Hinton - The Fourth Dimension.pdf

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212 THE FOURTH DIMENSION so we will let y go and draw w in its place. What will be our view of the cube? Evidently we shall have simply the square that is in z the plane of xz, the square ACDE. The rest of the cube stretches in w B D the y direction, and, as we have none of the space so determined, A C Fig. 4 (132). we have only the face of the cube. x This is represented in fig. 4. Now, suppose the whole cube to be turned from the x to the w direction. Conformably with out method, we will not take the whole of the cube into consideration at once, but will begin with the face ABCD. z Let this face begin to turn. Fig. 5 represents one of the positions it will w B D occupy; the line AB remains on the x axis. The rest of the face extends C between the x and the w direction. Now, since we can take any three A Fig. 5 (133). x axes, let us look at what lies in the space of xyw, and examine the turning there. We must now let the z axis disappear and let the w axis run in the direction in which the z ran. w Making this representation, what y E G A C Fig. 6 (134). do we see of the cube? Obviously we see only the lower face. The rest of the cube lies in the space of xyz. x In the space of xyw we have merely the base of the cube lying in the plane of xy, as shown in fig. 6. Now let the x to w turning take place. The square ACEG will turn about the line AE. This edge will remain alone the y axis and will be stationary, however far the square turns.
RECAPITULATION AND EXTENSION 213 Thus, if the cube be turned by an x to w turning, both the edge AB and the edge AC remain w stationary; hence the whole face ABEF in the yz plane remains fixed. y G The turning has taken place about E C the face ABEF. Suppose this turning to continue A x till AC runs to the left from A. Fig. 7 (135). The cube will occupy the position shown in fig. 8. This is the looking-glass image of the cube in fig. 3. By no rotation in three-dimensional space H F z can the cube be brought from the position in fig. 3 to that D shown in fig. 8. y B We can think of this turning G E as a turning of the face ABCD about AB, and a turning of each C A x x section parallel to ABCD round 2 the vertical line in which it intersects the face ABEF, the space in which the turning takes place being a different one from that in which the cube lies. One of the conditions, then, of our inquiry in the direction of the infinitely small is that we form the conception of a rotation about a plane. The production of a body in a state in which it presents the appearance of a looking-glass image of its former state is the criterion for a four-dimensional rotation. There is some evidence for the occurrence of such transformations of bodies in the change of bodies from those which produce a right-handed polarisation of light to those which produce a left-handed polarisation; but this is not a point to which any very great importance can be attached. Still, in this connection, let me quote a remark from nd position 1st position Fig. 8 (136).
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THE FOURTH DIMENSION
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THE FOURTH DIMENSION BY C. HOWARD H
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PREFACE I HAVE endeavoured to prese
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CONTENTS CHAP. PAGE. I. FOUR-DIMENS
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THE FOURTH DIMENSION ————
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FOUR-DIMENSIONAL SPACE 3 existence
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FOUR-DIMENSIONAL SPACE 5 Bringing i
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THE ANALOGY OF A PLANE WORLD 7 him.
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THE ANALOGY OF A PLANE WORLD 9 to a
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THE ANALOGY OF A PLANE WORLD 11 Aga
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THE ANALOGY OF A PLANE WORLD 13 sec
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CHAPTER III THE SIGNIFICANCE OF A F
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SIGNIFICANCE OF A FOUR-DIMENSIONAL
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CHAPTER IV THE FIRST CHAPTER IN THE
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THE FIRST CHAPTER IN THE HISTORY OF
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CHAPTER V THE SECOND CHAPTER IN THE
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SECOND CHAPTER IN THE HISTORY OF FO
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CHAPTER VI THE HIGHER WORLD IT is i
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THE HIGHER WORLD 63 dimensions. His
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THE HIGHER WORLD 65 If now he turns
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THE HIGHER WORLD 67 to him that, by
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THE HIGHER WORLD 69 simply the plan
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THE HIGHER WORLD 71 turn these rods
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THE HIGHER WORLD 73 revolution, the
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THE HIGHER WORLD 75 the mode of act
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THE EVIDENCES FOR A FOURTH DIMENSIO
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CHAPTER VIII THE USE OF FOUR DIMENS
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5 THE USE OF FOUR DIMENSIONS IN THO
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THE USE OF FOUR DIMENSIONS IN THOUG
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CHAPTER IX APPLICATION TO KANT’S
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APPLICATION TO KANT’S THEORY OF E
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3 APPLICATION TO KANT’S THEORY OF
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A FOUR-DIMENSIONAL FIGURE 123 The r
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A FOUR-DIMENSIONAL FIGURE 125 new d
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A FOUR-DIMENSIONAL FIGURE 127 pick
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A FOUR-DIMENSIONAL FIGURE 129 Now s
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A FOUR-DIMENSIONAL FIGURE 131 of me
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A FOUR-DIMENSIONAL FIGURE 133 note
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A FOUR-DIMENSIONAL FIGURE 135 These
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NOMENCLATURE AND ANALOGIES 137 to a
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NOMENCLATURE AND ANALOGIES 139 null
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NOMENCLATURE AND ANALOGIES 141 laye
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NOMENCLATURE AND ANALOGIES 143 a li
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NOMENCLATURE AND ANALOGIES 145 oppo
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NOMENCLATURE AND ANALOGIES 147 the
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NOMENCLATURE AND ANALOGIES 149 dime
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NOMENCLATURE AND ANALOGIES 151 larg
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NOMENCLATURE AND ANALOGIES 153 axes
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NOMENCLATURE AND ANALOGIES 155 lie
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CHAPTER XII THE SIMPLEST FOUR-DIMEN
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THE SIMPLEST FOUR-DIMENSIONAL SOLID
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Page 189 and 190: REMARKS ON THE FIGURES 179 ochre cu
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Page 213 and 214: CHAPTER XIV * A RECAPITULATION AND
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APPENDIX II: A LANGUAGE OF SPACE 26
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Sil Sit Sin Sil Sit Sin Silan Sitan
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TRANSCRIBER’S NOTE. Do what thou
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