Chapter 89: SPACE

INTRODUCTION

On the level of our everyday observations, space and time seem to be the obvious, the common, and the connected properties of physical things. We distinguish things from one another by their position in space, as we mark happenings by the date of their occurrence. The where and when of a thing is often used to identify it, for it is generally agreed that two bodies cannot occupy the same place at the same time, and that at the same time two distinct places cannot be occupied by the same body. According to a theologian like Aquinas, these limitations of space and time apply even to bodiless things, i.e., to angels.

“An angel and a body are said to be in a place,” he writes, “in quite a different sense.”

Whereas a body is in the place which contains it, “an angel is said to be in a corporeal place by application of the angelic power . . . not as being contained, but as somehow containing it.” It follows, nevertheless, that at a given time an angel “is not everywhere, nor in several places, but in only one place.” Nor does the incorporeality of angels permit more than one angel to be at the same time in the same place. According to the manner in which an angel is at a place—by the action of his power—“there can be only one angel in one place,” Aquinas declares, even as there can be only one body in one place at a time.

Location or position in space, and spatial relationships such as higher and lower, nearer and farther, are so familiar and intelligible that they provide terms of reference whereby men speak metaphorically of the moral hierarchy and spiritual distances. The whole of Dante’s Divine Comedy, for example, involves a spatial metaphor which sets forth the gradations of sin and the degrees of blessedness in terms of places beneath the earth and in the heavens above.

As he mounts from sphere to sphere in Paradise, Dante meets Piccarda Donati in the Heaven of the Moon. She explains to him that this place “which appears so far down,” is assigned to those who have violated their vows in some particular. Dante wonders why she and the others do not “desire a more exalted place, in order to see more.” Piccarda replies: “Brother, the virtue of charity quiets our will, and makes us wish only for that which we have, and quickens not our thirst for aught else. … So that as we are, from seat to seat throughout this realm, to all the realm is pleasing, as to the King who inwills us with His will; and His will is our peace.”

This speech of Piccarda’s makes it clear to Dante “how everywhere in Heaven is Paradise, even if the grace of the Supreme Good does not there rain down in one measure.” These different measures of beatitude in the diffusion of God’s love and light are represented by the celestial spheres from the earth-adjacent moon to the Crystalline Heaven, the outermost bound of the physical universe, of which Dante says that it “has no other Where than in the Divine Mind.”

When the whole expanse of physical space or the boundary of the universe is considered, Newton no less than Dante conceives the omnipresence and eternity of God as that which somehow encompasses all space and time. God “is not duration or space,” Newton writes at the end of the Principia, “but He endures and is present . . . and by existing always and everywhere, He constitutes duration and space.” In the concluding queries of the Optics, Newton appears to think of infinite space as the Divine Sensorium in which all things are at once present to God, who “being in all places is more able by His will to move the bodies within His boundless uniform sensorium, and thereby to form and reform the parts of the Universe, than we are by our will to move the parts of our own bodies.”

The physicist does not have to turn theologian, however, to be confronted with the mysteries of space. Even without the modern complication of the relation of its three dimensions to time as a fourth dimension, the physical concept of space raises difficulties for analysis.

In the tradition of western thought, conflicting definitions of space seem to result from a fundamental difference in the object being defined—whether it is an inseparable property of bodies, perhaps even identical with unformed matter, or a reality apart from the bodies which move and have their being in it. Sometimes this difference is signified by a difference between the meaning of the word “place” or “extension” and the meaning of “space.” It appears also to be involved in the contrast between filled space and empty space (i.e., the void or vacuum); and it bears some relation to Aristotle’s distinction between space and place, and to Newton’s distinction between absolute and relative space.

The controversial character of space in physical theory may be appreciated in terms of these oppositions in meaning, and the issues which they raise. In addition, physical theory is confronted with the problem of action-at-a-distance (i.e., action through a void or through an ethereal medium), the problem of the infinity of space (or the question of a bounded or unbounded universe), and the distinction between one physical space and the variety of geometrical spaces.

Space, which at first seems easily apprehended by sense and susceptible to measurement, becomes upon examination so subtle as to be almost a vanishing object. Reason finds it difficult to say precisely what space is in itself, and how it is related to matter and motion. Even the familiar space of ordinary sense-perception seems to have its puzzles. A psychologist like James is concerned with how the different fields of touch, vision, and hearing coalesce to form the single space of our experience; and in dealing with the process by which we learn to perceive the spatial manifold of positions and directions, he cannot avoid the issue of innate as opposed to acquired space-perception.

Plato’s theory of space is set forth in the Timaeus as part of “the likely story” which Timaeus tells about the production and constitution of the universe. The sensible things which come into being and pass away are, according to him, patterned after the eternal forms. To the eternal patterns and their copies in the world of change, Timaeus finds it necessary to add a third factor in order to account for the physical elements and their generation. This factor, he says, is “difficult to explain and dimly seen. … It is the receptacle, and in a manner the nurse, of all generation.” In contrast to the elements which are perpetually changing into and out of one another, the receptacle “never departs from her own nature, and never in any way assumes a form like that of any of the things which enter into her. . . . The forms which enter into and go out of her are the likenesses of real existences modelled after their patterns in a wonderful and inexplicable manner.”

Timaeus distinguishes the three principles as that which comes to be and passes away in the process of generation, that in which the generation takes place, and that which the generated thing resembles and which is its source. He likens the receptacle or “receiving principle to a mother, the source to a father, and the intermediate nature to a child,” and adds that “if the model is to take every variety of form, then the matter in which the model is fashioned will not be duly prepared, unless it is formless and free from the impress of any of those shapes which it is hereafter to receive from without. … Wherefore, that which is to receive all forms should have no form. … The mother and receptacle of all created, visible, and in any way sensible things, is not to be termed earth, or air, or fire, or water, or any of their compounds, or any of the elements from which these are derived, but is an invisible and formless being which receives all things, and in some mysterious way partakes of the intelligible, and is most incomprehensible.”

This third factor which Timaeus sometimes calls “matter” as well as “receptacle,” he also sometimes calls “space.” When matter and space are identified with each other under the conception of a receptacle for the forms, they have the characteristics of being absolutely formless and imperceptible to the senses. Nor are they, as are the forms, genuinely intelligible to reason. “The third nature, which is space, and is eternal,” Timaeus says, “admits not of destruction and provides a home for all created things, and is apprehended without the help of sense, by a kind of spurious reason, and is hardly real; which we beholding as in a dream, say of all existence that it must of necessity be in some place and occupy a space, but that which is neither in heaven nor on earth has no existence.”

The precise meaning of this conception of space is difficult to determine. Does it, for example, find an echo in Plotinus’ statement that “space is a container, a container of body; it is the home of such things as consist of isolated parts”? But he also says that space “in a strict sense is unembodied and is not, itself, body,” and that “body is not a void,” but rather that “the void must be that in which body is placed,” seeming thereby to imply that space is essentially the void. The statement in the Timaeus that “there can be no such thing as a vacuum,” may apply only to the filled space of the created heaven and earth. May it not also be said that space is a void when it is identified with the formless matter of the receptacle prior to creation?

This raises further questions. Is the receptacle space or matter? And is the conception of space in the Timaeus rightly interpreted by Aquinas, in commenting on Augustine’s reading of “the earth was void and empty” in Genesis 1:2? Augustine holds that by the word “earth” in this passage formless matter is to be understood. Because of its formlessness, Aquinas writes, “the earth is said to be void and empty, or invisible and shapeless,” and, he adds, “that is why Plato says matter is place.”

However these questions are answered, one thing seems to be clear. Space, functioning as receptacle, can be identified only with matter devoid of form, not with the matter of three-dimensional bodies. The relation of space to matter seems to be differently conceived by Descartes. Space is for him not an antecedent principle involved in the original production of sensible things, but rather—as the extension of bodies—it is inseparable in existence from them. It is a property which signifies the essence of material substances, as thinking signifies the essence of mind or soul. “By extension,” Descartes writes, “we understand whatever has length, breadth, and depth, not inquiring whether it be a real body or merely space.” Nevertheless, he goes on to say that “by extension we do not here mean anything distinct and separate from the extended object itself.”

