Fig. 21.
Fig. 21.
Next he would take the square section half-way between Moena and Murex. He knows that the line Alvus of this section is parallel to Arctos, and that the point Dos at one of its ends is half-way between Corvus and Cista, so that this line itself is the one he wants (because the sectional plane cuts Dos half-way between Corvus and Cista, and is parallel to Arctos). InFig. 21the two lines thus found are shown. a b is the line in Moena, and c d the line in the section. He must now find out how far apart they are. He knows that from the middle point of Cuspis to Corvus is half-a-unit, and from the middle point of Dos to Corvus is half-a-unit, and Cuspis and Dos are at right angles to each other; therefore from the middle point of Cuspis to the middle point of Dos is the diagonal of a square whose sides are half-a-unit in length. This diagonal may be written d (1⁄2)2. He would also see that from the middle point of Callis to the middle point of Via is the same length; therefore the figure is a parallelogram, having two of its sides, each one unit in length, and the other two each d (1⁄2)2.
He could also see that the angles are right, because the lines a c and b d are made up of the X and Y directions, and the other two, a b and d, are purely Z, and since they have no tendency in common, they are at right angles to each other.
Fig. 22.
Fig. 22.
If he wanted the figure made byZ0.X11⁄2.Y11⁄2it would be a little more difficult. He would have to take Moena, a section halfway between Moena and Murex, Murex and another square which he would have to regard as animaginarysection half-a-unit further Y than Murex (Fig. 22). He might now draw a ground plan of the sections; that is, he would draw Syce, and produce Cuspis and Dos half-a-unit beyond Nugæ and Cista. He would see that Cadus and Bolus would be cut half-way, so that in thehalf-way section he would have the point a (Fig. 23), and in Murex the point c. In the imaginary section he would have g; but this he might disregard, since the cube goes no further than Murex. From the points c and a there would be lines going Z, so that Iter and Semita would be cut half-way.
Groundplan of Sections shown in Fig. 22.Fig. 23.
Groundplan of Sections shown in Fig. 22.
Fig. 23.
He could find out how far the two lines a b and c d (Fig. 22) are apart by referring d and b to Lama, and a and c to Crus.
In taking the third order of sections, a similar method may be followed.
Fig. 24.
Fig. 24.
Suppose the sectional plane to cut Cuspis, Dos, and Arctos, each at one unit from Corvus. He would first take Moena, and as the sectional plane passes through Ilex and Nugæ, the line on Moena would be the diagonal passing through these two points. Then he would take Murex, and he would see that as the plane cuts Dos at one unit from Corvus, all he would have is the point Cista. So the whole figure is the Ilex to Nugæ diagonal, and the point Cista.
Now Cista and Ilex are each one inch from Corvus, and measured along lines at right angles to each other; therefore, they are d (1)2from each other. By referring Nugæ and Cista to Corvus he would find that they are also d (1)2apart; therefore the figure is an equilateral triangle, whose sides are each d (1)2.
Suppose the sectional plane to pass through Mala, cutting Cuspis, Dos, and Arctos each at unit from Corvus. To find the figure, the plane-being would have to take Moena, a section half-way between Moena and Murex, Murex, and an imaginary section half-a-unit beyond Murex (Fig. 24). He would produce Arctos and Cuspis to points half-a-unit from Ilex and Nugæ, and by joining these points, he would see that the line passes through the middle points of Callis and Far (a, b,Fig. 24). In the last square, the imaginary section, there would be the point m; for this is 11⁄2unit from Corvus measured along Dos produced. There would also be lines in the other two squares, the section and Murex, and to find these he would have to make many observations. He found the points a and b (Fig. 24) by drawing a line from r to s, r and s being each 11⁄2unit from Corvus, and simply seeing that it cut Callis and Far at the middle point of each. He might now imagine a cube Mala turned about Arctos, so that Alvus came into his plane; he might then produce Arctos and Dos until they were each unit long, and join their extremities, when he would see that Via and Bucina are each cut half-way. Again, by turning Syce into his plane, andproducing Dos and Cuspis to points 11⁄2unit from Corvus and joining the points, he would see that Bolus and Cadus are cut half-way. He has now determined six points on Mala, through which the plane passes, and by referring them in pairs to Ilex, Olus, Cista, Crus, Nugæ, Sors, he would find that each was d (1⁄2)2from the next; so he would know that the figure is an equilateral hexagon. The angles he would not have got in this observation, and they might be a serious difficulty to him. It should be observed that a similar difficulty does not come to us in our observation of the sections of a tessaract: for, if the angles of each side of a solid figure are determined, the solid angles are also determined.
