Lectures on Quaternions

LECTURES ON QUATERNIONS.
SIR WILLIAM ROWAN HAMILTON. 1853.
ERRATA.
PREFACE.
	[1.] The Method or Calculus of Quaternions.
	[2.] The doctrine of Negative and Imaginary Quantities in Algebra.
	[3.] Algebra as the Science of Order in Progression or the Science of Pure Time.
	[4.ERR] Equivalence B = A, and non-equivalence B > A, B < A of moments.
	[5.] B-A, to denote the difference between two moments.
	[6.] D-C = B-A, denoting equal time intervals; D-C > B-A, signifying D later to C than B to A.
	[7.] a = B-A; B = a+A, denotes moment B attained by step a from moment A.
	[8.] (B-A)+A = B, signifying difference between two moments and of applying that difference as a step.
	[9.] The algebraic ratio, or complex relation or quotient, B/A=q operates on A as qA=B to produce or generate B.
	[10.] Operations on algebraic numbers; interpreting the product of negative numbers as reversals.
	[11.] No Single Number can be the square root of a negative number; Theory of Couples of Numbers.
	[12.] (B1,B2)-(A1,A2) = (B1-A1,B2-A2), expressing the complex ordinal relation of one couple to another couple.
	[13.] Complex division: a(A1,A2) = (aA1, aA2); (aA1,aA2)/(A1,A2) = a; (B1,B2)/(A1,A2); (A1,A2) = (A1,0)+(0,A2).
	[14.] Complex multiplication: (a1,a2)(A1,A2)=(a1A1-a2A2,a2A1+a1A2).
	[15.] Primary Unit (1,0) and Secondary Unit (0,1); (1,0)(a,b) = (a,b); (0,1)(a,b) = (-b,a); (0,1)^2 = (-1,0) = -1.
	[16.] Operator of transposition composed with secondary reversal (0,1)=√-1.
	[17.] Solving the quadratic couple-equation (x,y)^2+(a,0)(x,y)+(b,0)=(0,0); x^2-y^2+ax+b=0, 2xy+ay=0.
	[18.] The General Logarithm of Unity.
	[19.] Essay quotation relating to Theory of Couples and Theory of Triplets.
	[20.] From couples to Quaternions; Addition and subtraction of number triads of the form (A1,A2,A3).
	[21.] Multiplication of a triad by a number; The quotient of proportional triads is a number.
	[22.] Linear combinations of unit-steps and unit-numbers, and their distributive multiplication.
	[23.] 27 arbitrary numerical constants of multiplication in the general theory of triplets.
	[24.ERR] Trials of arbitrarily assigning values to the constants of multiplication.
	[25.] The abstraction, or abridgment, of the step-triad.
	[26.] Zero divisors in multiplication, and indeterminate cases in division of triplets.
	[27.] Geometrical interpretation of triplet systems.
	[28.] In all triplet systems tried, at least one system of line and plane formed perpendicular lines whose products were zero.
	[29.] In trials of triplet systems, there would remain two undetermined arbitrary constants of multiplication.
	[30.] Extension of theory from couples and triplets to conceiving a system or set of n (A1,A2,..An) and their algebra.
	[31.] N independent unit-steps, n unit-numbers, and n^3 constants of multiplication.
	[32.] The plan for multiplication and division of sets of n numbers.
	[33.] Reference to essay entitled \'Researches respecting Quaternions. First Series.\' 1843
	[34.] Formula for the (symbolic) multiplication of two nmnber-sets.
	[35.] A new system of n^3 numerical coefficients.
	[36.] References to writings on the representations of √-1.
	[37.] A conjecture respecting such extension of the rule of multiplication of lines, from the plane to space.
	[38.] Another construction, of a somewhat similar character, and liable to similar objections, for the product of two lines in space.
	[39.] Mr. J. T. Graves\'s method of representing lines in space, and of multiplying such lines together.
	[40,] Professor Graves employed a system of two new imaginaries, i and j.
	[41.] Transformations of rectangular to polar co-ordinates.
	[42.] Mr. J. T. Graves\'s mode of representing quantity spherically.
	[43.] Attempts to extend to space, geometrical multiplication of lines.
	[44.] Ingenious and original papers by other able analysts.
	[45.] The endeavour to adapt triplets x+iy+jz to the multiplication of lines in space.
	[46.] The supposition ij=-ji: or that ij=+k, ji=-k, the value of k being still undetermined.
	[47.] QUATERNIONS a+ib+jc+kd, or (a,b,c,d), the symbol k denoting some new sort of unit operator.
	[48.] The fundamental assumptions for the multiplication of two quaternions.
	N.B.: In [48.], 1st mention of LEFT-HANDED i,j,k orientation, which Hamilton uses very often in examples and figures.
	N.B.: L-H:→i,j⊙,k↑; R-H:i←,j⊙,k↑; e.g., L-H:i=iₗ may be chosen geometrically inverse or opposed to R-H:i=iᵣ →←, while jₗ=jᵣ ⊙⊙, kₗ=kᵣ ↑↑.
	N.B.: Straighten hands, thumbs pointing up (as kₗ↑↑kᵣ); turn hands so fingertips touch (as iₗ→←iᵣ); curl lower-three fingers toward face (as jₗ⊙⊙jᵣ).
	N.B.: For any situation, L-H or R-H orientation can be initially chosen; the chosen must thereafter be used consistently in geometrical constructions.
	N.B.: All quaternion rules, i²=j²=k²=ijk=-1, work the same in either a chosen LEFT-HANDED or RIGHT-HANDED orientation applied consistently (try it).
	N.B.: In a figure or example, when a clockwise rotation is described, it usually implies that a L-H orientation for i,j,k is assumed.
	N.B.: In a figure or example, when an anticlockwise rotation is described, it usually implies that a R-H orientation for i,j,k is assumed.
	N.B.: The choice of L-H or R-H (i.e., clockwise ⥁ or anticlockwise ⥀ rotations) is by convenience, or by whichever seems to fit a problem easier.
	[49.] The new instrument for applying calculation to geometry.
	[50.] The product of two co-initial lines, or of two vectors from a common origin.
	[51.] The square of a given line must not be any line inclined to that given line.
	[52.] The product αβ of two mutually perpendicular lines, each length 1.
	[53.] Try whether we can consistently suppose αβ=mα+nβ+pγ, m,n,p numerical constants.
	[54.] The three principles.
	[55.] We are compelled to give up the commutative property of multiplication.
	[56.] Try whether we can connect two coefficients, to satisfy the associative principle a.βγ=αβ.γ.
	[57.] The FOURTH PROPORTIONAL, u.
	[58.] The same u be the fourth proportional to any three rectangular directions m,n,l.
	[59.] There is no objection against our supposing that u=+1,-u=-1.
	[60.] The calculus of quaternions were shewn (in 1844) to be consistent with a priori principles.
	[61.] A geometrical quotient β÷α operating as multiplier on divisor-line α produces (or generates) dividend-line β.
	[62.] About the book LECTURES ON QUATERNIONS.
LECTURES ON QUATERNIONS.
	CONTENTS. (with abridgment and commentary)
	CONTENTS./LECTURE I. Articles 1 to 36; Pages 1 to 32.
	LECTURES./LECTURE I. Articles 1 to 36; Pages 1 to 32.
		Introductory remarks (1848), Articles 1, 2, 3; Pages 1 to 4.
			Art.1: Astronomy and the successful application of the laws of Kepler and Newton.
			Art.2: The CALCULUS of QUATERNIONS have applied to the solution of many geometrical and physical problems.
			Art.3  With the admitted correctness of the results of this new Calculus, it seems the most expedient to adopt at present.
		CONT/§ I. Articles 4 to 14 ; Pages 4 to 14.
		LECT/§ I. Articles 4 to 14 ; Pages 4 to 14.
			Art.4: The four operations + - × ÷ are to be used in new senses in the Calculus of Quaternions.
			Art.5: SYNTHESIS(+) and ANALYSIS(-) of a STATE; SYNTHESIS(×) and ANALYSIS(÷) of a STEP in a Progression.
			Art.6: Primary Geometrical Operation -, or Minus; SPACE, the field of progression; POINTS, position; ordinal relation, GEOMETRICAL DIFFERENCE.
			Art.7.Fig.1: Ordinal Analysis (B-A) of ANALYZAND POINT B with respect to ANALYZER POINT A, is Synthesis of DISTANCE and DIRECTION of STEP from A to (B-A)+A.
			Art.8.Fig.2: RESULT (B-A) is analysis of position of B relative to A, and is the synthetic(+) rule|operator|step for transition|progression|convection from A to B.
			Art.9.Figs.3,4: Analysis (B-A) ←and→ (A-B) are ordinal relations that are opposite or inverse steps.
			Art.10: Heliocentric ☉ or Geocentric ♁ Analytic Position (B-A) is an analysis result of analyzand B compared relative to the Sun ☉ or Earth ♁ as analyzer A.
			Art.11: Minus, or -, denotes ordinal analysis, comparison of position, SIGN OF TRACTION; (B-A) denotes a SYMBOLICAL SUBTRACTION or DIFFERENCE, or straight line TRACTION conveying A to B.
			Art.12.Fig.5: Pure Mathematics primarily the science of ORDER in Time and Space; (B-A) abbreviates (BO-AO) when B and A are relative to a common or absolute origin O.
			Art.13: Notation (B-A) is a COMPLEX SYMBOL with component symbol \'minus\', or a symbolical expression \'Point minus Point\'.
			Art.14: RELATIVE POSITION, or GEOMETRICAL DIFFERENCE OF THE ABSOLUTE POSITIONS, denoted by sought point minus given point, or B-A.
		CONT/§ II.ERR.ix. Articles 15 to 26; Pages 15 to 25.
		LECT/§ II. Articles 15 to 26; Pages 15 to 25.
			Art.15: SYNTHETIC ASPECT of B-A as denoting the STEP, VECTOR, or RAY of final point B from initial point A.
			Art.16: Analytic and Synthetic interpretation of B-A; Geocentric ☉-♁ and Heliocentric ♁-☉ VECTORs.
			Art.17: Distinction between \'RADIUS-VECTOR\' polar co-ordinates and the new QUATERNION \'VECTOR\' TRINOMIAL FORM (ix+jy+kz) Cartesian rectangular co-ordinates.
			Art.18: EQUATION of equisignificant symbols B-A=a, where VECTOR symbol (a) is chosen to concisely denote the complex symbol B-A, the rectilinear Synthetic STEP in space from A to B.
			Art.19: Plus, or +, CHARACTERISTIC OF ORDINAL SYNTHESIS, SIGN OF VECTION; B=a+A, where vector (a) synthetically operates on A to REACH, CONSTRUCT, or TRANSITION by (a) from A to B.
			Art.20: Primary Signification of Plus in Geometry, (STEP, or VECTOR)+(Beginning of STEP, or VECTOR)=(End of STEP, or VECTOR).
			Art.21: DEFINITIONAL INTERPRETATION of \'Line plus point\'; LINE a = POINT B - POINT A; Point B = LINE a + POINT A; DIFFERENCE of two Points is a Line.
			Art.22: Distinctions between arithmetical sum and difference and algebraical addition of magnitudes, and GEOMETRICAL ADDITION as CHANGE OF POSITION in space.
			Art.23: SYMBOLICAL ADDITION B=(a+A), SYNTHESIS of VECTUM B by ACT OF VECTION by VECTOR (a) on VEHEND A, VECTUM=VECTOR+VEHEND; SYMBOLICAL SUBTRACTION, VECTOR=VECTUM-VEHEND.
			Art.24: A=(-a)+B, REVECTOR -a=A-B, SYNTHESIS of REVECTUM A by ACT OF REVECTION by REVECTOR (-a) on REVEHEND B, REVECTUM=REVECTOR+REVEHEND.
			Art.25: Eliminate VECTOR; Analysis a=B-A, Synthesis B=a+A; Synthesis∘Analysis B=(B-A)+A, or Vectum=(Vectum-Vehend)+Vehend.
			Art.26: Eliminate VECTUM; B=a+A, or Vectum=Vector+Vehend; a=B-A, or Vector=Vectum-Vehend; a=(a+A)-A, or Vector=(Vector+Vehend)-Vehend.
		CONT/§ III. Articles 27 to 29; Pages 25 to 27.
		LECT/§ III. Articles 27 to 29; Pages 25 to 27.
			Art.27: Triangular POINTS A,B,C; VECTOR B-A, PROVECTOR C-B; VECTION B=(B-A)+A and PROVECTION C=(C-B)+B; PROVEHEND B, PROVECTUM C.
			Art.28: GEOMETRICAL IDENTITY, Provectum=Provector+Vector+Vehend, or C=(C-B)+(B-A)+A, Equation of Provection.
			Art.29: Astronomical provection: Planet\'s Position = Planet\'s Heliocentric Vector + Sun\'s Geocentric Vector + Earth\'s Position.
		CONT/§ IV. Articles 30 to 35; Pages 27 to 31.
		LECT/§ IV. Articles 30 to 35; Pages 27 to 31.
			Art.30: One TRANSVECTION (C-A)+A = (C-B)+(B-A)+A or Two SUCCESSIVE VECTIONS; TRANSVEHEND=VEHEND=A, PROVEHEND=VECTUM=B, TRANSVECTUM=PROVECTUM=C.
			Art.31: Transvector+VEHEND=Provector+Vector+VEHEND; Transvector=Provector+Vector.
			Art.32: c=(C-A),b=(C-B),a=(B-A); C=c+A=b+a+A; c=b+a (compare geometrical identities Art.31,28,25)
			Art.33: Astronomical transvector: Planet\'s Geocentric Vector = Panet\'s Heliocentric Vector + Sun\'s Geocentric Vector.
			Art.34: Astronomical provector: Planet\'s Heliocentric Vector = Planet\'s Geocentric Vector - Sun\'s Geocentric Vector.
			Art.35: Geometrical Identities: b=C-B,C=c+A,B=a+A; b=(c+A)-(a+A); (C-B)=(B-A)-(C-A); A,B,C are POINTS; a,b,c are LINES.
		CONT/§ V. Article 36; Pages 31, 32.
		LECT/§ V. Article 36; Pages 31, 32.
			Art.36: PRIMARY SIGNIFICATIONS of + & - in geometry: ORDINAL Synthesis LINE+POINT & Analysis POINT-POINT; THIS +,- theory coincides (×,÷ extends) with OTHERS.
	CONTENTS./LECTURE II. Articles 37 to 78; Pages 33 to 73.