Descartes considers the significance of three statements: “extension occupies place, body possesses extension, and extension is not body.” The first statement, he thinks, means no more than “that which is extended occupies place.” The second statement seems to imply that “the meaning of extension is not identical with that of body; yet,” Descartes insists, “we do not construct two distinct ideas in our imagination, one of body, the other of extension, but merely a single image of extended body; and from the point of the view of the thing it is exactly as if I had said: body is extended, or better, the extended is extended.” Finally, in the statement that extension is not body, the word “extension,” according to Descartes, expresses a purely abstract conception—nothing which in itself has any sensible reality. So far as its existence is concerned, the thing conceived as extension cannot be separated from body. Those who think otherwise, Descartes asserts, are involved in “the contradiction of saying that the same thing is at the same time body and not body.”

The point is summarized in his Principles of Philosophy by the statement that “the nature of matter or of body in its universal aspect, does not consist in being hard, or heavy, or colored … but solely in the fact that it is a substance extended in length, breadth, and depth.” But, it may be asked, are the dimensions of a body the same as space? Descartes replies that “the same extension which constitutes the nature of a body constitutes the nature of space … not only that which is full of body, but also of that which is called a vacuum.”

If there were a vacuum, or empty space, extension might be separated from body. This Descartes flatly denies. “As regards a vacuum in the philosophic sense of the word, i.e., a space in which there is no substance, it is evident that such cannot exist, because the extension of space or internal place is not different from that of body.” And even “when we take this word vacuum in its ordinary sense,” Descartes goes on, “we do not mean a place or space in which there is absolutely nothing, but only a place in which there are none of those things which we expected to find there.”

These points made in the Principles confirm the identification of three-dimensional space or extension with body which appears in the Rules. They seem to be further confirmed in the Discourse by the reference to “a continuous body, or a space indefinitely extended in length, height or depth” which is “the object of the geometricians.” Descartes does not, however, neglect the distinction between space as the extension of body, and place as the position one body occupies in relation to another. According to common usage, he says, the word “place” signifies that “in virtue of which a body is said to be here or there.” He objects to those who, like Aristotle, mean by “place” the surrounding surface of a body. Local motion or change of place is not, he argues, a change in the body’s surrounding surface, but a change in its relative position.

It is place rather than space which Aristotle seeks to define, and place in the sense of the circumference of a body rather than its position in space. He rejects the notion that place is the extension of a magnitude, for that would, he thinks, identify it with matter. Place belongs to body, not as matter or a property of matter, but as its boundary. It is, Aristotle writes, “the innermost motionless boundary . . . a kind of surface and, as it were, a vessel, i.e., a container of the thing.” This boundary is itself made at the surface of a body by a surrounding body or bodies. “If a body has another body outside it and containing it,” Aristotle writes, “it is in place, and if not, not.”

The consequences of this conception of place are, first, a denial of space in the sense of void or empty place, since place is always “coincident with the thing” contained or bounded; second, a denial of any infinite place, since that would presuppose an actually infinite body—to Aristotle, an impossibility; and third, the conclusion that the whole universe itself does not have a place, for outside the outermost heaven which bounds the world, there can be no containing body by which the universe is bounded.

Aristotle explains that by “heaven” he means “the extreme circumference of the whole”—a whole “composed of all natural perceptible body.” The two words which Einstein uses when he discusses “the possibility of a ‘finite’ and yet ‘unbounded’ universe” seem to apply to Aristotle’s conception of the world—finite in body or matter, yet unbounded, i.e., without anything outside itself to determine or define its boundary.

Aristotle’s view of the world seems to be directly opposed to that of the ancient atomists. For them, the whole of matter is discontinuous, existing in indivisible units or atoms, each of which is a plenum—that is, a unit of matter absolutely continuous without void in it—but between which there is void or empty space. For Aristotle, the material world as a whole is a plenum, i.e., continuous body without void. Hence if by “space” is meant not place but void—a bodyless interval between or within bodies—there is no space. Aristotle considers the arguments of Democritus that without void local motion would be impossible, but he thinks “there is no necessity for there being a void if there is movement.”

Following Democritus, Lucretius gives another reason for positing void or empty space. As the indivisibility of the simple bodies or atoms consists in their absolute solidity—their lack of void—so the divisibility of composite bodies derives from their being constituted by both atoms and void. “Wherever space lies empty, which we call the void,” Lucretius writes, “body is not there; moreover wherever body has its station, there is by no means empty void. Therefore the first bodies are solid and free from void. . . . If there were nothing which was empty and void, the whole would be solid; unless … there were bodies determined, to fill all the places that they held, the whole universe would be but empty void space.”

For Aristotle, in contrast, the divisibility of matter seems to depend upon its being continuous. On his view, the composite body, constituted by atoms separated from one another by void, is not divisible, but is already actually divided; whereas the very thing which Lucretius regards as indivisible because it is continuous—the voidless atom—is for Aristotle divisible. To call an atom divisible is, of course, to deny that it is atomic or, in the language of Lucretius, an uncuttable bit of “solid singleness.”

Thus diametrically opposite theories of space and place seem to be connected with opposite theories of matter or body. Space as the empty interval or void between solid bodies goes along with atomism, whereas place as “the boundary of the containing body at which it is in contact with the contained body” goes along with the theory of the world as a material plenum.

The atomic theory and the plenum theory are opposed in one other fundamental respect concerning space. According to Aristotle, the impossibility of an actually infinite body makes the largest place finite. According to Lucretius, the infinite number of atoms requires an infinite space. Asking whether “the void that we have discovered, or room or space . . . is altogether bounded or spreads out limitless and immeasurably deep,” Lucretius answers that “the whole universe is bounded in no direction.” His argument seems to be like Aristotle’s for an “unbounded universe.” Since there can be “nothing outside the whole sum,” he writes, “it lacks therefore bound or limit.” But where Aristotle’s meaning seems to be that the universe has no place, since all places are inside it, Lucretius appears to mean that empty space extends infinitely in all directions.

Modern atomists like Newton and Locke hold a theory of space which accords with the view of matter existing in discontinuous units, separated by intervals of emptiness. Newton’s distinction, for example, between absolute and relative space acknowledges a space that is relative to bodies, but also affirms an absolutely independent space, which has being in separation from matter or bodies. “Absolute space, in its own nature, without relation to anything external,” he writes, “remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces, which our senses determine by its position to bodies and which is commonly taken for immovable space.” As for place in distinction from space, Newton holds that it is “a part of space which a body takes up, and is according to the space, either absolute or relative.” In opposing Aristotle’s view, he adds that place is “not the situation, nor the external surface of the body. For the places of equal solids are always equal; but their surfaces, by reason of their dissimilar figures, are often unequal.”

Locke also distinguishes between space and place, the one consisting in “the relation of distance between any two bodies or points,” the other in “the relation of distance betwixt anything and any two or more points which are considered as keeping the same distance one with another and, so considered, as at rest.” With this conception of place, he holds in apparent agreement with Aristotle that “we can have no idea of the place of the universe, though we can of all the parts of it.” Yet he goes on to say that what lies beyond the universe is “one uniform space or expansion, wherein the mind finds no variety or marks.”

This seems to indicate that Locke’s idea of space, like that of Lucretius, conceives an infinite void. “Those who assert the impossibility of space existing without matter must,” he writes, “make body infinite.” Furthermore, “those who dispute for or against a vacuum, do thereby confess that they have distinct ideas of vacuum and plenum, i.e., that they have an idea of extension void of solidity, though they deny its existence, or else they dispute about nothing at all. For they who so much alter the signification of words, as to call extension, body, and consequently make the whole essence of body to be nothing but pure extension, must talk absurdly whenever they speak of vacuum, since it is impossible for extension to be without extension: for vacuum, whether we affirm or deny it, signifies space without body, whose very existence no one can deny to be possible who will not make matter infinite, and take from God a power to annihilate any particle of it.”

