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 0 CONTENTS./LECTURE IV. Articles 121 to 174; Pages 130 to 185.
LECTURES./LECTURE IV. Articles 121 to 174; Pages 130 to 185.
CONT/§ XXVI. Articles 121 to 126; Pages 130 to 139.
LECT/§ XXVI. Articles 121 to 126; Pages 130 to 139.
Art.121: Recap. of prev. arts.; TAKING THE SCALAR Sq=w and VECTOR Vq=ρ PARTS of QUATERNION q.
Art.122: Quaternion β÷α degenerates into a SCALAR when β∥α, and a ⊥VECTOR (QUADRANTAL QUATERNION) when β⊥α.
Art.123: (vid.Fig.22) ε=|A-O|k, α=|A-O|i, γ=α÷ε=1j=|B-A|j=|C-O|j; δ=β÷ε=(α+γ)÷ε=(|A-O|i+j)÷|A-O|k=j-i/|A-O|; IF ∠AOB=60° THEN |B-A|=1=√3/√3,|A-O|=1/√3,|B-O|=2/√3, Tδ=2.
Art.124: Transfactor÷Factor=Profactor: (β÷ε)÷(α÷ε)=β÷α or δ÷γ=β÷α (art.103) EQUIVALENT BIRADIALS: same relative tensors Tδ/Tγ=Tβ/Tα, angle t, and versor ρ^t (or plane).
Art.125: Tδ/Tγ=Tβ/Tα or 2/1=(2/√3)/(1/√3) or 2=2; δ÷γ=(j-i/|A-O|)÷j=1+k√3=2[cos(60°)+ksin(60°)]=2e^(kπ/3)=2(-1)^(1/3)=2(kk)^(1/3)=2k^(2/3).
Art.126: ÷,× rules apply in general: 6÷2=3, or for UNIT a & angle t=0: 6a÷2a=3(a÷a)^0=3; 6a×2a=12(a÷a)^(2-0)=12; 2a×6a=12(a÷a)^(2+0)=12.
CONT/§ XXVII. Articles 127 to 130; Pages 139 to 144.
LECT/§ XXVII. Articles 127 to 130; Pages 139 to 144.
Art.127: SIGN OF COPLANARITY ⫴: β⫴α,γ; GIVEN: BIRADIAL β÷α & LINE γ, THEN: FOURTH PROPORTIONAL LINE is δ=β÷α×γ.
Art.128: δ=(β÷α)×γ=(β÷α)×(α÷ε)=β÷ε is tested okay for consistency in theory: ambiguity does not exist.
Art.129: Proportion between tensors Tα:Tβ∷Tγ:Tδ and versors Uα:Uβ∷Uγ:Uδ; GIVEN γ⫴α,β THEN δ⫴α,β also.
Art.130: For equiBIRADIALS (COPLANAR,Fig.22) δ÷γ=β÷α, the inverse γ÷δ=α÷β and alternate (triangles) pair δ÷β=γ÷α, β÷δ=α÷γ also hold.
CONT/§ XXVIII. Articles 131, 132; Pages 144 to 148.
LECT/§ XXVIII. Articles 131, 132; Pages 144 to 148.
Art.131.Fig.26: Arithmetic of tensors Tδ=Tβ÷Tα×Tγ; Concepts of the FOURTH PROPORTIONAL to 3 successive sides of a triangle inscribed in a circle.
Art.132.Figs.27,28: Illustrations of coplanar proportional lines β÷α=δ÷γ or p₂÷p₁=p₄÷p₃ inscribed in circles.
CONT/§ XXIX.ERR.xvii. Articles 133, 134; Pages 148 to 151.
LECT/§ XXIX. Articles 133, 134; Pages 148 to 151.
Art.133: (vid.Fig.26) THIRD∝ ε, α:γ∷γ:ε, to directed lines α,γ; continued or successive coplanar proportions (similar continued triangles).
Art.134: MEAN∝ γ=±(ε÷α)^(½)×α along bisector∠ between ε by α; RECTANGULAR 3RD∝ p₃=-p₁, (p₁=i)∝(p₂=j)∺(p₂=j)∝(p₃=-i); PARALLEL 3RD∝ p₃=p₁.
CONT/§ XXX. Articles 135, 136; Pages 151 to 153.
LECT/§ XXX. Articles 135, 136; Pages 151 to 153.
Art.135: nth-MEAN∝ (β÷α)^(n/m)×α, n=1…(m-1), along nth m-sector∠ between β by α; (T|U).(β÷α)ᵗ=[(T|U).β÷α]ᵗ=[(T|U)β÷(T|U)α]ᵗ; ∠.(β÷α)ᵗ=t×∠(β÷α).
Art.136: POWER OF A QUATERNION qᵗ: (T|U).qᵗ=[(T|U)q]ᵗ=(T|U)qᵗ; ∠.qᵗ=t∠q; qᴹqᴺ=qᴹ⁺ᴺ; RECIPROCAL q⁻¹q=qq⁻¹=q⁰=1.
CONT/§ XXXI.ERR.xviii. Articles 137 to 147; Pages 153 to 163.
LECT/§ XXXI. Articles 137 to 147; Pages 153 to 163.
Art.137.Fig.29: SIMPLE 1st & 2nd-MEAN∝: q=β÷α, ∠q=60° , ∠q^⅓=⅓∠q=+20° , ∠q^⅔=⅔∠q=+40° , γ=q^⅓×α, γ′=q^⅓×γ=q^⅔×α, β=q^⅓×γ′=q^⅔×γ=q×α, α∝γ::γ∝γ′::γ′∝β.
Art.138.Fig.29: < SIMPLE 1st & 2nd-MEAN∝: q=β÷α, ∠q=60°+360°, ∠q^⅓=⅓∠q=+140°, ∠q^⅔=⅔∠q=+280°, δ=q^⅓×α, δ′=q^⅓×δ=q^⅔×α, β=q^⅓×δ′=q^⅔×δ=q×α, α∝δ::δ∝δ′::δ′∝β.
Art.139.Fig.29: OPPOSITE 1st & 2nd-MEAN∝: q=β÷α, ∠q=60°-360°, ∠q^⅓=⅓∠q=-100°, ∠q^⅔=⅔∠q=-200°, ε=q^⅓×α, ε′=q^⅓×ε=q^⅔×α, β=q^⅓×ε′=q^⅔×ε=q×α, α∝ε::ε∝ε′::ε′∝β.