	LECTURES./LECTURE II. Articles 37 to 78; Pages 33 to 73.
		CONT/§ VI. Articles 37 to 44; Pages 33 to 39.
		LECT/§ VI. Articles 37 to 44; Pages 33 to 39.
			Art.37: Recapitulation on the ORDINAL ANALYSIS operation - of GEOMETRIC SUBSTRACTION.
			Art.38: Recapitulation on the ORDINAL SYNTHESIS operation + of GEOMETRIC ADDITION.
			Art.39: Complex RELATION OF LENGTH AND DIRECTION: CARDINAL ANALYSIS q=(β÷α), and SYNTHESIS β=q×α; Simple case: β=α+α=2α,β÷α=CARDINAL number 2.
			Art.40: Metrographic Relation: quotient q=β÷α, ANALYSIS(÷) of analyzand β BY analyzer α; q×α=β, SYNTHESIS(×) of factum β by factor q INTO faciend α.
			Art.41: GEOMETRIC DIVISION: FACTOR=FACTUM(by,÷)FACIEND; GEOMETRIC MULTIPLICATION: FACTUM=FACTOR(into,×)FACIEND, or ACT OF FACTION.
			Art.42: General principle of analysis and synthesis applies to ÷ and ×, analogous to - and +.
			Art.43: Compare identities q×α÷α=q to a+A-A=a, or Factor×Faciend÷Faciend=Factor to Vector+Vehend-Vehend=Vector.
			Art.44: RECIPROCAL cardinal relations or quotients: FACTOR β÷α, FACIEND α, FACTUM β; REFACTOR α÷β, REFACIEND β, REFACTUM α.
		CONT/§ VII. Articles 45 to 56; Pages 39 to 48.
		LECT/§ VII. Articles 45 to 56; Pages 39 to 48.
			Art.45: Successive acts of FACTION then PROFACTION, or single equivalent act of TRANSFACTION.
			Art.46: PROFACTOR r=γ÷β, PROFACTUM γ=r×β, PROFACIEND β=q×α, FACTOR q=β÷α, FACTUM β, FACIEND α.
			Art.47: Profactum=Profactor×Factor×Faciend, γ=r×q×α=(γ÷β)×(β÷α)×α.
			Art.48: TRANSFACTOR s=γ÷α, TRANSFACTUM γ=s×α, TRANSFACIEND α; Profactum=Transfactor×Faciend, γ=(γ÷α)×α.
			Art.49: FULL FORM: Transfactor×Faciend=Profactor×Factor×Faciend, ABRIDGED FORM: Transfactor=Profactor×Factor, INTO Faciend suppressed.
			Art.50: r=γ÷β=(s×α)÷(q×α)=s÷q, PROFACTOR=TRANSFACTOR÷FACTOR q; α,β,γ are VECTORS and q,r,s are FACTORS or RESULTS of CARDINAL ANALYSIS.
			Art.51: α=Faciend=Transfaciend, β=Factum=Profaciend, γ=Profactum=Transfactum.
			Art.52: PYRAMID OF FACTIONS αβγ ORIGIN D α=(A-D),β=(B-D),γ=(C-D); TRIANGLE OF VECTIONS ABC a=(B-A)=(β-α),b=(C-B)=(γ-β),c=(C-A)=(γ-α).
			Art.53.Figs.6,7: Selenocentric D=☽ VECTORS α=(♁-☽), β=(☉-☽), γ=(♀-☽) of Earth A=♁, Sun B=☉, Venus C=♀.
			Art.54: Significations MINUS-,PLUS+: (1) POINT-POINT,LINE+POINT (2) LINE-LINE,LINE+LINE; BY÷,INTO×: (1) RAY÷RAY,FACTOR×RAY (2) FACTOR÷FACTOR,FACTOR×FACTOR.
			Art.55: Vections C←B←A=C←A: C=(C-B)+(B-A)+A=(C-A)+A ≎ Factions γ⤺β⤺α=γ⤺α: γ=(γ÷β)×(β÷α)×α=(γ÷α)×α.
			Art.56: Decompositions on ABRIDGED FORMS: Secondary Ordinal Analysis provector=transvector-vector, Secondary Cardinal Analysis profactor=transfactor÷factor.
		CONT/§ VIII. Articles 57 to 64; Pages 48 to 58.
		LECT/§ VIII. Articles 57 to 64; Pages 48 to 58.
			Art.57.Fig.8: Examples of primary & secondary cardinal analysis & synthesis with rays β=α+α γ=β+β+β differing only in length.
			Art.58: REVECTOR = MINUS VECTOR; VECTOR = PLUS VECTOR; Double of ray α: 2×α=+2α, Double and opposite: -2×α=-2α.
			Art.59.Fig.9: RULE OF THE SIGNS: any real ± number w interpretted as Geometrical Quotient w=β÷α of rays β α having same(+) or opposite(-) directions.
			Art.60.Fig.10: INVERSION and NONVERSION: Factors or operators INVERSOR(-) β=(-1)×α=(-)×α=-1α=-α, and NONVERSOR(+) γ=(+1)×α=(+)×α=+1α=+α=α.
			Art.61: Cardinal quotients, INVERSOR (-)=β÷α & NONVERSOR (+)=γ÷α: RELATIONS OF OPPOSITION & SIMILARITY in directions of analyzands β & γ BY analyzer α.
			Art.62: As Factors, (-) and (+) are regarded as signs or characteristics of version and nonversion.
			Art.63: TENSION: TENSOR, or stretch/shrink FACTOR; TENSOR=SIGNLESS NUMBER; TRANSTENSOR=PROTENSOR×TENSOR; RE-TENSION: RE-TENSOR, or reciprocal TENSOR; NON-TENSOR=1.
			Art.64: SCALE: SCALAR=(±)×TENSOR, or REAL SIGNED NUMBER; composition of TENSOR and INVERSOR or NONVERSOR.
		CONT/§ IX. Articles 65, 66; Pages 58 to 61.
		LECT/§ IX. Articles 65, 66; Pages 58 to 61.
			Art.65: VERSION: VERSOR, or rotation or turning FACTOR; i,j,k rectangular VECTOR-UNITS, or Quadrantal Versors; VERSUM=VERSOR×VERTEND; TRANSVERSOR=PROVERSOR×VERSOR.
			Art.66: For equal length rays β=VERSUM α=VERTEND, cardinal quotient β÷α=VERSOR; VERSOR has a NON-TENSOR, or is unit or non-metric, and m̶e̶t̶r̶o̶graphically relates α to β.
		CONT/§ X. Articles 67 to 78; Pages 61 to 73.
		LECT/§ X. Articles 67 to 78; Pages 61 to 73.
			Art.67: Example of version, elevating a telescope.
			Art.68.Figs.11,12,13: Illustrations of telescope example of version and proversion.
			Art.69: VERTEND α=-i, VERSUM β=k, PROVERSUM γ=-j: VERSION k÷(-i)=j, j×(-i)=k; PROVERSION: (-j)÷k=i, i×k=-j.
			Art.70: TRANSVERSION: TRANSVERSOR=PROVERSOR×VERSOR or [(-j)÷k]×[k÷(-i)]=(-j)÷(-i)=i×j=k, TRANSVERSUM=γ=k×(-i)=-j.
			Art.71: i,j,k conceived as operators or axes of right-handed rotations.
			Art.72: Cyclic symbol permutations: i×j=k, j×k=i, k×i=j.
			Art.73: Axis j, as a versor, right-hand rotates the ki-plane and takes i TO -k; -k(BY÷)i=j; j(INTO×)i=-k.
			Art.74.Fig.14: ROTATION THEOREM: NOT generally commutative multiplication or composition of rotations: i×j=k,j×i=-k; j×k=i,k×j=-i; k×i=j,i×k=-j.
			Art.75: Combining i×j=k and i×k=-j=(-1)×j gives i×i×j=(-1)×j, ABRIGED i×i=-1; similarly j×j=k×k=-1; square of any quadrantal versor is -1.
			Art.76: Versor into RAY i×j = RAY k, or Proversor into Versor i×j = Transversor k; RAY by RAY k÷j = Versor i, or Transversor by Versor k÷j = Proversor i.
			Art.77: Every UNIT LINE is also a QUADRANTAL VERSOR, an operator of RIGHT HAND, RIGHT ANGLE rotation of the plane perpendicular to the line.
			Art.78: Multiplication of right lines, GRAPHIC OPERATION producing third line perpendicular to both, having non-commutative character.
	CONTENTS./LECTURE III. Articles 79 to 120; Pages 74 to 129.
	LECTURES./LECTURE III. Articles 79 to 120; Pages 74 to 129.
		CONT/§ XI. Articles 79 to 82; Pages 74 to 79.
		LECT/§ XI. Articles 79 to 82; Pages 74 to 79.
			Art.79: Principles of the Calculus of Quaternions w+xi+yj+zk; Extend NOTATIONS, SYMBOLS, RULES, and CONCEPTIONS.
			Art.80.ERR.76: FACTORS of classes I,II,III,IV VECTOR-UNITS,V,VI,VII,VIII, and IX.
			Art.81.Figs.15,16: Illustration of operator i,j,k of simultaneous quadrantal right-hand rotations of metric&graphic 1i&jk-planes,1j&ki-planes,1k&ij-planes, resp.
			Art.82: Multiplication of perpendicular LINES: Generally aι×bκ=ab×ικ for SCALARS a,b and VECTOR-UNITS ι,κ; Generally αβ=-βα, if LINES β⊥α.
		CONT/§ XII. Article 83; Pages 79, 80.
		LECT/§ XII. Article 83; Pages 79, 80.
			Art.83: Product of LINE or VECTOR α INTO SCALAR a: Suppose β⊥α, then α×a×β=a×α×β; ABRIDGED FORM gives α×a=a×α, or commutative αa=aα SCALAR-VECTOR products.
		CONT/§ XIII. Articles 84, 85; Pages 80 to 82.
		LECT/§ XIII. Articles 84, 85; Pages 80 to 82.
			Art.84: Product of parallel LINES ia×ix: Suppose jy⊥ix, then ia×ix×jy=ia×kxy=axy×ik=-ax×jy, ABRIGED ia×ix=-ax; VIEWED as successive METRIC 1i-plane rotations: i×i×ax=-ax.
			Art.85: EQUATION OF PARALLELISM aβ=+βα β∥α, PERPENDICULARITY αβ=-βα β⊥α OF LINES α,β; For any VECTOR α, square αα=α^2 equals NEGATIVE SCALAR -|α|^2; LINE α=√(-|α|^2).
		CONT/§ XIV. Article 86; Pages 82, 83.
		LECT/§ XIV. Article 86; Pages 82, 83.
			Art.86: Powers of unit vectors: For any VECTOR-UNIT ι, VERSOR ι^t into LINES⊥ι equals LINES⊥ι turned or right-hand rotated by SCALAR t quadrants (t90°) round axis ι.
		CONT/§ XV. Articles 87 to 89; Pages 83 to 87.
		LECT/§ XV. Articles 87 to 89; Pages 83 to 87; ÷,× of VECTORS λ⤺κ κ∢λ=t QUAD: VERSOR of λ÷κ,κ÷λ,λκ=(λ÷κ)κκ=-bb(λ÷κ),κλ=(κ÷λ)λλ=-cc(κ÷λ) rotates κλ-plane by t,-t,2+t,2-t.
			Art.87.Fig.17.ERR.85: GIVEN unit p in ik-plane (Z=k,N=α=-i,S=-α=i,W=β=-j,E=-β=j), p∢k=36° 36/90=2/5 QUAD, FIND p=j^(-2/5)×k, p×k=j^(-2/5)×k×k=j^(-2/5)×j×j=j^(2-2/5).
			Art.88: PRODUCT κλ OF VECTORS: λ÷κ=(c/b)×ι^t, κλ=bc×ι^(2-t) | TENSORS b=Tκ c=Tλ, VERSOR ι^t, ⊥VECTOR-UNIT ι of right-hand rotation λ⤺κ, angle t measured in QUADrants.
			Art.89: TAKING THE CONJUGATE Kq of q: For any unit-vector ι, K.ι^t=ι^(-t)=(-ι)^t; Conjugate products of lines K.κλ=λκ=cb×ι^(2+t),K.λκ=κλ=bc×ι^(2-t); κ÷λ=(b/c)×ι^(-t).
		CONT/§ XVI. Article 90; Pages 87 to 89.
		LECT/§ XVI. Article 90; Pages 87 to 89.
			Art.90: TAKING THE TENSOR Tq or VERSOR Uq of q: q=Tq×Uq; q=Uq×Tq; T.Uq=1 non-tensor; U.Tq=+ non-versor.
		CONT/§ XVII. Article 91; Pages 89, 90.
		LECT/§ XVII. Article 91; Pages 89, 90.
			Art.91: QUATERNION q=Tq×Uq=Tq×ι^t described by 4 numbers: (1) Tq, (2) angle t in QUADs, (3,4) direction of ι using 1 planar, and 1 polar angle.
		CONT/§ XVIII. Articles 92 to 95; Pages 90 to 95.
		LECT/§ XVIII. Articles 92 to 95; Pages 90 to 95.
			Art.92: METROGRAPHIC Analytic RELATION q=β÷α, and Synthetic AGENT q×α=β.
			Art.93: BIRADIALS: INITIAL RAY, ANGLE, FINAL RAY; BIRADIAL PLANE and ASPECT; ASPECTS: CONDIRECTIONAL, CONTRADIRECTIONAL; EQUIVALENT BIRADIALS.
			Art.94.Fig.18: Illustrative example of biradial ASPECT (rotation axis or versor direction), ANGLE (power of versor between radials), and RATIO (of tensors)
			Art.95: Equivalent biradials represent equal quotients δ÷γ=β÷α; CONDITIONS OF EQUALITY; MODES OF INEQUALITY.
		CONT/§ XIX. Articles 96 to 101; Pages 95 to 105.
		LECT/§ XIX. Articles 96 to 101; Pages 95 to 105.
			Art.96: A VECTOR is a natural TRIPLET; A BIRADIAL represents a QUATERNION.
			Art.97.Figs.19,20: Condition of equality of ordinal relations (B-A)=(D-C): ABDC must be a parallelogram (which may collapse onto a line).
			Art.98.Fig.21: (B-A)=(D-C) remains true when any two of the points are translated together.
			Art.99: For ▱ABDC, (B-A)=(D-C), (C-A)=(D-B) and D=(D-C)+C=(D-B)+B=(D-A)+A, then 4th point D = (B-A)+C=(C-A)+B ⇄ (B-A)+(C-A)+A for any 3 points A,B,C.
