An Introduction to The Geometry of N Dimensions

	Geometry of Four Dimensions by Henry P. Manning, 1914 (Dover 1956) (Archive, text, PDF) Pages photographed in links, and preface/table of contents/Bibliography OCR (minimal correction)
	CONTENTS (-2 to 0) PREFACE (-3) Introduction (1-22) Early References to Dimensions … 1 Beginnings of this Geometry … 4 Riemann, Grassmann … 6 Slow Recognition … 8 Extent and Variety of Applications … 10 Essential Part of Geometry … 12 Synthetic Method … 14 Foundations … 16 Collinear Relations … 18 Figure and its interior … 20 CHAPTER I: THE FOUNDATIONS OF FOUR-DIMEHSIONAL GEOMETRY (23-72) I. Points and Lines … 23 II. Triangles … 29 III. Planes … 35 IV. Convex polygons … 40 V. Tetrahedrons … 45 VI. Hyperplanes … 50 VII. Convex pyramids and pentahedroids … 55 VIII. Space of four dimensions … 59 IX. Hyperpyramids and hypercones … 63 CHAPTER II: perpendicularity and simple angles (73-104) Introductory … 73 I. Lines Perpendicular to a Hyperplane … 74 II. Absolutely Perpendicular Planes … 80 III . Simply Perpendicular Planes … 85 IV. Perpendicular Planes and Hyperplanes .… 90 V. Hyperplane Angles … 95 CHAPTER III ANGLES OF TWO PLANES AND ANGLES OF HIGHER ORDER (105-140) I. The Common perpendicular of Two Lines … 105 II. Point Geometry … 112 III. The Angles of Two Planes … 114 lV. POLYHEDROIDAL ANGLES … 126 V. Plano-Polyhedral angles … 133 CHAPTER IV: SYMMETRY, ORDER, AND MOTION (141-198) I. Rotation and Translation … 141 II. Symmetry … 146 III. Order … 153 IV. Motion in General … 167 V. Rectangular Systems … 179 VI. Isocline Planes … 180 CHAPTER V: Hyperpyramids, Hypercones, and the Hypheresphere (199-220) I. Pentahedrons and Hyperpyramids … 199 II. Hypercones and Double Cones … 204 III. The Hyper-sphere … 207 CHAPTER VI: (221-264) The Axiom of Parallels … 221 I. Parallels … 221 II. The "Hvperplane at Infinity" … 230 III. Hyperprisms … 235 IV. Double Prisms … 241 V. HVPERCYLINDERS … 253 VI. Prism Cylinders and Double Cylinders … 256 CHAPTER VII: Measurements of Volume and the Hypervolume in hyperspace (265-288) I. Volume … 264 II. HVPERVOLUME … 270 CHAPTER VIII THE REGULAR POLYHEDROIDS (289-326) I. The Four Simpler Regular Polyhedroids . … 289 II. The Polyhedroid Formula … 300 III. Reciprocal Polyhedroids and reciprical nets of polyhedroids … 303 IV. CONSTRUCTION OF THE REGULAR 600-HEDROID AND The Regular 120-hedroid … 317 TECHNICAI. TERMS (327-334) INDEX (335-348) Plates (349-351)
	PREFACE (-5) The object and plan of this book are explained in the Introduction (page 16). I had hoped to give some account of the recent literature, but this would have delayed work that has already taken several years. I have prepared a list of technical terms as found in a few of the more familiar writings, very incomplete, and, I fear, not without errors. The list may be of service, however, to those who wish to consult the authors referred to ; it will also indicate something of the confusion that exists in a subject whose nomenclature has not become fixed. It has been necessary for me to introduce a considerable number of terms, but most of these have been formed in accordance with simple or well-established principles, and no attempt has been made to distinguish thera from the terms that have already been used. I am indebted to the kindness of Mr. George A. Plimpton of New York for an opportunity to examine his copy of Rudolph's Coss referred to on page 2. I am also under many obhgations to Mrs. Walter C. Bronson of Providence, to Mr. Albert A. Bennett, Instructor at Princeton University, and to my colleagues, Professors R. C. Archibald and R. G. D. Richardson, from all of whom I have received valuable criticisms and suggestions. Many of the references in the first four pages were found by Professor Archibald ; several of these are not given in the leading bibliographies, and the reference to Ozanam I have not seen anywhere. HENRY P. MANNING. Providence, July, 1914-
	Technical Terms (327-334) In this list are some ot the temis of fout-dimensional geometry not used in the tiBtt and of B-dimensional geometry, also tenns equivalent to some that are used, and the principal abbreviations. In most cases a reference is added. For terms used or explained in the text, see Index. The iollowing are the authois most frequently mentioned, many of the refeiences being given in full in the preceding pages ; Cayley, Math. Papers (p. s) ; Clifford, Malh.Pap^sip.s); Cole (p. 14'); Dehn (p. 288); Enriitnes, EacyclapUie, vol. HI] (p. 15) ; Jouffret (p. ci) ; Loria (p. g) ; Pascal, Reperloriam der kSheren Mathe- taalik, Ger. trans, by Sehepp, vol. H, LdpMg, 1502; Poincat^, Frac. Leadon Math. So., vol. 33 {p. 13) ; Eiemann (p. 6) ; Schiafli (p. 21) ; Schoute (p. 5) ; Stringham (p. 289) ; Sylvester, 1851 and 1863 (p. s) ; Veronese, Gnatdiiise, etc. (p. g) ; Wil- son and Lewis (p. 11). A hlzell Zg rcgulare / perctb — net? Schoule II 20 4 See Zdi Allomorph a!h terph-c t o pol hedrons hav ng tl e same number of vert ces and edeea and the same number of facea of ea h knd that thej have amilarl of polj hedro da Scl o te 11 it 23 See Iso norpi Anki^el Aidcugelra im \nradms see Kigd Apothema (of a Mpercone) sla I iHgll S houte n 302 Ar4te erfjc Jnuiiret o5 Axe (of a piano polyhedral angle) tertex edge see Ka it Basis, base, of a pyramid, hyperpyra- mid, etc., Schoute, II, 35; — -raum, Schoute, II, 243; used also of the base of a linear system of spaces {e.g., the line commo the linear system formed from the equations of three hyperplani Ri), Schoute, I, 141. Bildraum, the apace of the figure of descriptive geometry, the space in which al! the different projections are placed together Schoute, I, 88, r24 fipiano (Ital ) R„ m R, Pascal 577-C^ C, C'S C C>2 C=" the SIX ¦egular fwh hedroids Jouffret 103 Case, ceil; hypercase corresponding term in space of fi\ e d mensions Poincar§, 278 See Jouffret 96 103. Cell, case, Cten raum SeilenrauiK Zell; (at a hjperpkne angle) Sckenkelraum cell, -hedrotd Maschke Am Jour Math., vol 18 181 Configuration Caylej Veronese (p s) Carver, Trans. Am. Math. So., vol. &, 534. Confine, polyhedrotd (^ n dimensions; face of a — , (_ii—i)-boundary; prime — , simplex rectangular prime — , with edg at on t equal and perpend la ne another; Clifford, 603 Cylinderraum, (Cy)i ( k dra a sions, hypercylind pha hwith spherical bas — zw Stufe, with cylinder f ba (At147) ; s-ter Stufe (,k + s)-ter Dimension, Cy[(Fo)t,RV(.s)],vlta,t a prism of this kind becomes when the bases, (Po)t, are no longer entirely linear (see Prisma) Schoute, II, 293. Kreiscylinde piwna-cylindrical hyp&'surface viith right directittg-drcle (Art. 148), Veronese, 557. [328] Decke, vertex-face, analogous to vertei-edge, Schoute, II, 4. Demi-, half-; — espace, half-hyptr-plane; Jouflret, 60. DiSdre d'espaces, kyperplane angle, Jouffcet, 60. Ditheme, surface, see Theme. Difiereutielte, gfiomStrie mStrique, restricted geometry, see Reslreint. Dreikant, trihedral angie, see Kant. Droite-sommet, vertex-edge, Joufftet,g2. Eben, flat; Ebene, plane; used by Pascal for an (» — i) -dimensional flat, E„_i, the same as Ra-i, 577 ; Dreieben, vierdimenaionale Eben- ttipel , plano-irihedral angle, S choute, 11, 8, 4. Eigentlidi, proper, not at infinity, UneigaiUich, Schoute, I, 30. Entendue, hyperspace (of four dimen- sions), Jouflret, I. EntgegengesetKte Punltte, opposite points (of the Double Elliptic Geometry), see Gegen-Funkle. Espace, hyperplane, Jouflret, 3. Face; — ^ deux dimensions, face angle (of a tetrahedroidal angle), haif-plane (of a piano-trihedral angle) ; — i trois dimensions, dihedral angle; — k quatre dimen- sions, hyperplane angle; Jouffret, 62-63. First, vertex-edge, Schoute, II, Flat noun and adjective), linear, konmloid (or omalotd), eben, flach. Fluchtpunkt (of a line), the point at infinity (the "vanishing point" ol perspective), Schoute, I, 2 ; Flucht- raum (of Rn), Schoute, I, 124. Fold: two-fold, «-fold, applied to figures, angles, boundaries, spheres, etc., to indicate the number of their dimensions, Stringham; see Manifold. A 4-fold relation in space of m- dimensions gives an (m — A) -dimensional locus, Cay- iey, VI, 458 ; see Omal. Funfzell, Z^, pentahedroid, Schoute, II,.¦e Zell.g„, line at infinity, Schoute, I, 2r. Gegen-Punkte or entgegengesetzte Punkte, opposite points (as in the Double Elliptic Geometry, see footnote, p. 215), Veronese, 237. Gegentlber, opposite (as in a tri- hedral angle each edge is opposite the face which contains the other two), Veronese, 449; see also Schoute, 1, 268. Gemischt, having both proper and improper points ; gemiachtes Sim- plex, Sj((i) ; Schoute, I, 29. Getade, right, Schoute, II, 108, 293; schief when not gerade. Gleichwinklige Ebene, isocline planes,, Veronese, 539. Grad von Pju^lelismus, Orthogonal- iiatsgrad, see Parallel, Orthogonal. Grenzraum, cell, Schoute, I, 10; Grenztetraeder, Schoute, II, 218. Half-, semi-, demi-, halb-. H6catonicosaSdroide, C", izo-ke- droid, Jouffret, 105, 169. -hedroid, Stringham. Hexacosi^droide, C™, Ooo-hedraid, Jouffret, 105, 169, Hexad6ca6droide, C", 16 -hedroid, Jouffret, 105, 128. Homaloid, fiat, represented by an equation or by equations of the first degree, Sylvester, 1851 ; writ- ten also omaloid; see Theme. Huf, polyhedron in which two faces are polygons of the ber of sides with ont mon [Fersp) and the remaining sides o£ one connected with the remaining sides of the other by two triangles and by quadri- laterals; in K„ a pnlyhedroid formed in a similar way. Partic- ular cases are the Prismenkei! and the Pyramidenkeil. Schoute, II, :!6, 41.43- Hundertzwanzigzell, Zim, l£0-he- droid, Schoute, II, 213. Hyper-: hyperlocus, Sylvester, 1851 pyramid, — pyramidal, — geome- try, ¦— theory, — ontological, Sylvester, Hyper. Hyper. >ne, kypercone de premiereine de premiere espJce, e; de seconds esptice, double cone; Jouffret, 92. HjTJercube, iessaract, oclaidroide, Acktseil, Masspolytop, Oktaschem (see schcm). Hypercylinder, CyJindemmm. Hj-perebene, ^_i (or E„.i, Pascal, 577). Hyperparallelopiped, paraUlllpipide d qtmlre dimetisions, Pardlelolop, Paralleloschem (see -schem). Hyperplane, Kneoid, quasi-plane, es- pace. plan, Hyperebene, Eaain Hyperplane angle, diddre d'espaces, Raurnmnkel. Hyperprism, Prisma. Hyperpyramid, Pyratmde. HypersoHd, Confine, Polytop. Hyperspace, 4-space, I'entendue, Hy- perraum (Pascal, 577). Hj^ersphere, quoH-sphere, Kugel- ravm, n-Spk&re, Polysphdre. (Hyper)'"' surface, of ^ ~ r dimen- sions in space of p dimensions; e.g., in space of five dimensions, hyper-hyper-surface. H. R. Greer, "Question 3503," Math. Questions Icosat^tcaSdroide, C^, s4'kedroid, Jouffret, 105, 137. Ideal, improper, uneigentlich. Inkugel, see Kugcl. Ineunt points, the points of a locus, Caylej-, VI, 469, In the same way he uses the espression, "tangent omals of an envelope." Inhalt, volume, kypervohime, etc, Schoute, II, 94. Iper- (Ital,), hyper-, Loria, 302. Isocline planes, plans d'angles 4gaiix, plans d une infimti d'angles, gleichmnUige Ebene. laomorph, isomorphic, allomorphic polyhedrons or polyhedroids are isomorphic when faces or cells which come together in one always correspond to faces or cells which come together in the other. Schoute. II, 2-2-23. 2^ Alio- Isomorphic geometries are different interpretations of the same ab- stract geometry (see p 15) Kt, Kugelraum, see this word. Kant, edge, Schoute, I, 9; Dreikant, trikedral angle, Schoute, I, 271, Vierkant, lelrahedraidal angle, Schoute, I, 267 ; n-Kant, Vieltant, Schoute, I, 270, 286; Scheitelkant, verlet-edge, Schoute, I, s68; Drei- kant zweiter Art, plano-lrihedrat angle. Axe, its VBrtei.-edge, Veronese, 540 544 reg dare Kant p ter Art Sdioute II 40 Kantenw nkel (ol a \ erkant) /a e angle Scho te I 68 Kegel Kre akegel erster \rt pla 10 antai hyper face of revel Ito zwe ter Art onical lypers rfa e of doTtble renolukon (Art. 112), Vero- nese, SS7- Kegelraum, iKe)t, of k dimensions, kypercone; — zweiter Stufe, double 33° cone, s-terSti>U{k + j)-Ler Dimen- sioE, Ke[{Poh, S(s)l what a pyramid of this kind becomes when the base, (Po)«, is no longer en- tirely linear (see Pyramide) ; tie Kegelraum (of any kind) is sphar- isch, if the base is spherical, gecade if also the centre of the base is the projection of the entire vertex- simplex upon the space of the base, regulaie if, further, the vertes- aimples is regular and its centre is the projection upoa its space of the centre of the base, schief if not gerade; Schoute, II, 292-293. Keil, dihedral angle, Veronese, 444; kyperplane angle (Keil von vier Dimensionen) , Veronese, 544 ; piano-polyhedral angle, Dehn, 57 j Prismenheil, Pyramidenkeii, see these words. Kiste, (Ki)i, of S dimensions, rec Umgtdar parallelopiped or hyper paralielopiped, Schoute, II, 94. Kontinuum, the aggregate of all solutions of an equatioa or of system of equations, locus ( spread, Schlafli. 