Art.140: NO NEW VARIETY of positions by multiple whole revolutions; Only variations are ∠q, ∠q+360°, and ∠q-360°.
Art.141: Cube-root (β÷α)^⅓ represents any one of THREE DISTINCT QUATERNIONS: 1. γ÷α (⅓∠); or 2. δ÷α (⅓∠+⅔π); or 3. ε÷α (⅓∠-⅔π).
Art.142: (β÷α)^(n/m), where m is prime to n (n/m in lowest terms), represents m distinct quaternion roots or values.
Art.143: If q̂=[smallest|SIMPLE ∠(q=β÷α)], then ∠q ≎ q̂+2lπ |l∈ℤ; ∠(qᵗ) = t×(q̂+2lπ)+2l′π = tq̂ +2(lt+l′)π ; ∠(qᵘ) = uq̂ + 2(mt+m′)π.
Art.144: ∠(qᵘ⁺ᵗ) = (u+t)q̂ + 2p(u+t)π +2p′π ; ∠(qᵘ×qᵗ) = (u+t)q̂ + 2(lt+mu+n)π ; FIVE whole numbers (INTEGERS) l,m,n,p,p′.
Art.145: qᵘqᵗ = qᵘ⁺ᵗ ; ∠(qᵘ⁺ᵗ)-∠(qᵘ×qᵗ)=2p(u+t)π+2p′π-2(lt+mu+n)π=2πp″,or p(t+u)-(lt+mu)+p′-n=p″,or (p-l)t+(p-m)u=p″+n=n′,or p(t+u)=lt+mu [if common ∠q, p=m=l n,primed=0].
Art.146: For GENERAL truth qᵘ⁺ᵗ=qᵘqᵗ, INTS p,l,m,n′ must be: p-l=p-m=n′=0,or p=l=m=n′=0,or p=m=l (same ∠q for qᵘ⁺ᵗ,qᵘ,qᵗ); Eg.IF a(t=√2)+b(u=√3)=c THEN a=b=c=0.
Art.147: GENERAL truth (qᵗ)ᵘ=qᵘᵗ requires GENERAL solution for ∠(qᵗ)ᵘ-∠(qᵘᵗ)=(l-m)ut+l′u=m″: m=l primed=0, a FIXED ∠(qᵗ) or FIXED ∠q.
CONT/§ XXXII. Articles 148 to 150; Pages 163 to 166.
LECT/§ XXXII. Articles 148 to 150; Pages 163 to 166.
Art.148: In arts.137-147, a chosen FIXED ∠q is required in GENERAL; the FIXED choice, by DEFINITION, is SIMPLE ∠q: ∠q=q̂ |0≤q̂≤π, supposing l=0 (art.143).
Art.149: At limit ∠q=0, q becomes SCALAR>0; 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 0 Art.153: If ∠q is acute 0<∠q<π/2, Then ∠(q²)=2∠q; If ∠q is obtuse π/2<∠q<π, Then ∠(q²)=2(π-∠q) Axis.q²=-Axis.q.
Art.154: (vid.Fig.29) Obtuse q=δ÷α Axis.q=-k ∠q=140°; ∠(q²)=∠(δ′÷α)=[2*SUPPLEMENT]=2(180°-140°)=80°≠280° Axis.q²=+k.
Art.155: Cubes β÷α=(γ÷α)³=(δ÷α)³=(ε÷α)³ Axis=-k; ∠(β÷α)=3∠(γ÷α)=3*20°=60°, ∠(δ÷α)³=3*140°-2π=60°, ∠(ε÷α)³=3(-100°)+2π=60°.
Art.156: β÷α=r³=r′³=r″³; by DEFINITION, the UNIQUE ROOT has the SIMPLE angle: r=γ÷α, ∠r=⅓∠(β÷α).
Art.157: (r³)^⅓=r IFF r is the UNIQUE ROOT; (r′³)^⅓=r=k^(-4/3)×r′, (r″³)^⅓=r=k^(4/3)×r″; Generally ᵗ√(qᵗ)≠q, but (ᵗ√q)ᵗ=q.
Art.158: Sqrt RULE: ?∠q<½π: (q²)^½=q; ?∠q>½π: (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′ CONT/§ LX. Articles 379 to 393; Pages 366 to 380.
LECT/§ LX. Articles 379 to 393; Pages 366 to 380: Focal Points of Polygons of Multiplication.
Art.379: Focal points E≘rq,F≘sr of conic inscribed in □ABCD≘srq (vid.Fig.65), denoted EF(‥)ABCD; (1/24)p(p-1)(p-2)(p-3) □inscr-conics, ½p(p-3) part-prod foci, per p-gonOF×.
Art.380: Table of Focal Relations for SPHERICAL HEXAGON OF CONTINUED MULTIPLICATION Q₁Q₂Q₃Q₄Q₅Q: FOCAL ENCHAINMENT of 15 AUX□ sharing 9 foci at 2ⁿᵈ⬡R₁R₂R₃R₄T₁T₂ △S₁S₂S₃.
Art.381.Fig.80: r₁=q₂q₁, r₂=q₃q₂,…; t₁=q₄q₃q₂q₁≅qq₅≅q₅q,…; ∠qₙ=∢R₁Q₁Q₂; ∠rₙ=∢Q₃R₁Q₂=π-∢Q₂R₁Q₁; At LIMIT,plane-regular⬡: ∠qₙ=30° ∠rₙ=60° ∠sₙ=90° ∠tₙ=120° ∠q=150° sₙ=s≘S.
Art.382: At LIMIT:(I.) Each 6□ has corner S, ⬭maj-axis=⬡side (II.) Each 6⬭ has foci S, ⬭min-axis=⬡side (III.) △S₁S₂S₃→S 3⬭ become 1○ in 2ⁿᵈ⬡R₁R₂R₃R₄T₁T₂ ○diameter=⬡side.
Art.383: Suppose plane-regular⬡Q₁Q₂Q₃Q₄Q₅Q of Fig.80 is made sensibly spherical-regular⬡; In what directions do points and angles deviate from the planar limiting case?
Art.384: Plane-regular⬡→spherical-regular⬡: Point deviations SS₁,SS₂,SS₃ are arcs of equal length toward Q₂,Q₃,Q₄ resp.