			Art.100: Addition of vectors is associative and commutative, α+(β+γ)=(γ+β)+α.
			Art.101: VECTOR TRINOMIAL FORM ix+jy+zk; 3 acts on D-C breaking vector equality ▱ABDC: stretch/shrink, turn within or out of plane ABC.
		CONT/§ XX. Articles 102 to 107; Pages 106 to 112.
		LECT/§ XX. Articles 102 to 107; Pages 106 to 112.
			Art.102: Referring to Fig.18 with versor vector-unit k into the page, angles are 60° or 2/3 quad, and ratios of lengths are 2/1.
			Art.103.Fig.22: Quotient β÷α=(Tβ/Tα)×ι^t of vectors depends only on RELATIVE length Tβ/Tα and RELATIVE direction or turn ι^t to β by α.
			Art.104: The angle t alone does not specify the axis ι of (right-handed) rotation required for full determination of the turn in space.
			Art.105: Equality of vector quotients δ÷γ=β÷α allows uniform translations, scalings, and rotations of the vectors.
			Art.106.Fig.23: Inequality of quotients δ÷γ≠β÷α for any difference in (1) relative lengths, (2) angles, or (3,4) planar,polar angles of biradial versor.
			Art.107: QUOTIENT OF TWO VECTORS, or a BIRADIAL, involves FOUR NUMBERS: TWO FOR SHAPE and TWO FOR PLANE; or a QUATERNION.
		CONT/§ XXI. Article 108; Pages 112, 113.
		LECT/§ XXI. Article 108; Pages 112, 113.
			Art.108: QUATERNIONS q″=(δ÷γ) q′=(γ÷β) q=(β÷α) MULTIPLICATION q″q′q is ASSOCIATIVE; POINT notation (art.112): q″q′.q=q″.q′q
		CONT/§ XXII. Articles 109 to 112 ; Pages 113 to 117.
		LECT/§ XXII. Articles 109 to 112 ; Pages 113 to 117; TENSOR: SIGNLESS NUMBER, METRIC ELEMENT of a FACTOR.
			Art.109: TENSOR OF SCALAR w: arithmetical magnitude, no algebraical sign; TENSOR OF VECTOR ρ: geometrical magnitude, no graphical direction.
			Art.110: VECTOR ρ=xi+yj+zk; TENSOR Tρ is LENGTH of ρ; TENSOR IDENTITIES: T.κλ=Tκ.Tλ⁣, T(λ÷κ)=Tλ÷Tκ.
			Art.111: QUATERNION q=w+ρ; TENSOR Tw=√(ww) OF SCALAR; TENSOR Tρ=√(-ρρ)=√(xx+yy+zz) OF VECTOR; TENSOR Tq=T(w+ρ)=√(ww+xx+yy+zz)=√(ww-ρρ) OF QUATERNION.
			Art.112: TENSOR T.ρσ OF PRODUCT ρσ OF LINES ρ⊥σ: T.ρσ=Tρ.Tσ=√(-ρρ)√(-σσ)=√(+ρ.ρσ.σ) & T.ρσ=√(-ρσρσ)=√(-ρ.σρ.σ) since ρσ=-σρ per opposite LINES (art.82).
		CONT/§ XXIII. Articles 113, 114; Pages 118, 119.
		LECT/§ XXIII. Articles 113, 114; Pages 118, 119.
			Art.113: VERSOR Uw of SCALAR w: Uw=w÷|w|=±1; VERSOR Uρ of VECTOR ρ=TρUρ: Uρ=ρ÷Tρ=ρ÷√(-ρρ); UρUρ=-1; U0=particular.
			Art.114: CONJUGATE of SCALER w Kw=w, VECTOR ρ Kρ=-ρ, QUATERNION q=w+ρ K(w+ρ)=w-ρ; CONJUGATE FACTORS have same TENSOR, opposite VERSOR.
		CONT/§ XXIV.ERR.xv. Articles 115 to 118; Pages 119 to 125.
		LECT/§ XXIV. Articles 115 to 118; Pages 119 to 125.
			Art.115: Power ρ^t, of any VECTOR ρ, is an AGENT (as a FACTOR) of t QUADs of rotation in any plane perpendicular to Uρ, and of scaling by (Tρ)^t.
			Art.116: IF ρ^t=(TρUρ)^t=(Tρ)^t(Uρ)^t THEN T.ρ^t=(Tρ)^t, U.ρ^t=(Uρ)^t=(+1|t=4x,x∊Z)=(-1|t=2x,x∊odd)=(⊥VECTOR-UNIT|t∊odd)=(QUATERNION|else).
			Art.117.Fig.24: Reciprocal ρ^-1=(TρUρ)^-1=(Tρ)^-1.(Uρ)^-1=(1/Tρ).(-Uρ)=(-Uρ/Tρ); Fig.24: ρ^-1=(ρ^-1÷1)×(1÷ρ)×ρ.
			Art.118: Notations for division, multiplication, and angles: β÷α=β/α=β×α^(-1); α^-1=1/α; Angle between α and β: ∠(β÷α)=∠(β×α^-1).
		CONT/§ XXV. Articles 119, 120; Pages 125 to 129.
		LECT/§ XXV. Articles 119, 120; Pages 125 to 129.
			Art.119.Fig.25: LOGARITHMIC SPIRAL σ=ρ^t×a: a=(A-O)i=i, d=(D-O)j=j√8, ρ=(d÷a)=k√8; ANGULAR STEP τ=ρ^(t+h)×a, CONST BIRADIAL τ÷σ=ρ^h, TANGENT RAY lim,h→0(τ-σ).
			Art.120.ERR.129: EVERY QUOTIENT β÷α=ρ^t of TWO RAYs β∦α is a \'QUADIANS\' EXPONENTIAL 00; At limit ∠q=π, q becomes SCALAR<0; At ∠q=π/2, q is a quadrantal versor, or VECTOR; Else, q is a QUATERNION.
			Art.150: qᵘqᵗ = qᵘ⁺ᵗ holds good in quaternions, as two successive rotations t∠q + u∠q, or single rotation (u+t)∠q round axis q.
		CONT/§ XXXIII. Articles 151 to 161; Pages 166 to 174.
		LECT/§ XXXIII. Articles 151 to 161; Pages 166 to 174.
			Art.151: ∠(qᵗ)=t∠q not confined to 0≤(t∠q)≤π; however, ∃ n∈ℤ ∣ -π≤(t∠q+2nπ)≤π holds; IF -π≤(t∠q+2nπ)<0, THEN reverse angle & axis.
			Art.152: For 0½π: (q²)^½=-q; E.G.∵ ∠q=135° ∠q²=270°=-90° ∴ Ax.q²=-Ax.q ∠q²=90°, Ax.(q²)^½=-Ax.q ∠(q²)^½=45°=π-∠q, (q²)^½=-q (cf.Art.183).
			Art.159: Since generally ᵗ√(qᵗ)≠q, then also generally (qᵗ)ᵘ≠qᵘᵗ; rⁿ=q is satisfied by n distinct values of r.
			Art.160: General powers: n,m∈ℤ, v̂=Ax.q, (qᵗ)ᵘ=v̂⁴ⁿᵘ.qᵗᵘ=v̂⁴ⁿᵘ.(v̂ˢ)ᵗᵘ=v̂⁴ⁿᵘ.v̂⁴ᵐᵗᵘ.v̂ˢᵗᵘ=v̂⁴⁽ⁿ⁺ᵐᵗ⁾ᵘ.v̂ˢᵗᵘ=v̂ᵘ⁽ᵗ⁽ˢ⁺⁴ᵐ⁾⁺⁴ⁿ⁾; multiple full rotations may add to powers.
			Art.161.ERR.174: (r³=[k^(2/9)]³=r′³=[k^(14/9)]³=r″³=[(-k)^(10/9)]³)^⅓=̂ᵥe₁=⅓(3(2/9+4m)+4n);Try n=∈{0,1,-1},m=0;n=-1: k^-4/3.r=k^-4/3.k^2/9=k^-10/9=(-k)^10/9.
		CONT/§ XXXIV. Articles 162 to 165; Pages 175 to 178.
		LECT/§ XXXIV. Articles 162 to 165; Pages 175 to 178.
			Art.162: q=TqUq; Uq=v̂ᵗ=cv̂s(tπ/2)=cos(tπ/2)+v̂sin(tπ/2), KUq=(-v̂)ᵗ=cv̂s(-tπ/2)=v̂⁻ᵗ=Uq⁻¹; q⁻¹=Tq⁻¹.KUq.
			Art.163: Tq=TKq, KUq=UKq=Uq⁻¹; Kq=TKq.UKq=Tq.Uq⁻¹; qKq=TqUqTqUq⁻¹=Tq²; Tq=√(qKq); Uq=q÷Tq=q÷√(qKq); Uq²=q÷Kq.
			Art.164.ERR.177: (q÷Kq)×(q÷q)=q²÷Tq²=Uq²; Acute:∠q²=2⦟q;Obtuse:∠q²=2(π-⦦q),Ax.q²=-Ax.q; L18:(q÷Kq)^½=∓Uq as ∠q≷½π.
			Art.165: IF ∠q=½π, THEN q is a VECTOR; Kq=-q; q÷Kq=Uq²=-1; √(q÷Kq)=√(Uq²)=√-1 is an INDETERMINATE VECTOR-UNIT.
		CONT/§ XXXV. Articles 166 to 174; Pages 178 to 185.
		LECT/§ XXXV. Articles 166 to 174; Pages 178 to 185.
			Art.166: INVERSOR (-1)ᵗ=ι²ᵗ: ∠(-1)=π, ∠(-1)ᵗ=tπ, Ax.(-1)=ι indeterminate and arbitrary.
			Art.167: Arbitrary VECTOR-UNIT ι=√-1: ∠ι=½π, Tι=1; e^(t½πι)=[e^(πι)]^(½t)=(-1)^(½t)=(ι²)^(½t)=ιᵗ=cιs(t½π)=cos(t½π)+ιsin(t½π).
			Art.168: ρ=P-O=√-1=ι, P=ι+O, Tι=1; ι represents any STEP from O to any point P on unit-radius SPHERICAL LOCUS from O, or UNIT-SPHERE ρ²+1=0.
			Art.169: Center β=B-O, Radius Tβ=b, any Sphere Point P=B+bι=P-O=ρ; ρ-β=bι; Sphere Eq: (ρ-β)²+b²=0; T(ρ-β)=T.bι=b=T(P-B); O is on Sphere.
			Art.170: ρ÷α=ι α is given; ∠ι=½π: ρ,α,ι perpendicular; Tρ÷Τα=1,Tρ=Τα; ρ=P-O: all P⟂α in great circle on sphere (origin O, radius Tρ=Τα).
			Art.171: ρ÷α=ι, (ρ÷α)²=ι²=-1, (ρ÷α)×ρ=-α: 3RD∝=-α; the versors Uρ,Uα,Uι are quadrantal.
			Art.172: For any Tρ, U.ρ÷α=ι is condition on ρ to be in PLANE⟂α thru P=O, or that ρ⟂α; U.(ρ-β)÷α=ι PLANE⟂α thru P=B, or that (ρ-β)⟂α.
			Art.173: ρ÷α=(-1)^⅓=(ι²)^⅓=ι^⅔, ∠ι^⅔=60°, Tρ=Tα: CIRCLE of all ρ 60° to α; U.ρ÷α=(-1)^⅓, no condition on Tρ: CONE, ∠.ρ÷α=⅓π, axis α.
			Art.174: (-1)ᵘ(-1)ᵗ=(-1)ᵘ⁺ᵗ is true IF the 3 arbitrary axes, represented by each -1=ι², are considered to coincide.
	CONTENTS./LECTURE V. Articles 175 to 250; Pages 186 to 240.
	LECTURES./LECTURE V. Articles 175 to 250; Pages 186 to 240.
		CONT/§ XXXVI. Articles 175 to 182; Pages 186 to 192.
		LECT/§ XXXVI. Articles 175 to 182; Pages 186 to 192.
			Art.175: Associative Principle of Multiplication: Is β.α⁻¹γ=βα⁻¹.γ, where α∣∥β,γ ?
			Art.176: (vid.Fig.22) and ε ⊗: βα⁻¹.γ=δ; αε⁻¹=γ, βε⁻¹=δ: α⁻¹γ=ε⁻¹, β.α⁻¹γ=βε⁻¹=δ=βα⁻¹.γ; assoc. mult. holds for α,β,γ.
			Art.177: α⟂γ: β∷γ:δ, δ=βα⁻¹.γ; inv+alt γ:α∷δ:β, K.βδ⁻¹=K.αγ⁻¹, γ⁻¹α=ε=δ⁻¹β, δ=βε⁻¹, ε⁻¹=α⁻¹γ; Now: βα⁻¹.γ=δ=β.α⁻¹γ.
			Art.178: α∥γ: γ=cα=αc, α⁻¹γ=c; βα⁻¹.γ=c(βα⁻¹.α)=cβ=βc=β.α⁻¹γ; γ=γ′+γ″ (γ′∥α,γ″⟂α), βα⁻¹.γ=βα⁻¹.γ′+βα⁻¹.γ″=β.α⁻¹γ′+β.α⁻¹γ″=β.α⁻¹γ.
			Art.179: IDENTITY:ζη.η⁻¹θ=ζθ ∣ η=TηUVη arbitrary, particular η chosen as needed; quaternions prod r.q=[xform to vectors prod]=ζη.η⁻¹θ=ζθ.
			Art.180: r=β=ζη, let η⟂β⫵ζ; Kq=K.α⁻¹γ=K.η⁻¹θ ∴ α:η∷γ:θ α:γ∷η:θ; K.β⁻¹rq=K.α⁻¹γ=K.β⁻¹ζθ ∴ α:β∷γ:ζθ α:γ∷β:ζθ;⊢ η:θ∷β:ζθ θ⟂ζθ, ⫴(η,θ,α,γ,β,ζθ=δ)⫵ζ.
			Art.181.Fig.30: Choose ±η ∋ UVζ=k, UVα=i, UVδ=k⁻⅑j; η:θ∷β:ζθ ⊢ β=ζη η:ζη∷θ:ζθ ζθ=δ, ζη=δθ⁻¹η; ζθ=ζη.η⁻¹θ=δθ⁻¹η.η⁻¹θ = β.α⁻¹γ = δ = βα⁻¹.γ ∎
			Art.182: Joining proportionals gives: α:γ∷ η:θ∷ζη:ζθ ∷β:δ or α:β∷ η:ζη∷θ:ζθ ∷γ:δ; rotation by ∠(β÷α) xform to rotation by ∠ζ=90°
		CONT/§ XXXVII. Articles 183 to 193; Pages 192 to 198.
