## The Fourth Dimension Simply Explained: A Collection of Essays Selected from Those Submitted in the Scientific American's Prize Competition |

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3-space ABCD ABCD-G absolutely perpendicular Algebra analogy analytical geometry axial plate axioms axis body boundary circle coincide conceive conception corresponding creatures cube curve cylinder dimensional figure dimensional space distance equal edges eight equation essays Euclidean Euclidean geometry existence faces final position fixed flatland four-dimensional geometry four-dimensional space fourth dimension Geometry of Four higher space hypercube hypersolid hyperspace hypersphere Hypervolume idea imagine infinite number intersect left glove length and breadth limited mathematical mathematician means measure motion move movement mutually perpendicular non-Euclidean geometries object one-dimensional space ordinary space parallel parallelopipeds pass perpendicular physical picture plane geometry polygon portion possible prisms properties pyramids relation represent right angles right glove rotation semaphore shadow side sion sional space solid geometry space of four space of three sphere square straight line suppose surface symmetrical with respect tesseract tetrahedron thickness third dimension three dimensions three-dimensional space three-space tion triangle two-dimen two-dimensional space variables wheel

### Popular passages

Page 162 - ... space, just as a line lies in a plane, or a plane lies in three-dimensional space. Just why one should stop at four dimensions is not made clear. In a brief way we have then shown how the term "fourth dimension" arose. We have shown how the efforts of mankind to tear himself away from the numbers i, 2, and 3 and to generalize have given rise to two classes of literature, one purely imaginative fiction for the general reader, and one mathematical for the mathematician. From these writings words...

Page 190 - I knew a man in Christ above fourteen years ago, (whether in the body, I cannot tell; or whether out of the body, I cannot tell: God knoweth;) such an one caught up to the third heaven. And I knew such a man, (whether in the body, or out of the body, I cannot tell: God knoweth;) How that he was caught up into paradise, and heard unspeakable words, which it is not lawful for a man to utter.

Page 168 - Y, and the Z axes at their intersection. Here the mathematicians, as the popular saying goes, found themselves up against it, for they could not draw four straight lines mutually perpendicular at a point. This limitation of our space prevented the geometric representation of equations of four variables, but it did not deter further study of the equations. Men are continually calculating what would happen if conditions were different from what they are. The student of history seeks to determine the...

Page 244 - Under the whip of external necessity their backward culture is compelled to make leaps. From the universal law of unevenness thus derives another law which, for the lack of a better name, we may call the law of combined development — by which we mean a drawing together of the different stages of the journey, a combining of separate steps, an amalgam of archaic with more contemporary forms.

Page 98 - ... and how of 4 space. Where is it? Go back to our first definition : Space is that which separates two portions of higher space from each other. Conceive of 2 space therefore as a vertical G plane, separating two portions of 3 space from each other. Now, in order that this separation should be effective, the plane must be something more than a mere geometrical abstraction, that is, if it is a "real" plane, it must have a very slight thickness. Its particles will have a free movement and circulation...

Page 237 - A spherical triangle is bounded by arcs of great circles (see p. 134). In two polar triangles, each angle in one is the supplement of the corresponding side in the other. In two symmetrical triangles, the sides and angles of one are equal to the corresponding sides and angles of the other, but arranged in the reverse order (like right-handed and left-handed gloves) . GEOMETRICAL CONSTRUCTIONS To Bisect a Line AB (Fig.

Page 15 - Stringham has given us a full account of the regular figures in space of four dimensions corresponding to the regular polyhedrons of our three-dimensional space. Others have written on the theory of rotations and on the intersections and projections of different figures. The great Italian geometer Veronese has an extensive work on Geometry of n Dimensions with theorems and proofs like those of the text-books studied in our schools. In the last few years there have been many articles in the popular...

Page 98 - Furthermore, if from our childhood phenomena had been of daily occurrence, requiring a space of four dimensions for their proper understanding, we would naturally grow up with the conception of a space of four dimensions.

Page 171 - ... microscope has, however, shown us that minute plants are as active as minute animals. Hence we cannot always assert that because we do not observe a phenomenon that it does not exist. If we insisted that everything were just as it appeared to be from our observation, we should be in the position of a child who believes that all people have enough to eat, and that all children have nurse-maids. The child reasons from uncontradicted experience, and so do we, usually. Although we cannot dogmatically...

Page 59 - For proof of his standpoint he referred to the existence of geometry. This argument was irrefutable until the discovery of nonEuclidean geometry. Another far-reaching conclusion is the following: Metaphysical axioms being mere Imitations of geometrical axioms will, like the latter, have to be discarded. It seems therefore fitting to conclude with the following words of the eminent German mathematician Hilbert: "The most suggestive and notable achievement of the last century is the discovery of non-Euclidean...