exact deductive science only by representing objects of experience through schematizing and idealizing concepts. The existence of straight lines, planes, and the angle-sum can be maintained only within limits of errors of observation. Physics sometimes replaces ideal assumptions with more general ones, and a similar procedure is permissible in geometry. This context illuminates the endeavors of Legendre, Lobachevski, and the two Bolyais.
Contributions of Lobachevski and Bolyai
Lobachevski’s works first became known in 1829, followed by the younger Bolyai’s publication in 1833, agreeing in essential points. Lobachevski undertook investigations hoping to become involved in contradictions by rejecting the Euclidean axiom, but after finding himself mistaken, drew all consequences from this fact. From a point outside a line, a perpendicular is dropped, and another line makes an acute angle with it. Assuming they do not meet but would on slight diminution of the angle, homogeneity of space forces the conclusion that a line with the same angle behaves similarly on the other side. All non-intersecting lines lie between the two boundaries, which are called parallels by Lobachevski. The parallel-angle is an inverse function of the perpendicular, designated Π(p). Parallels approach one another asymptotically toward convergence. A circle increasing indefinitely passes into the “boundary-line”; a sphere’s surface passes into the “boundary-surface.” Boundary-lines on a boundary-surface follow Euclidean rules. The ratio of boundary-arcs between parallels depends solely on distance, introducing exponentials and hyperbolic functions. For a right-angled Lobachevskian triangle, the formula passes into the spherical formula when the imaginary unit replaces the parameter, and into plane Euclidean geometry for very large values of that parameter.
Different Systems of Geometry
A self-consistent geometry can be constructed on the assumption of convergence of parallel lines, though no observation supports it. Our imagination, dominated by Euclidean concepts, grasps Lobachevski’s views only gradually. Different ideas can express the facts with equal exactness; facts must be distinguished from intellectual constructs. Riemann first intimated the possibility of an analogous geometry for three-dimensional space of positive curvature. Helmholtz, continuing Riemann’s researches physically, neglected in his first publication the Lobachevskian case of negative curvature. Both Euclidean geometry and the geometries of Lobachevski and Riemann can represent spatial facts precisely if the parameter is large enough in the latter cases. Physicists have found no reason to depart from the Euclidean assumption, adhering to simplest assumptions until facts force modification.
The Mathematical Extension of Geometry
When a mathematician tentatively modifies the simplest assumptions of geometrical experience and gains fresh insight, it is natural to pursue such research purely for mathematical interest. This leads to constructing analogues of familiar geometry on broader assumptions for any number of dimensions, with no pretension of being applied to reality. The text cites the advances made by Clifford, Klein, and Lie as support. It asserts that thinkers like Gauss, Lobachevski, Bolyai, and Riemann were clear that they were not imagining a space of more than three dimensions for our sense-space, nor departing appreciably from the Euclidean for representing it, and cannot be held responsible for the “grotesque fictions” constructed later in this domain. A physicist prefers to regard what is directly given as the source of his ideas, applicable to the inaccessible until obliged to change them. However, the physicist can be grateful for the discovery of several sufficing geometries and the abolition of conventional barriers of thought. An example is given: if we lived on a planet with a turbid, opaque atmosphere, assuming the earth’s surface was a plane and using only a square and chain for geodetic measurements, the increase in the excess of the angle-sum of large triangles would compel us to substitute a spherometry for planimetry. The physicist cannot reject the possibility of analogous experiences in three-dimensional space, though phenomena compelling acceptance of a Lobachevskian or Riemannian geometry would be so odd that their occurrence is not probable.
The Question of Straight Lines and Physical Objects
The text addresses the question of whether a physical object is a straight line or an arc of a circle, calling this question improperly formulated. A stretched chord or ray of light is neither. The real question is whether the object reacts spatially to conform better to one concept than the other, and whether it conforms at all to any geometric concept with sufficient exactitude. If not, one must determine and make allowance for the deviation, correcting the measurement. Since practical measurements depend on comparing physical objects, if direct investigation showed coincidence with geometric concepts but indirect results deviated more than errors permitted, we would be obliged to change our physico-metric notions. The physicist should await such a situation while the mathematician has free scope for speculation.