Precisely because he thinks no one can affirm an infinite body, and because he conceives space to be a void, distinct from bodies, Locke finds it necessary to affirm the infinity of space. “I would ask,” he says, “whether if God placed a man at the extremity of corporeal beings, he could not stretch his hand beyond his body. If he could, then he would put his arm where there was before space without body.” Furthermore, if “it be impossible for any particle of matter to move but into empty space, the same possibility of a body’s moving into a void space, beyond the utmost bounds of body, as well as into a void space interspersed amongst bodies, will always remain clear and evident. … So that, wherever the mind places itself by any thought, either amongst or remote from all bodies, it can, in this uniform idea of space, no where find any bounds, any end; and so must necessarily conclude it… to be actually infinite.”

It may seem paradoxical that pure space—space existing without matter—is denied by one who also denies the existence of matter. “When I speak of pure or empty space,” Berkeley writes, “it is not to be supposed that the word ‘space’ stands for an idea distinct from or conceivable without body or motion.” What is meant, he suggests, is merely that the resistance one body gives to another in motion is absent when space is relatively empty. But this is always relative. “In proportion as the resistance is lesser or greater,” Berkeley says, “the space is more or less pure.” There would be absolutely pure space only if all bodies other than his own were annihilated. “If that, too, were annihilated,” Berkeley concludes, “then there could be no motion, and consequently no Space.”

All these contradictions concerning space enter into Kant’s statement of the first cosmological antinomy, in which the thesis that the world is limited with regard to space and the antithesis that the world is infinite in space seem to be equally susceptible to proof—and so to disproof. Both alternatives violate our empirical concepts.

If space “is infinite and unlimited,” Kant writes, “it is too large for every possible empirical concept. If it is finite and limited, you have a perfect right to ask what determines that limit. Empty space is not an independent correlate of things, and cannot be a final condition, still less an empirical condition forming part of possible experience—for how can there be experience of what is absolutely void? But in order to produce an absolute totality in an empirical synthesis, it is always requisite that the unconditioned should be an empirical concept. Thus it follows that a limited world would be too small for your concept.”

Space itself, however, is for Kant “not an empirical concept which has been derived from external experience.” Rather it “is a necessary representation a priori forming the very foundation of all external intuitions” and, as Kant explains in his Prolegomena, it establishes geometry as an a priori science. “Space is nothing but the form of all phenomena of the external senses; it is the subjective condition of our sensibility, without which no external intuition is possible for us. … Nothing which is seen in space is a thing by itself,” nor is “space a form of things supposed to belong to them by themselves.” The external objects which we perceive in space “are nothing but representations of our senses, the form of which is space.”

So far as the experience of space is concerned, William James seems to take an opposite view. Time and space relations, he says, “are impressed from without” and “stamp copies of themselves within.” To the Kantian theory that space is “a quality produced out of the inward resources of the mind, to envelope sensations which, as given originally, are not spatial,” James replies that he can find “no introspective experience of mentally producing or creating space.”

He proposes two other alternatives: “either (1) there is no spatial quality of sensation at all, and space is a mere symbol of succession; or (2) there is an extensive quality given immediately in certain particular sensations.” The second seems to James best suited to explain the development of our perceptions of space, and he does not think it inconsistent with the a priori or non-empirical character of geometry, whose necessary truths refer to ideal objects, not to experienced things in physical space.

The chapter on MATHEMATICS considers the relation of the postulates of diverse geometries to the diversity of Euclidean and non-Euclidean spaces, such as that of the flat plane, the surface of a sphere, and the surface of a pseudo-sphere. Just as different parallel postulates select different spaces for geometrical construction, so a postulate like Euclid’s concerning the equality of all right angles seems to assume a uniformity of space which permits geometrical figures to be transposed without alteration. “If translation through space warped or magnified forms,” James remarks, “then the relations of equality, etc., would always have to be expressed with a position-qualification added.”

Confronted with a variety of purely mathematical spaces, the physicist is concerned with the problem of which geometry is, as Einstein says, in “correspondence with a ‘real’ object,” or true of the real world. “According to the general theory of relativity, the geometrical properties of space are not independent,” Einstein writes, “but are determined by matter.” It follows that our assumptions about the distribution of matter determine the character of the world’s space.

On the assumption of a world “not inhabited by matter everywhere,” in whose infinite space “the average density of matter would necessarily be zero,” Einstein says we can imagine “a quasi-Euclidean universe” analogous to “a surface which is irregularly curved in its individual parts, but which nowhere departs appreciably from a plane: something like the rippled surface of a lake.” But if the “average density of matter… differs from zero, however small may be that difference, then the universe cannot be quasi-Euclidean.” It would be spherical (or elliptical) if the matter were uniformly distributed; but “since in reality the detailed distribution of matter is not uniform,” Einstein concludes that “the real universe will deviate in individual parts from the spherical, i.e., the universe will be quasi-spherical. But it will be necessarily finite.”

The nature of the actual space of the universe thus seems to be related to the issue whether physical as opposed to mathematical space is a void or filled with matter. Defining a vacuum as “a space empty of all bodies known to the senses,” Pascal insists that “there is as much difference between nothingness and space, as there is between empty space and a material body,” so that “empty space occupies the mean between matter and nothingness.” Torricelli’s experiments seem to him complete proof against the disciples of Aristotle, for they upset the belief that “nature abhors a vacuum.”

Gilbert’s observations on magnetic influences, Newton’s observations on the transmission of light and heat as well as gravitational pull, and Faraday’s on electrical phenomena, all seem to admit the possibility of action at a distance, or through a vacuum. But the question remains whether the so-called physical vacuum is an absolute void or merely empty of “all bodies known to the senses.”

“Is not the heat of the warm room conveyed through the vacuum,” Newton asks, “by the vibrations of a much subtiler medium than air which after the air was drawn out remained in the vacuum? And is not this medium the same with that medium by which light is refracted and reflected, and by whose vibrations light communicates heat to bodies? … And is not this medium exceedingly more rare and subtile than the air, and exceedingly more elastic and active? And does it not readily pervade all bodies? And is it not (by its elastic force) expanded through all the heavens?”

Huygens also refers to an ethereal matter as the medium for the propagation of light. “One will see,” he writes, “that it is not the same that serves for the propagation of sound … It is not the same air, but another kind of matter in which light spreads; since if the air is removed from the vessel, the light does not cease to traverse it as before.” But this ethereal medium, without which bodies would act at a distance upon one another—gravitationally, magnetically, electrically—through an absolute void, seems to have contrary properties. It is not only “subtiler” than air, but, as Newton suggests, it may be “denser than quick-silver or gold,” since “planets and comets, and all gross bodies perform their motions more freely, and with less resistance in this aethereal medium than in any fluid, which fills all space adequately without leaving any pores.” And, in still another place, he asks: “What is there in places almost empty of matter, and whence is it that the Sun and Planets gravitate towards one another, without dense matter between them?”

Whatever may be thought of the ether as a physical hypothesis, the problem still remains whether action can take place at a distance through a void or must employ what Faraday calls “physical lines of force” through filled space. Faraday thinks the evidences support the latter alternative for both electricity and magnetism. He quotes a letter from Newton to Bentley to show that Newton was “an unhesitating believer in physical lines of gravitating force.”

In that letter, posthumously discovered, Newton says: “That gravity should be innate, inherent and essential to matter, so that one body may act upon another at a distance through a vacuum, without the mediation of anything else, by and through which their action and force may be conveyed from one to another, is to me so great an absurdity, that I believe no man who has in philosophic matters a competent faculty of thinking, can ever fall into it.”