Art.385: Plane-regular⬡→spherical-regular⬡: Angle deviations of ∠qₙ are decreasing in equal amounts from 30° such that ∠qₙ<30°.
Art.386: Plane-regular⬡→spherical-regular⬡: Angle deviations of ∠rₙ are decreasing in equal amounts from 60° such that ∠rₙ<60°, with more defect than that of ∠qₙ.
Art.387: Plane-regular⬡→spherical-regular⬡: Angle deviations of ∠sₙ are decreasing in equal amounts from 90° such that ∠sₙ<90°, with more defect than that of ∠rₙ.
Art.388: Plane-regular⬡→spherical-regular⬡: Point deviations TₙT′ₙ are arcs toward S; Angle deviations of ∠tₙ are decreasing such that ∠tₙ<120°, with more defect…
Art.389: Plane-regular⬡→spherical-regular⬡: Point deviation QQ′ (of quinary product) is arc toward S.
Art.390.ERR.377: Plane-regular⬡→spherical-regular⬡: Angle deviation of ∠q=∠q₅q₄q₃q₂q₁ is decreasing from 150° such that ∠q<150°, with most accumulated defect.
Art.391: The Table of Focal Relations for SPH.HEXAGON⬡ OF CONT.MULT. in Art.380 does not include the recently employed points S₁ S₂ S₃ T₁′ T₂′ Q′ M₂ P₁ M₃.
Art.392.ERR.379: Focal Relations for Pentagon ⬠ABCDE(vid.Figs.79,65); 5 ternary prods ≘ 2ⁿᵈ⬠FGHIK and 5 AUX-□ with 5 inscribed-⬭ (conics) having foci at corners of ⬠FGHIK.
Art.393.Fig.81: Plane-regular⬠ABCDE: Intersections of diagonals form 2ⁿᵈ⬠FGHIK; ∠qₙ=36° ∠rₙ=72°(FGH) ∠sₙ=108°(IK) ∠q=144°(E); 5□,5⬭ major-axis=length of exterior ⬠side.
CONTENTS./LECTURE VII. Articles 394 to 689; Pages 381 to 700
LECTURES./LECTURE VII. Articles 394 to 689; Pages 381 to 700
CONT/§ LXI. Articles 394 to 400; Pages 381 to 386.
LECT/§ LXI. Articles 394 to 400; Pages 381 to 386.
Art.394: Recap of Lecture I: B-A+A=B, a+A-A=a, TRANSVECTOR=PROVECTOR+VECTOR; Recap of Lecture II: β÷α×α=β, q×α÷α=q, TRANSFACTOR=PROFACTOR×FACTOR; i²=j²=k²=ijk=-1.
Art.395: Recap of Lecture III: Products of ⟂ and ∥ vectors; powers; biradials; conjugate K, tensor T, and versor U operations; κ÷λ λ÷κ κλ λκ angle relations of Art.87 &c.
Art.396: Recap of Lecture IV: ∝ lines, qᵗ, βα⁻¹γ, ∠q & Ax.q, √-1, loci; Recap of Lecture V: continued products of coplanar and non-coplanar lines.
Art.397: Recap of Lecture V: associative principle of multiplication of vectors, spherical constructions of versors as arcs of great circles, conjugates and reciprocals.
Art.398: Recap of Lecture V: 4ᵗʰ∝=δ=βα⁻¹γ; Auxiliary spherical-△DEF bisected by △ABC, and RULEs of Arts.223-240 for the SEMI-SUM ½[D+E+F] angle formulas.
Art.399: Recap of Lecture VI: SEMI-EXCESS angle formulas; s.rq sr.q Associative Principle; RULEs of Arts.268-271 EXT.VERT.∠; ∀△ γᶻβʸαˣ=-1; Conic rotation qrq⁻¹ 2∠q.
Art.400: Recap of Lecture VI: Plane & gauche polygons inscribed in circle|sphere; compositions of conical rotations; polygons of multiplication superscribed on sphere; foci.
CONT/§ LXII. Articles 401 to 405; Pages 387 to 391.
LECT/§ LXII. Articles 401 to 405; Pages 387 to 391.
Art.401: Objectives of Lecture VII: Addition and Subtraction of Quaternions; Distributive Property; Notations S,V taking scalar,vector parts; How to interpret the sum 1+k?
Art.402.Fig.82: (1i=i)+(ki=j)=(1+k)i=i+j; (1+k)=(i+j)÷i QUATERNION; T(1+k)=√2, U(1+k)=k^½, (1+k)=√(2k).
Art.403: Similar processes for determining meanings (or operation) of scalar + foward unit difference operator 1+Δ and quaternion 1+k (also as operator), by what each does.
Art.404.Fig.83: Partial operators 1,k of operator 1+k; Partial results 1i=i,ki=j and sum of partial results (1+k)i=1i+ki=i+j.
Art.405.Fig.84: [SCALAR w + VECTOR ρ (∠ρ=90°)]×[VECTOR α=OA (α⟂ρ)]=[COMMUTATIVE SUM OF MUTUALLY-⟂ LINES wα=OB∥α + ρ¹α=OC⟂(α,ρ)]; QUATERNION w+ρ=ρ+w=(ρα+wα)÷α=(wα+ρα)÷α.
CONT/§ LXIII. Articles 406 to 411; Pages 391 to 397.
LECT/§ LXIII. Articles 406 to 411; Pages 391 to 397.
Art.406.Fig.85: Decomposition of Biradial q=β÷α=OB÷OA into Scalar+Vector w+ρ=β÷α=q: β=β′+β″=OB′+B′B, β′∥α, β″⟂α, β′÷α=w, β″÷α=ρ, ρ⟂(β,α), wα=β′, ρα=β″, (w+ρ)α=β′+β″=β.
Art.407: TAKING SCALAR Sq=w=β′÷α and VECTOR Vq=ρ=β″÷α PARTS of q=w+ρ=β÷α=Sq+Vq=Vq+Sq; Sw=w Sρ=0=Vw Vρ=ρ SSq=Sq SVq=0=VSq VVq=Vq; Identities: 1=S+V=V+S S²=S SV=VS=0 V²=V.