There is another, and in some respects a better, way by which he might have found the sides of this figure. If he had noticed his plane-space much, he would have found out that, if a line be drawn to cut two other lines which meet, the ratio of the parts of the two lines cut off by the first line, on the side of the angle, is the same for those lines, and any other two that are parallel to them. Thus, if a b and a c (Fig. 25) meet, making an angle at a, and b c crosses them, and also crosses a′ b′ and a′ c′, these last two being parallel to a b and a c, then a b ∶ a c ∷ a′ b′ ∶ a′ c′.
Fig. 25
Fig. 25
If the plane-being knew this, he would rightly assume that if three lines meet, making a solid angle, and a plane passes through them, the ratio of the parts between the plane and the angle is the same for those three lines, and for any other three parallel to them.
In the case we are dealing with he knows that from Ilex to the point on Arctos produced where the plane cuts, it is half-a-unit; and as the Z, X, and Y lines are cut equally from Corvus, he would conclude that the X and Y lines are cut the same distance from Ilex as the Z line, that is half-a-unit. He knows that the X line is cut at 11⁄2units from Corvus; that is, half-a-unit from Nugæ: so he would conclude that the Z and Y lines are cut half-a-unit from Nugæ. He would also see that the Z and X lines from Cista are cut at half-a-unit. He has now six points on the cube, the middle points of Callis, Via, Bucina, Cadus, Bolus, and Far. Now, looking at his square sections, he would see on Moena a line going from middle of Far to middle of Callis, that is, a line d (1⁄2)2long. On the section he would see a line from middle of Via to middle of Bolus d (1)2long, and on Murex he would see a line from middle of Cadus to middle of Bucina, d (1⁄2)2long. Of these three lines a b, c d, e f, (Fig. 24)—a b and e f are sides, and c d is a section of the required figure. He can find the distancesbetween a and c by reference to Ilex, between b and d by reference to Nugæ, between c and e by reference to Olus, and between d and f by reference to Crus; and he will find that these distances are each d (1⁄2)2.
Thus, he would know that the figure is an equilateral hexagon with its sides d (1⁄2)2long, of which two of the opposite points (c and d) are d (1)2apart, and the only figure fulfilling all these conditions is an equilateral and equiangular hexagon.
Enough has been said about sections of a cube, to show how a plane-being would find the shapes in any set as inZI.XII.YIIorZI.XI.YII
He would always have to bear in mind that the ratio of the lengths of the Z, X, and Y lines is the same from Corvus to the sectional plane as from any other point to the sectional plane. Thus, if he were taking a section where the plane cuts Arctos and Cuspis at one unit from Corvus and Dos at one-and-a-half, that is where the ratio of Z and of X to Y is as two to three, he would see that Dos itself is not cut at all; but from Cista to the point on Dos produced is half-a-unit; therefore from Cista, the Z and X lines will be cut at2⁄3of1⁄2unit from Cista.
It is impossible in writing to show how to make the various sections of a tessaract; and even if it were not so, it would be unadvisable; for the value of doing it is not in seeing the shapes themselves, so much as in the concentration of the mind on the tessaract involved in the process of finding them out.
Any one who wishes to make them should go carefully over the sections of a cube, not looking at them as he himself can see them, or determining them as he, with his three-dimensional conceptions, can; but he must limit his imagination to two dimensions, and work through the problems which a plane-being would have to work through, although to his higher mind they may be self-evident. Thus a three-dimensional being can see at a glance, that if a sectional plane passes through a cube at one unit each way from Corvus, the resulting figure is an equilateral triangle.