		LECT/§ XXXVII. Articles 183 to 193; Pages 192 to 198.
			Art.183.Fig.31: ±BIRADIAL: +q=β÷α, -q=-1×q=-β÷α; T(-q)=Tq, 0<[∠(-q)=⦞q=(π-∠q)]<π, Ax.(-q)=-Ax.q; ≠ by DEF: 0≶[∠(-q)≠(±π+∠q)]≷π, Ax.(-q)≠Ax.q.
			Art.184: Negative ⇄ Conjugate: ∠Kq=∠q, Ax.Kq=-Ax.q; ∠(-q)=⦞q=(π-∠q)=(π-∠Kq)=⦞Kq; Ax.(-q)=-Ax.q=Ax.Kq.
			Art.185: Negative of Conjugate: T(-Kq)=Tq; ∠(-Kq)=⦞Kq=(π-∠Kq)=(π-∠q)=⦞q; Ax.(-Kq)=-Ax.Kq=-(-Ax.q)=Ax.q; KKq=q or KK=1.
			Art.186.Fig.32: Illustration of negative conjugate: γ÷α=Kq, -γ÷α=-Kq; Definition of Conjugate: ∠Kq=∠q, Ax.Kq=-Ax.q.
			Art.187: ∠K(-q)=⦞q=∠(-Kq), Ax.K(-q)=Ax.q=Ax.(-Kq): K(-q)=-Kq; IF -Kq=q THEN ∠(-Kq)=∠q=⦞q=(π-∠q),∠q=½π=t(½π),t=1, -Kq=qᵗ is a vector.
			Art.188: Products and quotients of VERSORS: U.κλ=Uκ.Uλ; U(λ÷κ)=Uλ÷Uκ; Uγ÷Uα=(Uγ÷Uβ)×(Uβ÷Uα), Tγ÷Tα=(Tγ÷Tβ)×(Tβ÷Tα); Tensors factor out.
			Art.189: Tension and Version are mutually independent acts; (T.rq)²=(Tr.Tq)²=Tr².Tq²; Tensors follow rules of ordinary arithmetic.
			Art.190: K.rq = Kq.Kr; Let qa=β,rβ=γ,rq.α=γ;Kr.γ=Kr.rβ=Tr².β: (Kq.Kr).γ=Tr²(Kq.β)=Tr²(Kq.qα)=Tr²Tq².α=(T.rq)².α=(K.rq×rq).α=K.rq.γ, ABRIDGE γ.
			Art.191: Version (& tension) α to γ: rq.α=γ; Reversion (& tension again) γ to α: Kq.Kr.γ=Κ.rq.γ=Tr²Tq².α; conjugates do only axis reversion.
			Art.192: (rq)⁻¹=q⁻¹r⁻¹; rq=γ÷α, (rq)⁻¹=α÷γ; q⁻¹=α÷β, r⁻¹=β÷γ; ε=γ⁻¹α, ε⁻¹=α⁻¹γ; inverses do reversion & retension by undoing steps backwards.
			Art.193: Vectors κ,λ, α:β∷γ:δ : K.κλ=Kκ.Kλ=-κ.-λ=κλ; Kα⁻¹=(-α)⁻¹=-α⁻¹, K(γα⁻¹.β)=Kβ.K.γα⁻¹=Kβ.Kα⁻¹Kγ=-β.α⁻¹γ=Kδ, -Kδ=δ=β.α⁻¹γ=βα⁻¹.γ (art.182).
		CONT/§ XXXVIII. Articles 194 to 200; Pages 198 to 203.
		LECT/§ XXXVIII. Articles 194 to 200; Pages 198 to 203.
			Art.194: ASSOC. MULT. holds; coplanar vectors α:β∷γ:δ or α:γ∷β:δ, partial-commutative CONTINUED PRODUCTS for 4TH∝ δ = βα⁻¹γ = γα⁻¹β.
			Art.195: Continued product-line of coplanar vectors μ⫴λ,κ (or 4th∝ to μ⫴λ⁻¹,κ): κλμ=μλκ; λ:κ∷μ:(κλμ=μλκ).
			Art.196: ∵ βαγ=βα²α⁻¹γ=β(-Tα²)α⁻¹γ=(-Tα²)βα⁻¹γ=α²βα⁻¹γ, ∴ βαγ=α².βα⁻¹γ=α².δ=-Tα².δ, 4th∝ δ reversed and scaled by TαTα.
			Art.197: (vid.Fig.26) α:β∷γ:δ or α:γ∷β:δ; α²βα⁻¹γ=α²γα⁻¹β = βαγ=γαβ = α²δ=-Tα².δ [tangent←|→ at A to ⥀|⥁ flux circle C⟲B⟲A|A⟳B⟳C]; TαTαTδ=TαTβTγ.
			Art.198: Any 3 POINTS ▷ABC inscribed in circle: ⥀↳ A◁[α=C-B] → B△[β=A-C] → C▽[γ=B-A] ↰⥀; βαγ=γαβ=α²δ is ⥀ tangent to circle at A.
			Art.199.Figs.33,34,35: 3 POINTS on a line: βαγ=TβTαΤγ.UVβUVαUVγ=±UVγ; →.→.↔ = ←.←.↔ =-1.↔; →.←.↔ = ←.→.↔ = +1.↔
			Art.200: (vid.Figs.27,28) 4 CONCIRCULAR POINTS ABCD: CONTINUED PRODUCT U.(γ=D-C)(α=C-B)(β=B-A)=±U(A-D)=-Uδ in direction of uncrossed 4th side; .
		CONT/§ XXXIX. Articles 201 to 210; Pages 203 to 208.
		LECT/§ XXXIX. Articles 201 to 210; Pages 203 to 208.
			Art.201: 3 non-coplanar lines α,β,γ: α⫵β,γ ~ β⫵α,γ ~ γ⫵α,β; Given: [γ=α÷ε]⟂α, ∴ ε⟂α ε⟂γ β⫵α,γ ε⌿β.
			Art.202: 4th∝ δ to non-coplanar lines: δ=β÷α.γ=β÷α.α÷ε=β÷ε, ε⌿β ∴ δ is a non-quadrantal (non-VECTOR) QUATERNION.
			Art.203: γ⫵α,β; γ=γ′+γ″, γ′∥α, γ″⟂α; βα⁻¹.γ = βα⁻¹.(γ′+γ″) = βα⁻¹.γ′ + βα⁻¹.γ″ = β.α⁻¹γ′ + β.α⁻¹γ″ = β.α⁻¹(γ′+γ″) = β.α⁻¹γ.
			Art.204: Suppose new line λ and β⟂α, β=λα, βα⁻¹=λ: β.α⁻¹γ=λα.α⁻¹γ=λγ=βα⁻¹.γ; λ⟂α λ⟂β γ⫵α,β λ⌿γ ∴ λγ (~ β÷ε) a QUATERNION.
			Art.205: 4th∝ δ to rectangular lines β⟂α γ⟂α γ⟂β: δ=β÷α.γ, Tδ=(TβΤγ÷Tα), Uδ=Uβ÷Uα.Uγ=±Uγ.Uγ=±(-1), if α:β:γ∷i:j:k then ±=+, else ±=-.
			Art.206: Example: 4th∝ δ to j,k,i: δ=k÷j.i=i.i=-1; 4th∝ δ to j,i,k: δ=i÷j.k=-k.k=+1.
			Art.207: Cont.product α²δ of rect.vectors α:β∷γ:δ α²δ=α²βα⁻¹γ=βαγ>0 if Ax.γ,∠(β÷α)=½π>0; For alt. α:γ∷β:δ α²γα⁻¹β=γαβ=(-1)³βαγ [3 commutes,~ kij=(-1)³jik].
			Art.208: Tensor of product is product of tensors: TΠ=ΠT; Tensors can always factor out by ordinary arithmetic, leaving only versor products.
			Art.209.ERR.208: βαγ=-βγα=+γβα=-γαβ=+αγβ=-αβγ=±TαTβTγ if α⟂β β⟂γ γ⟂α, ~ i⟂j j⟂k k⟂i or α:β:γ∷i:j:k (rectangular vectors).
			Art.210: The UNIT-CUBE edges are VECTOR-UNITS i j k: kji=j²kj⁻¹i=-kj⁻¹i=+1; ijk=j²ij⁻¹k=-ij⁻¹k=-1; i²=j²=k²=ijk=-1.
		CONT/§ XL. Articles 211 to 216; Pages 208 to 212.
		LECT/§ XL. Articles 211 to 216; Pages 208 to 212.
			Art.211: γ⫵α,β γ⌿α; biradial β÷α rotated in-plane to equiv β′÷α′ where γ⊥α′, γ=α′÷ε, α′⊥ε; β′÷α′.γ=β′÷ε = β÷α.γ=β÷α.α′÷ε; β′=β÷α.α′.
			Art.212: κ=ε λ=α′ μ=β′; γ=λ÷κ=α′÷ε, β÷α=μ÷λ=β′÷α′, β÷α.γ=μ÷κ=β′÷ε=β÷α.α′÷ε.
			Art.213: If lines α β γ are not coplanar, the product βα⁻¹γ is a quaternion; If coplanar, βα⁻¹γ is the 4th proportional coplanar line.
			Art.214.ERR.211: LINEs α,β,γ,ζ,η,θ: γ⫵α,β; Let r=β=ζη η⊥β ∴ ζ⊥(η,β); Let q=α⁻¹γ=η⁻¹θ ∴ α:η∷γ:θ ⫴(α,γ,η,θ); rq=βα⁻¹γ=ζθ=ζθ²θ⁻¹=-Tθ².ζθ⁻¹ ∴ ζ⌿θ.
			Art.215: T.μκ⁻¹ = TβTα⁻¹Tγ; Uγ=Uλ÷Uκ, Uβ÷Uα=Uμ÷Uλ; Uμ÷Uκ μ÷κ determined.
			Art.216: VECTOR-UNITs T*=1: α=A-O, β=B-O, γ=C-O, ι=I-O, η=H-O, θ=G-O, κ=K-O, λ=L-O, μ=M-O; βα⁻¹ with angle ∠AOB represented by great circle arc ⌒AB.
		CONT/§ XLI. Articles 217 to 222; Pages 212 to 217.
		LECT/§ XLI. Articles 217 to 222; Pages 212 to 217.
			Art.217.Fig.36: TRANSVECTOR-ARC KM (rq) = PROVECTOR-ARC LM (r) × VECTOR-ARC KL (q); rq=(M-O)÷(L-O) × (L-O)÷(K-O)=(M-O)÷(K-O).
			Art.218: Multiplication of versors -or- Addition of arcual vectors; ARCs as for LINEs, ARCUAL SUM: Transvector = Provector + Vector.
			Art.219.Fig.37: ARCUAL INEQUALITY of equal length arcs, but on different great circles: ⌒M′K′ ≠ ⌒KM, or qr≠rq; NON-COMMUTATIVE ×.
			Art.220: ◁LK′M shows rq⁻¹.q=r; rq⁻¹=r÷q; M÷L=r≘⌒LM, r⁻¹≘⌒ML, K′÷L=q≘⌒LK′, q⁻¹≘⌒K′L; ⌒K′M=⌒LM-⌒LK′≞(M-L)-(K′-L)=M-K′≘r÷q; (M-K′)+K′=M=(r÷q)×K′.
			Art.221: ⌒K′M≘(M÷L)×(L÷K′)=M÷K′=r÷q=rq⁻¹ ≠ q⁻¹r=(K÷L)×(L÷M′)=K÷M′≘⌒M′K; Arcs of different great circles on unit-sphere.
			Art.222.Fig.38: INVERSE q⁻¹=Kq÷Tq², REVERSOR Uq⁻¹=UKq; When ABSTRACTING TENSORS Tr≟1≟Tq: q⁻¹≟Kq CONJUGATE, K.rq = Kq.Kr ≟ q⁻¹r⁻¹=(rq)⁻¹;
		CONT/§ XLII. Articles 223 to 235; Pages 217 to 228; In this section, it is important to see that Hamilton uses LEFT-HANDED ijk with ⥁clockwise (retrograde) rotations.
		LECT/§ XLII. Articles 223 to 235; Pages 217 to 228; Celestial Sphere ∠s: °α(Ter.Lon/Right-Asc ∠⥀Cel.N-Pole/EQUATOR) °δ(Ter.Lat/Dec +∠[EQUA→Cel.N-Pole=k]) °λ(Cel.Lon ∠⥀Ecl.N-Pole/ECLIPTIC) °β(Cel.Lat +∠[ECLI→Ecl.N-Pole]).
			Art.223.Fig.39: ⥁LEFT-HANDED ijk; ⊙:C,♋; Ver.E ♈(Aries)=0°λ=-i ∩ ECLI=i^(ε=23.4°δ)×EQUA, Sum.S ♋(Cancer)=90°λ, Aut.E ♎(Libra)=180°λ=i, Win.S ♑(Capricorn)=270°λ; A=(100°λ,0°β) B=(70°λ,0°β) C=(90°α,0°δ)=j.
			Art.224.Fig.40: ⥁LEFT-HANDED ijk; ⊙:OC=γ=j; EQUA⤓0°δ ∩ ECLI⦨ε°δ; Cel.N-Pole K=k; L′=♈ Q=♋ L=♎; ⊾:⌒CL,⌒CK,⌒KL; γ=L÷K=C=j≘⌒KL; Ecl.N-Pole K′=k′=i^ε×k; β÷α=B÷A=k′^30°λ≘⌒AB=⌒LM≘M÷L; ⌒KM≘(M÷L).(L÷K)=β÷α.γ.
			Art.225: SPHERICAL-ANGLE ∢KND=∠[betw.gr.○](⌒NK,⌒ND)≤90° ?⊾NK=⊾ND=90°:N=pole,⌒KD=polar; ⌒CD=⌒LN ⊾ND=90°α ⌒QR=⌒LM ⊾MR=90°λ; ∵pole(MN)=D∴⊾MD=90°; ∵pole(DR)=M pole(DK)=N∴∢RDK=∡MN=∢MDN ~ ∢L′DR=∡KM=∢KDM.