Art.408: Conjugate of Quaternion q=w+ρ: Kq=w-ρ; Kw=+w Kρ=-ρ; Identity Kq=Sq-Vq; Identity abridged Kq=(S-V)q or K=S-V; SKq=+Sq, VKq=-Vq or SK=S, VK=-V [vid.Figs.85,32].
Art.409: TKq=Tq [cf.114,162] ∴ T(w-ρ)=T(w+ρ); qKq=Tq² [cf.163] ∴ T(w+ρ)²=(w+ρ)(w-ρ) Tq=T(w+ρ)=√(w²-ρ²) [cf.111]; Τ(β÷α)²=T(β′÷α)²+Τ(β″÷α)² Tq²=Tw²+Tρ²=+w²-ρ² [vid.Fig.85].
Art.410: Partial quotients scaled x T(xβ′÷α)²+T(xβ″÷α)²=T(xβ÷α)² or xw+xρ=xq; S.xq=xSq V.xq=xVq (Vx=0); S(-q)=-Sq V(-q)=-Vq; S(-Kq)=-Sq V(-Kq)=+Vq -Kq=-Sq+Vq -K=V-S.
Art.411: q=TqUq Sq=Tq.SUq Vq=TqVUq VUq=TVUq.UVUq, UVUq=UVq=Ax.q; q=Tq(SUq+VUq)=Tq(SUq+TVUq.UVq)=Tq(cos∠q+sin∠q.UVq); (UVq)²=-1 √(-1)=Ax.q=UVq, Uq=cos∠q+sin∠q.√(-1).
CONT/§ LXIV.ERR.xxxviii. Articles 412 to 415; Pages 397 to 402.
LECT/§ LXIV. Articles 412 to 415; Pages 397 to 402.
Art.412.Fig.86: Formulas of Perpendicularity, Parallelism: ∵ρ⟂α∴S(ρ÷α)=0=S([ρ′∥α+ρ″⟂α]÷α)=S([0+ρ″⟂α]÷α=VECTOR), ∵ρ∥α∴V(ρ÷α)=0=V([ρ′∥α+ρ″⟂α]÷α)=V([ρ′∥α+0]÷α=SCALAR).
Art.413: Fixed α=OA β=OB,var ρ=OP; ∥LOCUS-PLANES: S(ρ÷α)=0 ρ⟂α LOCUS(ρ)=[PLANE⟂α THRU O], S([ρ-β]÷α)=0 BP⟂α ρ∈[PLANE⟂α THRU B], ρ′=OP′ S(ρ÷α)=ρ′÷α=a ρ″⟂α ρ∈[PLANE⟂α THRU P′].
Art.414.Fig.87: S(α÷ρ)=1 or S([α-ρ]÷ρ)=0 LOCUS(ρ)=[SPHERE w/ DIAMETER=α=OA]: α=α′∥ρ+α″⟂ρ α′÷ρ=1 [α′=ρ=OP]⟂[α″=α-ρ=PA]; CENTER=C OC=½α; RADIUS=CP T(CP=ρ-½α)=½Tα=½DIAMETER.
Art.415.Figs.88,89: Co-initial bisector ½(α+β=OD)=OC=½OD of α=OA,β=OB: BC=½(α+β)-β=½(α-β)=α-½(α+β)=CA; T(ρ-½[α+β])=½T[α-β] or S([α-ρ]÷[ρ-β])=0 LOCUS(ρ)=[SPHERE w/ DIAMETER=BA].
CONT/§ LXV. Articles 416 to 421; Pages 402 to 407.
LECT/§ LXV. Articles 416 to 421; Pages 402 to 407.
Art.416: CIRCLE=∩(●-SPHERE,△-PLANE); CONE=PENCIL of all LINES that ∩ BASE-CIRCLE; CONIC-SECTION=∩(CONE,PLANE); SPHERICAL-CONIC|CURVE=∩(CONE,SPHERE).
Art.417: Coplanar ⫴(△QRS,A,P),fix QRSA∣OA⟂PLANE,var P; α=OA,ρ=OP=OA∥α+AP⟂α, EQ₁∶PLANE S(ρ÷α)=1 [cf.413]; SPH.DIAMETER β=OB, EQ₂∶SPHERE S(β÷ρ)=1 [cf.414]; CIRCLE=∩(EQ₁,EQ₂).
Art.418: EQ₃∶CONE = EQ₁∶PLANE×EQ₂∶SPHERE = S(ρ÷α).S(β÷ρ)=1, w/ vertex O, base-circle ∩(EQ₁,EQ₂); If EQ₃(ρ),Then also EQ₃(tρ) ∴ LINE tρ on CONE.
Art.419: The intersection (system of two equations) of another PLANE S(ρ÷γ)=1 and EQ₃∶CONE is the LOCUS(ρ) of a CONIC-SECTION; fixed αβγ, variable ρ.
Art.420: S(ρ÷α)=S.ρα⁻¹=SK.α⁻¹ρ=S.α⁻¹ρ=ρ²S(α⁻¹÷ρ), ρ²S(β÷ρ)=S.βρ=S.ρβ=S(ρ÷β⁻¹): S(ρ÷α)S(β÷ρ)=S(ρ÷β⁻¹)S(α⁻¹÷ρ)=1 SUBCONTRARY SECTION; ∵ PLANES⟂β ∴ S(ρ÷β⁻¹)=1 ∥ S(ρ÷β)=0.
Art.421: EQNS.SPHERE w/radius c: Tρ=c, ρ²+c²=0, S([ρ-γ][ρ+γ])=0 [cf.415 α=γ,β=-γ]; CYCLIC PLANES: S(ρ÷α)=S(αρ)=S(ρα)=0, S(ρ÷β)=~=0; CYCLIC ARCS: S.αρ=0 Tρ=c, S.βρ=0 Tρ=c.
CONT/§ LXVI. Articles 422 to 426; Pages 407 to 416.
LECT/§ LXVI. Articles 422 to 426; Pages 407 to 416.
Art.422.ERR.408.Fig.90: OS=σ OT=τ OA=α Tα=a=Tτ τ²=-a² S.στ⁻¹=S([OT∥τ+TS⟂τ]÷τ)=1; S.στ=-a² S.σ⁻¹τ=SΚ.σ⁻¹τ=S.τσ⁻¹=-a²σ⁻² PLANE⟂(σ@μ=OM=-a²σ⁻²σ); Uμ=Uσ Τμ=a²Tσ⁻¹ S.τμ⁻¹=1 μσ=α².