6. Kreiscylinder, Kreiskegel, see Cylh der, Kegel. Kxeuzea, used of lines not parallel and not intersecting, also of other spaces ; often used with senkrecht Schoute, I, 43-44- Kngei, hypersphere (of any number of dunensions), Veronese, 592, With Veronese Kugel denotes the in- terior, Kugeloberflache the hyper- surface; see Kugekaum; netz net of regular polygons on s sphere, Schoute, II, 154; Ankugel, sphere tangent to the edges regiUar polyhedron, Inki^el, scribed sphere, Umkugel, dr, scribed sphere; Ankugelraum, An- radius, etc.; Schoute, II, iji, 199, 145- Kugeira the hypersurface not the mterior. In space of » dimensions it is of n — r dimen- sions, and he writes K„-,, but later he writes Kn, the subscript denot- ing the number of dim the space. See foot-ni Linear, Jlal, spaces as defined in Art. 2, represented by equations of the first degree. Lineoid, hyperplaae, Cole, 192 ; col- lineoidal. Keyset, Bidl. Am. Math. So., vol. g; 86. bsunfc, point, used by SchlSfli to denote a set of \alues of the lari ables satisf\ug given equations and then for anv set of \alues like Cauchy a analytical point (p 6) Schlafli 6 Lot perpendicular hne Schoute I 44 Letiieleck simplei j^ith alt angles nght angles (in Elliptic Geometry) Schoute I 4.7 Manifold space of any number f iimensiims tariety Manmgfalhg keit Grassmarm Riemann and others Mantel lateral boundar} of a hipcr pyramid, etc., Schoute, II, 35. Masspolytop, hypercube, Schoute, n, 93- Netz, AcIitzoUnetz, etc., Schoute, II, 34J. Normal, perpendicular; (o£ planes) absotalely perpendicular; atereo- metrisch normale Ebenen, per- pettdicidar in a hyper plane; Schoute, I, 70-72 ; see Senkrecht. Octa^drolde, C, hypercube, Jou£fret, 83, 118. Omal or omaloid (both noun and adjective), line, plane, etc., Unear, the same as homaloid. If the relation is linear or omal, the locus is a S-fold or (w - S) -dimensional (in space of m dimensions) omaloid. Omaloid is used absolutely to de- note the onefold or (tM — t)-dimeii- sional omaloid, Cayley, VI, 463. Operationsraum, Rti, the space in which all the figures considered are supposed to lie, as if there were no space of higher dimensions (see Art. a6), Schoute, I, 4 Opposite, gegeatiber, entgegengeselzte Order, Richlung, Sinne. Orthogonal, teilweisc, Orthogonali tatsgrad, Schoute, I, 49 see Senkreckt. Orthogonal figure, cuhe, hypacule etc., StringlM,m, j. P„, point at infinity, Schoute I 21 Parallel: planes are parallel von der ersten Art" if they have the same line at infinity, "von d' Kweiten Art" if they have only point in common and this point is at infinity, Veronese, 516; planes are "parall61es a premier mode ou incomplStement paralitica" it their lines at infinity have a single point in "parallSles suivaiit le deuxiSme mode oti completeraent paraHMes' if their lines at infinity coincide, Jouffret, 31 (we should that these two writers use "first" and "second" in oppositi teilweise parallel or halb parallel are terms used by Schoute ; space of more than four dimensions lower spaces may be "ein Viertel, zwei Viertel or drei Viertel parallel," etc., Schoute, I, 34. Parall3fipip6de i quatre dimensions, Jouffret, S2. Paralleloschem, hyperparallelopiped. 39' Penta6dtoIde, C^, pentahedroid, Jouf- fret, 105, 132. Perpendicular, absolutely, simply. Cole, 195, 198; simplement ou in- compl6tement perpendiculaires, absolument ou completement — -, Jouffret, 34; see Normal, orthog- onal, senkreclil. Plagioschem, spherical simplest, see -schem. lan (Fr ) plane used by Cayley in 184& for hjperplane, demi-plan for ordmari plane (I, 321), see also plane plans d'angles £gaur ou plana i une mfinitS d'angles, tiochne planes Jouffret, 77. Plane used by Cayley in five-di- mensional geometry for a space of four dmiensions : " In five- dimensional geometry we have : space surface subsurface, super- curve, curve, and point-system ac- cording as we have between the six coordinates, o, i, 2, 3, 4, or 5 equa- tions: and so when the equations are linear, we have : space, plane, subplane, superline, line, and point," IX, 79. Piano- trihedral angle, triHre de sec- onds espice, Dreieben (see Eben). Pianoid, hyperpkme, Wilson and Lewis, 446. Point. T point (Fr.), the space of r — 1 dimensions determined by t independent points (Art. 2); bi- point, line, d'Ovidio, Math. An- nalfn. vol. 12, 403-404. Pohierkant, the edges perpendicular to the cells o£ a given Vierkant (analogous to supplementary tri- hedral angle), Schoute, I, 26S. (Po)t, of k dimensions, Polytop; when the boundaries are not all linear (i'o)* is used, Schoute, II, 28, 29. Poiyschem, polykedroid, see -schem. PolysphS-re, see Sphere 0/ n dimen- Polj top (Po)t a limited pott on of ¦uiy space the boundaries (PD)t_i; It la generalh understood that the boundaries are hnear then it IS a pohhedioid Simpleitpolytop, one bounded bv simple^es such lu ifi IS the Tetraederpolvtop or Vierflachzell Sthoute II 1, Prisma (Pr)t of k dimensions, prtsm hyper prism eti, — z welter Stuie vith prisms for bases (Art 130) — s ter btufe ( + r) ter D mension Pr[iP ); R\{s)\ the bases paraUel {Po)k, the lateral elements parallel Rs, ¦ Scho ite II 3 39 pj ramidales Prisma li3T»erprism with pyramid* for bases Schoute II 41 Prismenkei! the E f formed when i hvpetprism is cut into tito parts by a hyperplane (or by an Rn-i) which intersects a base Sthoute, II 43 Pnsmoid m diraension'U pohhedrold bounded by two (Pojn 1 1 1 parallel Rn-\, and a. Mantel of simpli S(«), Schoute, II, 44. Proper, see Eigentlich. Punktwert of a space, the number of independent points in it (Art. 2) ; for iftf it is d + 1 ; Schoute, I, 12, Pyramide, {Py)t, of k dimensions, pyramid, hyperPyramid, etc. ; — zweiter Stufe, doable pyramid; i-ter Stufe (k + s)-ter Dimenaon, Py[iPo)t, S(s)l with a polyht droid, (-Po)t, for base and a verto simplex. Sis); Schoute, II, 3$- 36. _ Pyramidenlteil, the Hiif formed when a hyperpjTamid is cut into two parts by a hyperplane (or by an Rn-i) which intersects the base, Schoute, n, 41. Quadri^dre k quatre dimensions, tetraedrolde, letrahedroidal Uilgle; it has six "fates k deus drnien- sions," four "laces 4 trnis dimen- sions," and fom' "trifidres de seconds espSce," Jouffret, 63. Quasi-plane, Ra-i in Rn, Sylvester. 1863, 173; quasisphere, sphere of any number of di7?tensions, Clif- ford, 604, 605. Ra, linear space of » dimensions ; tn Schoute Ra is usually the Opera- tionsraum, Rt any space in Ra, JU+i " ^ii but Ra is used for Rm, "w'' referring to the Punlit- wert; Schoute, I, 4, 13. RVls). the lateral elements, R,, of a pr an or ylmder s ter St Je Scho te II 39 used alone for Rj I Bild Cylnder Kegel K gel etc see these words Raumwnkel hvPerpla e angle Schentelraume ts cells Schoute I 26S Leg on limitfie restricted repon Restremt (Fr ) restricted reg -on I K die glomStrte n etr g e d f ferentielle Enriq e 48 11 Rchtung SmK order Veronese 'r 456 49Rotaton snple double <_ole 01 09 Rotat on von den (? � i) ten Pote z when each po nt descr bes aii.p(seeArt 116 Th 1 and Art 147, Th. 2), Schoute, II, 297. So, Si, Si, etc., point, line, plane, etc. (Stem), Veronese, 509. S(rf), simplex mtk d vertices, Schoute, I, 10; Su(d), improper simple^c, ¦^dWi gemischtes Simplex, Schoute, I, 28, 29. ^t(P, q), Simploiop. Scheitel, vertex, Scheitelkant, vertex- edge, Schoute, I, 268. Schein, contow of a figure as seen II. 1 Lside point, Schouschem, -hedroid, used by Schl fl with qualiiyicg prefixes to den t various polyliedroids : Polyschen ig; paiallelOBdiein, hyperpaToll I piped, 1.2 ; in space o£ four dim a siona the six regular polyhedro ds are, Pentaschem, Oktaschem, H k kaidekaschem, Eibositetraschem Hekatonkaieikosasdietn, and Heia losioschem, 46-53; a sphSrisdi Polyschem is a polyhedroid be longing to the geometry of the sphere, it is a Plagioschem wh it has n boundaries, the same simplex, the single paits of th boundary of a spharisches Poly- schem are Perrischeme, 38. Schenkel (of an angle) side, Schea- kelraum (of a hypetplane angle) oM, Schoute, I, 368. SiMtA, ohiique, Schoute, II, 293. Sechshundertzell, Zisa, 600-kedroid ; Schoute, II, 213. Sechszehnzell, Zu, 16-hedToid; Sechs- zehnzellneta, Schoute, II, Selten (of a polyhedroidal angle), ebeaea ^, face angles, — vo Dhnensionen, triliedral angk Tonese, 544; (of a pentahedroid) faces and cdk, Veronese, 547. Seitenraum, the J?„_i of a simplex S{n + j), Schoute, I, 142- Semi-parallel, half-paralld. Senkrecht, perpendiculat; — kreu- zen, — schneidea; — zugeordnet, polar, used of P^; halb — -, in R4 used of planes with oai right angle (see Art. 69) ; in space of more than four dimensions lower spaces may be "ein Viertel, zwei Viertel, or drei Viertel —," etc., Schoute, I, 40-49. Veronese uses senkrecht or "senkrecht von der ersten Art" for planes abso- lutely perpendicular in K4, and "senkrecht von der zweiren Art" for planes half-petpendicular, 521. Smpl S{d^\|/ ( ( / dron, and in p lyhed d whose ver- 11 dp dent (Art. 2). ) th unpl in Ra with Uces S houte, I, 9-1O, \pl St am f me confine. Simpl philnsch pkerical ielra-" ii ut H, agi. S mpl xp lyt p b nded by sim- pl S h n 28. S mpl hsmi m mp! c, Sylvester, Q n 8 4 M Ih. Question Ed I T •¦ S3- S (p q) obtained by f m g t mpl xes S(p + 1) nd S(g + ) w th one vertex m common from a set of ^ +1; + 1 independent points (and so lying in a space fijj+j), and then letting each move parallel to itself over the other. A particular case is the doubly triangular ptism (see Art. 143), Schoute, II, 44. Sinne, sense, order, see Richlumg. Situs (Lat.), used by Gauss for direc- tion, of a plane, like SteUung, " Disquisitiones generales circa superficies curvas," Werke, IV, 319. Space, 3-space, 4-space, etc., for ifj, Ri, etc. ; the word space is used by Cayley for the highest space con- sidered, like Schoute 's Opera- lionsraum, see Plane. Sphere of n dimensions, hypm-spitere, n-spkere; called by Schlafli Poly- spMre or n-sphare, DisphSre, cir- cle, Trisphare, sphere, 58. Spharisch Cylinderraum, Eegelraum, Simplex, see these terms. Spitze, verlcx (point, line, or higher space), Veronese, 466, 557, 606. Spread, surface, kypersttrface, repre- sented by an equation or by a system of equations, used with a number to denote the number of its dimensions as 2-sptead, 3-spread, etc. ; see articles in [334] Am. Jour. Math, by Carver, vol. 31, Sisam, 33, Eisenhart, 34, Eisbnd, 35. Stellung, direction of a plane, space, etc., Schoute, I, a6, 75- Stern, consists of all points collineir with the points of a apace and a point outside of the space, used by Veronese to define the different spaces. — erster Art., S3, — zweiter Art., Ri, etc., Veronese, 424, 507 ; Schoute, I, igo-igj. Strahl, Hne, Haupt— , Schoute, II, i6ri Halb--, Schoute, I, 85. Straight, fiat; — vector, line-veclar, plane-vector, etc. ; straight 3-space, E-acUdsan, Lewis, "Four Dimen- sional Vector Analysis," Proc. Am. Acad. Afis and Sci.,\. 46; 166,173. Stumpf (of an angle), cbtuse; (of a prism), truftcaied, etc., Schoute, II, 43. "7- Subplane, subsurface, see Plane. Superfine, supercurve, see Plane. Siurface (Cayley), see Plane. Teilweise parallel, orthogonal or senlcrecht, see these terms. Tessaract, hypercube, Hinton, The Fourth Dimension, London, 1904, I5g. Tessaract belongs to a ter- minology in which the name of i figure designates the number of iti axes: pentact, a figure with five axes, penta- tessaract, a regulai t6-hedroid, T. Proctor Hall, Am Jow. Math.,vo\. 15: I7g. Tetrahedroidd angle, qwdriidre t quatre dimensions, tetraedroide Vierkani. Theme, spread; mono-, Une, 01 curve, di-, surface, keno-, system of jiiitnf;, homaloid theme,^i[j,' te used by Sylvester in 1851, TotalitSt, hyp»rspace, Schlafli, 6. TriSdre . de seconde espSce, pU trihedral angle, it has three "faces k deux dimensions," three 4 tiois dimensions," and threel "faces a quatre dimensions," Jouf- ftet, 62. s-i, the infinitely distant (uneigent- _ lich) part of Ri, Schoute, I, 22. Ui - Ut-i (see Ra), Schoute, I, 45. Umkugel, see Kugel. UnabhSngig, independent (points) , Veronese, 256. Uneigentlich, improper, ideal, at infinity; — Punkt, Flachlpunkt; also unendlich; Schoute, I, 2, ai. Variety, varUti, Enrique, 66 ; VarielSl, Schoute, I, 209 ; manifold. Vertex-edge, droiie-sommet, First, Scheildkant. Vierflachzell, polyhedroid in R^ bounded by tetrahedrons, Tetrae- derpalytop, Schoute, II, a8. Vierkant, vierdimensionale — , letra- hedroidal angle, Schoute, I, 267. Vierundzwanzigzell, Z54, 24-hedroid; netz; Schoute, II, 203, 242. Volumeinheit, unit of volume, kyper- volume, etc., Schoute, II, gs- Volum, InkaU, Schoute, I, 156. Weg, line or curve, Schlafli, 6. Wuikel, angle, Kanten-, Flachen-, Raum-, see these words; kor- perlich — von vier Kanten, tetra- hedroidal angle, Veronese, 544. Zs, 2s, 2iE, Z-ii, Z120, 2600, the regular polyhedroids, Schoute, II, 203, 213. ;U, ceU, polyhedroid, Schoute, II, 196. Schoute uses the term Zell alone for polyhedroid of four dimen- sions, but with prefixes (Achtzel!, Ftlnfzell, etc.) it seems to refer to the three-dimensional boundaries. See Jouffret, 96. Zweiflach, used of a polygon in space regarded as having two faces, Schoute, II, 148, i8z. Zweiraum, hyper plane angle, Schoute, II, 8; used also of a polyhedron regarded as having two sides in Ri, Schoute, .II, 1S6-