OUTLINE OF TOPICS

Space, place, and matter 1a. Space or extension as the essence or property of bodies: space, the receptacle, and becoming 1b. Place as the envelope or container of bodies: place as a part of space or as relative position in space 1c. The tridimensionality of bodies: the indeterminate dimensions of pure space or prime matter 1d. The exclusiveness of bodily occupation of space: impenetrability
Space, void, and motion 2a. The role of space or place in local motion: the theory of proper places; absolute and relative space 2b. The issue of the void or vacuum (1) The distinction between empty and filled space (2) The indispensability of void or vacuum for motion and division: the absence of void in atoms (3) The denial of void or vacuum in favor of a plenum 2c. Space as a medium of physical action: the ether and action-at-a-distance; the phenomena of gravitation, radiation, and electricity
Space, quantity, and relation 3a. The finitude or infinity of space: the continuity and divisibility of space 3b. The relation of physical and mathematical space: sensible and ideal space 3c. Geometrical space, its kinds and properties: spatial relationships and configurations 3d. The measurement of spaces, distances, and sizes: trigonometry; the use of parallax
The knowledge of space and figures 4a. Space as the divine sensorium and space as a transcendental form of intuition: the a priori foundations of geometry 4b. The controversy concerning innate and acquired space-perception 4c. The perception of space: differences between visual, auditory, and tactual space; perspective and spatial illusions
The mode of existence of geometrical objects: their character as abstractions; their relation to intelligible matter
The spiritual significance of place, position, and space

REFERENCES

To find the passages cited, use the numbers in heavy type, which are the volume and page numbers of the passages referred to. For example, in 4 Homer: Iliad, Bk II [265-283] 12d, the number 4 is the number of the volume in the set; the number 12d indicates that the passage is in section d of page 12.

Page Sections: When the text is printed in one column, the letters a and b refer to the upper and lower halves of the page. For example, in 53 James: Psychology, 116a-119b, the passage begins in the upper half of page 116 and ends in the lower half of page 119. When the text is printed in two columns, the letters a and b refer to the upper and lower halves of the left-hand side of the page, the letters c and d to the upper and lower halves of the right-hand side of the page. For example, in 7 PLATO: Symposium, 163b-164c, the passage begins in the lower half of the left-hand side of page 163 and ends in the upper half of the right-hand side of page 164.

Author’s Divisions: One or more of the main divisions of a work (such as PART, BK, CH, SECT) are sometimes included in the reference; line numbers, in brackets, are given in certain cases; e.g., Iliad, Bk II [265-283] 12d.

Bible References: The references are to book, chapter, and verse. When the King James and Douay versions differ in title of books or in the numbering of chapters or verses, the King James version is cited first and the Douay, indicated by a (D), follows; e.g., OLD TESTAMENT: Nehemiah, 7:45—(D) II Esdras, 7:46.

Symbols: The abbreviation “esp” calls the reader’s attention to one or more especially relevant parts of a whole reference; “passim” signifies that the topic is discussed intermittently rather than continuously in the work or passage cited.

For additional information concerning the style of the references, see the Explanation of Reference Style; for general guidance in the use of The Great Ideas, consult the Preface.

1. Space, place, and matter

1a. Space or extension as the essence or property of bodies: space, the receptacle, and becoming

7 PLATO: Timaeus, 455c-458b
8 ARISTOTLE: Physics, BK I, CH 9 [192b9-14] 268a; BK IV, CH 2 [209b5-210a11] 288b-d; CH 4 [211b5-212a2] 290c-291a
17 PLOTINUS: Third Ennead, TR VI, CH 13 114c-115b; CH 19 118c-119a
19 AQUINAS: Summa Theologica, PART I, Q 66, A 1, REP 1 343d-345c
23 HOBBES: Leviathan, PART IV, 270d-271a
31 DESCARTES: Rules for the Direction of the Mind, XIV, 29b-32a / Discourse on the Method, PART IV, 52d-53a / Meditations on First Philosophy, II, 78c-d / Objections and Replies, DEF VII 130c-d; 135d-136a; 154a
33 PASCAL: The Vacuum, 370a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH XIII, SECT 11-27 150d-154d; CH XV, SECT 4, 162d-163a; CH XVII, SECT 20 172d-173c; BK III, CH VI, SECT 5, 269b-c; SECT 21 273c-d; CH X, SECT 6, 293b
42 KANT: The Critique of Pure Reason, 28b-33d

1b. Place as the envelope or container of bodies: place as a part of space or as relative position in space

8 ARISTOTLE: Physics, BK IV, CH 1-5 287a-292c esp CH 4 289d-291c / On the Heavens, BK I, CH 7 [275b5-12] 366d
17 PLOTINUS: Fourth Ennead, TR III, CH 20, 152b-c / Sixth Ennead, TR I, CH 14 260b-c
19 AQUINAS: Summa Theologica, PART I, Q 8, A 2, ANS and REP 3 35c-36b; Q 50, A 1, REP 3 269b-270a; Q 52 278d-280d; Q 53, A 1, ANS 280d-282a; Q 66, A 4, REP 5 348d-349d
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 83, A 2, ANS and REP 5 976c-978c; A 3, REP 3-4 978c-980d; A 5, ANS 981b-982c; Q 84, A 2, REP 1 984c-985d
23 HOBBES: Leviathan, PART II, 172b; PART IV, 270d-271c
31 DESCARTES: Rules for the Direction of the Mind, XII, 23c-d; XIII, 26b-c / Meditations on First Philosophy, II, 78c-d / Objections and Replies, 228c-229a
33 PASCAL: The Vacuum, 375a-376a
34 NEWTON: Mathematical Principles of Natural Philosophy, DEFINITIONS, SCHOL, 9a-10a-11a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH IV, SECT 2 129c-d; SECT 5 130d-131a; CH XIII, SECT 7-10 149d-150d; CH XV, SECT 5-8 163b-164b
35 BERKELEY: The Principles of Human Knowledge, SECT 111, 434c
42 KANT: The Critique of Judgement, 611d-612a
53 JAMES: The Principles of Psychology, 139a-140a; 626b

1c. The tridimensionality of bodies: the indeterminate dimensions of pure space or prime matter

7 PLATO: Timaeus, 455c-458b
8 ARISTOTLE: Topics, BK VI, CH 5 [142b20-29] 196b / Physics, BK IV, CH 1 [209a5-7] 287d / On the Heavens, BK I, CH 1 359a-c / Metaphysics, BK V, CH 6 [1016b25-31] 537b; CH 13 [1020a7-14] 541b; BK VII, CH 3 [1029a11-19] 551c-d
11 NICOMACHUS: Introduction to Arithmetic, BK II, 832c
17 PLOTINUS: Second Ennead, TR IV 50a-57c esp CH 7-9 52a-53b, CH 15 56c-57a
18 AUGUSTINE: Confessions, BK XII, PAR 3-4 99d-100a; PAR 15 102b-c
19 AQUINAS: Summa Theologica, PART I, Q 3, A 2, CONTRARY 15c-16a; Q 66, A 1, REP 1 343d-345c
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 79, A 1, REP 3 951b-953b; Q 80, A 5, REP 3 963a-964b; Q 83, A 2, ANS 976c-978c; A 3, REP 1-2 978c-980d; A 5, ANS 981b-982c
23 HOBBES: Leviathan, PART IV, 269d
31 DESCARTES: Rules for the Direction of the Mind, XIV, 29b-c; 31d / Meditations on First Philosophy, V, 93d
31 SPINOZA: Ethics, PART I, PROP 15, SCHOL, 360b
33 PASCAL: The Vacuum, 370a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH IV, SECT 5 130d-131a; CH XIV, SECT 26 160c-d; CH XV, SECT 1-8 162b-164b passim; CH XVII 167d-174a passim
35 BERKELEY: The Principles of Human Knowledge, SECT 117 436a
52 DOSTOEVSKY: The Brothers Karamazov, BK V, 121a-b