Art.423: OP=ρ; P in POLAR PLANE⟂@M S.ρμ⁻¹=1(or S.ρσ=-a²) of POLE S; S in POLAR PLANE SK.ρσ=S.σρ=-a² of POLE P; CONJUGATE POINTS P,S; S.ρσ=-ρ∙σ=TρTσ.cos(π-∠.ρσ⁻¹) ∠.ρσ=π-∠.ρσ⁻¹.
Art.424: S.βα=α²S.βα⁻¹=α².β′∥α.α⁻¹=β′α; S.βα⪋0∷∠.βα⁻¹⪋½π S.βα⁻¹⪌0∷∠.βα⁻¹⪋½π; U.βα=U.βα²α⁻¹=-U.βα⁻¹ ∠βα=π-∠βα⁻¹; TS.βα=T.β′α=ΤαTβ.cos(∠βα⁻¹); α′β=β′α,βα′=αβ′; β′=S.βα⁻¹×α=S.βα÷α.
Art.425: S.(σ-τ)σ=S.(MS-MT)OS=MS×OS=(σ-μ)σ=(σ-α²σ⁻¹)σ=σ²+a²=Ar.▭OSM=Ar.□ST²=(σ-τ)²=-Τ(σ-τ)²=-(Tσ²-Tτ²)=σ²+a², S.σ÷(σ-τ)=1; EQN(τ).ENVELOPING CONE {S.σ(τ-σ)}²=(τ-σ)⁴=(σ²+a²)(τ-σ)².
Art.426: CONE S(ρ÷α).S(β÷ρ)=1 [cf.418], MAJOR-AXIS thru FOCI&VERTEX=O @∩(CYCLIC PLANES) S.αρ=0,S.βρ=0 [cf.421]; RECIPROCAL POLARS PP′,SS′ μσ=ρ′∥σ.σ=-a²=σ′∥ρ.ρ &c.
CONT/§ LXVII. Articles 427 to 431; Pages 416 to 423.
LECT/§ LXVII. Articles 427 to 431; Pages 416 to 423.
Art.427.Fig.91: V.βα⁻¹=β″⟂α.α⁻¹ V.βα=β″α⁻¹α²=β″α; UV.βα=-UV.βα⁻¹Tα²=-UV.βα⁻¹ OPPOSITE DIRECTIONS; TV.βα=T.β″α=Tβ″Tα=Tβsin(∠βα⁻¹).Tα=base.height=Ar.▱AOB=2Ar.△AOB.
Art.428: U.βα=U.βα⁻¹α²=-U.βα⁻¹=-U[β′α⁻¹+β″α⁻¹]=-cos(∠βα⁻¹)+sin(∠βα⁻¹)U(-β″α⁻¹)]; S.βα=β′α⁻¹α²=-TβTαcos(∠βα⁻¹)=TβTαcos(π-∠βα⁻¹), TV.βα=ΤβTαsin(∠βα⁻¹)=ΤβTαsin(π-∠βα⁻¹).
Art.429: V.βα⁻¹α²=V.βα=V.(β′∥α+β″⟂α)α⁻¹α²=-Tα²β″α⁻¹=β″α ⟂(α,β,β″); β″=V.βα.α⁻¹=β″αα⁻¹=V.βα⁻¹.α=β″α⁻¹α=-K[V.βα.α⁻¹]=-K[V.βα⁻¹.α]=-α⁻¹V.βα=-αV.βα⁻¹; V.[αβ=K.βα]=-V.βα [cf.82,89].
Art.430: ρ=x₀α+β″, ρ-β″=ρ-(β-β′)=x₀α, ρ-β=x₀α-β′=xα; ρ=β+xα or V.(ρ-β)α=0 or V.ρα=β″α=V.βα LOCUS(ρ)=[LINE∥α THRU ANY β]; V.ρV.βα=0 ρ∥V.βα ρ⟂(α,β); S.ρV.βα=0 ρ⟂V.βα ρ⫴(α,β).
Art.431: If V.ρα=-V.βα=VK.βα=V.αβ=-β″α=γ″α=V.γα, ρ=γ+xα V.(ρ-γ)α=0; (V.ρα)²=(±V.βα)² or TV.ρα=TV.βα=Τ.β″α=a LOCUS(ρ)=[CYLINDER OF REV.w/axis α,radius aΤα⁻¹,arbitrary scalar a].
CONT/§ LXVIII. Articles 432 to 436; Pages 423 to 430.
LECT/§ LXVIII. Articles 432 to 436; Pages 423 to 430.
Art.432: q=ρβ⁻¹; CIR: CYL(ax.=β⁻¹,rad.=bTβ) TVq=b ∩ PLA⟂β Sq=a ∩ SPH(rad.=ΤqTβ) Tq=√(a²+b²)=√({Sq}²-{Vq}²); CIR: SPH Tρ=Τβ ∩ PLA⟂β -1<[S.q=x]<1 ∩ CYL(rad.=√(1-x²)Tβ) TVq=√(1-x²).
Art.433: ELLIPSE: PLA S.ρα⁻¹=a ∩ CYL TV.ρβ⁻¹=b; LOCUS(ρ)=ELLIPSOID(given lines α=OA,β=OB) generated as a system of elliptic sections of concentric cylinders.
Art.434.Fig.92: Construction of ELLIPSOID (elliptic sections of inner CYLs w/ variable major-axis L′N′ in PLANES⟂α@A…A′, center M′=B′…F′) inscribed in CYL (w/ axis β, radius Tβ).
Art.435: Triangular relations between point P (of sphere of radius Tβ) and point P′ (of related ellipsoid circumscribed by cylinder of radius Tβ and axis β).
Art.436: EQUATION OF ELLIPSIOD: (S.ρα⁻¹)²-(V.ρβ⁻¹)²=1, or (S.ρα⁻¹)²+(TV.ρβ⁻¹)²=1, or T(S.ρα⁻¹+V.ρβ⁻¹)=1.
CONT/§ LXIX. Articles 437 to 440; Pages 430 to 435.
LECT/§ LXIX. Articles 437 to 440; Pages 430 to 435.