	INDEX (335-344) [335] Absolutely mdepeEdent points, 24. Absolutely perpendicular planes, 81 ;see Ferpendkvlarity. Abstract geometry and different interpretations, 14; ?^t Geometries., examples of different kinds. Alembert, d', on time as the iourth dimension, 4. Analogies, thdr assistance, 17; hyf>erplane angles and dihedral angles, gg; tetrahedroidal angles and trihedral angles, i2g; pianopolyhedral angles and polyhedral an^es, 133; isocline planes and parallel lines, 194; piano-prismatic hypersurfaces and polj^ons, Analytic development of the higher geometry, 6. Analytical point, — locus, terms used by Cauchy, 6. Angle, hyperplane, 95; see HyperPlwne angle. Angle of a half-line and a hyperplane, 80; of a half-plane and a hyperplane, 104; nunimum between two Angles at infinity, 232-234. Angles, the two between two planes, 122; the associated rectangular system, the associated sense of rotation, 181. AppUcations of the higher geometries ;to a problem in probability (Clifford), 5 ; geometries with other elements, lines in space, spheres, etc., 10; in connection with comples variables, n; to mechanics, ii;in proofs of theorems in three-dimensional geometry, 13. Aristotle on the three dimensions of magnitudes, i. Axes of a hypercube and a i6-liedroid, AxiomofPasch, 30; of parallels, 221 ; Parallels, axitnn of. Axioms of collinearity, restrictions in Elliptic Geometry, 25, Aas of a sphere in a hjT>etsphere, 211; of a regular polyhedroidal angle, 312. Axis-element of a double pyramid, 204 ; of a double cone, 205. Rotation; of a circle in a hypetsphere, 211 ; of a piano-cylindrical hypersurface of revolution, 257, Asis-pknes of a conical hypersurface of double revolution, 206., of hyperprism Base, prismatic, with tetrahedral ei Beginnings of geometry of mote tlian three dimenaons, synthetic, 4; analytic, 6. Beltrami, Hyperbolic Geometry on certain surfaces, 7 ; kinematics. i, Loria, Sommerville, 9, Bflcher, our use of the term "infinity," 230. Boundaries of a hypersolid ate threedimensional, 64. Boundary-hypetsurf ace of Hyperbolic Geometry, 95, J12. Cauchy, appUed the language of geometry to analyas, 6. Cayley, early papers, s; "Memoir on Abstract Geometry," 8. [336] Cell of a, haU-hyperspace, Gr. Cells of pentahedroid, 57 ; polyhedroid, 63 ; double pyramid, 67 ;hyperplane angle, 95; polyhedroidal angle, 126; piano-polyhedral angle, 133 ; priamoidal hyperaurface, 235 ; piano-prismatic hypersuriace, 24 a. Centroid, 202 ; see Gramty, centre of. Classes of points constitute figures, 19, 23- Clifford, problem in probability, 5 ;"On the Classification of Loci," 8; kinematics, 13. Closed sphere, passing out of, 79. Collinear relation, 23; distinguished at first from "on a line," 19, 27;the two axioms, ig, 25- ColUoear with a segment, 25 \ triangle, 33; tetrahedron, 49; pentahedroid, 58. Complex variables represented in space of four dimensions, 11, 21Q. Cone, double, 70 ; see Double cotie. Cott^iurations of points (Cayley, Veronese), 5, 8. Conical hypersurface of double revolution inscribed in a hypersphere, intersectii^ it in the same surface as the inscribed cylinder of doublt revolution, 363. Conical, sometimes used for hyper conical, 6g. Conjugate series of isocline planes 1S3; see Isocline planes. Continuity of points on a line, 28. Coolidge, list of systems of geometry. Corresponding dihedral i parallel planes, 223. Craig, kinematics of four dimensions, 13- Curvature, Riemann, 7 ; of the hypersphere, constant, 21S. Cyclical order, 28. Cylinder, double, 262 ; see Double cylinder. Cylinder, prism, 259 ; see Prism cylinder. Cylindrical, sometimes used for hypercylindrical or for plano-cylindrical, 258, 266, 284. Datboux, hedtated to use geometry of four dimensions, 9. Delcagons, the seventy-two in a 600hedroid, 323. Density, of points on a line, 28. ~ escartes, useof "sursolid," 2 ; knew the polyhedron formula, 300. Descriptive geometry of four dimensions, iS. Detemune," meaning in geometry, ig. Diagrams only indicate relarions, iS. Dihedral angle, its plane angle the same at all points, proof independent of the axiom of parallels, 97;in a hypersphere, its volume, 209. Dimensions, early references to the number, i ; differences in spaces of an even number and of an odd numlier, 14 ; of a rectangular hyperparallelopiped, 239 ; only three regular figures in a space of five dimensions, 317. Diophantus Directing-curi. hypersurfao hypersurfac ing-ci square-square, of a plano-conicai 71; piano-cylindrical 256; similar direct-'. 257. Directing-polygon of a piano-polyhedral angle, 137 ; plano-prismaric hypersurface, 243; similar dicecting-polygons, 245. Directing polyhedral angle of a pianopolyhedral angle, 135. Directing-polyhedron of a polyhedroidal angle, 126; prismoidal hypersurface, 235. Directing-surface of a hyperconical hypersurface, 69; hypcrcylindrical hypersurface, 253. Direction on a line, opposite direc- Distance between a point and a liyperplane, 78; the minimiiin between two lines, 105 ; in a hypersphere, 208; at infinity, 232. Distances between two great circles in a hypersphere, 217; two lines at iniinity, 235. Dodekahedrons, the net, four at a point, 324. Double cone, 70; vertex-edge, base, ekments, end-cones, 71 ; cut from a plano-conlcal hypersiu^ace, 7 2 ;dcculat, axis-element, right, isosceles, generated by the rotation of a tetrahedron, 205- Double cylinder, directing-curves, interior, right, generated by the directing-curves and their interiors, spread out in a hyperplane, 262 ;cylinder of double revolution, inscribed in a hypersphere, 263 ;relation to infinity, 264; volume, 267; hypervolume, 285; ratio to circumscribed and to inscribed hypersphere, 287. Double Elliptic Geometry, 215 ; see Elliptic Non-Eudidean Geometry. Double prism, the two sets of prisms, 246 ; right, regular, its cells spread out in a h5T)erplane, 248; interior, directing-polygoas, 249; generated by the directing-polygons and their intetioTS, cut into two double prisms, 250; doubly triangular, 251 ; hyperprisms with prisms for bases as double prisms, relation to infinity, 252 ; volume, 265 ;doubly triangular double prisms cut into six equivalent pentahedroids, 280; hypervolume, 282. Double pyramid, 66; vertex base, elements, end-pyramids, lateral faces, lateral cells, intersecdon with a plane, 67; with a hyperplane, 68; cut from a plano-poly hedrai angle, 138; axis-element, right, isosceles, regular, 204; in a hyperplane, 318. Double revolution, conical hyper-vith plan surface of, 197 ; nected, 207 ; cylinder of, 263, see Double cylinder; surface of, in a hypersphere, its importance in the theory of functions, 219; the intersection of the hypersphere with an inscribed cylinder of double revolution and with an inscribed conical hypersurface of double revolution, 263. Double rotation, 145 ; in the hyper- Doubly triangular prism, 251. Duality in the hypersphere, recipro- Edge Geometry, the elements halfplanes with a common edge, 138;applied to the theory of motion with two points fixed, 173. Edge of a polyhedron, how defined, 21 ; of a half-plane, sg ; hemisphere, 209. Edges of a tetrahedron, 45 ; pyramid, 55 ; pentahedroid, S7 i polyhedroid, 63. Elements, linear, of two planes, 61. Elements of geometry, points, 19, 23 ; of Point Geometry, ti3. Elements of a double pyramid, 67 ; hyperconical hypersurface, hypercone, 69; piano-conical hypersurface, double cone, 71 ; polyhedroidal angle, 126; plano-polyhedral angle, 134; piano-prismatic hypersurface, 242; piano-cylindrical hypersurface, 256. Elliptic Non-Euclidean Geometry ;due to Riemaim, 7 ; restrictions to the axioms of coUinearity, 25 ; the points of a line are in cyclical order g modihca n p 00 n m mndan bwnwhn h h m h n n perpendicular plane, T12; Edge Geometryiselliptic, 173; themost general motion in liyperspace, 174, 177; volumeof a tetrahedron, 311 ;the geometry of Uie h3T>erspliere is the Double Elliptic, 215, 217;difference between the Single Elliptic and the Double Elliptic, 215 ;poles oi a hyperplane in four dimensions, 217; space of constant curvature, 218; the geometry at infinity is the Single Elliptic, 333;hsTJervolume of a pentahedroid, 2S7 ; the possible nets of polyhedrons m Elliptic Geometry, 306;a regular polyhedroid can be mscribed in a hypersphere, 309 ; the possible nets of polyhedroids, 316;see also NEnriques, the foundations of geometry, 15 ; definition of segment, 2i. [338] Euler's name usually associated with the polyhedron formula, 300. Face angles of a polyhedroidal angle, 126. Face of ahalT-hyperplane, 54; hyperplane angle, 95. Faces oi a tetrahedron, 45 ; pyramid, 55; pentahedroid, 57; polyhedroid, 63; piano-polyhedral angle, 133Figure is regarded as a class oi pomts. 19, 23; belong to, lie m, 23. Five dimensions, oidy three regulai figures, 317. Foundations, different ss^tems, 15 . definitions and intersections ol elementary figures particidarly conFour dimensions, space of, 34; oui Fourth dimension as time, 4, 11 ; Tlu Fourth Dimemion Simply Explained, 9. Geometries of i, 2, 3, . . . b, , , . dimensions, 34. Geometries of different kinds, different interpretations of an abstract geometry, 14, 15; the geometry of hal£-hyperpianes with a common face, 99; Point Geometry, 113;EJg Geom try, 138; ss^tem of so In pi n s, 197; the hyperph system of parallel as mann An delmtitigsUkre. 7. y nt of, memoir by Sylest t; fa pentahedroid, 201 ; Green, problen ,6. Half-hyperplane, "half-hyperplane A BC-D," face, opposite halth}T)erplanes, 54; half-hsTwrplanes with a common face form a onedimensional geometry, 99. Half-hyperspace, " half-hyperspace ABCD-E," cell, oppoate halfhyperspaces, 62. Half-hypersphere, 310. Half-line, "half-line AB," "AB produced," opposite half-hnes, 28. Half -parallel planes, 224; see Parallelism Half-perpendicular planes, 8s; see Perpendtciilarily. Half-plane, "half-plane AB-C," edge, opposite half-planes, 39. Halphen, geometry of n dimensions, 8. Halsted, use of the terms "sect" and "straight," 25; proof that a line divides a plane, 37. Hathaway, application of quaternians to geometry of four dimensions, 13. Hatzidakis, kinematics of four di- Hekatonikosahedroid, 326 ; s hedroid. Hexadekahedroid. 291 ; s hedroid. Hexakosiolhedroid. 321^ se hedroid. Hilbert, definition of Hyperbolic Non-Euclidean Geometry : planes with parallel elements, 95, 112 ; their common perpendioilai' plane, 112; boimdaryhypersurfaces, 95, 112 ; translation along a line, 146, along boundatyhyperplane around parallel axes, 172; the most general motion in hyperspace, 174; rotations around parallel axia-planes, 178; pentahedroids which have no point equidistant from the five vertices, 199 ;the possible nets of polyhedrons, 306 ; a regular polyhedroid can be inscribed in a hypersphere, 309;the possible nets of polyhedroids, 316; see also Non-Eudideim Ge- Hypercone, 69 ; intersections, interior, with a cone for base, different ways of regarding it, 70;axis, generated by the rotation of a half-cone, 204; lateral volume, 266 ; lateral volume of a frustum, 267 ; hypervolnme, 284 ; hypervolume of a frustum, 285 . Hyperconical hypersurface, directing-surf ace, elements, 69 ; of double revolution, 197 ; its plane elements, so6; its interior all coimected, 307;intersection with a hypersphere, spherical directing-surface, 220. Hypercube, the diagonal twice the edge,its cells spread out in a hyperplane, two forms of projection, 240; as a regular polyhedroid, cedprocal nets, 290; reciprocal relation to the i6-hedroid, 292 ;diagonals and axes form three rectangular systems, 293 ;associated 24-liedroid, 295. Hypercylinder, 254; lateral hypersurface, interior, spherical, generated by a rectangle, by don of a half-cylinder, 255; with cylinders for bases, different ways of regarding it, 255, 261 ; latei volume, 166 ; hypervolumc, 284. [339] Hypercylindrical hypersurface, directing-surface, elements, 353; interior, sections, 254; relation to infinity, 256. Hyperparallelopiped, its diagonals all bisect one another; rectangular, its dimensions, the square of the length of its diagonal equals the sum of the sqtmres of its four dimensions, 239 ; as a double prism, 352; hypervolumc when rectangular, 271, when oblique, Hyperplane, 24; "hyperplane ABCD," 50; figures wbidi determme it, only one cont^s fourgiven non-cqplanac points, 51 ;ordinary space a hyperplane, 5s ;divided by a plane, 53; intersection with a plane, 60 ; intersection of two h5T>erplanes, 52, 60; opposite sides o£ a hyperplane, 62;at infinity, 231. Hyperplane angle, face, cells, interior, divides the rest of hyperspace, 95 ; mtersecUon with a hyperplane perpendicular to its face, 96;plane angle, gb, gS; two hyperplane angles are congruent when they ha\e two equal plan 96 ; tile plan all points ol magnitude, ; angle ¦ , the 51 right hyperplane angles, the sum of two, 98 ; analogous to a dihedral angle, in the geometry of half-hj'perplanes, measured by the plane angle, 99 the bisecting half-hyperplane, 100 Hyperprism, lateral cells, etc., 237 its cells spread out tn a h}^rplane 238; with prisms for bases, different ways of regarding it, 252; lateral volume, 265;gruent and equivalent hyperprisms, 271 ; hypervolume when the bases are prisms, 273 ; when the bases are tetrahedrons, 274; hyperprism with tetiahedial ends, prismatic base and vertex edge, 274; hypervolurae of any hyperprism, 175. [340] Hyperp5T:amid, 63 ; base, interior, sections, 64; with a pyramid for base, difierent ways of regarding it, 66 ; cut from a polyhedroidal angle, 127; axis, regular, 203; lateral volume, of a frustum, 265 ; hypervolume, 276 ; frustum cut into pentahedroids, hypervolume, 278. Hypersolid, the interior of a pentahe droid as a hypersolid, 62 ; boun daries are three-dimensional, 64 hypervolume, ratio of two, equivalent hypersolids, 270. Hyperspace, term used to denote th space of four dimensions, 60; di vided by a hyperplane, not di vided by a plane, 62. Hypersphere, great spheres and small spheres, 207 ; great circles and small circles, 20S ; distance in hypersphere, tangent hyperplanes 20S; spherical dihedral angle, its volume, 2og ; tetrahedron, th sixteen associated tetrahedrons their volumes, 210; aais and poles of a sphere, axis-plane and polar dr cleof adrcle, 211; theirmotioni a rotation of the hypersphere, 212 duality, reciprocal figures, 212 the geometry of the hypersphere as an independent three-dimensional geometry, 212; it is the Double Elliptic Non-Eudidean Geometry, ajj; the Point Geometry at centre, 216; the distances between two great cirdes, parallel great drdes, 217; proof from Point Geometry that the geometry of the hypersphere is the Double Elliptic, 217; rotation, double rota' screw motion, paralld motion., section with a conical hypersurface of double revolution, "220; intersection with an inscribed cylindei of double revolution a surface of double revolution, 263 ; volume, 267 ; hypervolume, 28;. Hypersurface, 69 ; of a pentahedroid, 62. Hypervolume, 270; of a rectangular hyperparallelopiped, 271 ; any hyperparallelopiped, 272 ; hyperprism with prisms for bases, 273;with tetrahedral bases, 274; any hyperprism, 275 ; hyperpsramid, "76frustum "78double prism bo jl dncal d ra^al Ij p f 84 hype ph f th h -pe ph _ b d d dr bl jl d d lip 87 hype Idal pom 1 3 Ik sa t t hd d f)6 bed pe t h po hi h h 1 mfim y m 1 h p rpl 3d ta ngl 3 dih d I angl 3 th geom t y t mfin t h S gt Ellpt g lizat mad poss bl b th ^e! 33 Im la fin y f h p hypersurface, 241 ; hyperparallelopiped, 341, 253 ; pJano-prismatic hypersurface, double prism, 252 ;hypercylindrical hypersurface, 256 ;plano-cylindtical hypersurface, prism cylinder, double cylinder. Interior of a figure as distinguished from the figure itsdf , 20 ; see Segment, Tnangle, Polygon, etc. Intersect, intersection, 23. Isodinal angle (Stringham), 125. Isocline planes, 123, 180; have an infinite number of common perpendicular planes, 123, 182; series of isocline planes, 183 ; conjugate series, 183 ; the two senses in which planes can. be isocline, 184; conjugate series isocline in opposite senses, 185 ; through any hne pass two isocline to a given plane in opposite senses, i85; two intersecting planes are isocline to two pairs of planes, 187; when two ¦e isocline ti nthesa e the pendicular planes which they have with the given plane form a constant dihedral angle, 188; when two planes are isochne to a given plane in opposite senses, there is only one pair of common perpendicular planes, perpendiculM" to all three, 189; two planes isocline to a third m the same sense are isocline to each other in this sense also, 190; poles and polar series, all the planes of two conji^ate series are isocline at an angle of o a smgle pair of planes, the 193; system of planes isocline in a given sense any two series have of planes in common, 193. plane intersects two isocline planes in lines the angles are equal, analogy to parallel !, I94-I a system of p' isocline in a given sense forms two-dimensional geometry, 19', "ordinal" and "cardinal" system (Stringham), 198; a series ci hypersphere (with centre at ( a surface of double revolution with equal ladii, 220; projection from one upon the other of two isocline planes produces similar figures, 229. Isocline rotation, every plane retnams isocline to itself, tg6. Isosceles double pyramid, 204; double cone, 205. Kant, reference to the number of dimenaons of space, 3. Keyser. the fotw-dimensional geometry of spheres, 1 1 ; our intuition of hyjierspace, iG ; the angles of planes, 114; proof that two planes have a common perpendicular plane, iiS. Kinematics of four dimensions, articles by Clifford, Beltrami, Cra^, Hatzidakis, 13. Kwietnewski, complex variables represented in space of four dimen- Lagrange, time as the fourth dimen- Lateral edges, faces, cells, hypersurface, etc., see Pyramid, Hyperpyramid, Hypercone, etc. Layer, 241. Left, right and left in a plane, 154. Lewis, G. N., Wilson and Lewis on relativity, 12. Line, 24; "line AB," only contains two given points, 36; properties of its points, order, 27; density and continuity, 28; opposite sides in a plane, 38; at infinity, Z31, Linear elements of two planes, 61. Lobachevsky, Pangeomelry, 221. Loria, bibliography, g. McClmtock, interpretations of NonEuclidean Geometry, 10. Methods of studying the higher geometries, 12. Minkowski developed application to relativity, 12. Mebius, symmetrical figures, 4. Moore, E. H., definition of segment, Moore, R. L., properties of points on a line, aS; axioms of metrical geometry, 74. 342 iNi: More, Henry, spirits are (ourdimensional, 3- Motion in a plane does not change order in the plane, i6o; in a hyperplane, does not cliange order in the hyperplane, 163 ; in hyperspace, does not cliange order in hyperspace, 166; in, a plane vvitli one point ftsed, 167; tlie most general, 168; in a h}T>erplane, two equivalent if equivalent for three noE-coUinear points, motion with onepoint fixed, i6g; every motion in a hyperplane equivalent to a motion of a plane on itself or to a screw motion, 170; in hyperspace, two equivalent if equivalent for four with two points fixed, 173; with one point fixed, 174; every motion equivalent to a motion in which one on itself, 174. H dimensions, space of, 24. J7-hedroid and SJV-hedroid, 311; the same as 120-hedroid and 600hedroid, 321. Nets of hypercubes, 2f)o; 24-hedroids and 16-hedroids, 298 ; spherical pol}^ons, 303; polyhedral angles, polyhedrons, 304; polyhedroidal angles, 313; polyhedioids, 314;net of twenty tetrahedrons at a point, 317 ; four dodekahedrons at a pomt, 324. Non-Euclidean Geometry used in the theory of relativity, 12; not particularly considered in this text, " " s of 146; spherical poisons, 303 ;EUipUc Geometry, Hyperbolic Geomelry, ParalUlisM, RestricUans. NoH-Eudidean Geometry by the au-Nother, birational transformations, 8. Oppoate directions on a line, 27;half-lines, 38; in cyclical order, 2g ; sides of a line in a plane, 38; half-planes, 39; sides of a plane in a hyperplane, halt-hyperplaiies, 34; sides of a hyperplane, half -hsTJersp aces, 62; elements of Point Geometry, 113; elements of Edge Geometry, 13S; points in a hypersphere, no, 2r3. Order of points on aline, 27; "order AB," IS3 ; Veblen'a use of the term "order," 27; cyclical, 28; order in a plane, 153; two fundamental principles, right and left sides of Unes tlttoughapomt, 154; withrespect to a trianfele "order ABC," 156;unchanged by any motion in the plane 15b mdependent of any hyperplane 162 ; order in a hyperplane 161 with respect to a tetrahedron order A BCD," 162; unchanged by any motion in the hyperplane, 163; order in hyperspace, 164; with respect to a pentaiedroid, "order ABCDE," unchanged by any motion, 165; order in Point Geometry, 179, Ovidio, d', projective geometry, 12. Ozanam, higher products imaginary, Paciuolo, use of "prjmo relato," etc., Pangeometry, term used by Lobachevsky, i2s. Parallel axiom, proofs which do not depend on it, 77, 97, 105, ijS, 136, sion, 37, 78, 79, i03, 108, ir2, 138, 139, 153, 160; see EUiplic Geomdry and Restrictions, Par^elism taken up after many other subjects, 19 ; parallel great circles in a hypersphere, 217; parallel motion in a hypersphere, 218;axiom of parallels, 221 ; paraEel lines and planes, 221 ; half -parallel planes, their common perpendicular lines and planes . their n ment= 5 lues and pi. _ allel to a luperpUne par-dlel hvperplanes 2 Pasch Axiom 30 Pentahedroid edges idcei ceUi; 57 inter ection w th a plane 57 Oo interior coUinear with 58 pass ing from cell to cdl 59 mteriection with a line 60 the five half hy perspaces and the interior, 62 ; spread out in a hyperplane, 68;' the point equidistant from its vertices, ig(>; the point equidistant from its cells, 200 ; its centre of gravity, 201 ; pentahedroids with corresponding edges equal, 202 ; hypervolume in elliptic hyperspace, 287 ; regular, 203, 289 ;radii of drcumscribed and inscribed hyperspheres, reciprocal pentahedroids, sSg. Perpendicularity: lines perpendicular to a line at a point, 74; perpendicular hue and hyperplane, 75;planes perpendicular to a line at a point; two lines perpendicular to a hyperplane lie in a plai lines perpendicular to a plai point, 80; absolutely perpendicular planes, Si ; if two planes intersect in a line, their alwolutely perpendicular planes at any point of this Une intersect in a line, 83 ;two planes absolutely perpendicular to a third lie in a hyperplane, 83 ;perpendicular planes, simply perpendicular, half -perpendicular, perpendicular in a hsTJerpla a plane perpendicular to one of two absolutely perpendicular planes at their point of intersection is perpendicu^ to the other, 85 ; a plane intersecting two absolutely perpendicular planes in lines is perpendicular to both, 86 ; the common perpendicular planes of 'X 343 two planes intersecting in a line, 8;; perpendicular planes and hyperplanes, perpendicular along a line, go , the planes perpendicular or absolutely perpendicular to planes lying in the hyperplanes, gi ;lines lying in either and perpendicular to the other, 52 ; planes with linear elements all perpendicular to a hyperplane, Q4; perpendicular hj'ptrplanes, g8, lines or planes lying in one and perpendicular to liie other, toi ; the common perpendicular line of two lines not in one plane; lines with more than one common perpendicular line, 10&; the common perpendicular line of a line and plane; the common perpendicular plane of two planes n hich ha^ e a common perpendicular h^'p^plane, in ;the common perpendicular planes of two planes which intersi-ct onh in a point 118 planes with an infinite number of common perpendicular planes, iig, 182. Plan and object of this booit, 16, 73. Plane, 24; "plane ABC," only one contains three given non-collinear points, 35 ; divided by a line, 37 ;two planes with only odd point in two in a hyperplane, 53; opposite sides of a plane in a hyperplane, 54; intersection witha hyperplane, 60; linear elements of two planes, 61 ; absolutely perpendicular planes, 81 ; perpendicular, 85 ;see PerpmiictdaTity; if two not in a hyperplane have s pendicular line, they hav perpendicular hyperplane, g4 ;isocline planes, 1 23 ; see IsocUne planes; planes at infinity, 231. Plane ai^le ot a hyperplane angle, g6 ;see Hyperplane angle. Piano-conical hypersurface, vertexedge, directing-curve, elements, a with a hyperplane, 71 ;see Double cone. [344] Plano-cylindiical hj-persurface, direcUng-curve, elements, 256 ;interior, right directing-curvea, similar directing-curves, hjpersurface of revolution, axis-plane. 2S7 '> intersection, with plano-prisraalic hypersurface. the set of cylinders, 258; intersection of two pianocylindrical hypersurfaces, i6i ; the surface of intersection, 262 ; relaUon to infinity, 264; see Prism cylinder and Double cylinder. Piano-polyhedral angle, faces, vertexec^e. cells, 133; elements, simple, convex, its hyperplane angles, 134 ;vertical piano-polyhedral angles, 134; directing polyhedral angles, polyhedral angles which are right sections, 135 ; theorems proved by means of them, 136; dhectingpolj^ons, 137; Ulterior, 139, 140. Plano-prismaUc hypersurface, 341 ;faces, ceils, elements, simple, convex, sections, 242 ; directing-polygons, 243; triangular, similar directing-polygons, 245 ; intersection of two piano-prismatic hypersurfaces, the two sets of prisms, 246; intersection with a planocylmdrical hypersurface, 258; see Double prism and PHsfn cylinder. Piano-trihedral angle, r34. PHlcker, the four coordinates of a line Poincar^ avoided use of geometry of four dimen^ons, 10; on anal}^is situs, 12 ; double integrals, 219. Point, 23 ; independent and absolutely independent points, 24; at infinity, 231. Point Geometry, 113; theorems in regard to perpendicular planes stated in the language of Point Geometry, 114; applied to study of the aisles of two plai 114; piano-polyhedral angles and polyhedroidal angles, 136; Point Geometry of a rectangular system, 179 ; in the theory of isocline planes, 180; the same as the geometry of the hypersphere, ai6. Poles and polar series of isocline planes, 193; poles of a sphere and polar cirdes in a h5T)ersphere, 211 ;their motion m a rotation, 212 ; of a hyperplane in Elliptic Geometry of Four Dimensions, 217. Polygon, sides, diagonab, cyclical order, 40; simple, convex, intersection withaline,4i; dividedinto two polygons, 42; interior, 44; the half-planes and interior, 45. Polyhedral angle, 133 ; convex, can be cut in a convex polygon. 137;nets of polyhedral angles, 304. Polyhedroid, edges, faces, cells, interior, 63; regular, definition, 289;can be inscribed in a hypersphere, the associated net of hyperspherical polyhedrons, 309 ; reciprocal polyhedroids, 310; its polyhedroidal angles are regular, 313; nets of polyhedroids, 314; list of posable nets, 315; the nets in each of the non-Eudidean geometries, 316. Polyhedroid formula, 302 ; proved by Schlafli, 22 ; by Strmgham, 302. Polyhedroidal angle, elements, directing-polyhedton, face angles, polyhedral angles, cells, mterior, 126; vertical polyhedroidal angles, 127; regular, axis. 312; the polyhedroidal angles of a regular polyhedroid are regular, net, reciprocal Polyhedron, 63 ; regular scribed in a sphere ; tii net o£ spherical poisons ; reciprocal polyhedrons, 303; nets, 304;list of possible nets, 305 ; the nets m each of the non-Euclidean geometries, 306. Polyhedron formula, Descartes, Popular interest in the fourth dimen- Power'of a number :n cirlj algebra 2 Pnsm the two sets of pnhma in a doiible prism 146, »ee Double Prism cylinder, the set of c\!inders right, regular, spread out in a hy perplane, 359, the directing pol\ gons and the directing-Lurvea generated b} them and their interiors, cut into two prism cvl inders, 260, triangular hyper cy hnder with c\ Imders for bases as a pnsm cylinder gener'ited by the rotation of a pnsm, 2O1 volume 267 , hypervolume, 285. Prismatic base of a hyperprism with tetrahedral ends, 274. Prismoidal hypersurface, directingpolyhedron, e^^es, faces, cells, interior, sections, 135 ; with parallelopiped for directing-polyhedron, 336. Projectingline, 78; plane, 84; factor for area, J2g. Projection upon a hyperplane, 78;of a line is a line or a part of & line, 79, 84; upon a plane, 81; a line and its projection upon a plane not coplanar, 84; of a plane upon a hyperplane, 103; from a plane upon an isocline plane produces simitar figures, 22g. Projective geometry, points of a line in cyclical order, 2g. Ptolemy, the number of distances, i. Pyramid, base, edges, faces, intersection with a plane, 55 ; double pyramid, 66 ; see Double pyramid. Quaternions applied to the study of geometry of four dimensions by Hathaway and Stringham, 13. Ray or half-line, 28. Reciprocal figures in a hypersphere 213 ; pentahedroids, 289 ; hy percube and 16-h.edroid, 293 ; 24 hedroids, 297 ; polyhedrons, 303 polyhedroids, 309 ; nets, see Nets, Rectang ilar 1 perparallckp pei 230 Rei,taneular s\stem S 8q as a tetrahedroidal angle 128 wajs in which it IS congruent to itself 170 the different arrangements notation used in studying the an5,Ies of two planes 180 three beljiigmg to the hypercube and 16 hedroid 93 Regular hj perpyramid, 203, pen tahedroid, 203, 289; double pyramid, 204; hyperprism, 237; the hypercube is regular, 240, 289;regular double prism, 24S ; prism cylinder, 259; pjoiyhedroid, 289;octahedroid (hypercube), 290; 16hedroid, 291 ; 24-hedroid, 295 ;a regular polyhedroid can be inscribed in a hypersphere, the assodated