			Art.226: ⌒EC=⌒CD=⌒LN; ∢LSE=∢L′RD=90°; Symmetry about ⌒CK′: ∡DR=∡ES=∡TF, ⌒RL′=⌒LS, ⌒SQ=⌒QR=⌒LM=⌒AB, ⌒SR=2×⌒AB; ⌒AB=⌒SA+⌒BR≡⌒AT+⌒TB; △ABC bisects △DEF; Ax.(βα⁻¹γ)=D-O, ∠(βα⁻¹γ)=∢L′DR.
			Art.227.Fig.41: ∡DR=∡ES=∡T′F′; ∡PD=∡PE=∡PF′; ∢L′DR=∢CDP=∢CEP=a, ∢PDF′=∢PF′D=b, ∢PEF′=PF′E=c; D=π-(a+b) E=π-(a+c) F′=F=(b+c); π-½[D+E+F]=π-½[π-a-b+π-a-c+b+c]=a=∡KM.
			Art.228: The LEFT-HANDED orientation is explained: L=C×K ⥁; M=K′^30°×L ⥁; L-H longitude rotations are RETROGRADE A=100°λ → B=70°λ, -30°λ and approaching K; ∡KM<90°.
			Art.229: Symmetry about ⌒LK for αβ⁻¹γ: ⌒BA=⌒ML=⌒LM′, ⌒DC=⌒CE=⌒NL=⌒LN′, ∵⊾DN=⊾EN′=90° ⊾EK=90° ∴pole(KN′)=E, E-O=Ax.(αβ⁻¹γ); ∡KM+∡KM′=π, ∡KM′=π-a=π-∠(βα⁻¹γ)=∠(αβ⁻¹γ).
			Art.230: ∡KC=∡KE pole(CE)=K; ∵⌒QS=⌒ML=⌒LM′ ∡QL=∡SM′=90° E=pole(KN′)=pole(KM′) ∡EM′=90° ∴pole(ES)=M′ ∢CES=∡KM′=∠(αβ⁻¹γ)=π-a=½[D+E+F]; Supplementary: ∠(αβ⁻¹γ) ∠(βα⁻¹γ).
			Art.231: RULE of Arts.226,230: Ax.=D|E|F opp. 1st=A|B|C; ∠.= ½[D+E+F]∣(⥁1st∡2nd3rd>0) or⦞ π-½[D+E+F]∣(⥁1st∡2nd3rd<0); ∠.≞ ⥁Left-Handed (or ⥀Right-Handed if set up R-H).
			Art.232: RULE of Arts.226,230: βα⁻¹γ: ∵1st=α∴Ax.=D-Ο 2nd=β 3rd=γ ∵⥁α∡βγ<0∴∠.=π-½[D+E+F]; αβ⁻¹γ: Ax.E-Ο ∵⥁β∡αγ>0∴∠.=½[D+E+F]; γα⁻¹β: Ax.=D-O ∵⥁α∡γβ>0∴∠.=½[D+E+F].
			Art.233: △ABC being given, △DEF is determined without ambiguity under the conditions supposed (∀ ∠<90°).
			Art.234: AUXILIARY TRIANGLE △DEF; Supplemental representative spherical angles: ∠(γα⁻¹β) =½[D+E+F]=π-∠(βα⁻¹γ)= π-∢L′DR=∢RDC= ∠(αβ⁻¹γ)=∢CES; Poles: D,E.
			Art.235: Ax.(βα⁻¹γ)=Ax.(γα⁻¹β) ∠(γα⁻¹β)=π-∠(βα⁻¹γ); {βα⁻¹γ,γα⁻¹β}~{q,-Kq}: Ax.(-Kq)=-Ax.Kq=-(-Ax.q)=Ax.q ∠(-Kq)=⦞Kq=(π-∠Kq)=(π-∠q)=⦞q; q=-K(-Kq): βα⁻¹γ=-K(γα⁻¹β).
		CONT/§ XLIII. Articles 236 to 240; Pages 228 to 233.
		LECT/§ XLIII. Articles 236 to 240; Pages 228 to 233; In this section, rotations continue to be given in LEFT-HANDED ⥁ ijk orientation.
			Art.236.Fig.42: ⌒GH=⌒CA pole=X; L-H:θη⁻¹⥁⥀R-H:ηθ⁻¹⥀⥀L-H:η⁻¹θ (cf.Art.87:κλ,λκ rotate s,-s); B-O=β=ι×η; ιηη⁻¹θ=βα⁻¹γ=ιθ, ιθ×θ=-ι=I′ ιθ≘⌒GI′ by RULE: Ax.=D ∠.=π-½[D+E+F]
			Art.237: (vid.Fig.40) As ∡AB→90°: Ax.(βα⁻¹γ)=D→L′ E→L F→T→Q; ∠D,∠E→ε=∢TLC=∡QC ∠F=∢L′FL→∢L′QL=π; ∠(βα⁻¹γ)=π-½[D+E+F]→π-½[ε+ε+π]=½π-ε=∢KL′Q=∡KQ=½π-∡QC.
			Art.238: RULE of Art.230 altered when ∡AB>90°: D,E in 4th,3rd quads, TF↓, angles go negative: ∠(βα⁻¹γ)=½[D+E+F]-π=π-∠(αβ⁻¹γ), ∠(αβ⁻¹γ)=2π-½[D+E+F]=-½[D+E+F].
			Art.239: RULE: ∡AB,∡C′A,∡C′B>90°: γ′=(C′-O)=-γ; βα⁻¹γ′=-βα⁻¹γ; Ax.(-βα⁻¹γ)=-(D-O)=(O-D); ∠(-βα⁻¹γ)=π-(½[D+E+F]-π)=2π-½[D+E+F]; (cf.Art.183: ±BIRADIAL).
			Art.240: The Associative Property of Multiplication of THREE VECTORS is therefore fully proved, with assistance of Spherical Geometry.
		CONT/§ XLIV. Articles 241 to 244; Pages 233 to 237.
		LECT/§ XLIV. Articles 241 to 244; Pages 233 to 237; RULE of Arts.226,230 applied to triquadrantal vectors (using LEFT-HANDED ijk orientation).
			Art.241: When α,β,γ are triquadrantal: ∵ βα⁻¹γ=±NUMBER ∴ Ax.(βα⁻¹γ)=√-1=[arbitrary axis]; △DEF is arbitrary; ∠(βα⁻¹γ)=0|π (non-versor|inversor)
			Art.242.Fig.43: Construction of the arbitrary △DEF, when α,β,γ are triquadrantal; the particular axis D-O is chosen arbitrarily and represents Ax.=√-1.
			Art.243: Round G-O: 2π=(∢AGB=∢AFB=∢EFD=∠F)+(∢BGC=∢BDC=∢FDE=∠D)+(∢CGA=∢CEA=∢DEF=∠E)=4*½π=[D+E+F]; Ax.(βα⁻¹γ)=√-1, ∠(βα⁻¹γ)=π-½[D+E+F]=0; βα⁻¹γ=√-1⁰=-γγ=1.
			Art.244: ∵ α:β∷γ:(δ=βα⁻¹γ) LEFT-HANDED round 1stA, A-O: ⌒2ndB3rdC≘∡BC<0 ∴ ∠(βα⁻¹γ)=π-½[D+E+F]; ∵ αβ⁻¹γ L-H round B-O ∡AC>0 ∴ ∠(αβ⁻¹γ)=½[D+E+F]=π, αβ⁻¹γ=√-1²=γγ=-1.
		§ XLV. Articles 245 to 250; Pages 237 to 240.
		§ XLV. Articles 245 to 250; Pages 237 to 240.
			Art.245: (vid.Fig.43) Left-Handed: i=α=OA j=γ=OC k=β=OB; βα⁻¹γ=-γγ=k(-i)j=kji=-ii=+1; αβ⁻¹γ=γγ=i(-k)j=ijk=ii=-1.
			Art.246: Every MULTIPLICATION OF VERSORS corresponds to some COMBINATION OF VERSIONS (composition of rotations); VERTEND λ: μ=iλ ν=jμ=jiλ, VERSUM ξ=kν=kjμ=kjiλ.
			Art.247: LEFT-HANDED construction (cw rotations round VERSOR axes), λ=j=γ: (kBYj,cw rnd i) μ=iλ=ij=k=β, (iBYk,cw rnd j) ν=jμ=jk=i=α, (jBYi,cw rnd k) ξ=kν=ki=j=γ.
			Art.248: γ:β:α:γ or ξ=kjiλ=kjiγ=λ=γ; kji=+1 or ξλ⁻¹=+1; NONVERSOR kji.
			Art.249: Initial VERTEND λ can be chosen arbitrarily, but kjiλ=λ will still hold good since kji=+1=NONVERSOR, as before.
			Art.250: Still LEFT-HANDED, opposite order k,j,i: kλ=kj=-i=-α, j(-i)=k=β, ik=-j=-λ=-γ; ijkλ=-λ, ijk=-1=INVERSOR.
	CONTENTS./LECTURE VI. Articles 251 to 393; Pages 241 to 380.
	LECTURES./LECTURE VI. Articles 251 to 393; Pages 241 to 380.
		CONT/§ XLVI. Articles 251 to 257; Pages 241 to 247.
		LECT/§ XLVI. Articles 251 to 257; Pages 241 to 247.
			Art.251: Introductory remarks on continuing the study of the Associative Principle, and some expressions for rotations of solids.
			Art.252: ∵ Cross-way ∢DEF=π ∢FDE=∢EFD=0 SphArea(△DEF)=0=(D+E+F-π)r²=SphExcess∣r=1 ∴ ½[D+E+F]=½π=π-½[D+E+F] t=1, δᵗ=βα⁻¹γ=γα⁻¹β εᵗ=γβ⁻¹α=αβ⁻¹γ ζᵗ=αγ⁻¹β=βγ⁻¹α.
			Art.253.Fig.44: ∠α=a=½(y+z)=0 ∠β=b=½(z+x) ∠γ=c=½(y+x) ⇒ ∠δ=x=b+c-a ∠ε=y=c-b+a ∠ζ=z=b-c+a; (vid.Fig.40) ∢CLQ=0: a=100 b=70 c=90 ⇒ x=60 y=120 z=80.
			Art.254.Fig.45: ∵ Round-way ∢DEF=π ∢FDE=∢EFD=π SphArea(△DEF)=2π, ∡AB,∡CA,∡CB>90° ∴(cf.Arts.238,239) ∠(βα⁻¹γ)=π-(½[D+E+F]-π)=2π-½(3π)=½π, Ax.(βα⁻¹γ)=-D=D′.
			Art.255: ∵ ∡AB,∡CA,∡CB>90° ∴(cf.Arts.238,239) the angle taken is the explement 2π-½[D+E+F], and the axis taken is the negative (or conjugate) D′|E′|F′.
			Art.256.Fig.46: Modification of Figs.40,42, illustrating (polar arcs) ∢L′DR=∢ZDH′ corresponding to (pole arcs) ⌒KM=⌒GI′ round D, which represent βα⁻¹γ.
			Art.257.Figs.47,48,49: Illustrations of the case ∡AB>90°, applying RULEs of Arts.226,230,238,239 as needed (depending on acute-way or obtuse-way angles).
		CONT/§ XLVII. Articles 258 to 263; Pages 247 to 252.
		LECT/§ XLVII. Articles 258 to 263; Pages 247 to 252.
			Art.258: (vid.Fig.40) ∵ L-H:∢BAC<0 ∴ ∢L′DR=∡KM=∠(βα⁻¹γ)=π-½[D+E+F]; ∢MDN=∡MN ∡KM+∡MN=½π ∢MDN=(½π-∢L′DR)=½(D+E+F-π)=½(SPHERICAL EXCESS)=∠.(δγ⁻¹αβ⁻¹).Ax=δ=OD.
			Art.259: (vid.Figs.40,42) δβ⁻¹αγ⁻¹ or G:H H:J represented by ⌒GJ; L-H round D: ∵ ∡JG+∡GI′=½π; ∡GI′=∡KM=π-½[D+E+F] (cf.Art.256)∴ ∡GJ=½(D+E+F-π) Ax.(⌒GJ)=-δ.
			Art.260: (vid.Figs.47,48,49) ∡AB>90°; M:N=M:L:L:N or νμ⁻¹=νλ⁻¹λμ⁻¹=δγ⁻¹αβ⁻¹; ∠δγ⁻¹>½π ∠δε⁻¹=2∠δγ⁻¹=∠(δγ⁻¹)² or Ax.δε⁻¹=-Ax.δγ⁻¹ ∠δε⁻¹=2π-2∠δγ⁻¹ δγ⁻¹=-(δε⁻¹)^½ (cf.158,183).
			Art.261: ?∡AB<90°: νμ⁻¹=q (cf.258) Ax.q=δ ∠q=∢MDN=½π-∢KDM=½(D+E+F-π); ?∡AB>90°: νμ⁻¹=-q Ax.-q=-δ ∠-q=π-∢MDN=π-(½π-∢KDM)=π-(½π-(½[D+E+F]-π))=½(D+E+F-π),∢MDN=½(3π-[D+E+F]).
			Art.262: Conjugate Versors: Ax.q′=-Ax.q ∠q′=∠q Tq′=Tq=1 (also reciprocal) q′=Kq=q⁻¹; versor:q′=r″r′r reversor:q=Kq′=KrKr′Kr″
			Art.263: The associative principle of multiplication is shown again to hold good.
		CONT/§ XLVIII. Articles 264 to 272; Pages 252 to 261.
		LECT/§ XLVIII. Articles 264 to 272; Pages 252 to 261.
			Art.264.Fig.50: cw⥁,L-H: Given △QRS; meridian (or pole) directions α⟂⌒SQ β⟂⌒RQ γ⟂⌒RS; ∠Q=∢SQR=∠(-β÷-α)=∠(q=β÷α) ∠R=∢QRS=∠(r=γ÷β) ∠S=∢RSQ=∠(α÷-γ)=∠(-α÷γ).
			Art.265: ∵ ⊾[S→pole(⌒RS)]=⊾[S→pole(⌒SQ)]=½π ∴ Ax.(γα⁻¹)=S-O; ∠(α÷γ)=π-∠S Ax.(α÷γ)=-OS, ∠(γ÷α)=∠S-π Ax.(γ÷α)=-OS or ∠(γ÷α)=π-∠S=EXT.VERTICAL∠S Ax.(γ÷α)=OS.