The Concepts of Mathematics and Physics
Space and time are the simplest concepts. Spatial and temporal objects can be framed with great exactness, and deviations can be eliminated. Unlike perfect gas, fluid, or elastic bodies, which do not exist and whose deviations cannot be removed, we can conceive a sphere or plane with unlimited exactness without running counter to any fact. Therefore, when a new physical fact requires modifying concepts, the physicist sacrifices the less perfect concepts of physics rather than the simpler, more lasting concepts of geometry. However, the physicist can derive assistance from geometers regarding things of thought like atoms and molecules, which can never be objects of sensuous contemplation. There is no obligation to think of them in Euclidean three-dimensional spatial relationships. A footnote mentions the author’s 1863 idea that non-sensuous things did not have to be pictured in sensuous three-dimensional space, leading him to analogues of spaces of different dimensions, though the notion of finite spaces was then remote from him.
The Relativity of All Spatial Relations
Geometry traces its origin to practical needs. Recognizing spatial substantiality and invariability is a biological necessity. When physiological organization is insufficient, we use hands and feet for comparing spatial objects, entering the domain of physics. All physical determinations are relative, so all geometrical determinations are valid only relative to the measure. The concept of measurement is a relation containing nothing not in the measure. Geometry assumes the measure will always coincide with what it has coincided with before, but this determines nothing about the measure. Spatial physiological equality is replaced by a differently defined physical equality, not to be confounded with the former, just as thermometer indications are not identical to the sensation of heat. The practical geometer corrects for a heated measure’s dilatation, but to pure theory of space, assumptions about the measure are foreign. It is superfluous to assume bodies suffer alterations or remain unchanged on displacement, as this could only be determined by a new measure. This manifests the relativity of all spatial relations.
Introduction of the Notion of Number
Introducing number into geometry modifies the criterion of spatial equality, gaining a nicety of distinction congruence alone could not attain. Applying arithmetic to geometry leads to incommensurability and irrationality. Geometric concepts thus contain adscititious elements not intrinsic to space, representing space with greater precision than spatial observation could realize. This imperfect contact between fact and concept explains the possibility of different systems of geometry.
The Significance of the Metageometric Movement
The entire movement is characterized as sound and healthful, not terminated but expected to continue, advancing mathematics and geometry epistemologically and aiding other sciences. It sprang from a general need, not an individual, as seen in the different professions of its participants—mathematician, philosopher, educationist. Ideas uttered by Leibnitz recur in Fourier, Lobachevski, Bolyai, and H. Erb. The philosopher Ueberweg, opposing Kant and starting from Erb, anticipates much of Helmholtz’s labors.
Summary
The results are listed: first, the source of geometric concepts is experience. Second, the character of concepts satisfying the same geometrical facts is many and varied. Third, comparing space with other manifolds has freed geometric thought from conventional limitations.
Main Arguments
The central argument of the book is that space, as human beings experience it, is fundamentally different from space as geometry conceives it. Mach demonstrates through a wealth of physiological evidence that visual and tactual space are neither homogeneous nor isotropic. The space of vision is bounded, sometimes narrowly; it is organized around the three cardinal directions of the body—above and below, before and behind, right and left—which are not equivalent in experience. The space of touch is similarly anisotropic, with the skin’s sensitivity varying dramatically across its surface. These physiological spaces are not metrical; they do not present distances and sizes as quantities but as qualities, and they are shaped by biological function rather than by geometrical principle.
From this physiological foundation, Mach argues that geometry arises through a process of idealization from physical experience. The concept of the rigid body—a body whose spatial properties remain unchanged when it is moved—is not given in sensation but is abstracted from experience with material objects. Measurement, the comparison of bodies with bodies, is the fundamental operation from which geometrical concepts develop. The straight line is not a pure intuition but an idealization of physical experiences with stretched strings and straight edges. The angle-sum theorem of the triangle is not a truth of reason but an empirical generalization, known through physical manipulations and subject to the same kind of verification and potential revision as the laws of physics.