net of spherical polyhedrons, 309; regular polyhedroidal angle, 312; in space of five dimensions only three regular figures, 317;6oo-hedroid,3i7; 120-hedroid, 324. Relativity and the fourth dimension, Restricted geometry, 19. Restrictions to the second axiom oi coliinearity in Elliptic Geometry, Edge Geometry, piano-polyhedral angle is restricted, 139; in Point Geometry a rectangular system is restricted, 179; restrictions due to omission of the axiom of parallels, see Fardlel axiom; see also EUiptk Geometry. Revolution, surface of double revolution in a hypersphere of importance in the theory of functions, 219;see Double revolution. Rieiuinn on the foundations o£ geometry, Elliptic Geometry due Right and left in a plane, 154; see Rotation in a plane, in a hyperplane, figures remain invariable, 141 ; in hyperspace, the axis-plane, 142;^ures remain invariable, rotations around absolutely perpendicular planes commutative, 143 ;double rotation, 145 ; right and left inaplane,iS4; whentwc are equivalent to a singli hyperpkne, 171, 178; isocline, 196; of the hypersphere, the axis-circle and the circle of rotation, double rotation, screw motion, parallel motion, 218. Rudolph, use of terms representing powers of a number, 2. in hyperspace, the axis-plane, 142;^ures remain invariable, rotations around absolutely perpendicular planes commutative, 143 ;double rotation, 145 ; right and left inaplane,iS4; whentwc are equivalent to a singli hyperpkne, 171, 178; isocline, 196; of the hypersphere, the axis-circle and the circle of rotation, double rotation, screw motion, parallel motion, 218. Rudolph, use of terms representing powers of a number, 2. [346] SchlSfli, multiple mtegrals, 6; multiple continuity, 22. Schotten, definitions of segment, 21. Schonte, Mekrdimmisionale Geomeirie, 9; sections of a simplex, 14;descriptive geometry, 18 ; different kinds of perpendicularity, Sg ; the polyhedroid formula, 302. Schubert, eaumerative geometry, 1 2 . Screw motion, the translation and in the hypersphere, 218. Sect, used by Halsted for segment, 25. Sections, study of a figure by them, 18; divide a figure mto completely separated parts, 65, 245 ; of a pentahedroid, hyperpyramid, etc., see these terms ; of a piano-polyhedral angle, 135. Segment as d^ned by different writers, Hilbert, Enriques, E. H. Moore, Veblen, Schotten, 21 ;definition, "segment AB," collinear with, 25; interior, 28. Segre, the use of geometry of four dimensions, 10, 13. Semi-parallel, the same as half-parallel, 224. Separate, of cyclical order, 29, Series of isocline planes, 182 ; see Isocline planes. Sides of a polygon, how defined, 21 ;triangle, 29; polygon, 40; of a line in a plane a property of the plane, of a plane In a hyperplane a property of the hyperplane, 162. Similar figures produced by projection from one upon the other of two isocline planes, 229; directmgpolygons of a piano-prismatic hypersurface, 245 ; durecting-curves of a piano-cylindrical hypersurface, 257. Simplidus, reference to Aristotle and Ptolemy, 1. Solid, the interior of a tetrahedron as a solid, 54- SommervUle, bibliography, 9. Space of I, 2, 3, . . , B, . . . dimensions, 24; of four dimensions, 24, 59 ; ordinary space a hyperplani Spheri s geometry is elliptic closed, passing out of, 79 ; in a hypersphere, 207, Spherical sometimes used for hypetspherical, spherical dihedral angle, tetrahedron, 209. Stifel regards the higher powers as "against nature," 3. Straight, used by Halsted for line, 25. Stringham, application of quaternions to geometry of four dimensions, 13 ;on the angles of two planes, "4;use of the term "isoclinal angle," 125; "ordinal" and "cardinal ss^tems," of isocline planes, 19S;gave a proof of the polyhedroid formula, 302, Strip, the portion of a plane between two parallel lines, 235. Surface, the tetrahedron as a surface, 54 ; of double revolution in a hypersphere, 219; set Double revoluiioti. Sursolid in early algebra, sursolid Symmetrical figures congruent, MObius, 4, 149; defined as those that can be placed in positions of symmelcy with respect h 164. ipla« ahy- Symmetry, 146; in a pli perplane, 14; ' ' rotations whicli leave tlie symmetrical relation undisturbed plane, 147, in a hyperplane, which bring into coinddence figures symmetrical in a plane, 147, in a hyperplane, 148; symmetrical figures of ordinary geometry are really rongruent, 148; symmetry itt a hyperplane with respect to a point can be changed by rotation to symmetry with respect to a plane, 147 ; figures symmetrical in hyperspace with respect to a point or plane are congruent, 149; symmetry in hypecspace with respect to a line can be changed by rotation to symmetry with respect to a hyperplane, 150; in every kind of symmetry corresponding segments and aisles are equal, 153; figures symmetrical in a plane cannot be made to coincide by any motion in the plane, i5o; figures symmetrical in a hyperplane cannot be made coincide by any motion m the hyperplane, 1631 figures symmetrical in hyperspace cannot be made coincide by any motion in hyperSynthetic development of the highi geometry, 4; advantages of the synthetic method over the anal Tetrahedroidal angle, 127; two with corresponding face angles equal, i2g; the bisecting half-hyperplanef of its hyperplane angles have e Tetrahedron, edges, faces, intersection with a plane, 45 ; with a line, 48,33; interior, collinear with, 49 the four half-hyperplancs and the interior, 54; correspondeno 347 o, isS; spherical, its volume, D ; net of tetrahedrons, twenty a pomt, 317. Time as the fourth dimension, Lagrange, d'Alembert, 4; relativity, Translations along a Ime, figures rerawti invariable, 145; different kinds of translation in Non-Euclidean Geometry, 146. Triangle, sides, cyclical order, 29 ; intersection with a line, 30, 37 ; ulterior, 30; collinear with,32; the three half-planes and the interior, 39. Veblen, definition of segment, 21 ; use of the term"order," 27; iJie properties of points on a line, 28; Axiom of Pasch, 29 ; axioms of metrical geometry, 74. Vector analysis of Grassmann, 8. Veronese, Fondamenii, 5, 8, g; application of the higher geometry to theorems of ordinary geometry, 13 ;use of the dements at infinity, 230. Vertex-edge of a double pyramid, 67 ; double cone, piano-conical hypersurface, 71 ; piano-polyhedral angle, 133 ; hyperprism with tetrahedral ends, 274. Vertical polyhedroidal angles, 127;can be made to coincide, 167 ;piano-polyhedral angles, 134; cannot be made to coincide, 166. Vieta, use of terms representing powers of a number, 2. Volume of a spherical dihedral angle, 209 ; spherical tetrahedron, 210 ;at infinity, 233 ; lateral volume of a hyperprism, hyperpyramid, frustum of a hyperpyramid, 263 ;hypercylinder, hypercone, 266, frustum of a hypercone, 267 ; volume of a double prism, 265, 266, prism cylinder, double cylinder, 267 ;hypersphere, 267. [348] Wallis on the geometrical names of the higher powers, 3. Wilson and Lewis, relativity, 12. 16-hedroid or hexadekahedroid, 291 ;axes, reciprocal relation to the hypercube, 2^2 ; diagonals of the hypercube and the i6-hedroid form threerectangularsystems, 3g3; the assocjited 24-hedxoid, 295 ; redprocal nets of i6-hedtoid5 and 24-hedroid3, 298, 120-hedroid or ikosatetrahedrold, associated with a hypercube and a i6-hedroid, 295 ; reciprocal 24-hedroids, 297 ; reciprocal nets of i6-hedroids and 24-hedroids, or hexakosioihcdroid, 1 of the first half, 317;completed, 321; its seventy-two dekagons, 323 ; table of its parts, 324
	INTRODUCTION (1-22) [1] The geometry of more than three dimensions is entirely a modern branch of mathematics, going no farther back than the first part of the nineteenth century. There are, however, some early references to the number of dimensions of space. In the first book of the Heaven of Aristotle (384-322 B.c.) are these sentences : “ The line has magnitude in one way, the plane in two ways, and the solid in three ways, and beyond these there is no other magnitude because the three are all,” and “ There is no transfer into another kind, like the transfer from length to area and from area to a solid.”* Simplicius (sixth century, a.d.) in his Commentaries says, ” The admirable Ptolemy in his book Chi Distance well proved that there are not more than three distances, because of the necessity that distances should be defined, and that the distances defined should be taken along perpendicular lines, and because it is possible to take only three lines that are mutually perpendicular, two by which the plane is defined and a third measuring depth ; so that if there were any other distance after the third it would be entirely without measure and without definition. Thus Aristotle seemed to conclude from induction that there is no transfer into another magnitude, but Ptolemy proved it.” t Aristoteles, De Caclo, ed. Prantl, Leipzig, i88i, 26Sa, 7 and 30. t Simplicii in Aristotelis De Caelo Commentaria, e [2] There is also in the early history of algebra a use of terms analogous to those derived from the plane and solid geometry, but applicable only to geometry of more dimensions. With the Greeks, and then in general with the mathevmaticians that came after them, a number was thought of as a Urn (of definite length), the product of two numbers as a rectangle or plane, and the product of three numbers as a parallelopiped or solid; or, if the numbers were equal, the product of two was a square and of three a cube. When they began to study algebra, other terms were required for the higher powers, and so in Diophantus (third century) we find square-square, square-cube, and cube-cube In later times there was a variation in the use of these terms. Thus the square-cube came to mean the square of the cube, or sixth power, while with Diophantus it means the square times the cube, or fifth power. This change required the introduction of new terms for powers of prime orders, and, in particular, for the fifth power, which was finally called a sur solid. ^ The geometrical conception of equations and the geometrical forms of their solutions! hindered * Cantor, Vorlesungen itber Geschichfe der Mathematik, vol. I, 3d ed., Leipzig, 1907, p. 470. t In the edition of Rudolph’s Coss (algebra) revised by Stifel (Kbnigsberg, 1553, described by David Eugene Smith in Kara Arithmetica, Boston, 1908, p. 258) Sursolidum denotes the fifth power, Bsursolidum the seventh power, and so on (Part I, chap, s, fol. 63). Paciuolo (about 1445 1514) in his Summa de Arithmetica Geometria Proportioni ct Proportionalita, printed in 1494, uses the terms primo relate and secundo relate (Cantor, Vorleiungen, etc , vol. II, 2d ed , 1900, p. 317). On the other hand, Vieta (1540-1603) follows Diophantus. He expresses all the powers above the third by compounds of quadrate and cubo, cubo-cubo-cubus being the ninth power {Francisci Vieta opera mathematica, Leyden, 1646, p. 3 and elsewhere). The term sursoUd occurs several times in the geometry of Descartes (1596-1650). It is to be noted, however, that a product with Descartes always means a line of definite length derived from given lengths by proportions. Problems which lead to equations of the fifth or sixth degrees require for their geometrical solution curves “one degree more complicated than conics ” Conics were called by the Greeks solid loci, and these more complicated curves were called by Descartes sursolid loci (La Giomttrie. See pp. 20 and 29 of the edition published by Hermann, Paris, 1886), tt Such solutions are given in the second and sixth books of Euclid’s Elements. See Heath’s edition, Cambridge University Press, 1908, vol. I, p. 383. EARLY REFERENCES TO DIMENSIONS [3] the progress of algebra with the ancientsHigher equations than the third were avoided as unreal,* and when the study of higher equations forced itself upon mathematicians, it meant an impossible extension of geometrical notions, which met with many protests, and only in later times gave way to a purely numerical conception of the nature of algebraic quantities. Thus Stifel (i486 ?-i567), in the Algebra of Rudolph already referred to (footnote, preceding page), speaks of “going beyond the cube just as if there were more than three dimensions,” “which is,” he adds, “against nature.” f John Wallis (1616-1703) in his Algebra objects to the “ ungeometrical ” names given to the higher powers. He calls one of them a “Monster in Nature, less possible than a Chimaera or Centaure.” He says: “Length, Breadth and Thickness, take up the whole of Space. Nor can Fansie imagine how there should be a Fourth Local Dimension beyond these Three.”