			Art.266.Fig.51: cw⥁,L-H: ∢TSR=∠([pole(⌒SR)=-γ=OM]÷[pole(⌒ST)=-α=OK])=∠(γ÷α)=∠(rq)=∢KSM; ∢SQR=∠(-β÷-α)=∠(OL÷OK)=∠q; ∢QRS=∠(-γ÷-β)=∠(OM÷OL)=∠r.
			Art.267: The effect of the transversor rq: Points along ⌒ST (⌒QS extended away from base ⌒RQ) are rotated round pole S to become along ⌒SR toward base ⌒RQ.
			Art.268: RULE for rq represented by △QRS: T.rq=TrTq, ∵[∠(rq)→∢QRS+∢SQR=π-∢RSQ]∣(Q+R+S-π)→0 {~ eⁱ⁽ʳ⁺ᵗ⁾} ∴∠(rq)=EXTERIOR VERTICAL∠S=π-∢RSQ=∢TSR=∢QSU, Ax.(rq)=S.
			Art.269.Fig.52: qr represented by △RQS′: α′⟂⌒S′R β′⟂⌒QR γ′⟂⌒QS′; ∠R=∢S′RQ=∠(r=-β′÷-α′) ∠Q=∢RQS′=∠(q=γ′÷β′); ∠(qr)=∠(γ′÷α′)=∢RS′T′=π-∢QS′R=∢TSR Ax.(qr)=S′.
			Art.270: RULE for S|S′: 2nd=r⥁R 1st=q⥁Q 3rd=rq⥁S or 2nd=q⥁Q 1st=r⥁R 3rd=qr⥁S′; Ax.3rd=S|S′ ON HEMISPHERE WITH pole(⌒2nd1st), or ⌒1st3rd is positive round 2nd.
			Art.271: RULE: For product=multiplier×multiplicand, rotation round Ax.(multiplier), from multiplicand toward product, is positive on spherical surface.
			Art.272.Fig.53: rq:△QRS qr:△RQS′ ∠rq=EXT.VERT.∠S=∢TSR=∢QSU=∠qr Ax.rq=∵pole(⌒2nd1st=⌒RQ⥁)↑∴S Ax.qr=∵pole(⌒2nd1st=⌒QR⥁)↓∴S′ ~ sr:△RSQ rs:△RSQ′ qs:△SQR sq:△SQR′.
		CONT/§ XLIX. Articles 273 to 280; Pages 261 to 268.
		LECT/§ XLIX. Articles 273 to 280; Pages 261 to 268.
			Art.273.Fig.54: (-α)β⁻¹=(-α)β²β²β⁻¹=αβ ∠(αβ)=π-∠(βα⁻¹) Ax.(αβ)=Ax.(βα⁻¹)=OQ (cf.Art.88); ∠(OP÷OA=OQ)=π-∠(OA×OP=OQ)=½π ~ j÷i=k=i×j=(k÷j)×j.
			Art.274.ERR.262: cw,L-H: Tα=1 ∠α=∢BAQ ∠(-α=α⁻¹)=∢QA′B; △A′BQ: 3rd=Q 2nd=B 1st=A′ ∠(βα⁻¹)=∠(β×-α)=EXT.VERT.∠Q=∢AQB=∠(β÷α) (cf.RULE of Art.268).
			Art.275: OC=γ=βα⁻¹β; γ=REFLEXION(α) 180° round β, ⌒AB=⌒BC ∢AQB=∢BQC; △BQC: 3rd=C 2nd=Q 1st=B ∠b=½π ∠q=∠(β÷α) ∠(q.b)=∠(βα⁻¹.β=γ)=EXT.VERT.∠C=∢A′CQ=½π.
			Art.276: △CAQ: 3rd=Q 2nd=A 1st=C ∠(αγ)=EXT.VERT.∠Q=∢CQA′=π-∢AQC=π-∠(γα⁻¹)=π-∠(-γα)=π-(π-∠(γα))=∠(γα) [cf.Art.183], Ax.(αγ)=OQ, Ax.(γα)=Q′=QO (cf.△ACQ′).
			Art.277.Fig.55: β^(½=45°)α^½:△ABD α=OA β=OB δ=OD ∢DAB=∠A=∢ABD=∠B=45° Ax.(β½α^½)=δ ∠(β^½α^½)=EXT.VERT.∠D=∢TDB ~ β^(⅓=30°)α^(⅔=60°):△ABE, β^⅔α^⅓:△ABF.
			Art.278: Fractional powers t,s; βˢαᵗ:△ABX t=∢XAB/90° s=∢ABX/90°, Ax.(βˢαᵗ)=X=[⌒AX (@+∠s rnd B⥁) ∩ ⌒BX (@-∠t rnd A⥁)], ∠(βˢαᵗ)=EXT.VERT.∠X=π-∢BXA ⥁.
			Art.279: For any fixed sum u=t+s, βᵘ⁻ᵗαᵗ:△ABX, base AB is the major-axis of a fixed semi-ellipse ⌒AXB locus of X, or semi-(small)circle ⌒AXB when u=90°.
			Art.280.Fig.56: △ABC ∠A=x½π ∠B=y½π ∠C=z½π γ²⁻ᶻ=βʸαˣ γᶻβʸαˣ=αˣγᶻβʸ=βʸαˣγᶻ=γ²=-1 INVERSOR; ∢SUM(∢∈△ABC|rotations)=Cyc.Order (C+B+A=A+C+B=B+A+C)-2=0 ∢excess.
		CONT/§ L. Articles 281 to 292; Pages 268 to 277.
		LECT/§ L. Articles 281 to 292; Pages 268 to 277.
			Art.281.Fig.57: Lune BB′; △ABC:rq, △B′AD:qr⁻¹, △B′CE:∠(rq.r⁻¹)=EXT.VERT.∠E=∠(r.qr⁻¹):△DBE.
			Art.282: Rotation operator r()r⁻¹: ∠(rqr⁻¹)=∠q; Ax.(q)=OA Ax.(rqr⁻¹)=OE=[Ax.q=OA rotated POSITIVE round Ax.r=OB 2∠r]; ∠q=½π, q,q′=rqr⁻¹ VECTORS.
			Art.283: ∵q→r r→q⁻¹ r⁻¹→q ∴rqr⁻¹→q⁻¹rq=[Ax.r rotated NEGATIVE round Ax.q -2∠q, or rotated POSITIVE round Ax.q⁻¹ 2∠q⁻¹]; qᵗrq⁻ᵗ rotates 2t∠q.
			Art.284: (vid.Fig.37) q⁻¹r=(K÷L)×(L÷M′)=K÷M′=s; qs=r; qsq⁻¹=rq⁻¹=M÷K′=s′; qKq⁻¹=K′L⁻¹KKL⁻¹=K′ qM′q=(rq⁻¹)⁻¹K′=qr⁻¹K′=M″ K′÷M″=rq⁻¹=s′.
			Art.285: Rotation operation r′=q(r)q⁻¹: vertex.Ax.r′=[vertex.Ax.r rotated 2∠q round Ax.q] Tr′=Tr ∠r′=∠r; t∠r′=t∠r ∠r′ᵗ=∠rᵗ: (qrq⁻¹)ᵗ=qrᵗq⁻¹.
			Art.286: Vector rotation ρ′=qρq⁻¹: ∠ρ=½π=∠ρ′ (VECTORs); ∀ ρ=OP ∈ B (BODY) B′=qBq⁻¹=[B rot.rnd Ax.q=OQ 2∠q]; Successive rotations: s[r(qBq⁻¹)r⁻¹]s⁻¹.
			Art.287: α=A-O=O-B; Rot(2∠q⥁B:BC∥Ax.q)=Xlat(-α)∘Rot(2∠q⥁O:Ax.q)∘Xlat(α)=-α+q(α+ρ)q⁻¹; Rot(2∠q⥁O:Ax.q)∘Xlat(α)=q(α+ρ)q⁻¹=+α-α+q(α+ρ)q⁻¹=Xlat(α)∘Rot(2∠q⥁B:BC∥Ax.q).
			Art.288: Rotation operator round Ax.q=OQ by ∠q (!2∠q): q^½()q^⁻½=(βα⁻¹)^½()(αβ⁻¹)^⁻½; In plane of α,β thru O, rotation takes α→β.
			Art.289: Conical rotation (βα⁻¹)()(αβ⁻¹) by 2∠(βα⁻¹) round Ax.(βα⁻¹) thru O; (vid.Fig.41 R-H ccw) Ax.(βα⁻¹)=P 2∢APB=∢SPR=∢EPD=2∠(βα⁻¹) δ=βα⁻¹εαβ⁻¹ (⥀round dashed ◌).
			Art.290: Conical reflection round bisectors: ∠γ=½π Ax.γ=OC ρ′=γργ⁻¹=γ⁻¹ργ Tρ′=Tρ U.γρ⁻¹=U.ρ′γ⁻¹; (vid.Fig.40) △DEF δ=γεγ⁻¹ ∢DOC=∢COE U.γδ⁻¹=U.εγ⁻¹ ~ ζ=α⁻¹εα δ=βζβ⁻¹.
			Art.291: ρρ=-Tρ² ρ=-Tρ²ρ⁻¹; γ⁻¹γ⁻¹=-Tγ⁻² γ⁻¹=-γTγ⁻²; γ.ρ.γ⁻¹=γ.-Tρ²ρ⁻¹.-γTγ⁻²=Tρ²Tγ⁻².γρ⁻¹γ; γρ⁻¹γ=3RD∝ ρ∶γ∷γ∶3RD∝; T.γρ⁻¹γ=Tγ²Tρ⁻¹ T.γργ⁻¹=Tρ²Tγ⁻².Tγ²Tρ⁻¹=Tρ.
			Art.292: CONICAL rotation ε∶δ⥀γ EQUIVALENT to two successive PLANE rotations ε∶ζ⥀α ζ∶δ⥀β (vid.Fig.40): β(α⁻¹εα)β⁻¹=β(ζ)β⁻¹=δ=γεγ⁻¹=γ⁻¹εγ γ≠βα⁻¹≠γ⁻¹ (cf.Art.290).
		CONT/§ LI. Articles 293 to 304; Pages 277 to 290.
		LECT/§ LI. Articles 293 to 304; Pages 277 to 290.
			Art.293: ∠s≘⌒SS′ ∠r≘⌒RR′ ∠q≘⌒QQ′ [s.rq≘⌒SS′+(⌒RR′+⌒QQ′)=⌒TT′≘t]≟[sr.q≘(⌒SS′+⌒RR′)+⌒QQ′=⌒UU′≘u].
			Art.294.Fig.58: ∠s≘⌒EF=⌒HI ∠r≘⌒BC=⌒GH ∠q≘⌒AB=⌒KL; rq≘⌒BC+⌒AB=⌒AC=⌒DE s.rq≘⌒EF+⌒DE=⌒DF≘t; sr=⌒HI+⌒GH=⌒GI=⌒LM sr.q=⌒LM+⌒KL=⌒KM≘u.
			Art.295: ∠(rq)≘⌒AC=⌒DE ∠(sr)≘⌒GI=⌒LM; ∠t≘⌒DF ∠u≘⌒KM; ⌒DF≟⌒KM (& on gr.cir?), (t=s.rq)≟(u=sr.q)=srq [?=Yes]; (arcs⌒ on UNIT-SPHERE are equal to angles∠).
			Art.296: Sph.conics proof(?.295): ∵¶13 ⌒DK=⌒FM ∴⌒DF=⌒KM; ∵¶29 conic:cyc-arcs ◌BFH:(⌒CE,⌒GI)=◌BFK:(⌒AD,⌒LM) ◌HKB:(⌒GL,⌒CA)=◌HKF:(⌒IM,⌒ED) ∴gr.cir.⌒DF=⌒KM≘∠(srq).
			Art.297: Elementary proof of ⌒DF≟⌒KM rs.q≟s.rq shall next (Arts.298-301) be given, independent of theorems of spherical conics.
			Art.298.Fig.59: OB⇢P′P OH⇢Q′Q OF⇢R′R △P′Q′R′∥○GLIM △PQR∥○DAEC; Diacentric sphere ●OPQR and ●OBFH cut by ⌽GBHC (vid.Fig.58): ⎊OPQ PQ∥OC Q′P′∥OG ◌PQQ′P′.
			Art.299.Fig.60: Same spheres ●OPQR ●OBFH and planes △PQR △P′Q′R′ cut now by ⌽EHFI (vid.Fig.58): ⎊ORQ RQ∥OE Q′R′∥OI, ◌RQQ′R′ ◌PQQ′P′ on THIRD sphere ●PQRP′Q′R′.
			Art.300.Fig.61: OK⇢S′S; ●OPQRS ●OBFH △PQR ▭P′Q′R′S′ cut now by ⌽AKBL (vid.Fig.58): ⎊OPS PS∥OA S′P′∥OL, ◌PSS′P′ ◌PQRS ◌P′Q′R′S′ on THIRD sphere ●PQRSP′Q′R′S′.
			Art.301.Figs.62,63,64: Three spheres cut now by ⌽DKFM: ⎊ORS SR∥OM R′S′∥OD ◌RSS′R′ on THIRD sph; ∵∠…=∠… in ⫴gr.cir.⌽… betw. ●OBFH⇔●PQRSP′Q′R′S′ ∴⌒DF=⌒KM on gr.cir.⌽DKFM.
			Art.302.Fig.65: Proof ⌒DF=⌒KM by sph.conics: (cf.RULE Art.268) rep.axes&angles of rq,sr at foci E,F of conic ellipse in ABCD, and of q,r,s,(sr.q=s.rq) found at A,B,C,D.
			Art.303: IDENTITY for ANY 3 QUATERNIONS sr.q=s.rq=srq; rq,sr rep.by two foci of a conic in a quadrilateral, or by two cyclic arcs of another conic in another quadri.
			Art.304: Ordered× tsrq… (or arcualΣ) is ASSOCIATIVE, NOT COMMUTATIVE in general; WHEN all coplanar or ∥biradials, tsrq… (or arcual+ on a gr.cir.) IS COMMUTATIVE (~∈C).
		CONT/§ LII. Articles 305 to 316; Pages 290 to 303.
		LECT/§ LII. Articles 305 to 316; Pages 290 to 303.
			Art.305: ASSOC.THEOREM:(vid.Fig.58) Given spherical hexagon ⬡KLGHED; if ₁KL≘AB ₃GH≘BC ₅ED≘CA (1,3,5 ≘ ONE sph.△ABC), then ₂LG≘MI ₄HE≘IF ₆DK≘FM (2,4,6 ≘ ANOTHER sph.△MIF).