A related argument concerns the role of the a priori in geometry. Mach engages directly with Kant’s claim that space is a pure intuition a priori, acknowledging that Kant called attention to an important point: spatial sensations are independent of the quality of the stimulus that evokes them. We can make predictions about spatial relations without knowing what objects occupy the space. But Mach insists that this independence does not establish the a priori character of geometry. The concepts required for geometry—rigid bodies, congruence, measurement—are derived from experience, and the propositions of geometry are empirical hypotheses subject to empirical verification.
The book also argues for the legitimacy and potential applicability of non-Euclidean geometries. Mach traces the history of attempts to prove Euclid’s parallel postulate, showing how Saccheri and Lambert nearly discovered non-Euclidean geometry in the eighteenth century, and how Gauss, Lobachevski, and Bolyai finally recognized that the postulate could not be proved and that alternative geometries were logically consistent. He argues that both Euclidean and non-Euclidean geometries can represent spatial facts with equal exactness if appropriate parameters are chosen, and that the choice between them is a matter of empirical investigation and practical convenience rather than of a priori necessity.
A further argument concerns the relationship between the various “spaces” that human beings inhabit. Mach distinguishes between the physiological space of sensation, the metric space of physical measurement, and the abstract spaces of geometry. These are not the same, and confusion arises when they are conflated. Physiological space is bounded, anisotropic, and qualitative; metric space is established through physical operations of comparison and measurement; geometrical space is an idealization that abstracts from the peculiarities of both. The task of epistemology, as Mach conceives it, is to keep these distinctions clear while understanding how they relate to one another.
Key Concepts
Mach introduces and develops several specific concepts that structure his analysis. The distinction between primary and secondary sensations of space is fundamental. Primary sensations are those that arise from the direct irritation of sensory organs—the visual and tactual sensations that give us spatial information about objects in relation to our bodies. Secondary sensations are those that arise from movement, from the sensations of muscles, joints, and the labyrinth of the ear during locomotion. Primary space is bounded and organized around the body’s symmetry; secondary space becomes infinite and homogeneous through the possibility of free movement. Yet primary space is phylogenetically and ontologically older and stronger, and it reasserts itself whenever movement ceases.
The concept of space-sensation is central to Mach’s physiological account. He proposes that when an elementary organ is irritated, the resulting feeling has two components: one determined by the quality of the irritation (the sense-impression proper) and another determined by the individuality of the organ itself—a component that is the same for every stimulus affecting that organ but varies from organ to organ. This second component is the space-sensation. It is what allows us to distinguish the location of a stimulus without regard to its quality, and it provides the physiological basis for spatial perception. Mach is careful to note that this is a hypothesis, not an established fact, but he argues that it provides a fruitful framework for understanding how spatial perception could have evolved.
The notion of physiological space as a distortion of geometrical space is another key concept. Mach suggests that when we map the data of visual and tactual perception onto the framework of Euclidean geometry, the result is a systematic distortion—visual space is not merely an imperfect copy of geometrical space but a transformation of it that preserves certain properties (continuity, threefold manifoldness) while altering others (metrical relations, homogeneity). This concept allows Mach to explain both the similarities and the differences between perceived and geometrical space, and to account for the fact that physiological considerations continually intrude into geometrical thinking—in our tendency to distinguish right from left, above from below, near from far, even when these distinctions have no place in pure geometry.
Mach’s treatment of measurement emphasizes that all measurement is ultimately measurement by bodies. When we measure a length by applying a rod repeatedly, we are really counting the number of times a body of a certain size can be laid end to end; when we measure an area by covering it with unit squares, we are counting bodies of a certain shape. The abstraction involved in measuring a line by a line, or a surface by a surface, is a later development that conceals the physical basis of measurement. This concept of measurement as the enumeration of like bodies provides a unified account of how geometrical magnitudes are determined and explains why the concept of the rigid body is so fundamental to geometry.