! Ozanam (1640-1717), after speaking of the product of two letters as a rectangle and the product of three as a rectangular parallelopiped, says that a product of more than three letters will be a magnitude of “as many dimensions as there are letters, but it will only be imaginary because in nature we do not know of any quantity which has more than three dimensions.” § Again, we find in the writings of some philosophers references to a space of four dimensions. Thus Henry More (1614-1687), an English philosopher, in a book published in 1671, says that spirits have four dimensions,^ and Kant (1724-1804) refers in several places to the number of dimensions of space. H ? Matthiessca, GrutidzUgr der antiken und modrrnen Algebra, 2d ed., Leipzig, 1896, pp. 544 and q2i. f Part I, chap, i, fol. 9 recto, t London, 1685, p. 126. § Dictionaire maihemati^ue, Amsterdam, i6gi, p. 62. IT EnchiridioH metaphysicum, Pt. I, chap. 28, § 7, p. 584. II For example, he says in the Critique of Pure Rcaion, “For if the intuition [4] Finally, there is a suggestion made by certain writers that mechanics can be considered a geometry of four dimensions with time as the fourth dimension (see below, p. ii). This idea is usually credited to Lagrange (17361813), who advanced it in his Theoric des f mictions analyHques. first published in 1 797. * It is expressed, however, in an artideon Dimension” published in 1754 by d’Alembert (1717-1783) in the Encyclopedic edited by Diderot and himself. D’Alembert attributes the suggestion to “un homme d’esprit de ma connaissance.” f These are the only ways in which we have found our subject referred to before 1827. In the f)eriod beginning with 1827 we may distinguish those writings which deal with the higher synthetic geometry from those whose point of dew is that of analysis. In synthetic geometry our attention is confined at first chiefly to the case of four dimensions, while in analysis we are ready for n variables by the time we have considered two and three. So far as we know, the first contribution to the synthetic geometry of four dimensions is made by Mobius, who points out that symmetrical figures could be made to coincide if there were a space of four dimensions.J In 1846 Cayley were a concept gained a posteriori ... we should not be able to say any more than that, 90\|ar as hitherto observed, no space has yet been found having more than three dimeR^fbBa” (translation by F. Max Muller, 2d ed. revised, Macmillan, 1905, p. xq). C. H. Hinton finds in four-dimensional space illustration and interpretation of the ideas of PllliOit Aristotle, and other Greek philosophers (see Fourth Dimension, London, i\|llK)$\|hr«iiap. iv). * (Euvres, vol IX, Paris, 1881, p. 337. paper by R. C. Archibald, ‘"Time as a Fourth Dimension,” Bulletin of the 4 nii»§tan Matkentatkal Society, vol 20, 1014, pp. 409-412. ¦'t He states very clearly the analogy with symmetrical figures in a plane and .symmetrical groups of points on a line. Reasoning from this analogy, he says that * the coincidence of two symmetrical figures in space would require that wc should be able to let one of them malt^ a rotation in space of four dimensions. Then he adds, “Da aher ein solcher Raum nicht gedaebt werden kann, so ist auch die Coin- BEGINNINGS OF THIS GEOMETRY [5] makes use of geometry of four dimensions to investigate certain configurations of points, suggesting a method that is systematically developed by Veronese.* Cayley had already published a paper with the title “ Chapters in the Analytical Geometry of (n) Dimensions,” f but as this paper contains no actual reference to such a geometry, we may think of the paper of 1846 as the beginning of his published writings on this subject. Some of the most interesting examples of the direct study of these geometries were given by Sylvester. In 1851, in a paper on homogeneous f unctions, J he discusses tangent and polar forms in ^-dimensional geometry; in 185Q, in some lectures on partitions, § he makes an application of hyperspace; and in 1863, in a memoir '‘On the Centre of Gravity of a Truncated Triangular Pyramid,”^! he takes up the corresponding figures in four and n dimensions and proves his theorems for all of these figures, using analytic methods to some extent, but appealing freely to synthetic conceptions. Clifford also, about this time, makes a very interesting application of the higher geometry to a problem in probability. H cidenz in diescra Falle unmoglich ” (Der barycenlrische C'alcul, Leipzig, 1827, § X40, p 184). “Sur quelques thiJoremes dc la geometrie de position,” Crelie's Journal, vol. 31, pp. 213-226 (in particular, pp. 217-218); CoUcikd Malhemalical Papen, Cambridge, vol. 1, 1889, No. 50 See also V’^eronc&c, Fotidamtnh, etc. (the full title is given below on p. g), p. (jgo of the (lerman translation, and Veronese's memoir (mentioned on p. 8). In introducing this method of reasoning, Cayley says. "On peut en efiet, sans recourir k aucunc notion mt'taphy.sique k Tigard de ia possibilite de I'esp.'ice a qualre dimensions, raisonner comme suit (tout cela pourra aussi fitre traduit facilement en langut purement aualytique) . ” . . . t Cambridge Malhematkal Journal, vol. 4, 1844; Math. Papers, vol. I, No. ii, % Cambridge and Dublin Mathematical Journal, vol 6, p. i ; Collected Mathematical Papers, Cambridge, vol. I, 1904. No. 30. (Outlines of these lectures are published in the Proceedings of the London Mathematical Society, vol, 28, 1896, p. 33 ; Mathematical Papers, vol. II, 1908, No. 2b. If Philosophical Magazine, fourth series, vol, 26, Sept , 1863, pp. 167-183; Malkematical Papers, vol. II, No. 65, \\ Educational Times, Jan, 1866; Mathematical Reptrints, vol. 6, pp. 83—87; Mathematical Papers, Macmillan, 1882, p. 601. Quite independently of this beginning of its synthetic development, we find a notion of a higher geometry springing out of the applications of analysis. Certain geometrical problems lead to equations which can be expressed with any number of variables as well as with two or three. Thus, in 1833, Green reduces the problem of the attraction of ellipsoids to analysis, and then solves it for any number of variables, saying, “It is no longer confined as it were to the three dimensions of space.”* Other writers make the same kind of generalization, though not always pointing out so directly its geometrical significance, f It was but a step farther to apply the language of geometry to all the forms and processes of algebra and analysis. This principle is clearly announced by Cauchy in 1847, in a memoir on analytical loci, where he says, “We shall call a set of n variables an analytical point, an equation or system of equations an analytical locus,” etc.\| The most important paper of this period is that of Hicmann, “On the Hypotheses which Lie at the Foundations of Geometry.” § In this paper Riemann builds up the notion of multiply-extended manifolds and their measure-relations. He discusses the nature of the lineelement ds when the manifold is expressed by means of n variables. When ds is equal to the square root of the sum * MoHumatkal Papers of George Green, edited by N. M. Ferrers, Macmillan, 1871, p. 168. t C. G J. Jacobi, “De binis quibuslibet functioatbus homogeneis,” etc., Crete's Journal, vol. 12, 1834, p. i ; Cayley, two papers published in the Cambridge iiaikematical Journal, voi. 3, 1841 , Mathematnal Papers, vol. 1, Nos 2 and 3 ; SchULfli, “DdMar das Minimum des Integrals /(Vdxj* + dart* + . . + dxtf),'' etc., CreUe’s Journal, vol. 43. 1852, pp. 23-36; “On the Integral f^dxdy . . . da,” etc,, Quarterly Journal, vpls. 2 and 3, iSsSr-iSdo. X “Mftntnr sur les Ueux analytiques,” Comptes Rendus, vol. 34, p. 885. i‘Ucber die H3npothesen, welche der Geometric ®u Grunde Megen.” presented to the phdiosophical faculty at Gottingen in 1854, but not published till 1866; GtsanmeUe Werke, Ldpaig, 1892. No. xiii, pp 273-387; tranidated by Clifford in Nature, vol. 8, 1873, pp. 14 and 36 ; Mathematical Papers, No. g, pp. 55-6 q. RIEMANN, GRASSMANN [7] of the squares of the quantities dx, as in the ordinary plane and space, the manifold is flat. In general there is a deviation from flatness, or curvature; and the simplest cases are those in which the curvature is constant. Riemann points out that space may be unbounded without being infinite — that, in fact, it cannot be infim'te if it has a constant positive curvature differing at all from zero. We therefore attribute to Riemann the Elliptic NonEuclidean Geometry, which from this time on takes its place beside that other discovered by Bolyai and Lobachevsky. His paper has a bearing on our subject in two ways : in the first place, his manifold of n dimensions is a space of n dimensions, and geometrical concep^tions are clearly before the mind throughout the discussion; and then the notion of a curvature of space suggests at once a space of four dimensions in which the curved three-dimensional space may lie. Soon after, it w'as shown by Beltrami that the planimetry of Lobachevsky could be represented upon real surfaces of constant negative curvature just as the Elliptic Two-dimensional Geometry is represented upon the sphere, and the way was fully opened for the study of spaces of constant curvature and of curvature in general. Another work that has an important influence on recent developments of h>pergeometry, especially in its application to f)hysical theories, is the Ausdehnungslehre of Grassmann, first published in 1844, though little noticed at the * Beltrami, “Saggio di interpretazione della geomctria non-eudidea,” Giornale di matematiibe, vol 6, 1868; Opere, Milan, vol I, igo2, pp 374-405. .\nother memoir by Beltrami, “Teoria fondamentale degli spazii di curvatura costante," Annali di matematua pura ed appiUata, Ser. 2, vol. 2 , i868-i86q; Opere, vol. 1 , pp. 406-420. develops and explains much in Riemann 's paper that is difficult to understand. There are French translations of both memoirs by Houel, Annales Sctenitjiqjtes dt I' EcoU Normale Suptrieure, vol 6, i86q. Beltrami considers the representations of the three-dimensional geometries upon curved spaces as only analytic, while the representations of the two-dimensional geometries upon surfaces of constant curvature are real See Opere, vol. I, p. 396 and p. 427. time. His theory of estensive* magnitudes is a vector analysis, and the applications which he makes to plane geometry and to geometry of three dimensions can be made in the same way to geometry of any number of dimensions. The number of memoirs and books relating to geometry of four or more dimensions has increased enormously in recent years. We can mention only a few. In 1870, Cayley published his “Memoir on Abstract Geometry,” in which he lays dov-m the general principles of «dimensional geometry.* Another important contribution to the science was an unfinished paper “On the Classification of Loci” by Clifford. f An important paper by N’other on birational transformations was jmblishefl in 1870. \|! Other papers were published by Halphen in 1873 and by Jordan in 1875, § the latter giving a methodical generalization of metrical geometry by means of Cartesian coordinates. Perhaps the most important of all was a memoir by Veronese published in 1882,^ in which he takes up a study of the properties of configurations, the quadratic in any number of variables, the characteristics of curves, correspondence of spaces, etc. : he employed synthetic, not analytic methods, and inaugurated a purely synthetic method of studying these geometries. Veronese’s Fondamenti di geometria contains an elementary synthetic treatment of the geometry of four dimensions and the geometry of n dimensions; and the Mehrdimensionale Geometrie of * PkthsophualTransariionSfVol x6o , if aihemultral Papers, vol Vl.iSy,, No 413. f PhihsQpkical Transactions, vol. i6q, 1878; Mathemalical Papers, No 33, pp. 305-331- J“ZurThfioriedesemdeutigen Entsprechens algebraischer Gebildevon beliebig vieten Dimensionen,” Maihemaiische Annaien, vol 2, pp 293-316. {Haiphen, '^Kecbercbes de g6oni6trie k n dimensions,” Bulletin de la SodiU Ma^imatiqm ia France, vol. 2, pp. 34-52; Jordan, “Essai sur la gdom^trie il n dimennoBS,^^ idL vol. 3. PPioj-174. der projectivischen Verhaltnissc der RSume von verschiedeiien Dimeiiaoik«f\| jdurcb das Princip (lc.s Projicirens und Schneidens,” Maihematischc Aimaim, vqt 19, pp. 161-234. SX^W RECOGNITION [9] Schoute, employing a variety of methods, makes these subjects very clear and interesting. * A bibliography with nearly six hundred titles, up to 1907, is to be found Loria’s II passato ed il presente delle principali teorie geometriche. f The latest bibliography is that of Sommerville,t which contains 1832 references on n dimensions up to 1911 ; about onethird of these are Italian, one-third German, and the rest mostly French, English, and Dutch. § We see that the geometries of more than three dimensions were slow in gaining recognition. The general notion that geometry is concerned only with objective external space made the existence of any kind of geometry seem to depend upon the existence of the same kind of space. Consequent!}' some of our leading mathematicians hesitated to use the higher geometry.^ although the work- * V'eronesc, Fatidamenti di geomt'fria a ptu dimensinnx rd a piu ipezk di unita reitiliner f^posti in forma elrmcntarf, I'atiua, ihgi ; Gurmivn translation by Sthepp, Crundzugr der (ieomdrif von mrhrfri-n I)imtn<>wnem , etc , Leipzig, 1894. Schoute, Mchrdimensionale Geomdrir, Sammlung Schubert, XXXV and XXXVI, Leipzig, 1902 and 1905. Another elementary treatment of the subject is by Jouffret, Giomiitie d quatre dimensions, Paris, XQO.t t 3d ed., Turin, 1907. % Bibliography of Son-Euclidean Geomcirv, Including the Theory of Parallels^ the Foundations of Geometry, and Space of n Dimensions, L’niv'crsity of St. .Andrews, Scotland, 1911. § There is now a consideralxle pf>pular interest in the four-dimensional geometry, because of the many curious things alwut it, and because of attempts which have been made to explain certain mysterious phenomena by means of it. This interest has' producetl numerous articles Jind books written to descrii)c the fourth dimension in a non-mathematical way. In igoS a prize of $soo was offered through the Scientific American for the best non-raathematical essay on the fourth dimension. Two hundred and forty-five essays were submitted in this competition. Some of these have been published in a book, whose Introduction, by the present writer, gives quite a full discussion of the various questions connected with the subject (The Fourth Dimension Simply Explained, Munn and Company, New York, 1910). If Thus Darboux, in a memoir prcsenteil in >Sbo at the .Academy of Sciences ami published in 1873, speaks of a locum in geometry'of space as comf^ared with plane geometry, for certain plane curves can l>e studied with advantage as projections from space, but " Comme on n’a pas d'espace k quatre dimensions, les m£thodes de projection ne s’6tendcrit pas a la g^omdtrie dc I'espacc” (5Mr une elasse remarquaUe de courhes et de surfaces aigibriques, Paris, p. 164). F.ven in 1903. in his Rv‘ix>rl at lo introduction kg out of its details presented comparatively little difficulty to them. This objection has led some writers to emphasize those applications of four-dimensional geometry that can be made in three-dimensional space, interpreting it as a geometry four-dimensional in some other element than the point — just as we have interpretations of the non-Euclidean geometries, which cannot, however, take the place of their ordinary interpretation.