			Art.306: If(5 DIAGS ₁AB ₂MI ₃BC ₄IF ₅CA of ONE ⬡AMBICF)≘(5 SIDES of ANOTHER ⬡KLGHED),then(DIAG₆FM)≘(SIDE₆DK); ✡AMBICF inscribes △ABC,▽MIF assocd.with ⬡KLGHED.
			Art.307: ∀sph.△XYZ ⌒ZX+⌒YZ+⌒XY=0 (arcualΣ△=0); Given sph.⬡KLGHED: If (ONE SET of ALTERNATE SIDES) ₅⌒ED+₃⌒GH+₁⌒KL=0≘Σ△, then likewise (OTHER SET) ₆⌒DK+₄⌒HE+₂⌒LG=0≘Σ△.
			Art.308: 12 VECTORS: αβγ≘ABC δεζ≘DEF θηι≘GHI κλμ≘KLM; 6 EQNS: ₁q=β÷α=λ÷κ ₂r=γ÷β=η÷θ ₃s=ζ÷ε=ι÷η ₄rq=γ÷α=ε÷δ ₅sr=ι÷θ=μ÷λ ₆s.rq=ζ÷δ=sr.q=μ÷κ unaffected by Tensors arithmetic.
			Art.309: Comparison of (THEOREM) general assoc.× of QUATERNIONS to (DEFINITION) assoc.× of LINES: TRANSFACTOR=PROFACTOR×FACTOR γ=r×β=r×q×α=s×α rq.α=r.qα (cf.Art.108).
			Art.310: LINES γ,β,α: γ=rqα=(γ÷β)×(β÷α)×α; Planes of biradials r=γ÷β q=β÷α intersect in β; if planes r,q parallel, β be any line parallel; General assoc.× required PROOF.
			Art.311: q=β÷α r=γ÷β rq=γ÷α s=ζ÷ε s.rq=?; To multiply any 2 quaternions, prepare equiv.biradials with a common ray ε (like β) on the intersection of the biradial planes.
			Art.312: 6th EQN consequence of 5 EQNS: [₆μκ⁻¹=₅μλ⁻¹.₁λκ⁻¹=₅ιθ⁻¹.₁βα⁻¹=₃ιη⁻¹₂ηθ⁻¹.₁βα⁻¹=₃ζε⁻¹₂γβ⁻¹.₁βα⁻¹=₅sr.₁q]=[₃s.₄rq=₃ζε⁻¹.₂γβ⁻¹₁βα⁻¹=₃ζε⁻¹.₄γα⁻¹=₃ζε⁻¹.₄εδ⁻¹=₆ζδ⁻¹].
			Art.313: ∝ Substitutions: [₆ζδ⁻¹=ζε⁻¹γβ⁻¹βα⁻¹ ζδ⁻¹αβ⁻¹=ζε⁻¹γβ⁻¹ ⌒DF+⌒BA=⌒EF+⌒BC] ←if→ [δ∶ε∷α∶γ ₄εδ⁻¹=γα⁻¹ alt.δ∶α∷ε∶γ αδ⁻¹=γε⁻¹ ⌒DA=⌒EC δ⁻¹α=ε⁻¹γ] (vid.Fig.58).
			Art.314: ₄⁻¹δε⁻¹=αγ⁻¹; ₆δ∶ζ∷κ∶μ ₆ᵄμ∶ζ∷κ∶δ δκ⁻¹=ζμ⁻¹; ∵β⫴κ,λ β⫴θ,η ∴∃α,γ ∋κλ⁻¹θη⁻¹=αβ⁻¹βγ⁻¹=αγ⁻¹; ∵ι⫴ε,η ι⫴θ,λ ∴∃ζ,μ ∋εη⁻¹θλ⁻¹=ζι⁻¹ιμ⁻¹=ζμ⁻¹ (vid.Fig.58).
			Art.315: If δε⁻¹=κλ⁻¹θη⁻¹,Then ζμ⁻¹ = δκ⁻¹=εη⁻¹θλ⁻¹ (cf.Art.314); or L←R× κ⁻¹δ=λ⁻¹θη⁻¹ε ⊢ εη⁻¹θλ⁻¹=δκ⁻¹; If ⌒ED=⌒LK+⌒HG,Then ⌒KD=⌒HE+⌒LG (vid.Fig.58).
			Art.316: Formula for CONTINUED PRODUCT of 5 VECTORS equal to a 6TH δ= κλ⁻¹θη⁻¹ε=εη⁻¹θλ⁻¹κ (L←R×); Generally εδγβα=ζ,if αβγδε=ζ (equal in OPPOSITE ORDER).
		CONT/§ LIII. Articles 317 to 322; Pages 303 to 309.
		LECT/§ LIII. Articles 317 to 322; Pages 303 to 309.
			Art.317: K.rq=KqKr, K(…tsrq any n quaternions)=KqKrKsKtK…; Kα=-α (vector α), K(…γβα ⁺even|₋odd n vectors)=±αβγ…, If vector δ=(…γβα odd n),Then -Kδ=-K(…γβα odd n)=δ=αβγ…
			Art.318: If vector δ=(…γβα even n vectors),Then -Kδ=-K(…γβα even n)=δ=-αβγ…; (…γβα any n vectors or quaternions)⁻¹=α⁻¹β⁻¹γ⁻¹…
			Art.319: If (vectors αβ)=(scalar a),Then α∥β K.αβ=Ka=a=KβKα=(-β)(-α)=βα; Generally If(…αβ even n vectors)=(scalar a),Then …αβ=βα…=a.
			Art.320.Fig.66: △ABC:β=CA α=BC γ=AB βαγ=γαβ=(-Tα².δ tan@A)∝AT,AE⥁ (vid.Fig.26); △ACD:(DA×CD×AC)∝AT⥁; △:(D′A×CD′×AC)∝AT′⥀; ₁□:(△ DA×CD×AC)(△ CA×BC×AB)∝AT×AT<0.
			Art.321: Crossed-₂□:(△ D′A×CD′×AC)(△ CA×BC×AB)∝AT′×AT>0; △Seg.=∠,□Opp.⦞ in cir:(₁□,₂□)⊢U.BC×AB=U.DC×DA=U.D′C×AD′⊢U.BC÷BA=U.DC÷AD=U.D′C÷D′A=U.AC÷AT′≘π-∠CDA; U∏=∏U.
			Art.322: ⭔ABCDE:(△ EA.DE.AD∝AT)(□ DA.CD.BC.AB<0)∝AT′; Crossed-⭔ABCDE′⇇⭔ABCD′E∝AT; □×□=⬡=scalar; POLYgons: ODDgon:tan-vector@A EVENgon:scalar ∀gon:…βαγ=γαβ…
		CONT/§ LIV. Articles 323 to 328; Pages 309 to 315.
		LECT/§ LIV. Articles 323 to 328; Pages 309 to 315.
			Art.323.Fig.67: (△ABC: CA.BC.AB∝AT)⫵(△ACD: DA.CD.AC∝AU); Gauche-□ABCD: U.△ACD×△ABC=U.DA×CD×BC×AB=U.AU×AT=[×of tan-vectors@A to circles&sphere ◌&●].
			Art.324.Fig.68: Ax.(U.AU×AT=U.AU÷TA)=[normal@A ●]=±U.OA ⁺⥁L-H,₋⥀R-H; ∠(U.AU÷TA)=∠(U.AU×AT)=π-∠UAT=∠TAU′=\'LUNULE\'⦡BAD=∠[pole(◌ACD)÷pole(◌ABC)] ⥁.
			Art.325.Fig.69: Gauche-⭔ABCDE:U.△ADE×□ABCD=U.AV×AU×AT=U.AW=△:U.V′A×T′V′×AT′=[tan-vector@A ○AT′V′&●]; Gauche O-gon:U.×UV.=tan-UV.@A, E-gon:UV.÷UV.=normU.@A.
			Art.326: If AB.BC.CD.DE.EA=(AW line),Then ABCDE HOMOSPHAERIC,Gauche-⭔ABCDE INSCRIPTIBLE in ●ABCD; AW=AV×AU×AT/(AD.DA×AC.CA) are lines in tan-plane@A,AW=4th∝
			Art.327: ●Eq.Homosphaericism: AB.BC.CD.DE.EA=EA.DE.CD.BC.AB (=iff tan-VECTOR@A ●); ○Eq.Concircularity: AB.BC.CD.DA=DA.CD.BC.AB (=iff SCALAR=tan×tan@A ○).
			Art.328: ●Eq→○Eq: Gauche-⭔ABCDE Let E→A,DE→DA,AB.BC.CD.DE→[Q Gauche-□:Ax.Q=normal@A],EA→tanV.@A,Q×EA=[another(or∥)tanV.@A]; Gauche→Concir.□ABCD∷Q→scalar.
		CONT/§ LV. Articles 329 to 340; Pages 315 to 325.
		LECT/§ LV. Articles 329 to 340; Pages 315 to 325.
			Art.329: ⍜Eq.TAN-PLANE@A(●ABCD,AP): AB.BC.CD.DA.AP=AP.DA.CD.BC.AB [3+1 applns.of△ βαγ=γαβ,&iff AP⫴(Gauche-□ tans@A)]; ●Eq,⍜Eq both hold iff P=E=A [AE=∅ tan@A, homo●].
			Art.330.Fig.70: Gauche-□ABCD: U.(DA×CD×BC×AB)=U.(OM×OL×OK×OI)=U.(OM×-OL×OK×-OI)=U.(OM÷OL×OK÷OI)=U.(OH÷OG×OG÷OF)=U.(OH÷OF)=U.(OH×-OF)=U.(AU×AT tans@A); A|A′=pole(⌒FH).
			Art.331: U.(OM×OL×OK×OI)=U.(DA×CD×BC×AB)|U.(D′A′×C′D′×B′C′×A′B′) [Gauche-□ABCD|□A′B′C′D′]; ●-inscribed n-gon: nᵗʰ homo●-point,chord∥chord-parallel OPₙ,Pₙ₋₁Pₙ∥OIₙ.
			Art.332.Fig.71: α=OA β=OB γ=OC|OC′; ∴By conical rotation -β=γαγ⁻¹,β=-γαγ⁻¹ ∴expressing(or finding) chord-point B in terms of chord-point A and a chord-parallel γ∥AB.
			Art.333.Fig72: β=-γαγ⁻¹ is independent of radius r=Tα=Tβ, and Tγ is arbitrary for ANY γ∥AB: OB=-AD×OA÷AD=-AE×OA÷AE=-AF×OA÷AF, even when AB,D,E,F are collinear.
			Art.334: OPₙ=ρₙ ρ₀=α₀=OA₀ Tρₙ=r; Pₙ₋₁Pₙ∥OIₙ=(ρₙ-ρₙ₋₁)∥ιₙ ι₀=1; ρₙ=-ιₙρₙ₋₁ιₙ⁻¹; If qₙ=ιₙιₙ₋₁‥ι₃ι₂ι₁ι₀,Then nᵗʰ homosphaeric●-point ρₙ=(-)ⁿqₙρ₀qₙ⁻¹.
			Art.335.Fig.73: Let OAₙ=αₙ be any point (≠ρₙ₋₁) collinear to nᵗʰchord ρₙ-ρₙ₋₁,chord-parallel ιₙ=αₙ-ρₙ₋₁ ι₀=1; If again qₙ=ιₙιₙ₋₁‥ι₃ι₂ι₁ι₀,Then again ρₙ=(-)ⁿqₙρ₀qₙ⁻¹.
			Art.336.ERR.321: POLYGON CLOSURE, final side n ρₙ=ρ₀; EVEN-n: qₙ=scalar or Ax.qₙ∥ρ₀[∴qₙ normU.@A]; ODD-n: (qₙ=vector=-Kqₙ)⟂ρ₀[∴qₙ tan@A] ∋(-)ⁿqₙ(ρ₀)qₙ⁻¹=ρ₀ [2reversals].
			Art.337: ●Gauche-□ABCD closure condition: ∵ρ₄=ρ₀=(-)⁴q₄(ρ₀)q₄⁻¹=+(OH÷OF)(ρ₀)(OH÷OF)⁻¹ [cf.Art.330] ∴ ρ₀∥Ax.(OH÷OF) [no rotation] & ρ₀ on sphere ∋ ρ₀=pole(⌒FH)=OA|OA′.
			Art.338: ●Gauche-⭔ABCDE closure: ρ₀≠pole(⌒FH), ρ₄≠ρ₀; ρ₅=ρ₀=(-)⁵ι₅q₄(ρ₀)q₄⁻¹ι₅⁻¹=-ι₅(OH÷OF)(ρ₀)(OH÷OF)⁻¹ι₅⁻¹=-ON(OH÷OF)(ρ₀)(OH÷OF)⁻¹ON⁻¹ ON⟂pole(⌒FH) [vid.Figs.70,71].
			Art.339: 13-gon: 6-gon.A′B′C′D′E′F′ 7-gon.ABCDEFG 1…5ᵗʰside:AB∥A′B′‥EF∥E′F′ 6ᵗʰside:FG∥F′A′ 7ᵗʰ:GA∥tan-plane@A′ or 7…11ᵗʰ:GH∥A′B′‥LM∥E′F′ 12ᵗʰ:MN∥F′A′ 13ᵗʰ:NG∥tan-plane@A′.
			Art.340: 4n+1-gon: 2n-gon.A₀A₁…A₂ₙ₋₁ 4n+1-gon.P₀P₁…P₄ₙ, 1…2nᵗʰ:P₀P₁∥A₀A₁‥P₂ₙ₋₁P₂ₙ∥A₂ₙ₋₁A₀, 2n+1…4nᵗʰ:P₂ₙP₂ₙ₊₁∥A₀A₁‥P₄ₙ₋₁P₄ₙ∥A₂ₙ₋₁A₀, 4n+1ᵗʰ:P₄ₙP₀, P₀P₂ₙP₄ₙ∥tan-plane@A₀.
		CONT/§ LVI.ERR.xxxii. Articles 341 to 349; Pages 325 to 334.
		LECT/§ LVI. Articles 341 to 349; Pages 325 to 334.
			Art.341: COMPOSITION OF CONICAL ROTATIONS: srqB(srq)⁻¹=s.r.qBq⁻¹.r⁻¹.s⁻¹: 1*resultant Ax.srq 2∠.srq, or 3*successive Ax.q 2∠q, Ax.r 2∠r, Ax.s 2∠s conical rotations.