1d. The exclusiveness of bodily occupation of space: impenetrability

8 ARISTOTLE: Physics, BK IV, CH 1 [209a5-7] 287d; CH 6 [213b4-12] 293a
17 PLOTINUS: Fourth Ennead, TR II, CH 1, 139d
19 AQUINAS: Summa Theologica, PART I, Q 8, A 2, ANS 35c-36b; Q 52, A 3, REP 1 280a-d
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 83, AA 2-4 976c-981b; A 5, ANS and REP 2 981b-982c; A 6, ANS and REP 2 982c-983b
23 HOBBES: Leviathan, PART I, 54b-c; PART IV, 271b-c
28 GALILEO: Two New Sciences, FIRST DAY, 153a; 156d-157b
30 BACON: Novum Organum, BK II, APH 36, 167b; APH 48, 179d-180a
31 DESCARTES: Meditations on First Philosophy, II, 78c-d
33 PASCAL: Pensées, 512 262a / The Vacuum, 370a
34 NEWTON: Mathematical Principles of Natural Philosophy, BK III, RULE III, 270b
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH IV 129b-131a; CH XV, SECT 11 162a-b; BK IV, CH VII, SECT 5 338b
45 FARADAY: Experimental Researches in Electricity, 854c-855a,c

2. Space, void, and motion

2a. The role of space or place in local motion: the theory of proper places; absolute and relative space

7 PLATO: Timaeus, 453c; 455c-458b; 460c-d / Laws, BK X, 762b-c
8 ARISTOTLE: Physics, BK III, CH 5 [205a10-206a8] 283b-284b; BK IV, CH 1 [208b1-24] 287a-c; CH 2 [210a2-4] 288d; CH 4 [211a4-6] 290a; [212a20-28] 291c; CH 5 [212b29-213a11] 292b; CH 8 [214b13-18] 294b; [215a1-10] 294c-d; BK VI 312b,d-325d passim / On the Heavens, BK I, CH 2 [268b11]-CH 3 [270a13] 359d-361b; CH 7 [274b30-33] 366a; CH 7 [275b30]-CH 8 [277b25] 367a-369a; CH 9 [278b22-279a8] 370a-b; BK II, CH 2 376b-377c; BK III, CH 5 [304b11-23] 395d-396a; CH 6 [305a22-28] 396c; BK IV, CH 1 399a-d; CH 3-5 401c-404d / Metaphysics, BK V, CH 13 [1020a25-33] 541c; BK XI, CH 10 [1067a8-33] 595c-596a / On the Soul, BK I, CH 3 [406b12-29] 635c-d
9 ARISTOTLE: Nicomachean Ethics, BK X, CH 4 [1174a29-b5] 428d
11 NICOMACHUS: Introduction to Arithmetic, BK II, 832c
12 LUCRETIUS: On the Nature of Things, BK I [988-1007] 13b; [1052-1082] 14a-c; BK II [184-250] 17b-18b
16 PTOLEMY: Almagest, BK I, 10b-11b
16 COPERNICUS: On the Revolutions of the Heavenly Spheres, BK I, 517b-518a; 519b-520b
16 KEPLER: Epitome of Copernican Astronomy, BK IV, 855b; 931b-932a
17 PLOTINUS: Third Ennead, TR VII, CH 8, 123d-124a
19 AQUINAS: Summa Theologica, PART I, Q 53 280d-284d
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 84, A 2, REP 1,4-5 984c-985d; A 3, ANS and REP 2-3 985d-989b
21 DANTE: The Divine Comedy, PURGATORY, XVIII [22-33] 80a; PARADISE, I [94-142] 107b-d; IV [73-87] 111b-c; XXI [40-45] 141d
23 HOBBES: Leviathan, PART I, 50a; 61b; PART II, 173a; PART IV, 271a-b; 271d
28 GILBERT: On the Loadstone, BK VI, 110b-c
28 GALILEO: Two New Sciences, FIRST DAY, 162d; 164a-c; THIRD DAY-FOURTH DAY 197a-260a,c esp THIRD DAY, 197b-d, 200b-c, 203a-b, 205b-208c
30 BACON: Novum Organum, BK II, APH 35, 163a-d; APH 36, 166b-c; APH 45 176a-177c; APH 48, 179d-180d; 181d
34 NEWTON: Mathematical Principles of Natural Philosophy, DEFINITIONS-LAWS OF MOTION 5a-24a esp DEFINITIONS, SCHOL 8b-13a; BK I-II 25a-267a passim, esp BK I, PROP 1-17, SCHOL 32b-50a, PROP 32-39 81a-88a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH XIII, SECT 7-10 149d-150d; SECT 27, 154c; CH XV, SECT 5-8 163b-164b
35 BERKELEY: The Principles of Human Knowledge, SECT 110-117 434b-436a
42 KANT: The Critique of Pure Reason, 29c-d; 31d-32a; 55c-56a; 84b-c; 135d [fn 2]
51 TOLSTOY: War and Peace, BK XI, 469a-d

2b. The issue of the void or vacuum

2b(1) The distinction between empty and filled space

8 ARISTOTLE: Physics, BK IV, CH 1 [208b25-26] 287c; CH 6-9 292c-297c esp CH 7 [213b30-214a16] 293b-c / On the Heavens, BK I, CH 9 [279a12-18] 370b-c; BK III, CH 2 [301b32-302a9] 393b
12 LUCRETIUS: On the Nature of Things, BK I [329-369] 5b-c; [418-448] 6b-c; [503-550] 7b-d
17 PLOTINUS: Fourth Ennead, TR III, CH 20, 152c
19 AQUINAS: Summa Theologica, PART I, Q 46, A 1, REP 4 250a-252d
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 83, A 2, ANS 976c-978c
31 SPINOZA: Ethics, PART I, PROP 15, SCHOL, 361b-d
33 PASCAL: The Vacuum, 359b-361a; 363b-365b; 370a; 373b-376a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH IV, SECT 2-5 129c-131a; CH XIII, SECT 11-27 150d-154d esp SECT 21-27 152d-154d; CH XIV, SECT 26 160c-d; CH XV, SECT 1-4 162b-163b passim; CH XVII, SECT 4 168b-d; SECT 20 172d-173c
35 BERKELEY: The Principles of Human Knowledge, SECT 116-117 435d-436a
42 KANT: The Critique of Pure Reason, 71b-72a; 84b-c; 135d [fn 2]; 152c-d

2b(2) The indispensability of void or vacuum for motion and division: the absence of void in atoms

8 ARISTOTLE: Physics, BK IV, CH 6 292c-293b; CH 7 [214a16]-CH 8 [216b26] 293d-295d / On the Heavens, BK I, CH 7 [275b30-276a5] 367a; BK III, CH 2 [301b32-302a9] 393b; BK IV, CH 2 [308b29-310a13] 400b-401c / On Generation and Corruption, BK I, CH 8 [325a24-b11] 423d-424b
12 LUCRETIUS: On the Nature of Things, BK I [329-397] 5b-6a; [418-448] 6b-c; [483-550] 7a-d; [988-1007] 13b; [1052-1082] 14a-c; BK VI [998-1041] 93c-94a
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 84, A 3, REP 2 985d-989b
28 GALILEO: Two New Sciences, FIRST DAY, 138b-141d; 151c-153a; 156d-160a passim
30 BACON: Novum Organum, BK II, APH 48, 187c
31 SPINOZA: Ethics, PART I, PROP 15, SCHOL, 361b-d
33 PASCAL: The Vacuum, 359a-381b / The Weight of the Mass of the Air, 405b-415b
34 NEWTON: Mathematical Principles of Natural Philosophy, BK III, PROP 6, COROL II-IV 281b / Optics, BK III, 541b
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH XIII, SECT 23 153c-d; CH XVII, SECT 4 168b-d
45 LAVOISIER: Elements of Chemistry, PART I, 9a-b
45 FARADAY: Experimental Researches in Electricity, 850b,d-855a,c esp 851a-b