Art.437.Fig.93: EQUILATERAL HYPERBOLIOD OF REVOLUTION and its ASYMPTOTIC CONE; HYPERBOLIOD OF TWO SHEETS and its ASYMPTOTIC CONE OF 2ND DEGREE.
Art.438: EQUILATERAL HYPERBOLOID OF 2 SHEETS OF REV. (S.ρβ⁻¹)²+(V.ρβ⁻¹)²=1 in CONE (S.ρβ⁻¹)²+(V.ρβ⁻¹)²=0; HYPERBOLOID OF 2 SHEETS (S.ρα⁻¹)²+(V.ρβ⁻¹)²=1 in CONE (S.ρα⁻¹)²+(V.ρβ⁻¹)²=0.
Art.439: EQUILATERAL HYPERBOLOID OF 1 SHEET OF REV. (S.ρβ⁻¹)²+(V.ρβ⁻¹)²=-1; HYPERBOLOID OF 1 SHEET (S.ρα⁻¹)²+(V.ρβ⁻¹)²=-1.
Art.440: ELLIPTIC PARABOLOID OF REV. S.ρβ⁻¹+(V.ρβ⁻¹)²=0, OF NON-REV. S.ρα⁻¹+(V.ρβ⁻¹)²=0; HYPERBOLIC PARABOLOID S.ρα⁻¹S.ρβ⁻¹=S.ργ⁻¹; SURFACE OF REV. TV.ρβ⁻¹=f(S.ρβ⁻¹), SYS.ELLIPSES TV.ρβ⁻¹=f(S.ρα⁻¹).
CONT/§ LXX. Article 441; Pages 435 to 437.
LECT/§ LXX. Article 441; Pages 435 to 437.
Art.441
CONT/§ LXXI. Article 442; Pages 437 to 439.
LECT/§ LXXI. Article 442; Pages 437 to 439.
Art.442
CONT/§ LXXII. Articles 443 to 447; Pages 439 to 444.
LECT/§ LXXII. Articles 443 to 447; Pages 439 to 444.
Art.443: ADDITION OF QUATERNIONS; COMMON OPERAND α; CASES OF ADDITION: scalar+scalar y+x, vector+vector.
Art.444: ALGEBRAICAL SUM of scalars (y+x)α=yα+xα, y+x=(yα+xα)÷α; β=xα γ=yα, (γ÷α)+(β÷α)=(γ+β)÷α if β∥α,γ∥α.
Art.445.Fig.94: Also (γ÷α)+(β÷α)=(γ+β)÷α when β⟂α,γ⟂α.
Art.446: DISTRIBUTIVE PRINCIPLE of multiplication (r+q)α=rα+qα; SUM OF ANY TWO QUATERNIONS r+q.
Art.447: RULE OF THE COMMON OPERAND α.
CONT/§ LXXIII. Articles 448, 449; Pages 444 to 447.
LECT/§ LXXIII. Articles 448, 449; Pages 444 to 447.
Art.448: Addition of two quaternions is a COMMUTATIVE OPERATION r+q=q+r; S(r+q)=Sr+Sq; V(r+q)=Vr+Vq; K=S-V, K(r+q)=Kr+Kq.
Art.449: S∑=∑S, V∑=∑V, K∑=∑K; SΔ=ΔS, VΔ=ΔV, KΔ=ΔK; q+Kq=2Sq, ½(αβ+βα)=S.αβ=S.βα; q-Kq=2Vq, ½(αβ-βα)=V.αβ=-V.βα.
CONT/§ LXXIV. Article 450; Pages 447 to 449.
LECT/§ LXXIV. Article 450; Pages 447 to 449.
Art.450: QUADRINOMIAL FORM q=w+ix+jy+kz=w+ρ; Sq=w, Vq=ix+jy+kz=ρ; If q=q′,Then w=w′ x=x′ y=y′ z=z′.
CONT/§ LXXV. Articles 451 to 455; Pages 449 to 455.
LECT/§ LXXV. Articles 451 to 455; Pages 449 to 455.
Art.451: Rigorous examination of GENERAL DISTRIBUTIVE PRINCIPLE (r+q)s=rs+qs; GENERAL ASSOCIATIVE PRINCIPLE s.rq=sr.q.
Art.452: The vector part of a scalar is a null line of indeterminate direction that is coplanar with any two lines.
Art.453: Scalar and Vector Parts of quaternion product rq: S.rq=SrSq+S.VrVq, V.rq=VrSq+VqSr+V.VrVq.
Art.454: γα+βα=(γ+β)α; K.(γ+β)α=α(γ+β)=αγ+αβ=K.(γα+βα)=K.γα+K.βα; (β±α)²=β²±βα±αβ+α²=β²+α²±2S.βα; (β+α)(β-α)=β²-βα+αβ-α²=β²-α²+2V.αβ.
Art.455: DISTRIBUTIVE PRINCIPLE holds good generally in the MULTIPLICATION OF ANY TWO SUMS OF QUATERNIONS; ∑r.∑q=∑.rq.
CONT/§ LXXVI. Articles 456 to 459; Pages 455 to 460.
LECT/§ LXXVI. Articles 456 to 459; Pages 455 to 460.
Art.456: SPHERE(diameter α) S.αρ⁻¹=1 or S.(α-ρ)ρ⁻¹=S.(αρ⁻¹)-1=0 or T(ρ-½α)=½Tα, T(ρ-½α)²=¼Tα²=-(ρ-½α)²=-(ρ²-S.αρ+¼α²)=-¼α², ρ²=S.αρ, S.αρ⁻¹=1.
Art.457.Fig.95: ρ²-S.(α+β)ρ+¼(α+β)²=¼(α-β)², ρ²-S.(α+β)ρ+¼α²+½S.αβ+¼β²=¼α²-½S.αβ+¼β², ρ²-S.(α+β)ρ+S.αβ=0, S(ρ²-αρ-ρβ+αβ)=S.(ρ-α)(ρ-β)=T(ρ-β)²S.(α-ρ)(ρ-β)⁻¹=0.
Art.458: Central Point or Centre of Gravity.
Art.459.Fig.96: Apollonius circle of spheric surface T(ρ-β)=c.
CONT/§ LXXVII. Articles 460 to 464; Pages 460 to 466.
LECT/§ LXXVII. Articles 460 to 464; Pages 460 to 466.
Art.460.ERR.460: Intersections of the right line and sphere.
Art.461: Equation of the enveloping cone.
