A New Era of Thought

by Charles Howard Hinton

APPENDIX D.

The following list gives the colours, and the various uses for them. They have already been used in the foregoing pages to distinguish the various regions of the Tessaract, and the different individual cubes or Tessaracts in a block. The other use suggested in the last column of the list has not been discussed; but it is believed that it may afford great aid to the mind in amassing, handling, and retaining the quantities of formulae requisite in scientific training and work.

Colour.	Region of Tessaract.	Tessaract in 81 Set.	Symbol.
Black	Syce	Plebs	0
White	Mel	Mora	1
Vermilion	Alvus	Uncus	2
Orange	Cuspis	Moles	3
Light-yellow	Murex	Cortis	4
Bright-green	Lappa	Penates	5
Bright-blue	Iter	Oliva	6
Light-grey	Lares	Tigris	7
Indian-red	Crux	Orcus	8
Yellow-ochre	Sal	Testudo	9
Buff	Cista	Sector	+ (plus)
Wood	Tessaract	Tessara	- (minus)
Brown-green	Tholus	Troja	± (plus or minus)
Sage-green	Margo	Lacerta	× (multiplied by)
Reddish	Callis	Tibicen	÷ (divided by)
Chocolate	Velum	Sacerdos	= (equal to)
French-grey	Far	Scena	≠ (not equal to)
Brown	Arctos	Ostrum	> (greater than)
Dark-slate	Daps	Aer	< (less than)
Dun	Portica	Clipeus	∶		-	signs of proportion
Orange-vermilion	Talus	Portio	∷
Stone	Ops	Thyrsus	· (decimal point)
Quaker-green	Felis	Axis	∟ (factorial)
Leaden	Semita	Merces	∥ (parallel)
Dull-green	Mappa	Vulcan	∦ (not parallel)
Indigo	Lixa	Postis	π⁄2(90°) (at right angles)
Dull-blue	Pagus	Verbum	log. base 10
Dark-purple	Mensura	Nepos	sin. (sine)
Pale-pink	Vena	Tabula	cos. (cosine)
Dark-blue	Moena	Bidens	tan. (tangent)
Earthen	Mugil	Angusta	∞ (infinity)
Blue	Dos	Frenum	a
Terracotta	Crus	Remus	b
Oak	Idus	Domitor	c
Yellow	Pagina	Cardo	d
Green	Bucina	Ala	e
Rose	Olla	Limen	f
Emerald	Orsa	Ara	g
Red	Olus	Mars	h
Sea-green	Libera	Pluma	i
Salmon	Tela	Glans	j
Pale-yellow	Livor	Ovis	k
Purple-brown	Opex	Polus	l
Deep-crimson	Camoena	Pilum	m
Blue-green	Proes	Tergum	n
Light-brown	Lua	Crates	o
Deep-blue	Lama	Tyro	p
Brick-red	Lar	Cura	q
Magenta	Offex	Arvus	r
Green-grey	Cadus	Hama	s
Light-red	Croeta	Praeda	t
Azure	Lotus	Vitta	u
Pale-green	Vesper	Ocrea	v
Blue-tint	Panax	Telum	w
Yellow-green	Pactum	Malleus	x
Deep-green	Mango	Vomer	y
Light-green	Lis	Agmen	z
Light-blue	Ilex	Comes	α
Crimson	Bolus	Sypho	β
Ochre	Limbus	Mica	γ
Purple	Solia	Arcus	δ
Leaf-green	Luca	Securis	ε
Turquoise	Ancilla	Vinculum	ζ
Dark-grey	Orca	Colus	η
Fawn	Nugæ	Saltus	θ
Smoke	Limus	Sceptrum	ι
Light-buff	Mala	Pallor	κ
Dull-purple	Sors	Vestis	λ
Rich-red	Lucta	Cortex	μ
Green-blue	Pator	Flagellum	ν
Burnt-sienna	Silex	Luctus	ξ
Sea-blue	Lorica	Lacus	ο
Peacock-blue	Passer	Aries	π
Deep-brown	Meatus	Hydra	ρ
Dark-pink	Onager	Anguis	σ
Dark	Lensa	Laurus	τ
Dark-stone	Pluvium	Cudo	υ
Silver	Spira	Cervix	φ
Gold	Corvus	Urna	χ
Deep-yellow	Via	Spicula	ψ
Dark-green	Calor	Segmen	ω

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APPENDIX E.
A Theorem in Four-space.

If a pyramid on a triangular base be cut by a plane which passes through the three sides of the pyramid in such manner that the sides of the sectional triangle are not parallel to the corresponding sides of the triangle of the base; then the sides of these two triangles, if produced in pairs, will meet in three points which are in a straight line, namely, the line of intersection of the sectional plane and the plane of the base.

Let A B C D be a pyramid on a triangular base A B C, and let a b c be a section such that A B, B C, A C, are respectively not parallel to a b, b c, a c. It must be understood that a is a point on A D, b is a point on B D, and c a point on C D. Let, A B and a b, produced, meet in m. B C and b c, produced, meet in n; and A C and a c, produced, meet in o. These three points, m, n, o, are in the line of intersection of the two planes A B C and a b c.

Now, let the line a b be projected on to the plane of the base, by drawing lines from a and b at right angles to the base, and meeting it in a′ b′; the line a′ b′, produced, will meet A B produced in m. If the lines b c and a c be projected in the same way on to the base, to the points b′ c′ and a′ c′; then B C and b′ c′ produced, will meet in n, and A C and a′ c′ produced, will meet in o. The two triangles A B C and a′ b′ c′ are such, that the lines joining A to a′, B to b′, and C to c′, will, if produced, meet in a point, namely, the point on the base A B C which is the projection of D. Any two triangles which fulfil this condition are the possible base and projection of the section of a pyramid; therefore the sides of such triangles, if produced in pairs, will meet (if they are not parallel) in three points which lie in one straight line.

A four-dimensional pyramid may be defined as a figure bounded by a polyhedron of any number of sides, and the same number of pyramids whose bases are the sides of the polyhedron, and whose apices meet in a point not in the space of the base.

If a four-dimensional pyramid on a tetrahedral base be cut by a space which passes through the four sides of the pyramid in such a way that the sides of the sectional figure be not parallel to the sides of the base; then the sides of these two tetrahedra, if produced in pairs, will meet in lines which all lie in one plane, namely, the plane of intersection of the space of the base and the space of the section.

If now the sectional tetrahedron be projected on to the base (by drawing lines from each point of the section to the base at right angles to it), there will be two tetrahedra fulfilling the condition that the line joining the angles of the one to the angles of the other will, if produced, meet in a point, which point is the projection of the apex of the four-dimensional pyramid.

Any two tetrahedra which fulfil this condition, are the possible base and projection of a section of a four-dimensional pyramid. Therefore, in any two such tetrahedra, where the sides of the one are not parallel to the sides of the other, the sides, if produced in pairs (one side of the one with one side of the other), will meet in four straight lines which are all in one plane.

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