2b(3) The denial of void or vacuum in favor of a plenum

7 PLATO: Timaeus, 460c-d; 470d-471c
8 ARISTOTLE: Physics, BK IV, CH 6-9 292c-297c / On the Heavens, BK I, CH 9 [279a12-18] 370b-c; BK III, CH 2 [301b32-302a9] 393b; CH 6 [305a14-22] 396b-c
10 GALEN: On the Natural Faculties, BK I, CH 6, 188c-189c
12 LUCRETIUS: On the Nature of Things, BK I [370-383] 5c-d
19 AQUINAS: Summa Theologica, PART I, Q 46, A 1, REP 4 250a-252d; Q 52, A 3, REP 2 280a-d
23 HOBBES: Leviathan, PART III, 172b; PART IV, 269d
30 BACON: Novum Organum, BK II, APH 8 140b; APH 48, 180a; 187c
31 DESCARTES: Meditations on First Philosophy, VI, 100d
31 SPINOZA: Ethics, PART I, PROP 15, SCHOL, 361b-d
33 PASCAL: The Vacuum, 376a-b; 379a-380a
35 BERKELEY: The Principles of Human Knowledge, SECT 116-117 435d-436a
42 KANT: The Critique of Pure Reason, 71b-72a; 84b-c; 135d [fn 2]
45 FARADAY: Experimental Researches in Electricity, 850b,d-855a,c esp 854a-855a,c

2c. Space as a medium of physical action: the ether and action-at-a-distance; the phenomena of gravitation, radiation, and electricity

8 ARISTOTLE: Physics, BK VIII, CH 10 [266b25-267b20] 354b-d / On the Heavens, BK III, CH 2 [301b16-31] 393a-b / On the Soul, BK II, CH 7 [418b27]-CH 8 [420a26] 649b-651c; BK III, CH 12 [434b22-435a10] 667c-668a / On Dreams, CH 2 [459b28-34] 703b
12 LUCRETIUS: On the Nature of Things, BK I [440-448] 6c; BK VI [906-1041] 92b-94a
16 KEPLER: Epitome of Copernican Astronomy, BK IV, 897b-905a esp 900b-901b; 906a-b; 922a-b; 934b; BK V, 965a-b
19 AQUINAS: Summa Theologica, PART I, Q 8, A 1, REP 3 34d-35c
28 GILBERT: On the Loadstone, BK II, 26d-40b esp 30a-32c; 43a-c; 45d-47b; 51a-c; 54d-55c; BK V, 102d-104b; BK VI, 112d
28 GALILEO: Two New Sciences, THIRD DAY, 202d
30 BACON: Novum Organum, BK II, APH 36, 167b-c; APH 37 168d-169c; APH 45 176a-177c; APH 48, 183a-c; 186a
31 DESCARTES: Rules for the Direction of the Mind, IX, 15c
33 PASCAL: The Vacuum, 366a-367a
34 NEWTON: Mathematical Principles of Natural Philosophy, DEF V-VIII 6a-8a; BK I, PROP 69, SCHOL 130b-131a; BK III, GENERAL SCHOL, 371b-372a / Optics, BK III, 516a-b; 520a-522b esp 521a-b; 525b-529a; 531b-542a passim, esp 531b
34 HUYGENS: Treatise on Light, CH 1, 553b-560b
35 BERKELEY: The Principles of Human Knowledge, SECT 102-108 432d-434a passim
35 HUME: An Enquiry Concerning Human Understanding, SECT VII, DIV 57, 475d [fn 2]
36 SWIFT: Gulliver’s Travels, PART III, 118b-119a
42 KANT: The Critique of Pure Reason, 8d [fn 2]; 227b
45 LAVOISIER: Elements of Chemistry, PART I, 9b-c
45 FARADAY: Experimental Researches in Electricity, 441a-442b; 451a-454a; 463d-465d; 513d-514c; 521a-524a; 528c-532a; 604b-c; 631b-c; 648b-d; 685d-686c; 816b,d-819a,c; 819a-d; 824a-b; 832a-c; 840c-842c; 855a,c
51 TOLSTOY: War and Peace, EPILOGUE II, 687d; 695c

3. Space, quantity, and relation

3a. The finitude or infinity of space: the continuity and divisibility of space

8 ARISTOTLE: Categories, CH 6 [5a6-23] 9b-c / Physics, BK III, CH 5 [205a10-206a8] 283b-284b; BK IV, CH 5 [212b11-21] 292a / On the Heavens, BK I, CH 9 [279a12-b4] 370b-d / Metaphysics, BK V, CH 13 [1020a25-33] 541c; BK XI, CH 10 [1067a8-33] 595c-596a
11 ARCHIMEDES: The Sand-Reckoner 520a-526b passim
12 LUCRETIUS: On the Nature of Things, BK I [951-1007] 12d-13b; BK II [89-94] 16a-b; [1048-1063] 28b-c; BK VI [647-652] 89a
16 PTOLEMY: Almagest, BK I, 10b
16 COPERNICUS: On the Revolutions of the Heavenly Spheres, BK I, 516a-517b
16 KEPLER: Epitome of Copernican Astronomy, BK IV, 882a-886b
18 AUGUSTINE: The City of God, BK XI, CH 5 324d-325c
19 AQUINAS: Summa Theologica, PART I, Q 8, A 4, REP 3 37c-38c; Q 46, A 1, REP 8 250a-252d; Q 52, A 1 278d-279b; Q 53, A 1, ANS and REP 1 280d-282a; A 2, ANS and REP 1 282a-283b
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 84, A 3, CONTRARY 985d-989b
23 HOBBES: Leviathan, PART I, 54b-c; 61b; PART II, 162b; PART IV, 271b
30 BACON: Novum Organum, BK I, APH 48 110d-111a
31 DESCARTES: Objections and Replies, 112b
33 PASCAL: Pensées, 121 195a; 205-206 211a / On Geometrical Demonstration, 434b-439b
34 NEWTON: Mathematical Principles of Natural Philosophy, DEFINITIONS, SCHOL, 8b-11a / Optics, BK III, 543a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH XIII, SECT 4 149b; SECT 21 152d-153b; CH XV, SECT 2-6 160c-d; CH XV, SECT 1-8 162b-164b passim; CH XVII, SECT 8 167c; CH XVII 167d-174a passim, esp SECT 3-4 168b-d, SECT 11 170c
35 BERKELEY: The Principles of Human Knowledge, SECT 117 436a
42 KANT: The Critique of Pure Reason, 24c-d; 28b-29c; 135a-137a,c; 152a-d; 160b-163a esp 162b / The Critique of Judgement, 501a-b
51 TOLSTOY: War and Peace, EPILOGUE II, 693c-694a passim
53 JAMES: The Principles of Psychology, 631a
54 FREUD: A General Introduction to Psychoanalysis, 562c-d

3b. The relation of physical and mathematical space: sensible and ideal space

8 ARISTOTLE: Physics, BK IV, CH 1 [209a5-7] 287d; CH 8 [216b2-11] 296a-b / Metaphysics, BK XIV, CH 5 [1092a18-21] 625a
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 83, A 2, ANS 976c-978c; A 3, REP 1-2 978c-980d
31 DESCARTES: Rules for the Direction of the Mind, XIV 28a-33b / Discourse on the Method, PART IV, 52d-53a / Objections and Replies, 169c-170a
33 PASCAL: The Vacuum, 370a; 373a-374a
34 NEWTON: Mathematical Principles of Natural Philosophy, DEFINITIONS, SCHOL 8b-13a esp 8b-9a, 12a-b
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH IV, SECT 5 130d-131a; CH XIV, SECT 26 160c-d; CH XV, SECT 1-8 162b-164b passim
35 BERKELEY: The Principles of Human Knowledge, SECT 110-117 434b-436a; SECT 124-128 437d-438d
42 KANT: The Critique of Pure Reason, 24a-26b; 161d-163a / The Critique of Judgement, 574b-575a
53 JAMES: The Principles of Psychology, 877b

3c. Geometrical space, its kinds and properties: spatial relationships and configurations