The concept of mental experiment is developed to explain how geometrical knowledge can appear to be a priori even though it is derived from experience. Mach argues that when we “see” that the sum of the angles of a triangle is two right angles, or that two sides of a triangle are together greater than the third, we are not intuiting a necessary truth but reproducing in imagination an experiment that we have performed physically. The mental experiment is a copy of the physical experiment, and it is possible only because physical experience has taught us about the behavior of rigid bodies. This concept allows Mach to acknowledge the psychological force of geometrical conviction while denying that it has any non-empirical source.
The historical concept of the parallel postulate and its vicissitudes provides Mach with a concrete case study of how geometrical knowledge develops. He traces the futile attempts of Euclid’s successors to prove the postulate, the near-discoveries of Saccheri and Lambert, and the final recognition by Gauss, Lobachevski, and Bolyai that the postulate is an empirical assumption rather than a necessary truth. The concept of the angle of parallelism, introduced by Lobachevski, and the associated ideas of boundary-lines and boundary-surfaces, are explained as consequences of rejecting the Euclidean postulate. Mach uses this history to illustrate his general thesis that geometrical propositions are empirical hypotheses, subject to the same kind of verification and revision as the laws of physics.
Themes
A recurring theme throughout the book is the biological grounding of human cognition. Mach repeatedly emphasizes that space perception exists because it serves the needs of organisms—that we perceive space where spatial information is biologically useful and fail to perceive it where it is not. The skin’s varying sensitivity, the eye’s limited field, the organization of space around the body’s cardinal directions—all are understood as adaptations to the conditions of life. This biological perspective leads Mach to a kind of naturalism about knowledge: our concepts, including our most abstract geometrical concepts, are products of our nature as embodied organisms interacting with a physical environment.
The theme of the unity of science appears in Mach’s insistence that geometry is continuous with physics rather than separate from it. Geometrical propositions, like physical laws, are idealizations of experience; they hold within limits of observational error; they can be modified or replaced if facts demand it. The distinction between geometry and physics is one of degree of abstraction rather than of kind. This theme is developed through the historical account of non-Euclidean geometry, which shows that geometry can be treated as an empirical science without losing its logical rigor.
The relationship between sensation and concept is another pervasive theme. Mach argues that sensations provide the raw material for all knowledge but that concepts are required to organize and interpret them. He quotes Kant’s dictum that thoughts without contents are empty and intuitions without concepts are blind, but he modifies it: concepts without intuitions are blind, intuitions without concepts are lame. The development of geometry illustrates this interdependence. Geometrical concepts—rigid body, straight line, congruence—are abstracted from experience, but they are not given in experience; they require an act of idealization that goes beyond what the senses directly provide.
The theme of intellectual economy appears in Mach’s account of why geometry developed as it did. Geometry, he suggests, is a device for answering recurring questions economically—for codifying the results of physical experience in a form that can be applied without repeating the experiments. The selection of certain propositions as fundamental, the emphasis on triangles and their properties, the axiomatic organization of the subject—all are understood as economic devices that make geometrical knowledge more usable. This economic perspective is characteristic of Mach’s broader philosophy of science, which views scientific theories as economical summaries of experience.
A final theme concerns the limits of human cognition. Mach is impressed by the fact that some of our most confident intellectual constructions—such as the non-Euclidean geometries—strain against the limits of what we can actually imagine. Lobachevski’s geometry, he notes, can be grasped only gradually because our imagination is dominated by Euclidean concepts. The mathematician may construct geometries of any number of dimensions, but these constructions may correspond to nothing that is real or representable to the senses. Mach warns against the illusion of operating with symbols that have no intuitive content, while acknowledging that such symbolic manipulation can be fruitful even when it outruns immediate understanding.
The Author’s Conclusions
Mach concludes that space perception and geometry must be understood through a synthesis of physiological, psychological, and physical inquiry. The physiological investigation reveals that the space of sensation is a product of the organism’s biological organization—bounded, anisotropic, organized around the body’s cardinal directions, and shaped by the needs of survival. The psychological investigation shows that geometrical concepts develop from physical experience through a process of idealization and abstraction, and that what we call “mental experiments” in geometry are really reproductions of physical experiments. The physical investigation demonstrates that geometry is an empirical science, grounded in the behavior of rigid bodies and subject to verification through measurement.