If he wished to prove it, he would show that the three bounding lines are the diagonals of equal squares. This is all a two-dimensional being would have to do; but it is not so evident to him that two of the lines are the diagonals of squares.
Moreover, when the figure is drawn, we can look at it from a point outside the plane of the figure, and can thus see it all atonce; but he who has to look at it from a point in the plane can only see an edge at a time, or he might see two edges in perspective together.
Then there are certain suppositions he has to make. For instance, he knows that two points determine a line, and he assumes that three points determine a plane, although he cannot conceive any other plane than the one in which he exists. We assume that four points determine a solid space. Or rather, we say thatifthis supposition, together with certain others of a like nature, are true, we can find all the sections of a tessaract, and of other four-dimensional figures by an infinite solid.
When any difficulty arises in taking the sections of a tessaract, the surest way of overcoming it is to suppose a similar difficulty occurring to a two-dimensional being in taking the sections of a cube, and, step by step, to follow the solution he might obtain, and then to apply the same or similar principles to the case in point.
A few figures are given, which, if cut out and folded along the lines, will show some of the sections of a tessaract. But the reader is earnestly begged not to be content withlookingat the shapes only. That will teach him nothing about a tessaract, or four-dimensional space, and will only tend to produce in his mind a feeling that “the fourth dimension” is an unknown and unthinkable region, in which any shapes may be right, as given sections of its figures, and of which any statement may be true. While, in fact, if it is the case that the laws of spaces of two and three dimensions may, with truth, be carried on into space of four dimensions; then the little our solidity (like the flatness of a plane-being) will allow us to learn of these shapes and relations, is no more a matter of doubt to us than what we learn of two- and three-dimensional shapes and relations.
There are given also sections of an octa-tessaract, and of a tetra-tessaract, the equivalents in four-space of an octahedron and tetrahedron.
A tetrahedron may be regarded as a cube with every alternate corner cut off. Thus, if Mala have the corner towards Corvus cut off as far as the points Ilex, Nugæ, Cista, and the corner towards Sors cut off as far as Ilex, Nugæ, Lama, and the corner towards Crus cut off as far as Lama, Nugæ, Cista, and the corner towards Olus cut off as far as Ilex, Lama, Cista, what is left of the cube is a tetrahedron, whose angles are at the points Ilex, Nugæ, Cista, Lama. In a similar manner, if every alternate corner of a tessaract be cut off, the figure that is left is a tetra-tessaract, which is a figure bounded by sixteen regular tetrahedrons.
Fig. 26.Fig. 27.Fig. 27.Fig. 26.(i)
(i)
Fig. 28.Fig. 29.Fig. 30.(ii)
(ii)
Fig. 31.Fig. 32.(iii)
(iii)
Fig. 33.Fig. 35.Fig. 34.(iv)
(iv)
Fig. 36.Fig. 37.Fig. 38.(v)
(v)
Fig. 39.Fig. 41.Fig. 40.(vi)
(vi)
The octa-tessaract is got by cutting off every corner of the tessaract. If every corner of a cube is cut off, the figure left is an octa-hedron, whose angles are at the middle points of the sides. The angles of the octa-tessaract are at the middle points of its plane sides. A careful study of a tetra-hedron and an octa-hedron as they are cut out of a cube will be the best preparation for the study of these four-dimensional figures. It will be seen that there is much to learn of them, as—How many planes and lines there are in each, How many solid sides there are round a point in each.
The above are sections of a tessaract.Figures 33to35are of a tetra-tessaract. The tetra-tessaract is supposed to be imbedded in a tessaract, and the sections are taken through it, cutting the Z, X and Y lines equally, and corresponding to the figures given of the sections of the tessaract.
Figures 36,37, and38are similar sections of an octa-tessaract.
Figures 39,40, and41are the following sections of a tessaract.
It is clear that there are four orders of sections of every four-dimensional figure; namely, those beginning with a solid, those beginning with a plane, those beginning with a line, and those beginning with a point. There should be little difficulty in finding them, if the sections of a cube with a tetra-hedron, or an octa-hedron enclosed in it, are carefully examined.
Model 1.MALA.Colours: Mala, Light-buff.
Model 1.MALA.
Colours: Mala, Light-buff.