8 ARISTOTLE: Categories, CH 6 [5a1-23] 9b-c / On the Heavens, BK I, CH 1 359a-c / Metaphysics, BK V, CH 6 [1016b25-31] 537b
11 EUCLID: Elements, BK I, DEFINITIONS, 1-2,5 1a; 14 1b; POSTULATES esp 5 2a; COMMON NOTIONS, 4 2a; PROP 4 4a-b; PROP 8 6b-7a; PROP 16 10b-11a; PROP 26 16a-17b; PROP 29 18b-19a; BK X, PROP 1 191b-192a; BK XI, DEFINITIONS, 1 301a
11 ARCHIMEDES: On the Sphere and Cylinder, BK I, ASSUMPTIONS esp 5 404b / On Spirals, 484b / The Quadrature of the Parabola, 527b
11 NICOMACHUS: Introduction to Arithmetic, BK II, 832b-d
16 KEPLER: Epitome of Copernican Astronomy, BK IV, 865a-b
17 PLOTINUS: Sixth Ennead, TR III, CH 13-14 287d-289a
19 AQUINAS: Summa Theologica, PART I, Q 85, A 8, REP 2 460b-461b
31 DESCARTES: Rules for the Direction of the Mind, XIV 28a-33b esp 30d-32a / Discourse on the Method, PART IV, 52d-53a / Meditations on First Philosophy, V, 93a-d; VI, 96b-d / Objections and Replies, 217a-c; 228c-229a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH XIII, SECT 5-6 149b-d
35 BERKELEY: The Principles of Human Knowledge, SECT 123-132 437c-439c
35 HUME: An Enquiry Concerning Human Understanding, SECT XII, DIV 124-125 506a-507a passim
42 KANT: The Critique of Pure Reason, 24a-26b
52 DOSTOEVSKY: The Brothers Karamazov, BK V, 121a-b
53 JAMES: The Principles of Psychology, 550b-551b [fn 1-2]; 876b-878a esp 877b

3d. The measurement of spaces, distances, and sizes: trigonometry; the use of parallax

16 PTOLEMY: Almagest, BK V, 165a-176a; BK IX, 270a-b
16 COPERNICUS: On the Revolutions of the Heavenly Spheres, BK I, 516a-b; 521b-529a; BK IV, 705a-714b
16 KEPLER: Epitome of Copernican Astronomy, BK IV, 861a-863b; 868b-887a / The Harmonies of the World, 1016b-1018a
30 BACON: Novum Organum, BK I, APH 109, 129b; BK II, APH 39, 170b-c; APH 45 176a-177c
31 DESCARTES: Rules for the Direction of the Mind, XIV, 31b-33b
34 NEWTON: Mathematical Principles of Natural Philosophy, BK III, LEMMA 4 333a-337b
41 GIBBON: The Decline and Fall of the Roman Empire, 299b-c
53 JAMES: The Principles of Psychology, 551a-b; 673b

4. The knowledge of space and figures

4a. Space as the divine sensorium and space as a transcendental form of intuition: the a priori foundations of geometry

34 NEWTON: Mathematical Principles of Natural Philosophy, BK III, GENERAL SCHOL, 370a-371a / Optics, BK III, 529a; 542a-543a
42 KANT: The Critique of Pure Reason, 16a-c; 17d-18d; 23a-26b; 31b-d; 46a-c; 55c-56a; 62a-d; 68a-69c; 86b-c; 87b-c; 94b-95a; 99a-101b; 110a; 131a-c; 135d [fn 2]; 136c-137c [antithesis]; 153c-155a; 186d-187a; 211c-218d esp 211c-212a, 213d-215a / The Critique of Practical Reason, 307d-308b; 312c-313d / The Critique of Judgement, 471b-c; 551a-553c; 574b-575a
53 JAMES: The Principles of Psychology, 629a-631a; 874a; 876b-878a

4b. The controversy concerning innate and acquired space-perception

42 KANT: The Critique of Pure Reason, 23a-24a; 25c-26a; 31d-32c / The Critique of Practical Reason, 307d-308b
53 JAMES: The Principles of Psychology, 474a-475a; 627a-635a esp 628b-631a, 633a-635a; 852b-853a; 860b-861a

4c. The perception of space: differences between visual, auditory, and tactual space; perspective and spatial illusions

8 ARISTOTLE: On Sense and the Sensible, CH 6 [445b4-18] 683b-d / On Memory and Reminiscence, CH 2 [452b7-18] 694b-c
12 LUCRETIUS: On the Nature of Things, BK II [308-332] 19a-b; BK IV [230-255] 47b-c; [269-291] 47d-48a; [353-452] 48d-50a
17 PLOTINUS: Second Ennead, TR VIII 64c-65c
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH XIII, SECT 2 149a; SECT 5 149b-c
35 BERKELEY: The Principles of Human Knowledge, SECT 42-44 420c-421a
53 JAMES: The Principles of Psychology, 318b; 400a; 406b; 410a-b; 471b-479a esp 474a-477b; 540a-635a esp 541a, 548b-552a, 560a-575a, 627a-635a

5. The mode of existence of geometrical objects: their character as abstractions; their relation to intelligible matter

7 PLATO: Republic, BK VI, 387b-c; BK VII, 394b-c / Philebus, 636b-c / Seventh Letter, 809c-810b
8 ARISTOTLE: Posterior Analytics, BK I, CH 13 [79a6-10] 108c; CH 18 [81b3-5] 111b-c / Topics, BK VI, CH 6 [143b11-33] 197b-c / Physics, BK II, CH 2 [193b23-194a11] 270a-c; BK III, CH 5 [204a17-19] 282a-b; BK IV, CH 1 [208b19-24] 287b-c / On the Heavens, BK III, CH 1 [299a1-17] 390b-c / Metaphysics, BK I, CH 6 [987b14-19] 505d; CH 8 [989b29-33] 508a; CH 9 [991b9-992a18] 509d-511a; BK III, CH 1 [995b13-18] 514a; [996a13-15] 514c; CH 2 [997b12-998a19] 516b-d; CH 5 [1001b26]-CH 6 [1002b25] 520c-521c; BK VII, CH 2 [1028b18-28] 551a-b; CH 10 [1035b32-1036a12] 559b-c; CH 11 [1036b32-1037a4] 560b-c; BK XI, CH 2 [1060b36-1061a19] 588c-d; CH 3 [1061a29-b4] 589c; BK XII, CH 1 [1069a30-37] 598b; BK XIII, CH 1-3 607a-610a; CH 6 [1080b23-30] 612c; CH 9 616d-618c; BK XIV, CH 3 622d-623d passim; CH 5 [1092a18-21] 625a; CH 6 [1093b24-29] 626d / On the Soul, BK III, CH 7 [431b3-19] 664b
9 ARISTOTLE: On the Motion of Animals, CH 1 [698a24-b1] 233b-c / Nicomachean Ethics, BK VI, CH 8 [1142a16-19] 391b
11 NICOMACHUS: Introduction to Arithmetic, BK I, 811a-812a
18 AUGUSTINE: Confessions, BK X, PAR 19 76a-b
19 AQUINAS: Summa Theologica, PART I, Q 5, A 3, REP 4 25a-d; Q 11, A 3, REP 2 49a-c; Q 44, A 1, REP 3 238b-239a; Q 85, A 1, REP 2 451c-453c
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 83, A 2, ANS 976c-978c; A 3, REP 2 978c-980d
31 DESCARTES: Rules for the Direction of the Mind, XIV 28a-33b esp 30d-32a / Discourse on the Method, PART IV, 52d-53a / Meditations on First Philosophy, I, 76b-c; V, 93a-d; V-VI, 96a-d / Objections and Replies, 169c-170a; 217a-d; 218c; 228c-229a
35 LOCKE: An Essay Concerning Human Understanding, BK II, CH XIII, SECT 5-6 149b-d; BK III, CH III, SECT 19 259c-260a; BK IV, CH IV, SECT 5-8 324d-325c
35 BERKELEY: The Principles of Human Knowledge, INTRO, SECT 12-16 408a-409d; SECT 12-16 415b-416a; SECT 123-128 437c-438d esp SECT 125-126 438a-c
35 HUME: An Enquiry Concerning Human Understanding, SECT IV, DIV 20 458a-b; SECT XII, DIV 122 505c-d; DIV 124-125 506a-507a esp DIV 125, 507b [fn 1]
42 KANT: The Critique of Pure Reason, 16a-c; 17d-18d; 24d-25b; 31b-d; 35b-36a; 46a-c; 55c; 62a-d; 68a-69c; 86b-c; 87b-c; 91c-d; 94b-95a; 211c-213c; 217c-d / The Critique of Practical Reason, 312c-d / The Critique of Judgement, 551a-552c
53 JAMES: The Principles of Psychology, 876b-878a; 880b-881a