The three perspectives are not in conflict but complement one another. Physiological space provides the raw material from which geometrical concepts are abstracted; psychological analysis reveals the processes by which this abstraction occurs; physical inquiry establishes the empirical content of geometrical propositions and the limits within which they hold. Each perspective corrects the errors to which the others are prone. The physiological perspective corrects the illusion that geometrical space is directly given in perception; the psychological perspective corrects the illusion that geometrical truths are a priori; the physical perspective corrects the illusion that geometry is independent of experience.
Mach’s final position is a form of empirical realism about geometry. Geometrical propositions are true of the physical world, but they are true in the way that physical laws are true—as idealizations of experience that hold within limits of observational error and may be modified if new facts require it. The choice between Euclidean and non-Euclidean geometry is not a matter of a priori necessity but of empirical adequacy and practical convenience. Physicists have found no reason to depart from Euclidean geometry, adhering to the simplest assumptions until facts force modification. But the possibility of alternative geometries remains open, and the history of the subject shows that what seemed like necessary truths were in fact empirical hypotheses.
The book ends with a reflection on the mathematical extension of geometry. When a mathematician modifies the simplest assumptions of geometrical experience and gains fresh insight, it is natural to pursue such research for its own sake, constructing analogues of familiar geometry on broader assumptions for any number of dimensions, with no pretension of being applied to reality. This purely mathematical activity is valuable in itself, and it may eventually prove useful in ways that cannot be foreseen. The thinkers who created non-Euclidean geometry—Gauss, Lobachevski, Bolyai, Riemann—were engaged in such exploratory work, and their discoveries transformed our understanding of space even though they had no immediate empirical application.
Significance
The significance of Mach’s work lies in its comprehensive synthesis of physiological, psychological, and physical approaches to a single fundamental question. By insisting that space perception must be understood biologically, that geometry must be understood empirically, and that the relationship between them must be understood historically, Mach provided a framework that transcended the limitations of each approach taken separately. His analysis of physiological space anticipated later developments in perceptual psychology and neuroscience, while his treatment of geometry as an empirical science contributed to the philosophical discussion of the foundations of mathematics that continued through the twentieth century.
The book is also significant for its historical method. Mach’s account of the development of geometry—from primitive measurement through Euclid’s axiomatization to the non-Euclidean geometries of the nineteenth century—demonstrated that understanding how a science developed is essential to understanding its logical foundations. This historical approach influenced later philosophers of science who argued that scientific concepts must be understood in terms of their origins and functions rather than as timeless truths.
Mach’s treatment of non-Euclidean geometry is particularly noteworthy. By explaining clearly how Lobachevski’s geometry arises from rejecting the parallel postulate, and by arguing that both Euclidean and non-Euclidean geometries can represent spatial facts with equal exactness, Mach helped to make these revolutionary ideas accessible to a wider audience. His insistence that the choice between geometries is an empirical matter, not a matter of a priori necessity, anticipated the role that non-Euclidean geometry would later play in Einstein’s theory of general relativity.
The book’s significance also lies in its philosophical naturalism. Mach’s refusal to accept any sharp distinction between the a priori and the empirical, his insistence that all knowledge arises from experience, and his willingness to treat even the most abstract mathematical concepts as products of human intellectual evolution, all contributed to a naturalistic approach to epistemology that has remained influential. His warning against the illusion of operating with symbols that correspond to nothing real or representable to the senses, and his recognition that such symbolic manipulation can nevertheless be fruitful, anticipated later discussions of the role of abstraction and idealization in scientific thought.
The work stands as a model of interdisciplinary inquiry, demonstrating how questions that might seem to belong exclusively to one discipline—physiology, psychology, physics, mathematics, philosophy—can be illuminated by bringing multiple perspectives to bear on them. Mach’s synthesis remains valuable not only for its specific conclusions but for its demonstration of how such a synthesis might be achieved.