Art.462: Equation of the polar plane; harmonic property; harmonic conjugates.
Art.463.Fig.97: Harmonic mean between any two vectors; harmonically conjugate; CIRCULAR HARMONIC GROUP; conjugate chords.
Art.464: The tangent vector β⁻¹-α⁻¹ at O of circle OAB with direction DO (or OD′).
CONT/§ LXXVIII. Articles 465 to 470; Pages 466 to 475.
LECT/§ LXXVIII. Articles 465 to 470; Pages 466 to 475.
Art.465: EQUATION OF THE ELLIPSOID.
Art.466.Fig.98: Illustration of EQUATION OF THE ELLIPSOID; guide chord; conjugate guide chord.
Art.467.ERR.469.Fig.99: Conjugate guide-point; GENERATING TRIANGLE; DIACENTRIC SPHERE; axes and summits
Art.468: Properties of the ellipsoid.
Art.469: Ellipsoid cut in circles by the two cyclic planes; mean sphere.
Art.470
CONT/§ LXXIX. Articles 471 to 474; Pages 476 to 479.
LECT/§ LXXIX. Articles 471 to 474; Pages 476 to 479.
Art.471
Art.472
Art.473
Art.474
CONT/§ LXXX. Articles 475 to 480; Pages 480 to 485.
LECT/§ LXXX. Articles 475 to 480; Pages 480 to 485.
Art.475
Art.476
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CONT/§ LXXXI. Articles 481 to 486; Pages 485 to 491.
LECT/§ LXXXI. Articles 481 to 486; Pages 485 to 491.
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Art.482
Art.483
Art.484
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Art.486
CONT/§ LXXXII. Articles 487, 488; Pages 491, 492.
LECT/§ LXXXII. Articles 487, 488; Pages 491, 492.
Art.487
Art.488
CONT/§ LXXXIII. Articles 489, 490; Pages 493 to 495.
LECT/§ LXXXIII. Articles 489, 490; Pages 493 to 495.
Art.489
Art.490
CONT/§ LXXXIV. Articles 491 to 495; Pages 495 to 502.
LECT/§ LXXXIV. Articles 491 to 495; Pages 495 to 502.
Art.491
Art.492
Art.493.Fig.100:
Art.494
Art.495
CONT/§ LXXXV. Article 496; Pages 502, 503.
LECT/§ LXXXV. Article 496; Pages 502, 503.
Art.496
CONT/§ LXXXVI. Articles 497 to 500; Pages 503 to 509.
LECT/§ LXXXVI. Articles 497 to 500; Pages 503 to 509.
Art.497
Art.498
Art.499
Art.500.ERR.508
CONT/§ LXXXVII. Articles 501 to 503; Pages 509 to 511.
LECT/§ LXXXVII. Articles 501 to 503; Pages 509 to 511.
Art.501
Art.502
Art.503
CONT/§ LXXXVIII. Articles 504, 505; Pages 511 to 513.
LECT/§ LXXXVIII. Articles 504, 505; Pages 511 to 513.
Art.504
Art.505
CONT/§ LXXXIX. Articles 506 to 512; Pages 513 to 621.
LECT/§ LXXXIX. Articles 506 to 512; Pages 513 to 621.
Art.506
Art.507
Art.508
Art.509
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CONT/§ XC. Articles 513 to 518; Pages 521 to 526.
LECT/§ XC. Articles 513 to 518; Pages 521 to 526.
Art.513
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CONT/§ XCI. Articles 519 to 523; Pages 526 to 529.
LECT/§ XCI. Articles 519 to 523; Pages 526 to 529.
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CONT/§ XCII. Articles 524 to 526; Pages 529 to 532.
LECT/§ XCII. Articles 524 to 526; Pages 529 to 532.
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CONT/§ XCIII. Articles 527 to 529; Pages 532 to 537.
LECT/§ XCIII. Articles 527 to 529; Pages 532 to 537.
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CONT/§ XCIV. Articles 530 to 536; Pages 537 to 545.
LECT/§ XCIV. Articles 530 to 536; Pages 537 to 545.
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Art.532
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CONT/§ XCV. Articles 537 to 550; Pages 545 to 557.
LECT/§ XCV. Articles 537 to 550; Pages 545 to 557.
Art.537.ERR.545
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CONT/§ XCVI. Articles 551 to 553; Pages 557 to 559.
LECT/§ XCVI. Articles 551 to 553; Pages 557 to 559.
Art.551
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CONT/§ XCVII. Articles 554 to 567; Pages 559 to 569.
LECT/§ XCVII. Articles 554 to 567; Pages 559 to 569.
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CONT/§ XCVIII. Articles 568 to 573; Pages 569 to 572.
LECT/§ XCVIII. Articles 568 to 573; Pages 569 to 572.
Art.568
Art.569
Art.570
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CONT/§ XCIX. Articles 574 to 578; Pages 572 to 575.
LECT/§ XCIX. Articles 574 to 578; Pages 572 to 575.
Art.574
Art.575
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CONT/§ C. Articles 579 to 590; Pages 575 to 584.
LECT/§ C. Articles 579 to 590; Pages 575 to 584.
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CONT/§ CI. Articles 591 to 597; Pages 584 to 588.
LECT/§ CI. Articles 591 to 597; Pages 584 to 588.
Art.591
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CONT/§ CII. Articles 598 to 601; Pages 588 to 592.
LECT/§ CII. Articles 598 to 601; Pages 588 to 592.
Art.598
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CONT/§ CIII. Articles 602 to 606; Pages 592 to 596.
LECT/§ CIII. Articles 602 to 606; Pages 592 to 596.
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Art.605.ERR.595
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CONT/§ CIV. Articles 607 to 612; Pages 596 to 601.
LECT/§ CIV. Articles 607 to 612; Pages 596 to 601.
Art.607
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CONT/§ CV. Articles 613 to 615; Pages 601 to 604.
LECT/§ CV. Articles 613 to 615; Pages 601 to 604.
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Art.614
Art.615
CONT/§ CVI. Articles 616 to 619; Pages 604 to 609.
LECT/§ CVI. Articles 616 to 619; Pages 604 to 609.
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Art.619
CONT/§ CVII. Article 620; Pages 609 to 611.
LECT/§ CVII. Article 620; Pages 609 to 611.