			Art.342.Fig.74: Conical rotation round VECTORS α,β: (β÷α)ρ(α÷β)=β.α⁻¹ρα.β⁻¹: 1* Ax.(β÷α) ○ 2∠(β÷α) , or 2*REFLEXIONS Ax.(α⁻¹) ◌ 2∠(α⁻¹)=2½π=π, Ax.β ◌ 2∠β=2½π=π.
			Art.343.Fig.75: Vectors α=OA,β=OB,γ=OC: (γ÷β)(β÷α)ρ(α÷β)(β÷γ)=γβ⁻¹βα⁻¹ραβ⁻¹βγ⁻¹=γα⁻¹ραγ⁻¹, 1*.[P∶S]=2*.[P∶Q∶R,R∶Q∶S], 1*.[⌒◌PS⋕2⌒○AC]=2*.[⌒◌RS⋕2⌒○BC + ⌒◌PR⋕2⌒○AB].
			Art.344: NO EFFECT: △ABC ∅=2(⌒CA+⌒BC+⌒AB) ≘ αγ⁻¹γβ⁻¹βα⁻¹ραβ⁻¹βγ⁻¹γα⁻¹=ρ, 6*◊. ⌒◌QP⟂α+⌒◌SQ⟂γ⁻¹+⌒◌QS⟂γ+⌒◌RQ⟂β⁻¹+⌒◌QR⟂β+⌒◌PQ⟂α⁻¹,or 3*◍ ⌒◌SP⋕2⌒○CA+⌒◌RS⋕2⌒○BC+⌒◌PR⋕2⌒○AB.
			Art.345.Fig.76: NO EFFECT round any sph.n-gon: ∅=2[⌒GA+…+⌒CD+⌒BC+⌒AB] ≘ αθ⁻¹…δγ⁻¹γβ⁻¹βα⁻¹ραβ⁻¹βγ⁻¹γδ⁻¹…θα⁻¹=ρ, 2n*REFLEXIONS or n*CONICAL-ROTATIONS having null resultant.
			Art.346: △ABC [vid.Fig.56] 2∢CAB=πx 2∢ABC=πy 2∢BCA=πz; γᶻβʸαˣ=γᶻγ²⁻ᶻ=γ²=-1 [RULE.Art.268], βʸαˣ=-γ⁻ᶻ, α⁻ˣβ⁻ʸ=-γᶻ; βʸαˣρα⁻ˣβ⁻ʸ=γ⁻ᶻργᶻ ≘ ⌒◌(∠πy)⟂β+⌒◌(∠πx)⟂α=-⌒◌(∠πz)⟂γ.
			Art.347.Fig.77: 2∢CAB=πx=∢CAD αˣγα⁻ˣ=δ ≘⌒◌CD⟂α, 2∢ABC=πy=∢DBC=∢ABE βʸδβ⁻ʸ=γ ≘⌒◌DC⟂β, 2∢ACB=-πz γ⁻ᶻαγᶻ=βʸαˣαα⁻ˣβ⁻ʸ=βʸαβ⁻ʸ=ε; 2[π-∢BCA] γ⁻².γ²⁻ᶻαγᶻ⁻².γ²=βʸαˣαα⁻ˣβ⁻ʸ=ε.
			Art.348: TRIQUADRANTAL △ABC [vid.Fig.43] Reflexions: αεα⁻¹=ζ ⌒◌EF⟂α, βζβ⁻¹=δ ⌒◌FD⟂β, γεγ⁻¹=δ ⌒◌ED⟂γ; βαεα⁻¹β⁻¹=γεγ⁻¹=δ; γβαεα⁻¹β⁻¹γ⁻¹=ε; ijkρk⁻¹j⁻¹i⁻¹=kjiρi⁻¹j⁻¹k⁻¹=ρ.
			Art.349: Any △ABC, NO EFFECT: γᶻβʸαˣ=-1, α⁻ˣβ⁻ʸγ⁻ᶻ=-1⁻¹=-1, γᶻβʸαˣρα⁻ˣβ⁻ʸγ⁻ᶻ=ρ; Any SPH.N-GON, NO EFFECT:[cf.Art.345] Successive rotations round all corner Ax. by 2∠. is ∅.
		CONT/§ LVII. Articles 350 to 357; Pages 334 to 343.
		LECT/§ LVII. Articles 350 to 357; Pages 334 to 343.
			Art.350: △DβFαEγ (cf.258)⥁: δγ⁻¹=(δε⁻¹)^½,εα⁻¹=(εζ⁻¹)^½,ζβ⁻¹=(ζδ⁻¹)^½; αζα⁻¹=ε, U.q=U.δγ⁻¹εα⁻¹ζβ⁻¹=U.δγ⁻¹αβ⁻¹≘⌒MN, ∠q=½π-∠(βα⁻¹γ≘⌒KM)=½π-(π-½[D+E+F])=½[D+E+F-π], Ax.q=δ.
			Art.351: qρq⁻¹=ρ‴ 2∠q=D+E+F-π [SPH.EXCESS]; ρ′=(ζδ⁻¹)^½.ρ(δζ⁻¹)^½≘⌒DF, ρ″=(εζ⁻¹)^½.ρ′(ζε⁻¹)^½≘⌒FE, ρ‴=(δε⁻¹)^½.ρ″(εδ⁻¹)^½≘⌒ED; ½⌒ED+½⌒FE+½⌒DF=⌒MN, pole.⌒MN=D ∠⌒MN=∠q.
			Art.352: U.q=U.δγ⁻¹αβ⁻¹, qρq⁻¹=δγ⁻¹αβ⁻¹ρβα⁻¹γδ⁻¹ 4 REFLEXIONS or 2 CONICAL ROTATIONS shewn by VECTOR-ARCs: ⌒ED+⌒FE+⌒DF or 2⌒CD+2⌒BA=2(⌒LN+⌒ML)=2⌒MN=⌒NM`+⌒MN (vid.Fig.40).
			Art.353.Fig.78: Ar.Lune.AA′=2Ar² Ar⥀.△DEF=½[(2D+2E+2F)-2π]r²=½[(2△+◖)-◖]r²=+[D+E+F-π]r²; If qρq⁻¹ shewn by ⌒XA+…+⌒BC+⌒AB,Then Ax.q=OA, 2∠q=∑Ar.△A‥=Ar.SphOPyramid.A∣r=1.
			Art.354: q=(α÷ζ)^½‥(ε÷δ)^½(δ÷γ)^½(γ÷β)^½(β÷α)^½; Ar.⭔ABCDE=Ar.△ABC+Ar.△ACD+Ar.△ADE=A+B+C+D+E-3π=2∠q,Ax.q=Uα; If⥁ & ⥀A∶B∶C,-Ar.△ABC; Ar.n-gon=2∠q=(∑Aₙ)-(n-2)π≘2∑½⌒Aₙ₋₁Aₙ.
			Art.355: If q=(α÷ζ)^½‥inf.‥(ε÷δ)^½(δ÷γ)^½(γ÷β)^½(β÷α)^½, qρq⁻¹[inf.,infinites.succ.conical rotns.],Then 2∠q=(∑Aₙ)-(n-2)π=TOTAL AREA OF FIGURE≘2∑½⌒Aₙ₋₁Aₙ=2[⌒∑SEMI-SIDES].
			Art.356: (α÷γ)(γ÷β)(β÷α)=1; OTHER PRODUCT: q`=(β÷α)(γ÷β)(α÷γ)≘⌒AB+⌒BC+⌒CA=⌒AB+⌒CB`+⌒A`C=⌒AB+⌒A`B`=⌒LM+⌒M`L=⌒M`M=2⌒NM, ∠q`=Ar⥁.△EFD=∓[D+E+F-π] Ax⥁.q`=±δ [vid.Fig.40].
			Art.357: UVα,β,γ; q`=βα⁻¹γβ⁻¹αγ⁻¹=-βα⁻¹γ(-β⁻¹)(-α)(-γ⁻¹)=-(βα⁻¹γ)² ≘ ⌒π-2⌒KM Ax.q`=-δ [cf.Arts.227,258,183] ∠(βα⁻¹γ)²=2∠(βα⁻¹γ)=2π-[D+E+F], ∠q`=π-(2π-[D+E+F])=[D+E+F-π].
		CONT/§ LVIII. Articles 358 to 364; Pages 343 to 350.
		LECT/§ LVIII. Articles 358 to 364; Pages 343 to 350.
			Art.358: sr.q=s.rq ≘ ⌒GI+⌒AB=⌒DF (vid.Fig.58); Double CO-ARCUALITY DAEC: ⌒DA=⌒EC ⌒DE=⌒AC; Given 3 DAEC,CHBG,EHFI: 3 more points K,L,M and AKBL,GLIM,DKFM can be determined.
			Art.359: ∵For chord OP, ∠Q=∠(POT|TPO tan@O|P)=⌒CH=⌒BG ∴OG tan@O[Fig.59] &~OI[Fig.60] ∋ ○GLIM tan@O, &c.
			Art.360: Versor depends on 3 angles (θ,φ,t); Versor arcs: CO-ARC MNM‵ ≎ ⌒MN=⌒NM‵ (1 arc eqn)∶(3 scalar eqns;=θ,=φ,=t); DBL.CO-ARC DAEC ≎ ⌒DA=⌒EC ⌒DE=⌒AC [6 scalar eqns].
			Art.361.Fig.79: Total product of factors u=tsrq; Partial products v=rq w=sr x=ts y=srq z=tsr; 6products+4factors≘10points; Ax.v=⌒BF∩⌒FA=OF ∠v=π-BFA=EXT.VERT.∠F,&c.[cf.268]
			Art.362: 1△∶1product 6products≘6△,6constructed points; rq∶△ABF sr∶△BCG ts∶△CDH, s.rq∶△FCI sr.q∶△AGI, t.sr∶△GDK ts.r∶△BHK, t.srq∶△IDE ts.rq∶△FHE tsr.q∶△AKE.
			Art.363: 10triangles; NUMBERS of △ representations|usages of angles: q∶3 r∶3 s∶3 t∶3, rq∶3 sr∶3 ts∶3, srq∶3 tsr∶3, tsrq∶3; Other possible triangles are not CONSTRUCTIVE.
			Art.364: Each of ABCDEFGHIK a common corner of 3of10 triangles using 1of3 angle reps.; ∠rep1=∠rep2=∠rep3 (2eqns), 2×(4+6)=20 angle eqns [the given 8 determine other 12].
		CONT/§ LIX. Articles 365 to 378; Pages 351 to 366.
		LECT/§ LIX. Articles 365 to 378; Pages 351 to 366: Polygons of Multiplication.
			Art.365: Ranks n-x…1: n-1 rₙ₋₁…r₁=qₙqₙ₋₁…q₂q₁, n-2 sₙ₋₂=qₙqₙ₋₁qₙ₋₂, n-3 tₙ₋₃=qₙqₙ₋₁qₙ₋₂qₙ₋₃,…, 2 z₂=qₙqₙ₋₁‥q₃q₂ z₁=qₙ₋₁qₙ₋₂‥q₂q₁, 1 q=qₙqₙ₋₁‥q₂q₁;∑ranks+factors=½n(n-1)+n.
			Art.366: ½n(n-1)+n=½n(n+1) TOTAL points determined by given 3n numbers qₙ(θₙ,φₙ,tₙ); 2co-ords(θ,φ)/pt n(n+1)-3n=n(n-2) unknown co-ords determined by as many relations.
			Art.367: 1prod∶1△ ∷ [½n(n-1) prods & constructed points (each FOUND ONCE) by SPHERICAL TRIANGULATION]∶[SYSTEM OF ½n(n-1)△]; n=4 ½4(4-1)=2(3)=6△ [not 10△ since ONLY ONCE].
			Art.368: PARTIAL PRODUCT used ONCE AGAIN (but NOT if last-in-rank prodₙ₋ₓ) as MULTIPLICAND; Points R₁…Rₙ₋₂ used TWICE (prod.vertex&1st.corner∶2△),last Rₙ₋₁ ONCE (vertex∶1△).
			Art.369: By number of times used, the ½n(n-1) prods supply ½n(n+1)-(n-1)=½n(n-1)(n-2) ∠eqns, & the n factors supply another ½(n+1)(n-2) ∠eqns; Total CONSTR. ∠eqns: n(n-2).
			Art.370: SPHERICAL POLYGON OF CONTINUED MULTIPLICATION Q₁Q₂Q₃…Qₙ₋₁QₙQ of quaternion factors q=qₙ…q₁; Last corner Q: Ax.q=OQ ∠q=[angle round OQ].
			Art.371: Number of PARTIAL PRODUCTS ½n(n-1)-1=½(n+1)(n-2)=½p(p-3)= NUMBER OF INSERTED|AUXILIARY POINTS related to complete SPH. POLYGON OF CONT.MULT. having p=n+1 sides.
			Art.372: The mᵗʰ given factor qₘ is used as MULTIPLIER m-1 times, and as MULTIPLICAND n-m times, for total n-1 times in all possible CONSTRUCTIVE + ASSOCIATIVE triangles.
			Art.373: The same angle ∠qₘ used in common corner of the n-1 triangles, supplying n-2 angle eqns; Each partial product also enters n-1 times and supplies n-2 angle equations.
			Art.374: Each of ½n(n+1) POINTS of figure common corner of (n-1)△ and supplies n-2 ∠eqns, for ½n(n+1)(n-2) total ∠eqns, n(n-2) CONSTR.∶½n(n-1)(n-2) ASSOC. or 2∶(n-1).
			Art.375: Total AUXILIARY TRIANGLES OF MULTIPLICATION (CONTINUED BINARY MULTIPLICATIONS) in figure: ½n(n+1)×(n-1)×⅓=⅙(n+1)n(n-1); ⬡ OF MULTIPLICATION:n=5 ⅙(5+1)5(5-1)=20 △.
			Art.376: Total AUXILIARY QUADRILATERALS OF MULTIPLICATION (CONTINUED TERNARY MULTIPLICATIONS): (1/24)(n+1)n(n-1)(n-2); ⬡ OF MULTIPLICATION:n=5 (1/24)(5+1)5(5-1)(5-2)=15 □.
			Art.377: Total AUXILIARY PENTAGONS OF MULTIPLICATION (CONTINUED QUATERNARY MULTIPLICATIONS): (1/120)(n+1)n(n-1)(n-2)(n-3); ⬡OF×:n=5 6⬠; HeptagonOF×(7-gon):n=6 21⬠.
			Art.378.ERR.366: Tot AUX SPH POLYGONS OF× of INFERIOR DEGREE p′