Points: Corvus, Gold. Nugæ, Fawn. Crus, Terra-cotta. Cista, Buff. Ilex, Light-blue. Sors, Dull-purple. Lama, Deep-blue. Olus, Red.
Lines: Cuspis, Orange. Bolus, Crimson. Cadus, Green-grey. Dos, Blue. Arctos, Brown. Far, French-grey. Daps, Dark-slate. Bucina, Green. Callis, Reddish. Iter, Bright-blue. Semita, Leaden. Via, Deep-yellow.
Surfaces: Moena, Dark-blue. Proes, Blue-green. Murex, Light-yellow. Alvus, Vermilion. Mel, White. Syce, Black.
Model 2.MARGO.Colours: Margo, Sage-green.
Model 2.MARGO.
Colours: Margo, Sage-green.
Points: Spira, Silver. Ancilla, Turquoise. Mugil, Earthen. Panax, Blue-tint. Felis, Quaker-green. Passer, Peacock-blue. Talus, Orange-vermilion. Solia, Purple.
Lines: Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple. Opex, Purple-brown. Pagus, Dull-blue. Onager, Dark-pink. Vena, Pale-pink. Lixa, Indigo. Tholus, Brown-green. Calor, Dark-green. Livor, Pale-yellow. Lensa, Dark.
Surfaces: Silex, Burnt-sienna. Sal, Yellow-ochre. Portica, Dun. Crux, Indian-red. Lares, Light-grey. Lappa, Bright-green.
Model 3.LAR.Colours: Lar, Brick-red.
Model 3.LAR.
Colours: Lar, Brick-red.
Points: Spira, Silver. Ancilla, Turquoise. Mugil, Earthen. Panax, Blue-tint. Corvus, Gold. Nugæ, Fawn. Crus, Terra-cotta. Cista, Buff.
Lines: Luca, Leaf-green. Mappa, Dull-green. Mensura, Dark-purple. Opex, Purple-brown. Ops, Stone. Limus, Smoke. Offex, Magenta. Lis, Light-green. Cuspis, Orange. Bolus, Crimson. Cadus, Green-grey. Dos, Blue.
Surfaces: Lotus, Azure. Olla, Rose. Lorica, Sea-blue. Lua, Bright-brown. Syce, Black. Lappa, Bright-green.
Model 4.VELUM.Colours: Velum, Chocolate.
Model 4.VELUM.
Colours: Velum, Chocolate.
Points: Felis, Quaker-green. Passer, Peacock-blue. Talus, Orange-vermilion. Solia, Purple. Ilex, Light-blue. Sors, Dull-purple. Lama, Deep-blue. Olus, Red.
Lines: Tholus, Brown-green. Calor, Dark-green. Livor, Pale-yellow. Lensa, Dark. Lucta, Rich-red. Pator, Green-blue. Libera, Sea-green. Orsa, Emerald. Callis, Reddish. Iter, Bright-blue. Semita, Leaden. Via, Deep-yellow.
Surfaces: Limbus, Ochre. Meatus, Deep-brown. Mango, Deep-green. Croeta, Light-red. Mel, White. Lares, Light-grey.
Model 5.VESPER.Colours: Vesper, Pale-green.
Model 5.VESPER.
Colours: Vesper, Pale-green.
Points: Spira, Silver. Corvus, Gold. Cista, Buff. Panax, Blue-tint. Felis, Quaker-green. Ilex, Light-blue. Olus, Red. Solia, Purple.
Lines: Ops, Stone. Dos, Blue. Lis, Light-green. Opex, Purple-brown. Pagus, Dull-blue. Arctos, Brown. Bucina, Green. Lixa, Indigo. Lucta, Rich-red. Via, Deep-yellow. Orsa, Emerald. Lensa, Dark.
Surfaces: Pagina, Yellow. Alvus, Vermilion. Camoena, Deep-crimson. Crux, Indian-red. Croeta, Light-red. Lua, Light-brown.
Model 6.IDUS.Colours: Idus, Oak.
Model 6.IDUS.
Colours: Idus, Oak.