* As long ago as 1846 it was pointed out by Plucker that four variables the Congress at St. Louis, he says, “Une seulc objection pouvait fttre faitc . . . I'absence de toute base r^ele, de tout substratum,’* etc. (Bulletin des sciences mathifttaliques, ser. 2, vol. 28, p. 261, Congress of Arts and Sciences, edited by H. J. Rogers, Houghton, Mifflin and Co., Boston, vol. I, iqos, P557) But Darboux himself has made important contributions to the geometry of n dimensions see, for example, his Lefonssur les systemes orthogonaux, 2d ed., Paris, igio; in particular, Bk. I, chap. t>, and Bk. II, chap. i. Poincare, in speaking of the representation of two complex variables in space of four dimensions, says, “On cst expos6 k rebuter la plupart dcs lectcurs ct de plus on ne posskde que I'avantage d'un langage commode, moLs incapable dc parler aux sens.” Acta Matkemaiica, vol. 9, 1880-1887, p. J24 On the other hand, we have the fbUowing from Sylvester : “There are many who regard the ^eged notion of a generalized space as only a disguised form of algebraic formulization ; but the same might be said with equal truth of our notion of infinity, or of impossible lines, or lines making a zero angle in geometry, the utility of dealing with which no one will be found to dispute Dr. .Salmon in his extension of Chasles' theory of characteristics to surfaces, Mr Clifford in a question of probability, and my’self in my theory of partitions, and also in my paper on barycentric projection, have all felt and given evidence of the practical utility of handling .space of four dimensions as if it were conceivable space ” {“ \ Plea for the Mathematician,” Nature, vol. i, 1869, p. 237 ; Mathematical Papers, vol. II, p 716). A statement of Cayley's has lieen given in a presHous footnote (p. 5) For other expres^ons of his views we may refer to the first paragraph of the “Memoir on Abstract Geometry ” mentioned above, and to a statement quoted by Forsyth in his “ Biographical Notice," Cayley’s Malhemalical Papers, vol. VIII, 1895, p. xxxv. As to the existence of a higher space. Gauss also is said to have considered it a possability (W, Sartorias von Wakershausen, “Gauss zum Gedhehtniss,'' Gauss IFerke, Gdttingen, vol. VIII, 1900, p, 267). Sogre, referring to the first of the two remarks that we have quoted from Darboux, says, “Maitttcnant nous faisons usage de I'espacc k quatre dimensions sans nous prfocettper de la question de son existence, que nous regardons comme une question tout'k'fait seoQStdaire, et personne ne pense qu'on vienne ainsi k [lerdre de la rigeur " Ma^temalbcke lltimlrn, vol. 24, 1884. p. 318. ?See Emory McClintock, “On the Non-Euclidean tieometry,” Bulletin of the New York Mathematical Society, vol, 2, 892, \|>p. 21-33. EXTENT AND VARIETY OF AFPUCATIONS can be regarded as the coordinates of a line in ^ace. Another four-dimensional geometry that has been suggested is that of spheres, t But this higher geometry is now recognized as an indispensable part of mathematics, intimately related to many other branches, and with direct applications in mathematical physics. The most important application for the mathematician is the application as analytic geometry to algebra and analysis : it furnishes concise terms and expressions, and by its concrete conceptions enables him to grasp the meanings of complicated formulae and intricate relations. This is true of all the geometries as well as the geometry of four dimensions. The latter is of special use in connection with two complex variables, both in the study of one as a function of the other, and when it is desired to study functions of both considered as independent variables.}: Another very important application of geometry of four dimensions is that mentioned by d’Alembert, making time the fourth dimension : within a few years this idea has been developed veiy^ fully, and has been found to furnish the simplest statement of the new physical principle of relativity. § * Systrm der Geometrir des Dusseklorf, p. 322. t See article by Profe&sor KeV'^er. “.\ Sensuous Representation of Paths that Lead from the Inside to the Outside of a Sphere in Space of Four Dimensions, BuUftin of the American Afathcmalical Society, vol. i8, iqii, pp 18-22. t .See reference given on the preceding' page to Poincare’s memoir in the Acta Maikematica; also Kwietnewski, Vetter Flikhen des vierdimensumaltn Raumes, deren sdmtliche Tangentialehrnen untereinander gleicku’inklig sind, und ikre Beaekung Ku den ebenen Kurven, Zurich, IQ02. § The theory has been developed somewhat as follows : If time is represented by a coordinate I measured on an axis perpendicular to the hyperplane of the spaceaxes, the ^axis itself or any parallel line will represent a stationary point, and uniform motion will be represented by lines oblique to the /-axis, forming an angle with the /-axis which depends on the rate of the motion. A certain velocity (the v'elodty of light) is taken as the greatest po.ssible velocity and the same for all systems of measurement. The lines through the origin, or through any point, representing this velocity are the elements of a conical hypersutface. All lines not parallel to With these various applications have been developed many methods of studying the higher geometries, besides the ordinary synthetic and analytic methods. We now have the synthetic and analytic projective geometries, including the projective theories of measurement ; we have the theories of transformations and transformation groups ; the geometry of algebraic curves and algebraic functions ; the geometry associatetl with the representation of two complex variables; differential geometry and the transformation of differential expressions ; analysis situs, enumerative geometry, kinematics, and descriptive geometry ; the extensive magnitudes of Grassmann and different kinds of vector geometry ; the application of quaternions to four dimensions; and the very' recent application of fourdimensional vector analysis to the principles of relativity.* these elements are diWded into two classo'* the linet. of one cJa>>s, let..s inclined to the l-axis, represent possible motions, while the tines of the other class can reprc-^ent only ima^naiy motions. The system may be regarded as a nun-Ku\.lidean gcometr> in whidti the conical hypersurfa.ee piaya the part of ab-solute for angles, white distances along lines of the two classes are independent and cannot i>e comijaresl. Nuw a point mosnog uniformly may be regarded as stationary, and the points which are really stationary as moving uniformly in the opposite spacc-direttion. This chaitge of view is represented by a transformation of coordinates, the new t-axi.s being the line representing the given uniform rar>tion. In this theory the angles of plane'play an important part, and line and plane vectors are freely used. Thisapplicatkm of four-dimensional geometry wa.s develojwd by Minkowdii. For further elaboration see article by E, B Wilson and Ci. N. Lewis, “The Spatetime Manifold of Relativity. The NonEuclidean Geometry of Mechanics and Electro magnetics,” Proceedings of the American Academy of Arts and Sciences, voi. 4^, No. It, Nov., tors. * On the projeedve theory of measurement sec d'C)vi«iio, “ Le funzione metrtche fondanientaJi nei^ spazii di quantesivogliano dimenstoni c di curvatura costante,' AM deUa Aecademia de lancet, ser. s, vol. i, i 87 <^, pp abstract in the Matkmatische Annakn, vol, la, 1877, pp. 40^-418. On analysia tdtus there is an important series of ractnoirs by Poinau^ : Jowrnat de pSeok PdykadHaique, vol. 100, 1804; RendUonti del Cvedo Matemaiico dr Palermo, vci. IS, tSw; Pmaedingi of Ike London HathemaHtal Society, vol, ja, 1900; BuHetin de h Sidm JfdtiMnatigne de Prance, vol. 30, 190? ; Journal de nusthimotiques pures d apfltepUMk^5, vol. 8, igoj; RtndieorUi di Palermo, vol, 18.1904; Comptes Mkidni, vdl. IJ3, igot. The enumerative geometry has been devdk>ped chiefly by Schubert. He hac ESSENTIAL PART OF GEOMETRY [13] All these interpretations and methods that have been applied to the study of the higher geometries, and all these uses to which they have been put, are interesting and valuable to a greater or less degree ; but the greatest advantage to be derived from the study of geometry of more than three dimensions is a real understanding of the great science of geometry. Our plane and solid geometries are but the beginnings of this science. The four-dimensional geometry is far more extensive than the three-dimensional, and all the higher geometries arc more extensive than the lower. The number and variety of figures increases more and more rapidly as wc mount to higher and higher spaces, each space extending in a direction not existing in the lower spaces, each space only one of an infinite number of such spaces in the next higher. A study of the four-dimensional geometry, with its h>'pcrplanes like our three-dimensional space, enables us to prove theorems in geometry of three dimensions, just as a consideration of the latter enables us to prove theorems in plane geometry. Such theorems may come from much simpler theorems relating to the four-dimensional figures of which the given figures are sections or projections.* articles in the if athematiseke Annalm, vols. aft, 38. and 45, iSSt>, i8qi and iSv4'. i! the Acta Mathrmaika, vol. S, iSSo; and elsewhere In kinematics we may mention ¦ CUtTord, “On the Free Motion under No Forces of a Rigid System in an .V-fold U fondamentales de cinemati Quaternions have been applied to geometry of four dimensions by Hathaway, Bulletin of the .imeruaH Matkemaiiiai Socirly, vol 4, iHg?. pp. S 4 ~ 57 1 actions of the Amrriain M alhematkai .WiVly, vol. 3, igo2. pp. 46-59; and by Stringham, Transactions, vol. j, 1001, pp. 183-214; Bulletin, vol. ii. 1905, pp. « 7 “ 439 . Other methods are illustrated in memoirs already iTfcrrt [14] Indeed, many theorems and processes are seen only partially or not at all in the lower geometries, their true nature and extent appearing in the higher spaces. Thus in space of four dimensions is found the first illustration of figures which have two independent angles, and of different kinds of parallelism and different kinds of perpendicularity. Another example is the general theorem of which a particular case is given in Art 31, namely, that a section of a simplex of n dimensions is one of the two parts into which a simplex of « — i dimensions (that is, its interior) may be di\'ided by a section.* There are also many properties in which spaces of an even number of dimensions differ from spaces of an odd number of dimensions, and these differences would hardly be recognized if we had only the ordinary geometries. Thus in spaces of an even number of dimensions rotation takes place around a point, a plane, or some other axis-space of an even number of dimensions, while in spaces of an odd number of dimensions the axis of a rotation is always of an odd number of dimensions (see chap. TV). The study of these geometries gives us a truer view of the nature of geometrical rea.soning, and enables us to break away from intuition. This is especially true if we adopt the synthetic method. The analytic geometry may seem to be free from difficulty, and many feel a higher degree of certainty in the results of their algebraic processes. But we are apt to attach the terms of geometry to our algebraic forms without any attempt at a realization of their significance. There is, indeed, an abstract geometry in which the terms are regarded as meaningless symbols; but the interest and usefulness of geometry depend on the clearness c# pur perception of the figures to which it may be applied^ ijtid so we prefer to study some concrete geometry, * See Sdboute, Mehriimensumaie Gumetne, voi. II, 1 1 , Nr. 6. SYKTHETIC METHOD [15] some interpretation of the abstract geometry which we could have obtained by giving a particular interpretation to its terms. And then the abstract geometry and other interpretations can all be obtained from the concrete geometry.* There is really the same absolute certainty to synthetic geometry if it is developed logically from the axioms, and in the synthetic study of four-dimensional geometry we are forced to give up intuition and rely entirely on our logic, t Although it is doubtful whether we can ever picture to ourselves the figures of hyperspace in the sense that we can picture to ourselves the figures of ordinary space, yet we can reason about them, and, knowing that the validity of our geometry depends only on the logical accuracy of our reasoning, we can proceed to build it up without waiting for a realization of it ; and then we may in time acquire such facility in handling the geometrical proofs of the theorems and in stating precisely the forms and properties of the figures that it is almost as if we could see them. For * Some portions of our study arc treated by themsflvc-' as new interpretations of /?