6. The spiritual significance of place, position, and space

7 PLATO: Timaeus, 455b-c
8 ARISTOTLE: On the Heavens, BK I, CH 3 [270a1-12] 361c-d; BK II, CH 1 375b,d-376a; CH 13 [293a15-b15] 384d-385a
16 COPERNICUS: On the Revolutions of the Heavenly Spheres, BK I, 520b; 526a-528a
16 KEPLER: Epitome of Copernican Astronomy, BK IV, 853b-854a; 857b-860b; 915b-916a / The Harmonies of the World, 1080b-1085b
18 AUGUSTINE: Confessions, BK VII, PAR 20-21 49d-50a
19 AQUINAS: Summa Theologica, PART I, Q 3, A 1, REP 5 14b-15b; Q 8, A 1, REP 3 34d-35c; Q 61, A 4 316d-317c; Q 66, A 3 347b-348d; Q 102 523d-527a,c
20 AQUINAS: Summa Theologica, PART III SUPPL, Q 84, A 2, REP 5 984c-985d
21 DANTE: The Divine Comedy, HELL, V [1-15] 7a; VI [77-87] 9a-b; IX [16-33] 12b-c; PURGATORY, IV [1-96] 57c-58c; XVIII [22-33] 80a-b; XX [124]-XXI [72] 84c-85d; PARADISE, I [64-142] 106d-107d; III [34-90] 109d-110b; IV [28-48] 111a; XXVII 148d-150b; XXX-XXXIII 151d-156a
32 MILTON: Paradise Lost, BK II [850-1009] 129b-133a; BK III [523-539] 146b-147a; BK VII [114-130] 234b-235a
33 PASCAL: Pensées, 194, 207b; 205-206 211a
42 KANT: The Critique of Practical Reason, 360d-361d
46 HEGEL: The Philosophy of History, INTRODUCTION, 186c-d
54 FREUD: A General Introduction to Psychoanalysis, 562c-d

CROSS-REFERENCES

For other discussions of the theory of the receptacle, see FORM 1d(1); MATTER 1; WORLD 4b.
For other discussions of extension as a property of bodies, see BEING 7b(4); FORM 2d; MATTER 2a.
For other discussions of the doctrine of prime matter, see BEING 7c(3); CHANGE 2a; FORM 2c(3); INFINITY 4c; MATTER 1a.
For the role of space or place in local motion, see CHANGE 7a; RELATION 6a.
For discussions bearing on the measurement of space, see MATHEMATICS 5a; MECHANICS 3a; QUANTITY 6b-6c.
For matters relevant to space as a transcendental form of intuition and to the related problem of the foundations of geometry, see FORM 1c; MATHEMATICS 1c; MEMORY AND IMAGINATION 6c(2); MIND 1e(1), 4d(3); TIME 6c; and for other discussions bearing on space-perception, see QUALITY 2b; SENSE 3c(3).
For the problem of the mode of being which is possessed by the objects of geometry, see BEING 7d(3); IDEA 2g; MATHEMATICS 2a-2b; MATTER 1c.
For the issue concerning a void or vacuum, and for the related problem of action-at-a-distance, see ASTRONOMY 3b; CHANGE 7a; ELEMENT 5c; MECHANICS 5d, 6d(2).
For other considerations of the infinity or continuity of space, see INFINITY 3d; MATHEMATICS 2c; ONE AND MANY 3a(4); QUANTITY 2, 5a; and for discussions of astronomical space and the size of the universe, see ASTRONOMY 5, 9c; WORLD 6a, 7.
For the analysis of geometrical space, spatial relationships and configurations, see QUALITY 3b; QUANTITY 3-3c(2).
For the relation of spiritual being or action to place or space, see ANGEL 3f; ASTRONOMY 6; GOD 7g; IMMORTALITY 5g; SOUL 3e, 4d(3).

ADDITIONAL READINGS

Listed below are works not included in Great Books of the Western World, but relevant to the idea and topics with which this chapter deals. These works are divided into two groups:

I. Works by authors represented in this collection. II. Works by authors not represented in this collection.

For the date, place, and other facts concerning the publication of the works cited, consult the Bibliography of Additional Readings which follows the last chapter of The Great Ideas.

DESCARTES. The Principles of Philosophy, PART II, 4-19, 21
HOBBES. Concerning Body, PART II, CH 7
HUME. A Treatise of Human Nature, BK I, PART II
BERKELEY. An Essay Towards a New Theory of Vision
———. Siris
KANT. On the First Grounds of the Distinction of Regions in Space
———. De Mundi Sensibilis atque Intelligibilis Forma et Principiis (Inaugural Dissertation)
———. Metaphysical Foundations of Natural Science
JAMES, W. Collected Essays and Reviews, XXI

II.

SEXTUS EMPIRICUS. Against the Physicists, BK I, CH 1
———. Outlines of Pyrrhonism, BK III, CH 1-20
SUÁREZ. Disputationes Metaphysicae, XXX (7), XXXIX, XL (7), LI-LIII
LEIBNIZ. New Essays Concerning Human Understanding, BK II, CH 13
———. Correspondence with Clarke
D’ALEMBERT. Traité de dynamique
VOLTAIRE. “Space,” in A Philosophical Dictionary
WHEWELL. The Philosophy of the Inductive Sciences, VOL I, BK II, CH 2-6
RIEMANN. Über die Hypothesen, welche der Geometrie zu Grunde liegen (The Hypotheses of Geometry)
HODGSON. Time and Space
HELMHOLTZ. Popular Lectures on Scientific Subjects, II
CLIFFORD. The Common Sense of the Exact Sciences, CH 2, 4
LOTZE. Metaphysics, BK II, CH 1-2; BK III, CH 4
STALLO. Concepts and Theories of Modern Physics, CH 13-15
BRADLEY. Appearance and Reality, BK I, CH 4; BK II, CH 18
HILBERT. The Foundations of Geometry
MACH. Space and Geometry in the Light of Physiological, Psychological and Physical Inquiry
POINCARÉ. Science and Hypothesis, PART II
———. The Value of Science, PART I, CH 3-4
———. Science and Method, BK II, CH 1
CASSIRER. Substance and Function, PART I, CH 3; SUP IV-V
YOUNG, J. W. Lectures on Fundamental Concepts of Algebra and Geometry, LECT XVI-XVII
ROBB, A. A Theory of Time and Space
WEYL, H. Space—Time—Matter
ALEXANDER, S. Space, Time, and Deity
EDDINGTON. Space, Time, and Gravitation
WHITEHEAD. The Organization of Thought, CH 8
———. An Enquiry Concerning the Principles of Natural Knowledge, CH 14
———. The Concept of Nature, CH 5-6
BERGSON. Time and Free Will
———. Durée et simultanéité, à propos de la théorie d’Einstein, CH 6
EINSTEIN. Relativity: The Special and the General Theory
———. Sidelights on Relativity
———. The Meaning of Relativity
LEWIS, G. N. The Anatomy of Science, ESSAY II
SANTAYANA. The Realm of Matter, CH 4
LENZEN. The Nature of Physical Theory, PART II, CH 11
BORING. The Physical Dimensions of Consciousness, CH 4
RUSSELL, B. An Essay on the Foundations of Geometry
———. Principles of Mathematics, CH 44-52
———. The Analysis of Matter, CH 28-29, 32, 36
———. Human Knowledge, Its Scope and Limits, PART II, CH 6; PART IV, CH 6-7