Art.620
CONT/§ CVIII. Articles 621 to 624; Pages 611 to 620.
LECT/§ CVIII. Articles 621 to 624; Pages 611 to 620.
Art.621.ERR.612
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CONT/§ CIX. Articles 625 to 630; Pages 620 to 627.
LECT/§ CIX. Articles 625 to 630; Pages 620 to 627.
Art.625
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CONT/§ CX. Articles 631 to 635; Pages 627 to 631.
LECT/§ CX. Articles 631 to 635; Pages 627 to 631.
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CONT/§ CXI. Articles 636 to 650; Pages 631 to 643.
LECT/§ CXI. Articles 636 to 650; Pages 631 to 643.
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CONT/§ CXII. Articles 651 to 668; Pages 643 to 664.
LECT/§ CXII. Articles 651 to 668; Pages 643 to 664.
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Art.661
Art.662
Art.663
Art.664
Art.665
Art.666
Art.667
Art.668
CONT/§ CXIII. Articles 669 to 675; Pages 664 to 674.
LECT/§ CXIII. Articles 669 to 675; Pages 664 to 674.
Art.669.ERR.665
Art.670
Art.671
Art.672
Art.673
Art.674.ERR.672
Art.675
CONT/§ CXIV. Articles 676, 677; Pages 674 to 678.
LECT/§ CXIV. Articles 676, 677; Pages 674 to 678.
Art.676
Art.677
CONT/§ CXV. Articles 678, 679; Pages 678 to 688.
LECT/§ CXV. Articles 678, 679; Pages 678 to 688.
Art.678
Art.679.ERR.687
CONT/§ CXVI. Articles 680 to 688; Pages 688 to 700.
LECT/§ CXVI. Articles 680 to 688; Pages 688 to 700.
Art.680
Art.681.Fig.102:
Art.682
Art.683
Art.684
Art.685
Art.686
Art.687
Art.688
CONT/§ CXVII. Article 689; Page 700.
LECT/§ CXVII. Article 689; Page 700.
Art.689
END OF THE CONTENTS.
END OF THE LECTURES.
APPENDIX A. (referred to in § CXIII.), Pages 701 to 710.
APPENDIX B. (respecting the results of § CXIV.), Pages 717 to 730.
APPENDIX C. (containing an additional account of the analysis by which some of those results were obtained) Pages 731 to 736.
REFERENCES TO THE FIGURES.
Figure.1 Article.7 Page.6
Figure.2 Article.8 Page.8
Figure.3 Article.9 Page.9
Figure.4 Article.9 Page.9
Figure.5 Article.12 Page.12
Figure.6 Article.53 Page.44
Figure.7 Article.53 Page.44
Figure.8 Article.57 Page.49
Figure.9 Article.59 Page.52
Figure.10 Article.60 Page.53
Figure.11 Article.68 Page.62
Figure.12 Article.68 Page.62
Figure.13 Article.68 Page.62
Figure.14 Article.74 Page.68
Figure.15 Article.81 Page.77
Figure.16 Article.81 Page.77
Figure.17 Article.87 Page.85
Figure.18 Article.94 Page.93
Figure.19 Article.97 Page.97
Figure.20 Article.97 Page.98
Figure.21 Article.98 Page.99
Figure.22 Article.103 Page.107
Figure.23 Article.106 Page.110
Figure.24 Article.117 Page.123
Figure.25 Article.119 Page.125
Figure.26 Article.131 Page.144
Figure.27 Article.132 Page.147
Figure.28 Article.132 Page.147
Figure.29 Article.137 Page.154
Figure.30 Article.181 Page.190
Figure.31 Article.183 Page.193
Figure.32 Article.186 Page.194
Figure.33 Article.199 Page.201
Figure.34 Article.199 Page.202
Figure.35 Article.199 Page.202
Figure.36 Article.217 Page.213
Figure.37 Article.219 Page.214
Figure.38 Article.222 Page.217
Figure.39 Article.223 Page.218
Figure.40 Article.224 Page.218
Figure.41 Article.227 Page.222
Figure.42 Article.236 Page.228
Figure.43 Article.242 Page.235
Figure.44 Article.253 Page.243
Figure.45 Article.254 Page.244
Figure.46 Article.256 Page.245
Figure.47 Article.257 Page.246
Figure.48 Article.257 Page.246
Figure.49 Article.257 Page.246
Figure.50 Article.264 Page.253
Figure.51 Article.266 Page.255
Figure.52 Article.269 Page.258
Figure.53 Article.272 Page.261
Figure.54 Article.273 Page.261
Figure.55 Article.277 Page.265
Figure.56 Article.280 Page.267
Figure.57 Article.281 Page.268
Figure.58 Article.294 Page.278
Figure.59 Article.298 Page.282
Figure.60 Article.299 Page.283
Figure.61 Article.300 Page.284
Figure.62 Article.301 Page.285
Figure.63 Article.301 Page.286
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Figure.65 Article.302 Page.287
Figure.66 Article.320 Page.306
Figure.67 Article.323 Page.309
Figure.68 Article.324 Page.310
Figure.69 Article.325 Page.312
Figure.70 Article.330 Page.316
Figure.71 Article.332 Page.318
Figure.72 Article.333 Page.319
Figure.73 Article.335 Page.320
Figure.74 Article.342 Page.327
Figure.75 Article.343 Page.329
Figure.76 Article.345 Page.330
Figure.77 Article.347 Page.332
Figure.78 Article.353 Page.337
Figure.79 Article.361 Page.347
Figure.80 Article.381 Page.369
Figure.81 Article.393 Page.380
Figure.82 Article.402 Page.387
Figure.83 Article.404 Page.389
Figure.84 Article.405 Page.390
Figure.85 Article.406 Page.391
Figure.86 Article.412 Page.398
Figure.87 Article.414 Page.400
Figure.88 Article.415 Page.401
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Figure.90 Article.422 Page.408
Figure.91 Article.427 Page.416
Figure.92 Article.434 Page.425
Figure.93 Article.437 Page.430
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Figure.95 Article.457 Page.457
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Figure.97 Article.463 Page.464
Figure.98 Article.466 Page.467
Figure.99 Article.467 Page.470
Figure.100 Article.493 Page.499
Figure.101 Article.530 Page.538
Figure.102 Article.681 Page.691
ERRATA.
THE END.