Points: Ancilla, Turquoise. Nugæ, Fawn. Crus, Terra-cotta. Mugil, Earthen. Passer, Peacock-blue. Sors, Dull-purple. Lama, Deep-blue. Talus, Orange-vermilion.
Lines: Limus, Smoke. Bolus, Crimson. Offex, Magenta. Mappa, Dull-green. Onager, Dark-pink. Far, French-grey. Daps, Dark-slate. Vena, Pale-pink. Pator, Green-blue. Iter, Bright-blue. Libera, Sea-green. Calor, Dark-green.
Surfaces: Pactum, Yellow-green. Proes, Blue-green. Orca, Dark-grey. Sal, Yellow-ochre. Meatus, Deep-brown. Olla, Rose.
Model 7.PLUVIUM.Colours: Pluvium, Dark-stone.
Model 7.PLUVIUM.
Colours: Pluvium, Dark-stone.
Points: Spira, Silver. Ancilla, Turquoise. Nugæ, Fawn. Corvus, Gold. Felis, Quaker-green. Passer, Peacock-blue. Sors, Dull-purple. Ilex, Light-blue.
Lines: Luca, Leaf-green. Limus, Smoke. Cuspis, Orange. Ops, Stone. Pagus, Dull-blue. Onager, Dark-pink. Far, French-grey. Arctos, Brown. Tholos, Brown-green. Pator, Green-blue. Callis, Reddish. Lucta, Rich-red.
Surfaces: Silex, Burnt-Sienna. Pactum, Yellow-green. Moena, Dark-blue. Pagina, Yellow. Limbus, Ochre. Lotus, Azure.
Model 8.TELA.Colours: Tela, Salmon.
Model 8.TELA.
Colours: Tela, Salmon.
Points: Panax, Blue-tint. Mugil, Earthen. Crus, Terra-cotta. Cista, Buff. Solia, Purple. Talus, Orange-vermilion. Lama, Deep-blue. Olus, Red.
Lines: Mensura, Dark-purple. Offex, Magenta. Cadus, Green-grey. Lis, Light-green. Lixa, Indigo. Vena, Pale-pink. Daps, Dark-slate. Bucina, Green. Livor, Pale-yellow. Libera, Sea-green. Semita, Leaden. Orsa, Emerald.
Surfaces: Portica, Dun. Orca, Dark-grey. Murex, Light-yellow. Camoena, Deep-crimson. Mango, Deep-green. Lorica, Sea-blue.
Model 9.SECTION BETWEEN MALA AND MARGO.Colours: Interior or Tessaract, Wood.
Model 9.SECTION BETWEEN MALA AND MARGO.
Colours: Interior or Tessaract, Wood.
Points(Lines): Ops, Stone. Limus, Smoke. Offex, Magenta. Lis, Light-green. Lucta, Rich-red. Pator, Green-blue. Libera, Sea-green. Orsa, Emerald.
Lines(Surfaces): Lotus, Azure. Olla, Rose. Lorica, Sea-blue. Lua Bright-brown. Pagina, Yellow. Pactum, Yellow-green. Orca, Dark-grey. Camoena, Deep-crimson. Limbus, Ochre. Meatus, Deep-brown. Mango, Deep-green. Croeta, Light red.
Surfaces(Solids): Pluvium, Dark-stone. Idus, Oak. Tela, Salmon. Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red.
Model 10.SECTION BETWEEN LAR AND VELUM.Colours: Interior or Tessaract, Wood.
Model 10.SECTION BETWEEN LAR AND VELUM.
Colours: Interior or Tessaract, Wood.
Points(Lines): Pagus, Dull-blue. Onager, Dark-pink. Vena, Pale-pink. Lixa, Indigo. Arctos, Brown. Far, French-grey. Daps, Dark-slate. Bucina, Green.
Lines(Surfaces): Silex, Burnt-sienna. Sal, Yellow-ochre. Portica, Dun. Crux, Indian-red. Pagina, Yellow. Pactum, Yellow-green. Orca, Dark-grey. Camoena, Deep-crimson. Moena, Dark-blue. Proes, Blue-green. Murex, Light-yellow. Alvus, Vermilion.