<omctries already sludicfl As s<�on is the fundamental propositions which correspond to the a\ioins of some suih KcoraclTy have been established, so as to justify this motie of pniceduro, wc have only to translate its theorems in accordance with these projxjsitions to have in our pos.session a complete development of the particular subject < onsidere t We do not seek to know which of several geomctric-s is the true geometry, and in laying the foundations we tlo not 'seek for the true system of axioms, or even the true system of elements and relations All geometries are equally true, and sometimes a particular geometry may be built up etiually well in several different ways. A complete treatise on geometry should consider not only' the different geometries, but different melhorls of building up each geometry. An example of such a treatment is the first volume of Fragen der FJr mentor gfcmctrir, cdit<d by Einriques (Leipzig, iQH , (icrman translation by H Thicme of Questioni riguardanti la grametria elemetUare, Bologna, iqoo). See also the chapters on this subject by Enriques and others in the French and German Encyciopedi-os {F.ncyklapddie der math. H'r'jrr., vol. nil, Leipzig, igo7 ; Eruyclopfdie des sci. math., xol. IlL, Leipzig, iqiiL A list of different systems of fundamental elements and relations is given in a footnote at the beginning of Coolidge�s Non~Euclidean Geometry, 0 .xford., iQog. [l6] in stud3ang the geometry synthetically our attention is fixed upon the figures themselves, and this takes us directly to the heart of the difficulty and keeps it before us until we have mastered it. Thus in its results this geometry greatly increases our power of intuition and our imagination* The following pages have been written with the object of meeting as far as possible the difficulties of the subject. No knowledge of higher mathematics is necessary; yet we do not believe that the simplest way is to avoid a mathematical treatment. The confidence gained from a study of the proofs, if they can be made clear and precise, will do more for the student than a mere description of fourdimensional space. We will indicate how this purpose has influenced us in our choice of subject-matter and the form of presentation. We have adopted the synthetic method and made no use of analytic proofs, feeling, as we have already explained, that this study of the figures themselves wdll serve best to help us understand them. We have confined ourselves to the fourth dimension, although it would have been easy to cover a much wider field, t We hope that in this way the four-dimensional space will be made to appear as a concrete matter to be studied by itself, and not as one of an indefinite series of spaces, each understood only in a vague general way. We have wished to give to these pages a familiar appearance, and so have endeavored to follow the popular textbooks and build up a structure that will rest on the foundations laid in the schools. Our geometry might have been adapted to the axioms of some modem investigation, or C. j ''Mathematical Emandpations.” Momst, vol. i6, igo6, tPpartkularijr pp t See, for example, the MeMimtHmmak Gometrie of Schoute. [17] FOUNDATIONS have attempted to establish a system of axioms, but either course would have raised questions quite different from those of four-dimensional geometry. The methods employed in this book are methods which the student has used freely in the past, even though he may be ignorant of their true significance and justification : there is nothing new in their application here, and their employment without question leaves him free to fix his attention upon the difficulties inherent in the subject. There is, however, one part of the foundations which has been presented w'ith considerable care, namely, that which relates to the definitions and the intersections of certain elementary figures. It is here that the four-dimensional geometry begins to contradict our experience, declaring, for examf>le, that two complete planes may have only a point in common, and that a line can pass through a point of the interior of a solid without passing through any other of its points. It is true that these facts and many others nut easy to realize are easily proved, and require only a few of the theorems given in this connection. On the other hand, the theorems for which most of these details are needed arc so “evident” that they are usually ignored altogether. Now a statement of these theorems, with a realization of what is assumed and of what is to be proved, and a logical working out of the proofs themselves, wdll give the student more confidence in all the results of his study. Similar considerations have led us in the fourth chapter to take up symmetry, order, and motion in space of two dimensions and in space of three dimensions. Great assistance comes from the analogies that exist in geometry, and so we have gone back in some cases and given proofs which are not well known, and to which more difficult proofs that follow are analogous; and we have * Sec, for example, the theorems of Arts. 6t and 6a. 18 tried to facilitate the comparison of chapters and sections analogous to one another by adopting in them the same arrangement of paragraphs and the same phraseology. Not much use can be made of diagrams, and so far as they are given they must be regarded as indicating the relations of different parts of a figure rather than as showing in any way its appearance. A figure can be accurately determined by its projections, and the descriptive geometry of four dimensions will be helpful to those who are familar with the methods of descriptive geometry.* Much can also be learned by studying the sections of a figure. A section of a four-dimensional figure is that part which lies in a three-dimensional space or hyperplane, and is, therefore, like the figures of our space. We can suppose that we are able to place ourselves in any hyperplane, and so to examine any hyperplane section : in connection with the diagrams we shall sometimes call attention to those parts which lie in any one section, speaking of them as “what we can see in a hjqx^rplane.” One way of studying a figure is to let it pass across our space, giving us a continuously varying section, as if time were the fourth dimension, Another way is to let it turn, or our section of it, so that the direction of our view changes. It is along these lines, if at all, that we are to acquire a perception of hyperspace and its figures. Some explanation should be made in regard to the arrangement, the particular form chosen for the foundations so far as they are considered, and the fundamental conceptions as we have presented them. We have given only the Euclidean Geometry, except that the geometry of the hypersphere, and of the hyperplane at infinity^ and the geometry in a few less important cases, * See Schoute, Mekrdimensional^ Geomeirie, vol. I, f s. COLLINEAR RELATION [19] arc themselves non-Euclidean. It has been found, however, that several chapters can be completed before we make any hypothesis in regard to parallels, and that, too, without much variation from the usual treatment. Perpendiculars and all kinds of angles, symmetry and order, and those hypersurfaces (the hyperpyramid, the hypercone, and the hypersphere) which do not involve parallels — in fact, all of “restricted” geometry — can be taken up before the introduction of parallels.* In the chapter on the hypersphere, its geometry, being elliptic, is stated as such, and a group of theorems is given from the non-Euclidean geometry; and in the last chapter the non-Euclidean properties of the hypersphere are used quite freely. Although these portions of the book may be omitted, the student will find it an advantage to make himself familiar with the Hyperbolic and Elliptic geometries, f We have started w'ith iwints only as elements, regarding all figures as classes of points, and .so defining a figure simply by stating w’hat points constitute the class. To do this we assume first a relation by which wdth any two points certain points are said to be collinear. Then for line we take two points and the class of points collinear with them, add to the group all points collinear with any two that we now have, and thus continue, at each step adding to our class of points all that are collinear with any tw^o already in the group, so that the line includes every point which it is possible to get in this w'ay. Thus any tw'o points determine a class consisting of the points which are collinear with them, and any two points determine a class of the kind which we c.all a line.J By the axioms of Art. 3 the * Sec the autharW on-EucfulroH Grometry, Ginn ami (« , Boston, igoi, chap. I; in particular, p 6. t The Hyperbolic and Elliptic geometries nre the only non -Euclidean geometries that we have referred to at all. t That two points determine a line does not mean, as in some of our text-books. [20] two classes are identical : the line consists only of the points coUinear with the two given points, and there are no additional points to be obtained by taking any two of these points. In fact, any two points of a line determine the same class of points as coUinear with them, and the same line. But until we have adopted these axioms we must suppose that the line might be a much more extended class , that, if we have the points coUinear with two given points, the class of p>oints coUinear with any two of these might be quite different; and that, while a lihe must contain every point of the line determined by any two of its points, the latter might not contain every point of the former Thus we make a distinction at the beginning between the notion of coUinear points and the notion of points of a line, and this distinction makes line analogous to plane and h)q)erplane, and to spaces of more than three dimensions. But after we have adopted our first two axioms we are able to employ the word coUinear in its commonly accepted sense, and thus to avoid the introduction of a new term for one of these two relations. A careful distinction has been made between the points of a closed figure and th(? points of its interior. Thus a triangle is made to consist of three vertices and the points of its sides, a tetrahedron of its vertices and the points of its edges and faces, and so on. This is only carrying to the limit the tendency to regard a circle as a cur\='e rather than as the portion of the plane enclosed by the curve, and a sphere as a surface. The figure of one-dimensional geometry corresponding to the triangle and tetrahedron, the one-dimensional simplex, is the segment. Therefore, we have defined segment as consisting of two points, and let that the line contains the two points, or that no other line contains them A figure may be in various ways 'thus a line in (he ordinary plane geometry may be deUiWitettd by two points asJthe locus of ptnnts equidistant from them FIGURE AND ITS INTERIOR [21] the points bet\!oints of a closed figure and the points of its interior is of great importance, and has been carefully observed. Hilbert defines sesment (Strccke) as a “system of two points/’ but he speaks of the points between A and B a-. ‘ txnnt>. of the segment AB," although he also speaks of them as jxnnts “situateil within the segment” ^Grundlacen der Geamctrie, I^eipzig, i8gg, p (\ 4th ed . igi 3. p <;) In the Eticydopidte drs \nem-)s mathfmaiiques, vol II Ii, p 21, Ennques dehnes segment ujwn a lim as "has mg its eslreme points at two given jxiints A and B of the line and tontaining the intermediate points ” More definitely, in the FJementi di Rfomfiria ol CnrKiut'^ aiul Vnidl.li (Bologna, igii), half-line is defined so as to include its extremitj, and then the segment AB the part common to the halflinev AB and BA (p 0 K H Mcxjre defines the segment AB as consisting of (Xiints “distinct from A and B,” etc ; that is, A and B are not included among the points of his segment (“On the Projective .\)Eioma of (Ieonietr>,’’ Tranuictions of the American Mathematical Society, vol t, iijoi, p 147^ \xiom j) See also Veblcn, “A S>'stem of Axioms for Goometrj',” Transoilums, vol 5, igo4, j> 354, Definition i, and “The Foundations of Geometry,” Monographs on Modertt Mathematics, edited by J, W A. Young, New York, iqh, p s Most writers who use the word segment in this connection regard a segment as an entity, a piece of a line, without considering whether the end-points are included or not. Many writers s\|X‘ak of the segment as the “measure of the distance” betwccji the two points (see Schotten, Inhall und Methode des Planimelrischen Unlerrtchis, Leipzig, vol IT, 1803, chap i, § 2) Veblen, in the “ Foundations of Geometry” just referred to, defines triangle and tetrahedron in the same way that we have defined them (pp. 29 and 45). [22] * A remarkable memoir on geometry o! n dimensions is Tkeorie der vidjachen Kontinuitdt, by L. Schlafli. edited by J. H. Graph, Bern, 1911. This was written in the years 1850-1852, but the author did not succeed in getting it published, apparently on account of its length, and it remained among his papers for fifty years, until after his death (see Vorbemerkimg). Among other things he works out the theory of perpendicularity and all kinds of angles, giving, in particular, a generalization of the theorems which we have given in Arts. 66 and 67 (§15). He proves the polyhedroid formula and the corresponding formula for any number of dimensions, and he constructs the six regular convex polyhedroids and the three regular figures which exist in each of the higher spaces^ proving that these are the only regular figures of this kind (§ 17). He makes an extensive study of the hypervolume of a spherical simplex, showing the difference between the cases of an even number and of an odd number of dimensions, and giving the formula for a pentahedroid to which we have referred at the end of Art. 165 (§ 22). In the third part of the memoir he bikes up quadratic hypersurfaces, the classification of these hypersurfaces, confocal hyj)ersurfaces, etc. The methods are analytical, but the language and conceptions are purely geometrical. ?This note was. written after the te5,t of the Introduction was m tyf»e.
	Index (193-196)

(Created September 6, 2024)