Surfaces(Solids): Pluvium, Dark-stone. Idus, Oak. Tela, Salmon. Vesper, Pale-green. Mala, Light-buff. Margo, Sage-green.
Model 11.SECTION BETWEEN VESPER AND IDUS.Colours: Interior or Tessaract, Wood.
Model 11.SECTION BETWEEN VESPER AND IDUS.
Colours: Interior or Tessaract, Wood.
Points(Lines): Luca, Leaf-green. Cuspis, Orange. Cadus, Green-grey. Mensura, Dark-purple. Tholus, Brown-green. Callis, Reddish. Semita, Leaden. Livor, Pale-yellow.
Lines(Surfaces): Lotus, Azure. Syce, Black. Lorica, Sea-blue. Lappa, Bright-green. Silex, Burnt-sienna. Moena, Dark-blue. Murex, Light-yellow. Portica, Dun. Limbus, Ochre. Mel, White. Mango, Deep-green. Lares, Light-grey.
Surfaces(Solids): Pluvium, Dark-stone. Mala, Light-buff. Tela, Salmon. Margo, Sage-green. Velum, Chocolate. Lar, Brick-red.
Model 12.SECTION BETWEEN PLUVIUM AND TELA.Colours: Interior or Tessaract, Wood.
Model 12.SECTION BETWEEN PLUVIUM AND TELA.
Colours: Interior or Tessaract, Wood.
Points(Lines): Opex, Purple-brown. Mappa, Dull-green. Bolus, Crimson. Dos, Blue. Lensa, Dark. Calor, Dark-green. Iter, Bright-blue. Via, Deep-yellow.
Lines(Surfaces): Lappa, Bright-green. Olla, Rose. Syce, Black. Lua, Bright-brown. Crux, Indian-red. Sal, Yellow-ochre. Proes, Blue-green. Alvus, Vermilion. Lares, Light-grey. Meatus, Deep-brown. Mel, White. Croeta, Light-red.
Surfaces(Solids): Margo, Sage-green. Idus, Oak. Mala, Light-buff. Vesper, Pale-green. Velum, Chocolate. Lar, Brick-red.
Transcriber’s NotesLay-out and formatting have been optimised for browser html (available at www.gutenberg.org); some versions and narrow windows may not display all elements of the book as intended, depending on the hard- and software used and their settings.Inconsistencies in spelling (Mœnas v. Moenas; Praeda v. Proeda), hyphenation (Deep-blue v. Deep blue, etc.) have been retained.Page 197, row starting Sophos: the last letter of Blue has been assumed.Changes made:Footnotes, tables, diagrams and illustrations have been moved outside text paragraphs. Indications for the location of illustrations (To face p. ...) have been removed; the illustrations concerned have been moved to where they are discussed.Some minor obvious typographical errors have been corrected silently.Page 42: ... the flat, being ... changed to ... the flat being ...Page 127: Cube itself: considered to be the table header rather than a table elementPage 175: is all Ana our space changed to is all Ana in our spacePage 187: Clipens changed to Clipeus; legend Y added to right-hand side grid axesPage 219: Part II. Appendix K. changed to Appendix K. cf. other Appendices.
Lay-out and formatting have been optimised for browser html (available at www.gutenberg.org); some versions and narrow windows may not display all elements of the book as intended, depending on the hard- and software used and their settings.
Inconsistencies in spelling (Mœnas v. Moenas; Praeda v. Proeda), hyphenation (Deep-blue v. Deep blue, etc.) have been retained.
Page 197, row starting Sophos: the last letter of Blue has been assumed.
Changes made:
Footnotes, tables, diagrams and illustrations have been moved outside text paragraphs. Indications for the location of illustrations (To face p. ...) have been removed; the illustrations concerned have been moved to where they are discussed.
Some minor obvious typographical errors have been corrected silently.
Page 42: ... the flat, being ... changed to ... the flat being ...
Page 127: Cube itself: considered to be the table header rather than a table element
Page 175: is all Ana our space changed to is all Ana in our space
Page 187: Clipens changed to Clipeus; legend Y added to right-hand side grid axes
Page 219: Part II. Appendix K. changed to Appendix K. cf. other Appendices.